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abstract: 'An approximate analytical solution of the Thomas-Fermi equation for neutral atoms is obtained, using the Ritz variational method, which reproduces accurately the numerical solution, in the range $0\leq x\leq50$, and its derivative at $x=0$. The proposed solution is used to calculate the total ionization energies of heavy atoms. The obtained results are in good agreement with the Hartree-Fock ones and better than those obtained from previously proposed trial functions by other authors.'
author:
- 'M. Oulne'
title: 'Analytical solution of the Thomas-Fermi equation for atoms'
---
INTRODUCTION
============
Since the first works of Thomas and Fermi [@1], there have been many attempts to construct an approximate analytical solution of the Thomas-Fermi equation for atoms [@1]. E.Roberts [@2] suggested a one-parameter trial function: $$\begin{aligned}
\phi_{1}(x)=(1+\eta\sqrt{x})e^{-\eta\sqrt{x}},\end{aligned}$$ where $\eta=1.905$ and Csavinsky [@3] has proposed a two-parameters trial function: $$\begin{aligned}
\phi_{2}(x)=(a_{0}e^{-\alpha_{0}x}+b_{0}e^{-\beta_{0}x})^2,\end{aligned}$$ where $a_{0}=0.7218337$, $\alpha_{0}=0.1782559$, $b_{0}=0.2781663$ and $\beta_{0}=1.759339$. Later, Kesarwani and Varshni [@4] have suggested: $$\begin{aligned}
\phi_{3}=(ae^{-\alpha x}+be^{-\beta x}+ce^{-\gamma x})^2,\end{aligned}$$ where $a=0.52495$, $\alpha=0.12062$, $b=0.43505$, $\beta=0.84795$, $c=0.04$ and $\gamma=6.7469$. The equations (2) and (3) are obtained by making use of an equivalent Firsov’s variational principle [@5]. The equation (1) has been modified by Wu [@6] in the following form: $$\begin{aligned}
\phi_{4}=(1+m\sqrt{x}+nx)^2e^{-m\sqrt{x}},\end{aligned}$$ where $m=1.14837$ and $n=4.0187.10^{-6}$. Recently, M. Desaix et al.[@7] have proposed the following expression: $$\begin{aligned}
\phi_{5}=\frac{1}{(1+(kx)^\beta)^\alpha},\end{aligned}$$ where $k=0.4835$, $\alpha=2.098$ and $\beta=0.9238$. Moreover, other attempts have been conducted to solve this problem [@8; @10]. But, all of these proposed trial functions cannot reproduce well the numerical solution of the Thomas-Fermi equation [@11] and its derivative at $x=0$. They didn’t prove efficient when used to calculate the total ionization energy of heavy atoms. In the present work, we propose a new trial function, constructed on the basis of the Wu [@6] function, which reproduces correctly the numerical solution of the Thomas-Fermi equation [@11]. It also gives more precise results for the total ionization energies of heavy atoms in comparison with the previously proposed approximate solutions.
THEORY
======
The Thomas-Fermi method consists in considering that all electrons of an atom are subject to the same conditions: each electron, subject to the energy conservation law, has a potential energy $e\Phi(r)$ where $\Phi(r)$ is the mean value of the potential owed to the nucleus and all other electrons. The electronic charge density $\rho(r)$ and the potential $\Phi(r)$ are related via the Poisson equation: $$\begin{aligned}
\frac{1}{r}\frac{d^{2}}{dr^{2}}(r\Phi(r))=-4\pi\rho(r),\end{aligned}$$ assuming that $\rho(r)$ and $\Phi(r)$ are spherically symmetric. The energy conservation law applied to an electron in the atom gives the following relation: $$\frac{p^{2}}{2m}-e\Phi(r)=E,$$ From the equation (7), we can obtain the maximum of the electron impulsion: $$\begin{aligned}
p=\sqrt{2me\Phi(r)},\end{aligned}$$ where $\Phi(r)$ has to satisfy the boundary conditions: $$\begin{aligned}
\Phi(R)=0,\left(\frac{d\Phi(r)}{dr}\right)
_{R}=\left[\frac{d}{dr}(\frac{eZ}{r})\right]_{R}=-\frac{eZ}{R^{2}},\end{aligned}$$ where R is the radius of a sphere representing the atom. By considering that the contribution of the electrons situated near the nucleus to the potential $\Phi(r)$ is null, we obtain another boundary condition: $$\begin{aligned}
r\Phi(r)\rightarrow eZ \;for\; r\rightarrow 0,\end{aligned}$$ The electronic charge density is defined by the relation: $$\begin{aligned}
\rho=-\frac{8\pi e}{3}\left(\frac{p}{h}\right)^{2},\end{aligned}$$ where p is the electron impulsion and h the Planck’s constant. By combining the relations (8) and (11), we obtain the following expression for the charge density: $$\begin{aligned}
\rho=-\frac{8\pi}{3}\frac{e}{h^{3}}\lbrack2me\Phi(r)\rbrack^{3/2},\end{aligned}$$ To get rid of the numerical constants in the equations, one can perform the following changes: $$\begin{aligned}
x=\frac{r}{a},\phi(x)=\frac{1}{Ze}r\Phi(r),\end{aligned}$$ with $a=a_{B}(\frac{9\pi^{2}}{128Z})^{1/3}$, where $a_{B}=\frac{h^{2}}{4\pi^{2}me^{2}}$ is the first Bohr radius of the hydrogen atom and r is the distance from the nucleus. With these changes, we get from the equations (6) and (13) the differential equation of Thomas-Fermi [@1]: $$\begin{aligned}
\frac{d^{2}\phi}{dx^{2}}=\sqrt{\frac{\phi^{3}}{x}},\end{aligned}$$ with the boundary and subsidiary conditions, obtained from the equations (9) and (10): $$\begin{aligned}
\phi(0)=1,\phi(\infty) = 0,\left(
\frac{d\phi}{dx}\right)_{x\rightarrow\infty}=0,\end{aligned}$$ In this case, the charge density becomes: $$\begin{aligned}
\rho=\frac{Z}{4\pi a^{3}}\left(\frac{\phi}{x}\right)^{3/2},\end{aligned}$$ and must satisfy the condition on the particles number: $$\begin{aligned}
\int\rho dv=Z,\end{aligned}$$ where Z is the number of electrons in neutral atom and dv is the volume element. The use of the variational principle to the lagrangian: $$L(\phi)=\int_{0}^{\infty} Fdx,$$ where: $$F(\phi,\phi^{'},x)=\frac{1}{2}\left(\frac{d\phi}{dx}\right)^{2}+\frac{2}{5}
\left(\frac{\phi^{5/2}}{\sqrt{x}}\right),$$ is equivalent to the equation (14) since substitution of the functional (19) into the Euler-Lagrange equation: $$\begin{aligned}
\frac{d}{dx}\left(\frac{\partial
F}{\partial\phi^{'}}\right)-\frac{\partial F}{\partial\phi}=0,\end{aligned}$$ leads to the Thomas-Fermi equation (14). While solving the Thomas-Fermi problem by using the variational principle, we can assume an infinite number of trial functions which depend on different variational parameters. In this paper, we propose a trial function which depends on three parameters $\alpha$, $\beta$ and $\gamma$: $$\phi(x)=(1+\alpha\sqrt{x}+\beta
xe^{-\gamma\sqrt{x}})^{2}e^{-2\alpha\sqrt{x}},$$
![\[fig:epsart\] Comparison of $\phi$ from Eqs.(1), (2), (3), (4), (5) and (21).](fig_1){height="10cm" width="8cm"}
After inserting the equation (21) into the equations (19) and (18), the lagrangian $L(\phi)$ transforms into an algebraic function $L(\alpha, \beta,\gamma)$ of the variational parameters $\alpha$, $\beta$ and $\gamma$ and the Thomas-Fermi problem turns into minimizing $L(\alpha, \beta,\gamma)$ with respect to these parameters subject to the constraint (17) which is taken into account through a Lagrange multiplier. All calculations, in this work, are performed with the software Maple Release 9.
RESULTS
=======
The optimum values of the variational parameters $\alpha$, $\beta$ and $\gamma$, obtained by minimizing the lagrangian (18) taking into account the subsidiary condition (17), are respectively equal to 0.7280642371, -0.5430794693 and 0.3612163121. The obtained trial function (Eq.(21)), with these universal parameters, reproduces accurately the numerical solution [@11] of the Thomas-Fermi equation (14), in the range $0\leq x\leq50$, in comparison with the equations (1), (2), (3), (4) and (5) as it is shown in Fig. 1 and Tab. I. The mean error of our calculations, calculated on 67 points in the range $0\leq x\leq50$ with respect to the numerical solution, is about 2 % , while the other calculations have a mean error greater than 17 %.
![\[fig:epsart\] Comparison of $\phi$ from Eqs.(1), (2), (3), (4), (5) and (21) in the main region of the screening potential.](fig_2){height="10cm" width="8cm"}
x
------- ---------- ---------- -------- ---------- -------- ---------- -------- ---------- -------- ---------- -------- ---------- -------- --
0 1 1 0.00 1 0.00 1 0.00 1 0.00 1 0.00 1 0.00
0.001 0.9985 0.9983 -0.02 0.9988 0.03 0.9986 0.01 0.9987 0.02 0.9982 -0.03 0.9984 -0.01
0.002 0.9969 0.9966 -0.03 0.9975 0.06 0.9972 0.03 0.9975 0.06 0.9966 -0.03 0.9969 0.00
0.003 0.9955 0.9949 -0.06 0.9963 0.08 0.9958 0.03 0.9962 0.07 0.9950 -0.05 0.9954 -0.01
0.004 0.994 0.9933 -0.07 0.9951 0.11 0.9944 0.04 0.9950 0.10 0.9935 -0.05 0.9939 -0.01
0.005 0.9925 0.9917 -0.08 0.9939 0.14 0.9930 0.05 0.9938 0.13 0.9920 -0.05 0.9924 -0.01
0.006 0.9911 0.9901 -0.10 0.9926 0.15 0.9917 0.06 0.9926 0.15 0.9906 -0.05 0.9910 -0.01
0.007 0.9897 0.9886 -0.11 0.9914 0.17 0.9903 0.06 0.9914 0.17 0.9891 -0.06 0.9895 -0.02
0.008 0.9882 0.9870 -0.12 0.9902 0.20 0.9889 0.07 0.9902 0.20 0.9877 -0.05 0.9881 -0.01
0.009 0.9868 0.9855 -0.13 0.9890 0.22 0.9876 0.08 0.9890 0.22 0.9863 -0.05 0.9867 -0.01
0.01 0.9854 0.9840 -0.14 0.9878 0.24 0.9862 0.08 0.9878 0.25 0.9849 -0.05 0.9853 -0.01
0.05 0.9352 0.9314 -0.41 0.9412 0.64 0.9357 0.06 0.9451 1.06 0.9359 0.07 0.9348 -0.05
0.09 0.8919 0.8874 -0.51 0.8983 0.72 0.8914 -0.05 0.9076 1.76 0.8933 0.16 0.8913 -0.06
0.4 0.6596 0.6609 0.19 0.6557 -0.59 0.6607 0.16 0.6972 5.71 0.6598 0.03 0.6601 0.08
0.8 0.4849 0.4920 1.47 0.4816 -0.68 0.4867 0.36 0.5268 8.63 0.4821 -0.57 0.4858 0.19
1.5 0.3148 0.3233 2.70 0.3276 4.06 0.3136 -0.38 0.3476 10.43 0.3116 -1.01 0.3147 -0.03
5 0.0788 0.0743 -5.71 0.0877 11.25 0.0861 9.30 0.0749 -4.96 0.0838 6.34 0.0774 -1.73
10 0.0243 0.0170 -30.06 0.0147 -39.33 0.0247 1.73 0.0150 -38.11 0.0304 25.01 0.0247 1.44
15 0.0108 0.0052 -51.53 0.0025 -77.04 0.0074 -31.56 0.0041 -62.31 0.0157 45.66 0.0116 7.78
20 0.00578 0.00190 -67.13 0.00042 -92.78 0.00221 -61.72 0.00130 -77.51 0.00965 66.93 0.00653 12.98
37.5 0.00131 0.00011 -91.70 8.14E-07 -99.94 3.25E-05 -97.52 5.03E-05 -96.16 3.17E-03 141.61 0.00140 6.87
45 8.28E-04 3.88E-05 -95.31 5.62E-08 -99.99 5.32E-06 -99.36 1.54E-05 -98.14 2.27E-03 174.18 7.92E-04 -4.37
50 6.32E-04 2.04E-05 -96.77 9.45E-09 -100 1.59E-06 -99.75 7.36E-06 -98.84 1.87E-03 196.02 5.50E-04 -12.90
In the main region of the screening potential of Thomas-Fermi $(0\leq x\leq10)$, our function is even more precise than all other proposed functions as one can see from Fig. 2 and Tab. I. The mean error of our calculations, calculated on 47 points in this region, is equal to 0.28 %, while the Eq.(2) has a mean error equal to 1.13 % and the Eqs.(1), (3), (4) and (5) have a mean error greater than 2.5 %.
The derivative of our function (Eq.(21)) at $x = 0$ is equal to -1.61623647 which is close to the numerical derivative: -1.58807102 [@11]. The relative error is less than 2 %, while the equations (1), (2), (3) and (4) give a result with an error greater than 11 % with respect to the numerical derivative and the Eq.(5) has an infinite derivative at x = 0.\
To test the efficiency of the different trial functions, given by the equations (1), (2), (3), (4) and (21), we have calculated the total ionization energy of heavy atoms following the relation [@12]: $$E=\left(\frac{12}{7}\right)\left(\frac{2}{9\pi^{2}}\right)^{1/3}
\left(\frac{d\phi}{dx}\right)_{x=0}Z^{7/3},$$ in hartrees $(e^{2}/a_{B})$ and the obtained results, presented in Tab. II, are compared with those of Hartree-Fock (HF) [@13]. The Eq.(5) cannot be used because of its infinite derivative at $x=0$. From Tab. II, one can see that our results are fairly better than those obtained from the Eqs.(1), (2), (3) and (4). The precision of our calculations rises with the atomic number Z, on the contrary of the other calculations performed with the Eqs.(1), (2), (3) and (4), so our trial function is more suited for heavy atoms.\
CONCLUSION
==========
The proposed new trial function (Eq.(21)) provides a more satisfactory approximation for the solution of the Thomas-Fermi equation for neutral atoms than all other previousely proposed analytical solutions. The results obtained for the total ionization energies of heavy atoms agree with the Hartree-Fock data and are more precise than those calculated with the Eqs.(1), (2), (3) and (4). The proposed solution (Eq.(21)) can be used to calculate, with high precision, other atomic characteristics of heavy atoms.
Z
----- ------- ------- ------ ------- ------- ------- ------- ------- ------- ------- -----
92 28070 33562 19.6 22864 -18.5 25972 -7.5 24392 -13.1 29894 6.5
93 28866 34419 19.2 23448 -18.8 26636 -7.7 25015 -13.3 30658 6.2
94 29678 35289 18.9 24040 -19.0 27309 -8.0 25647 -13.6 31433 5.9
95 30506 36171 18.6 24641 -19.2 27992 -8.2 26288 -13.8 32219 5.6
96 31351 37066 18.2 25251 -19.5 28684 -8.5 26938 -14.1 33015 5.3
97 32213 37973 17.9 25869 -19.7 29386 -8.8 27598 -14.3 33823 5.0
98 33093 38893 17.5 26495 -19.9 30098 -9.1 28266 -14.6 34643 4.7
99 33990 39825 17.2 27130 -20.2 30819 -9.3 28944 -14.8 35473 4.4
100 34905 40770 16.8 27774 -20.4 31550 -9.6 29631 -15.1 36315 4.0
101 35839 41727 16.4 28426 -20.7 32292 -9.9 30327 -15.4 37168 3.7
102 36793 42698 16.0 29088 -20.9 33042 -10.2 31032 -15.7 38032 3.4
103 37766 43681 15.7 29757 -21.2 33803 -10.5 31746 -15.9 38908 3.0
104 38758 44677 15.3 30436 -21.5 34574 -10.8 32470 -16.2 39795 2.7
105 39772 45686 14.9 31123 -21.7 35355 -11.1 33203 -16.5 40694 2.3
106 40806 46707 14.5 31819 -22.0 36145 -11.4 33946 -16.8 41604 2.0
107 41862 47742 14.0 32524 -22.3 36946 -11.7 34698 -17.1 42525 1.6
108 42941 48790 13.6 33238 -22.6 37757 -12.1 35459 -17.4 43458 1.2
109 44042 49850 13.2 33960 -22.9 38578 -12.4 36230 -17.7 44403 0.8
[90]{} E. D. Grezia and S. Esposito, DSF-6/2004, Physics/04606030. E. Roberts, Phys. Rev. , 8 (1968). P. Csavinsky, Phys. Rev.A , 1688 (1973). R. N. Kesarwani and Y. P. Varshni, Phys. Rev. A , 991 (1981). O. B. Firsov, Zh. Eksp. Teo. Fiz., 696 (1957)\[Sov. Phys.-JETP, 1192 (1957); , 534 (1958)\]. M. Wu, Phys. Rev. A , 57 (1982). M. Desaix, D. Anderson and M. Lisak, Eur. J. Phys. 24 (2004) 699-705. N. Anderson, A. M. Arthurs and P. D. Robinson, Nuovo Cimento , 523 (1968). W. p. Wang and R. G. Parr, Phys. Rev. A , 891 (1977). E. K. U. Gross and R. M. Dreiler, Phys. Rev. A , 1798 (1979). Paul S. Lee and Ta-You Wu, Chinese Journal of Physics, Vol.35, N°6-11, 1997. P. Gombas, *Encyclopedia of Physics*, edited by S. Flügge ( Springer, Berlin, 1956), Vol. XXXVI. L. Visschen and K. G. Dyall, Atom. Data. Nucl. Data. Tabl., (1997) 207.
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abstract: 'Data science and informatics tools have been proliferating recently within the computational materials science and catalysis fields. This proliferation has spurned the creation of various frameworks for automated materials screening, discovery, and design. Underpinning all these frameworks are surrogate models with uncertainty estimates on their predictions. These uncertainty estimates are instrumental for determining which materials to screen next, but there is not yet a standard procedure for judging the quality of such uncertainty estimates objectively. Here we present a suite of figures and performance metrics that can be used to judge the quality of such uncertainty estimates. This suite probes the accuracy, calibration, and sharpness of a model quantitatively. We then show a case study where we judge various methods for predicting density-functional-theory-calculated adsorption energies. Of the methods studied here, we find that the best performer is a model where a convolutional neural network is used to supply features to a Gaussian process regressor, which then makes predictions of adsorption energies along with corresponding uncertainty estimates.'
author:
- Kevin Tran
- Willie Neiswanger
- Junwoong Yoon
- Eric Xing
- 'Zachary W. Ulissi'
bibliography:
- 'uncertainty\_benchmarking.bib'
title: Methods for comparing uncertainty quantifications for material property predictions
---
Introduction
============
The fields of catalysis and materials science are burgeoning with methods to screen, design, and understand materials.[@Medford2018; @Gu2019; @Schleder2019; @Alberi2019] This research has spurned the creation of models to predict various material properties. Unfortunately, the design spaces for these models are sometimes too large and intractable to sample completely. These undersampling issues can limit the training data and therefore the predictive power of the models. It would be helpful to have an for a model so that we know when to trust the predictions and when not to. More specifically: would enable various online, active frameworks for materials discovery and design (e.g., active learning,[@Settles2012] online active learning,[@Chu2011] Bayesian optimization,[@Frazier2018] active search,[@Garnett2012] or goal oriented design of experiments[@Kandasamy]).
Such active frameworks have already been used successfully in the field of catalysis and materials informatics. For example: @Peterson2016 has used a neural network to perform online active learning of calculations, reducing the number of force calls by an order of magnitude. @Torres2018 have also used online active learning to accelerate calculations, but they used a model instead of a neural network. @Jinnouchi2019 have used online active learning to accelerate molecular dynamics simulations. Each of these active methods are underpinned by models with , which have garnered increasing attention themselves.[@Peterson2017; @Musil2019]
The goal of in predictive models is to quantify the relative likelihood of potential true outcomes associated with a predicted quantity, and to accurately assess the probabilities of these outcomes. For example, given an input for which we wish to make a prediction, a predictive method might return a confidence interval that aims to capture the true outcome a specified percentage of the time, or might return a probability distribution over possible outcomes. Performance metrics for predictive methods aim to assess how well a given quantification of the probabilities of potential true outcomes adheres to a set of observations of these outcomes. Some of the performance metrics for predictive are agnostic to prediction performance—they provide an assessment of the uncertainty independent of the predictive accuracy (i.e. a method can predict badly, but could still accurately quantify its own uncertainty).
To our knowledge though, we have seen few[@Janet2019] comparisons of different methods for within the field of catalysis and materials informatics. Here we attempt to establish a protocol for comparing the performance of different modeling and methods (Figure \[fig:overview\]). We then illustrate the protocol on a case study where we compare various models’ abilities to predict calculated adsorption energies. We also offer anecdotal insights from our case study. We acknowledge that such insights may not be transferable to other applications, but we still find value in sharing these results so that others can build their own intuition from them.
![Overview of proposed procedure for judging the quality of models with uncerainty estimates. First and foremost, the models should be accurate. Second, the models should be “calibrated”, which means that their uncertainty estimates should be comparable with their residuals. Third, the models should be “sharp”, which means that their uncertainty estimates should be narrow. This study will show how to visualize and quantify all three of these characteristics so that different methods of can be compared objectively.[]{data-label="fig:overview"}](intro/intro.pdf){width="75.00000%"}
Methods
=======
Data handling
-------------
All regressions in this paper were performed on a dataset of calculated adsorption energies created with [@Tran2018; @Tran2018a]. These data included energies from 21,269 different H adsorption sites; 1,594 N sites; 18,437 CO sites; 2,515 O sites; and 3,464 OH sites; totaling in 47,279 data points. performed all calculations using [@Kresse1993; @Kresse1994; @Kresse1996; @Kresse1996a] version 5.4 implemented in [@HjorthLarsen2017]. The functionals[@Hammer1999] were used along with ’s pseudopotentials, and no spin magnetism or dispersion corrections were used. Bulk relaxations were performed with a $10\times10\times10$ k-point grid and a 500 cutoff, and only isotropic relaxation were allowed during this bulk relaxation. Slab relaxations were performed with k-point grids of $4\times4\times1$ and a 350 cutoff. Slabs were replicated in the X/Y directions so that each cell was at least 4.5 Å wide, which reduces adsorbate self-interaction. Slabs were also replicated in the Z direction until they were at least 7 Å thick, and at least 20 Å of vacuum was included in between slabs. The bottom layers of each slab were fixed and defined as those atoms more than 3 Å from the top of the surface in the scaled Z direction.
To split the data into train/validate/test sets, we enumerated all adsorption energies on monometallic slabs and added them to the training set manually. We did this because some of the regression methods in this paper use a featurization that contains our monometallic adsorption energy data[@Tran2018], and so having the monometallic adsorption energies pre-allocated in the training set prevented any information leakage between the training set and validation/test sets. After this allocation, we performed a 64/14/20 train/validate/test split that was stratified[@Thompson2012] by adsorbate. We then used the validation set’s results to tune various hyperparameters manually. After tuning, we calculated the training set results and present them in this paper exclusively. Note that the test results were obtained using models that were trained only using the training set, not the validation set. This is acceptable because we only seek to compare methods here, not to optimize them.
Regression methods
------------------
We explore a number of predictive methods that aim to quantify the uncertainty for regression procedures, where the predicted quantity is a continuous variable. In order to standardize the assessment of performance, we ensure that each method returns predictive uncertainty results in a consistent format: a distribution over possible outcomes of the predicted quantity for any specified input point. This result format allows us to compute all of the predictive uncertainty performance metrics which we will introduce in subsequent sections. Figure \[fig:methods\] illustrates all of the methods we investigate in this study, and we describe each method in detail below.
![Overview of the various methods we investigated in this study. $\Delta E$ represents -calculated adsorption energies; $\Delta \hat{E}$ represents -predicted adsorption energies; $UQ$ represents -predicted uncertainty quantifications; $\mu$ represents the mean of a sample of points; $\sigma$ represents the standard deviation of a sample of points; $\epsilon$ represents the residuals between and ; and $\hat{\epsilon}$ represents the residuals between -predicted $\epsilon$ and the actual $\epsilon$. []{data-label="fig:methods"}](methods/methods.pdf){width="80.00000%"}
**:** To establish a baseline for predictive accuracy, we re-trained a previously reported [@Xie2018; @Back2019] on this study’s training set. This model projects a three-dimensional atomic structure into a graph, which is then fed into a convolutional neural network to predict various properties. In this case, we predict -calculated adsorption energies, $\Delta$E. Reference @Back2019 for additional details.
**Ensemble:** We created an ensemble of s by K-fold subsampling the training data into five different folds and then training individual models on the K sets of K-1 folds, where we used a unique random initialization for each model. For the final prediction of the ensemble we computed the mean of the set of models’ predictions, and for the ensemble’s estimate of uncertainty we computed the standard deviation of the set of predictions.
**:** The aim of is to determine the posterior distribution of model parameters rather than a single optimal value of the parameters. In practice, inferring true posterior is very difficult and even infeasible in most cases. Thus, we approximate the model posterior to be as close as possible to the true posterior by doing variational inference. This could be done by training the to minimize the Kullback-Leibler divergence between the variational distribution and the true posterior. We sampled the model parameters K times from the approximated posterior, and used the mean of these predictions as the final prediction and the standard deviation of these predictions as the estimation of uncertainty. We implemented the and performed variational inference using Pyro.[@Bingham2018]
**:** Suppose we have trained a . We may aim to empirically fit an additional mapping that predicts the error of the first . Here we show , which trains a secondary to predict the residuals of the initial . When training the first , we hold out 10% of the training data. Afterwards, we use the residuals of the initial on the held-out portion as training data for the second . After the secondary training, this second can predict residuals for the first on some new set of input data. The predictions of the second can then be used as uncertainty estimates.
**:** s are one of the most common regression methods for producing s, and so we use them here as a baseline. We fit a standard using the same exact features that we used in previous work.[@Tran2018] These features are defined by the elements coordinated with the adsorbate and by the elements of its next-nearest neighbors. Specifically: We use the atomic numbers of these elements, their Pauling electronegativity, a count of the number of atoms of each element near the adsorbate, and the median adsorption energy between the adsorbate and the elements. To ensure that these features interacted well with the ’s kernel, we normalized each of the features to have a mean of zero and standard deviation of one. Reference @Tran2018 for additional details. To define the , we assumed a constant mean and used a Matern covariance kernel. We trained the length scale of the Matern kernel using the method. All training and predictions were done with GPU acceleration as implemented in GPyTorch.[@Gardner2018].
**:** s are Bayesian models in which a prior distribution is first specified and then updated given observations to yield a posterior distribution. The mean of this posterior distribution is used for regression, and the covariance matrix is used for . Typically, in lieu of any additional prior knowledge, practitioners will take the prior distribution to have zero-mean. However, we could instead supply an alternative curve for the prior mean, and then perform the usual Bayesian updates to compute the posterior of this given observations. Here, for the prior mean, we supply the prediction given by a single pre-trained . We call this method . For the , we used a Matern covariance kernel, where we fit the kernel hyperparameters using . All training and predictions were done with GPU acceleration as implemented in GPyTorch.[@Gardner2018].
**:** A limitation of using this formulation of a with -predicted mean is that it requires the use of hand-crafted features for the . This requirement reduces the transferability of the method to other applications where such features may not be readily available. To address this, we formulated a different method whereby we first train a neural network on the learning task (i.e., predict adsorption energies), and then we use the pooled outputs of the convolutional layers of the network as features in a new . The would then be trained to use these features to produce both mean and uncertainty predictions on the adsorption energies. We call this a . In this case, we used the baseline as the network from which we obtained the convolution outputs. We also normalized the convolution outputs of the so that each output would have a mean of zero and a standard deviation of one. To define the , we assumed a constant mean and used a Matern covariance kernel. We trained the length scale of the Matern kernel using the method. All training and predictions were done with GPU acceleration as implemented in GPyTorch.[@Gardner2018].
Performance metrics
-------------------
We used six different metrics to quantify the accuracy of the various models: , , , , and . We used because is insensitive to outliers and is therefore a good measure of accuracy for the majority of the data. We used because it is sensitive to outliers and is therefore a good measure of worst-case accuracy. We used because it lies between and in terms of sensitivity to outliers. We used and because they provide normalized measures of accuracy that may be more interpretable for those unfamiliar with adsorption energy measurements in . values were calculated with Equation \[eq:marpd\] where $n$ is the index of a data point, $N$ is the total number of data points, $x_n$ is the true value of the data point, and $\hat{x}_n$ is the model’s estimate of $x_n$. In this case, $x_n$ is a DFT-calculated adsorption energy and $\hat{x}_n$ is the surrogate-model-calculated adsorption energy. The ensemble of these metrics provide a more robust view of accuracy than any one metric can provide alone.
$$\label{eq:marpd}
MARPD = \frac{1}{N} \sum_{n=1}^{N} \abs{100 \cdot \frac{\hat{x}_n - x_n}{\abs{\hat{x}_n} + \abs{x_n}}}$$
To assess the calibration (or “honesty”) of these models’ UQs, we created calibration curves. A calibration curve “displays the true frequency of points in each \[prediction\] interval relative to the predicted fraction of points in that interval”, as outlined by @Kuleshov2018. In other words: We used the standard deviation predictions to create Gaussian-shaped prediction intervals around each test point, and then we compared these intervals to the models’ residuals at these points. If the residuals tended to fall outside the prediction intervals too often, then the UQs were considered overconfident. If the residuals tended to fall inside the prediction intervals too often, then the UQs were considered underconfident. Thus “well-calibrated” models had residuals that created a Gaussian distribution whose standard deviation was close to the model’s predicted standard deviations. We discuss calibration curves in more detail in the Results section alongside specific examples.
As @Kuleshov2018 also pointed out, well-calibrated models are necessary but not sufficient for useful UQs. For example: A well-calibrated model could still have large uncertainty estimates, which are inherently less useful than well-calibrated and small uncertainty estimates. This idea of having small uncertainty estimates is called “sharpness”, and @Kuleshov2018 define it with Equation \[eq:og\_sharpness\]
$$\label{eq:og_sharpness}
sha = \frac{1}{N} \sum_{n=1}^{N} var(F_n)$$
where $var(F_n)$ is the variance of the cumulative distribution function $F$ at point $n$. This is akin to the average variance of the uncertainty estimates on the test set. Here we propose and use a new formulation (Equation \[eq:sharpness\]) where we add a square root operation. This operation gives the sharpness the same units as the predictions, which provides us with a more intuitive reference. In other words: Sharpness is akin to the average of the -predicted standard deviations.
$$\label{eq:sharpness}
sha = \sqrt{\frac{1}{N} \sum_{n=1}^{N} var(F_n)}$$
We also assessed the performance of each predictive uncertainty method by comparing their values on a held out set of data. For each test point, we established a Gaussian probability distribution using the mean and uncertainty predictions of each model. Then we calculated the conditional probability of observing the true value of the test point given the probability distribution created from the ; this is the likelihood of one test point. We then calculated the product of all the likelihoods of all test points, which yielded the total test likelihood. It follows that better methods yield higher total likelihood values. Equivalently, we could calculate the natural logarithms of each likelihood, sum them, and then take the negative of this value; this is . Equation \[eq:nll\] shows how we calculated , where $y_i$ is the true value of a test point, $\hat{y}_i$ is a model’s predicted mean value at that test point, $\hat{\sigma_i}^2$ is the model’s predicted variance at that test point, and $N$ is the set of all test points.
$$\label{eq:nll}
NLL = - \sum_{i=1}^{N} \ln{P(y_i | N(\hat{y_i}, \hat{\sigma_i}^2))}$$
Note how the value depends on the size and location of the test set. This means that the absolute value of changes from application to application, and so a “good” value must be contextualized within a particular test set. In general though, a lower value indicates a better fit. We also note that we assumed Gaussian distributions for our methods’ predictions. This assumption does not necessarily need to be applied, meaning that the normal distribution in Equation \[eq:nll\] may be replaced with any other appropriate distribution.
We use because it provides an overall assessment that is influenced by both the predictive accuracy of a method as well as the quality of its . Previous work [@Gneiting2007; @Dawid2014] has shown the to be a strictly proper scoring rule, which intuitively means that it provides a fair quantitative assessment (or score) for the performance of the method, and that it can be decomposed into terms that relate to both calibration and sharpness. is also a popular performance metric that has been used to quantify uncertainty in a variety of prior work [@Lakshminarayanan2017] and provides an additional single score for methods.
Results
=======
Illustrative examples
---------------------
Let us first discuss the results of our ensemble for illustrative purposes. Figure \[fig:results\_example\] contains a parity plot, calibration curve, and predicted-uncertainty distribution of our ensemble model. The parity plot shows the accuracy of the model; the calibration curve shows the honesty of the model’s uncertainty predictions; and the uncertainty distribution shows the sharpness of model’s uncertainty predictions. Accurate models have parity plots whose points tend to fall near the diagonal parity line. Calibrated models have calibration curves that approach the ideal diganoal line. Sharp models have uncertainty distributions that tend towards zero. Note that sharpness should not be won at the cost of calibration.
[0.32]{} ![Results of the ensemble. Each figure here was created with the test set of 8,289 points.[]{data-label="fig:results_example"}](NN_ensemble/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Results of the ensemble. Each figure here was created with the test set of 8,289 points.[]{data-label="fig:results_example"}](NN_ensemble/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Results of the ensemble. Each figure here was created with the test set of 8,289 points.[]{data-label="fig:results_example"}](NN_ensemble/sharpness.pdf "fig:"){width="\textwidth"}
The calibration curve was created by first normalizing all test residuals by their respective uncertainty values, i.e. we divided the residuals by their predicted standard deviations. If we assume that the normalized residuals follow a Gaussian distribution, then 68% of the normalized residuals would fall between \[-1, 1\], 95% of them would fall between \[-2, 2\], 99% of them would fall between \[-3, 3\], etc. To challenge this assumption, we plotted the actual fraction of points within each prediction interval against the expected fraction of points. This plot is a calibration curve. Thus a perfectly calibrated model would have normalized residuals that are perfectly Gaussian, which would yield a diagonal calibration line. Therefore, models’ calibration could be qualified by the closeness of their calibration curves to this ideal, diagonal curve. We quantified this closeness by calculating the area between the calibration curve and the ideal diagonal. We call this the miscalibration area, and smaller values indicate better calibration.
The shape of a calibration curve could also yield other insights. If a model’s s were too low/confident, then the normalized residuals would be too large and they would fall outside their expected prediction intervals too frequently. This would result in a lower fraction of points observed within the expected prediction intervals, which would correspond to a calibration curve that falls below the ideal diagonal. Therefore, overconfident models yield calibration curves that fall under the ideal diagonal, and underconfident models yield calibration curves that fall over the ideal diagonal. Figure \[fig:error\_bars\] illustrates this point by plotting calibration curves of various models alongside their parity plots that contain error bars corresponding to $\pm$2 standard deviations.
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](NN_ensemble/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](GP/Matern/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](CFGP/Matern/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](NN_ensemble/error_bar_parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](GP/Matern/error_bar_parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves and parity plots of an overconfident ensemble, an underconfident , and better-calibrated . The vertical uncertainty bands in the parity plots indicate $\pm$2 standard deviations in the uncertainty predictions of each model. For clarity, we sampled only 20 points of the 8,289 test points to put in the parity plots. It follows that relatively overconfident models would have more points with uncertainty bands that do not cross the diagonal parity line; relatively underconfident models would have more points that cross the diagonal parity line; and a well-calibrated model would have *ca.* 19 out of 20 points cross the parity line.[]{data-label="fig:error_bars"}](CFGP/Matern/error_bar_parity.pdf "fig:"){width="\textwidth"}
Summary results
---------------
Figure \[fig:parity\] contains parity plots for all methods studied here; Figure \[fig:calibration\] contains all calibration curves; and Figure \[fig:sharpness\] contains all distribution plots of the -predicted s. These figures illustrate the accuracy, calibration, and sharpness of the different methods, respectively. Table \[tab:results\] lists their performance metrics.
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](NN_ensemble/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](BNN/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](NNdNN/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](GP/Matern/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](NNdGP/Matern/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Parity plots for all methods used in this study. Shading plots were used in lieu of scatter plots because the large number of test points (8,289) obfuscated patterns. Darker shading indicates a higher density of points. Logarithmically scaled shading was used to accentuate outliers. The dashed, diagonal lines indicate parity.[]{data-label="fig:parity"}](CFGP/Matern/parity.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](NN_ensemble/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](BNN/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](NNdNN/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](GP/Matern/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](NNdGP/Matern/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Calibration curves for all methods used in this study. Dashed, blue lines indicate perfect calibration while solid orange lines indicate the experimental calibration of the test set. The blue, shaded area between these lines is defined as the miscalibration area. []{data-label="fig:calibration"}](CFGP/Matern/calibration.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](NN_ensemble/sharpness.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](BNN/sharpness.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](NNdNN/sharpness.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](GP/Matern/sharpness.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](NNdGP/Matern/sharpness.pdf "fig:"){width="\textwidth"}
[0.32]{} ![Distribution plots of the -predicted standard deviations for each method. Sharpness values are indicated by vertical lines.[]{data-label="fig:sharpness"}](CFGP/Matern/sharpness.pdf "fig:"){width="\textwidth"}
Method MisCal Sha \[$\cdot$10^3^\]
---------- ------ ------ ------ ---- ------ -------- ------ ------------------
0.11 0.19 0.34 61 0.80 N/A N/A N/A
ensemble 0.11 0.18 0.32 59 0.82 0.23 0.14 192.08
0.11 0.19 0.31 59 0.83 0.41 0.03 669.61
0.11 0.19 0.34 59 0.80 0.10 0.16 18.61
0.11 0.21 0.39 61 0.73 0.28 0.65 6.41
0.11 0.19 0.33 59 0.81 0.05 0.21 6.09
0.11 0.19 0.33 59 0.80 0.06 0.24 2.80
: Performance metrics for all methods used in this study, which include: , , , , , miscalibration area (MisCal), sharpness (Sha), and . The units of , , , and sharpness are all in . The units of are in %. The miscalibration area and are unitless.[]{data-label="tab:results"}
Regarding accuracy: All methods’ results are virtually identical, and their results are within 10% of each other. This suggests that all methods have comparable predictive accuracies for inliers. The plain has a higher value than the rest of the methods though. Thus our results suggest that our -based methods are more accurate at outlier prediction than our sole non--based method. Each of our -based methods may not have practically different accuracies between each other though.
Regarding calibration: The ensemble and are overconfident; the is underconfident; and the , , and models are relatively calibrated. The three more calibrated methods all share a characteristic that the other methods do not: They all start with a that is dedicated for prediction alone, and then they end with some other in-series method to estimate uncertainty. Interestingly, this in-series method of learning predictions and then learning uncertainties is similar in spirit to how deep networks “learn in stages” at each subseqent hidden layer.
Regarding sharpness: The ensemble and yield the sharpest uncertainties, although both do so at the cost of calibration. Among the three more calibrated models, the yields the lowest sharpness of 0.16 eV while the and yield sharpnesses of 0.21 and 0.24 eV, respectively. Note how -based methods tend to yield less sharp uncertainties than methods based purely on s. This suggests that s may yield more conservative s.
Regarding : The method yields the best (i.e., lowest) value of *ca.* 2,800 while both the and models yield relatively moderate values of *ca.* 6,000. Note how the under-confident model has a worse miscalibration area and sharpness than the but a better value. Simultaneously, the two most over-confident models ( ensemble and ) yield the worst results. This shows that better values correlate with relatively conservative estimates of , but not with relatively liberal estimates. In other words: If we use as our main performance metric, then we will favor under-confident estimates in lieu of over-confident estimates.
Given the performance metrics for accuracy, calibration, sharpness, and , we expect the method to yield the best performing model for our dataset. It yields the best while maintaining a relatively competitive accuracy and calibration with a moderate sharpness.
When choosing methods for different applications though, other factors should be considered. For example: Although the method performed relatively well, it required the use of a hand-crafted set of features. If future researchers wish to use the method to predict other properties from atomic structures, they may have to define their own set of features. This process of feature engineering is non-trivial and varies from application to application. In some cases, it may be easier to use a method that does not require any additional features beyond the input, such as or .
Another factor to consider is the overhead cost of implementation. For example: The ensemble method is arguably the simplest -based method used here and may be the easiest or fastest method to implement. Conversely, ensembles also have a higher computational training cost than some of the other methods used here, such as or . This high training cost is exacerbated if the ensemble is meant to be used in an active framework where the model needs to be trained continuously. As another example: The method yielded perhaps the worst results of all the methods studied here. It could be argued that further optimization of the could have resulted in higher performance. But creation and training of s is still an active area of research with less literature and support than non-Bayesian s or s. This lack of support led to us spending nearly twice as long creating a compared to the other methods. It follows that further optimization of the would be non-trivial and may not be worth the overhead investment.
Conclusions
===========
We outlined a procedure for comparing different methods for . This procedure considers the accuracy of each method, the honesty of their uncertainty estimates (i.e., their calibration), and the size of their uncertainty estimates (i.e., their sharpness). To assess accuracy, we outlined a common set of error metrics such as or , among others. To assess calibration, we showed how to create, interpret, and quantify calibration curves. To assess sharpness, we showed how to calculate and plot sharpness. To assess all three aspects simultaneously, we suggest using the as a performance metric. The ensemble of all these metrics and figures can then be used to judge the relative performance of various UQ methods in a holistic fashion.
As a case sutdy, we tested six different methods for predicting calculated adsorption energies with . The best performing method was a , which used a pre-trained convolutional output from a as features for a subsequent to make uncertanity estimates. Our studies also showed that the -based methods we tested tended to yield higher and more conservative uncertainty estimates than the methods that used only s and derivatives. We also found that in-series methods tended to yield more calibrated models—i.e., methods that use one model to make value predictions and then a subsequent model to make uncertainty estimates were more calibrated than models that attempted to make value and uncertainty predictions simultaneously. These results are limited to our dataset though. Results may vary for studies with different applications, different models, or different hyperparameters. But the underpinning procedure we used to compare each of these models is still broadly applicable.
Note that it would be possible to recalibrate[@Kuleshov2018] each of the models in this study to improve their uncertainty estimates. We purposefully ommitted recalibration in this study to (1) simplify the illustration of the assessment procedure; (2) assess the innate performance of each of these UQ methods without confounding with recalibration methods; and (3) reduce overhead investment. Future work should consider recalibration if the feasible UQ methods provide insufficiently calibrated uncertainty predictions
Future work may also consider uncertainty propagation from the underlying training data. For example: If we used the ,[@Wellendorff2012] then our calculated adsorption energies would have been distributions rather than than single point estimates. Such distributions could be propagated to certain surrogate models, e.g., as a variable-variance kernel in a -type method.
Code availability {#code-availability .unnumbered}
=================
Visit `https://github.com/ulissigroup/uncertainty_benchmarking` for the code used to create the results discussed in this paper. The code dependencies are listed inside the repository.
Author information {#author-information .unnumbered}
==================
Corresponding author email: zulissi@andrew.cmu.edu. The authors declare no competing financial interest.
Acknowledgements {#acknowledgements .unnumbered}
================
This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. We also acknowledge the exceptional documentation and clear examples in the GPyTorch[@Gardner2018] repository, which formed the basis on much of the code used for this work.
|
---
abstract:
- |
We investigate the liquid-vapor interface of the restricted primitive model for an ionic fluid using a density functional approach. The applied theory includes the electrostatic contribution to the free energy functional arising from the bulk energy equation of state and the mean spherical approximation for a restricted primitive model, as well as the associative contribution, due to the formation of pairs of ions. We compare the density profiles and the values of the surface tension with previous theoretical approaches.
density functional, adsorption, chains, crystals 68.08.-p, 68.43.Fg, 82.35.Gh, 68.43.-h
- |
=3000Ми досліджуємо міжфазну границю рідина-пара обмеженої примітивної моделі іонного плину, використовуючи теорію функціоналу густини. Застосована теорія включає електростатичний вклад у функціонал вільної енергії, який виникає з енергетичного рівняння стану для об’єму, вклад від середньо-сферичного наближення для обмеженої примітивної моделі, а також асоціативний вклад, який виникає в результаті врахування утворення іонних пар. Ми порівнюємо профілі густини і значення поверхневого натягу з результатами попередніх теоретичних підходів.
функціонал густини, адсорбція, ланцюжки, кристали
author:
- 'A. Patrykiejew, S. Sokołowski, O. Pizio[^1]'
- 'А. Патрикеєв, С. Соколовскі, О. Пізіо'
date:
- 'Received September 13, 2010, in final form November 23, 2010'
- 'Отримано 13 вересня 2010р., в остаточному вигляді — 23 листопада 2010р.'
title:
- 'The liquid-vapor interface of the restricted primitive model of ionic fluids from a density functional approach'
- 'Міжфазна границя рідина-пара обмеженої примітивної моделі іонних плинів з теорії функціоналу густини'
---
Introduction
============
According to the simplest model of ionic fluids, called the restricted primitive model (RPM), the fluid consists of charged hard spheres (at total density $\rho$) of equal diameter, $d$, half of which carry a charge of $+q$, while the other half carry a charge of $-q$. These spheres are immersed in a dielectric medium of dielectric constant $\epsilon$. Reduced thermodynamic quantities, temperature and density, appropriate to this model are defined as follows $T^*=kT\epsilon d/q^2$ and $\rho^*=\rho d^3$.
Perhaps, the first successful statistical-mechanical description of this model was proposed by Debye and Hückel [@DH] about ninety years ago. The Debye-Hückel approach predicted the existence of the first-order transition for RPM and was subsequently applied in several works, e.g., to describe phase behavior of three-component ionic fluids [@NNN]. More recently, the bulk RPM was studied by means of the Ornstein-Zernike (OZ) equation supplemented by the mean spherical approximation (MSA) [@1; @2; @2a], which permits to obtain an analytical solution for the pair correlation functions. The MSA correlation functions can be used to evaluate an equation of state [@WL; @WL1]. In spite of quite reasonable results obtained for the structure and thermodynamics in a general sense, neither virial nor compressibility equations of state permit to obtain gas-liquid transition for the RPM, which contradicts the computer simulations [@3; @4; @5; @6; @SYM; @7; @new7; @new8]. By contrast, the equation of state resulting from the *energy* route predicts the first-order phase transition between an ionic vapor and a dense ionic liquid [@WL; @WL1]. However, there is a big discrepancy between the values of critical temperature and critical density of MSA theory and the critical parameters resulting from computer simulations. Moreover, a more sophisticated generalized mean-spherical approximation [@8], which was designed to reconcile thermodynamic inconsistency of the MSA, does not lead to a better agreement of the critical parameters with simulations as well.
Since both MSA and generalized MSA theories underestimate the critical density and overestimate the critical temperature, attempts were made to include *ad hoc* the effect of ion pairing (association) into the RPM, which is most pronounced along the vapor branch of the coexistence envelope and near the critical point [@7; @9]. The inclusion of the association into the theory improves the agreement of theoretical predictions for the critical constants with simulations (cf. table I of [@7]). However, the overall shape of the coexistence envelope coming from several theoretical approaches is not well described compared to simulation data. In particular, the liquid densities along coexistence are poorly described for different temperatures. We should bear in mind that the Hamiltonian of the model with association used in the theoretical developments differs from the genuine RPM used in simulations. Recent developments in the modelling of different types of associations and clustering in liquids and solutions have been comprehensively and critically reviewed by Holovko in [@hol1].
The liquid-vapor interface is one of the simplest examples of nonuniform systems. However, compared to the case of Lennard-Jones fluids [@LJ], much less is known about the liquid-vapor interface in the RPM. Theoretical work on the gas-liquid interface for RPM was pioneered by Telo da Gama et al. [@Gama], that used the gradient expansion method [@Gradient]. The problem of describing an ionic liquid-ionic vapor interface has been also studied by Groh et al. [@Groh], who proposed a density functional approach to evaluate the surface tension and the density profiles through the interface. Their approach involved a local density approximation for the hard-sphere part of the free-energy functional and a nonlocal treatment of Coulombic contributions. The latter term was evaluated approximating the inhomogeneous pair correlation functions by their bulk counterparts. There are significant differences between the theory used by Groh et al. [@Groh] and the density functional approaches that were proved successful in the description of electrical double layers at charged walls [@13; @17; @18; @19; @21; @21a; @21b; @21c; @21d]. Namely, the theories of [@13; @17; @18; @19; @21] treat the electrostatic part of the free energy by means of a second-order density expansion about the density of a reference fluid, which was taken to be the homogeneous bulk fluid far from the surface. Although such an approach can be appropriate for many purposes, but it is problematical when it comes to a liquid-vapor interface, where two bulk fluids are involved. Moreover, it is known that in the case of Lennard-Jones fluids, the corresponding second-order expansion with respect to a homogeneous fluid fails to account for the liquid-vapor coexistence. More recently, the description of a liquid-vapor interface was the subject of investigations of Weiss and Schröer [@23]. Using a square-gradient type theory they computed the density profiles and the interfacial tension at different temperatures using Debye-Hückel theory and its extension to ion-pair formation, as well as adding the free energy term describing correlations between ion pairs as entities and free ions, as developed by Fisher and Levin [@9].
Any density functional theory determines thermodynamic properties of an inhomogeneous fluid from the Helmholtz free energy $F$ and its functional dependence on the local densities $\{\rho _{i}(\mathbf{r})\}$. The free energy functional is commonly decomposed into the sum of three contributions, namely into the ideal, the hard-sphere, and the electrostatic terms. Of course, one can also include here an additional term, due to possible association of ions. Various formulations of the DFT have been discussed in the literature. As we have already stressed, a majority of the DFTs have followed perturbational second-order expansion of the electrostatic free energy functional with respect to a bulk homogeneous fluid [@13; @17; @18; @19; @21].
Recently, Gillespie et al. [@24; @25] proposed a version of the electrostatic free energy functional that replaces a uniform reference system with a suitably chosen position-dependent reference fluid. The inhomogeneous reference fluid densities are then computed from the local densities by a self-consistent iteration procedure. Actually, Gillespie et al. [@24; @25] proposed a reference fluid density functional which permits to construct a reference model that locally satisfies electroneutrality condition and has the same ionic strength at every point as the inhomogeneous fluid in question. Such a construction of the reference fluid permits to apply the expression for the electrostatic contribution to the free energy, which results from the equation of state for a bulk ionic system and makes it unnecessary to employ the direct correlation functions as input quantities. This kind of approach was proposed by us to study liquid-vapor transitions in RPMs confined to slit-like pores [@21d; @pizio]. The results of our approach reasonably well reproduce the structure of ionic fluids at a charged wall at sufficiently high temperatures and correctly predict [@reszko] the temperature dependence of the double layer capacitance, in agreement with experiments and simulation data [@boda]. Quite recently, the theory was also extended to include the effects of association between unlike ions [@pizio1]. However, the applications of the approach described above would not be complete without exploring liquid-vapor interface of ionic fluid. Therefore, in this communication we intend to apply the theory of [@pizio; @reszko; @pizio1] to the study of liquid-vapor interface of the RPM. We calculate the density profiles and the interfacial tension. Moreover, we would like to investigate the effect of association leading to the formation of ion pairs and whether this type of effects improves the description of the interfacial properties similarly to the bulk structure and thermodynamics.
Theory
======
We consider a binary mixture of ionic species. The symbols $d$, $Z_{i}=q_i/e$ and $\mu _{i}$ denote, respectively, the hard-sphere diameter, valence of ions and the chemical potential of species $i=1,2$, and $e$ is the electron charge. The interaction between the ions is $$u_{ij}(r)=\left\{
\begin{array}{ll}
\infty , & r<1, \\
{e}^{2}Z_{i}Z_{j}/\varepsilon r, & r>1,
\end{array}
\right. \label{1}$$ where $r$ is the distance between a pair of ions. We also assume that $Z_{1}=-Z_{2}=1$ and that the dielectric constant $\varepsilon $ is uniform throughout the entire system.
The DFT we use in this work is identical to that described in [@pizio; @pizio1] and therefore we present only its brief description. If there is no external potential field the grand potential of the system is written in the form, $$\Omega =\int \rd\mathbf{r}f[\{\rho _{i}\}]-\sum_{i=1,2}\int \rho _{i}(z)\mu _{i}\rd\mathbf{r}.$$ According to the usual density functional treatment, the free energy density functional, $f[\{\rho _{i}\}],$ is decomposed into ideal, hard sphere, electrostatic and associative terms $ f[\{\rho
_{i}\}]=f_{\mathrm{id}}[\{\rho _{i}\}]+f_{\mathrm{hs}}[\{\rho
_{i}\}]+f_{\mathrm{el}}[\{\rho _{i}\}] + f_{\mathrm{as}}[\{\rho
_{i}\}]$. The ideal term is $ f_{\mathrm{id}}[\{\rho
_{i}\}]=\sum_{i=1,2} [\rho _{i}(z)\ln \rho _{i}(z)-\rho _{i}(z)]$, whereas for the hard-sphere contribution we apply an expression resulting from a recent version [@28] of the Fundamental Measure Theory [@15; @16], with the free energy density consisting of terms dependent on scalar and on vector weighted densities, $n_{\alpha }(\mathbf{r})$ ($\alpha =0,1,2,3$) and $\mathbf{n}_{\alpha }(\mathbf{r})$ ($\alpha =V1,V2$) [@28], $$f_{\mathrm{(hs)}}=-n_{0}\ln (1-n_{3})+\frac{n_{1}n_{2}-\mathbf{n}_{V1}\cdot \mathbf{n}%
_{V2}}{1-n_{3}}+n_{2}^{3}(1-3\xi ^{2})\frac{n_{3}+(1-n_{3})^{2}\ln (1-n_{3})%
}{36\pi n_{3}^{2}(1-n_{3})^{2}}\;, \label{eq:7}$$ where $\xi
(\mathbf{r})=|\mathbf{n}_{V2}(\mathbf{r})|/n_{2}(\mathbf{r})$. The definitions of weighted densities, $n_{\alpha }(\mathbf{r}),$ $\alpha =0,1,2,3,V1,V2$, are given in [@15; @16].
The electrostatic contribution is evaluated from the approach described in [@pizio; @pizio1], according to which $$f_{\mathrm{el}}[\{\rho _{i}\}]/kT=-\frac{e^{2}}{T^{\ast }}[Z_{1}^{2}\bar\rho_1
+Z_{2}^{2}\bar\rho _{2}]\frac{\Gamma(\{\bar\rho_i\}) }{1+\Gamma(\{\bar\rho_i\}) d }
+\frac{\Gamma ^{3}(\{\bar\rho_i\})}{3\pi }\,,$$ where $\bar{\rho}_{i}(z)$ denote suitably defined inhomogeneous average densities of a reference fluid. The form for $f_{\mathrm{el}}[\{\bar{\rho}_{i}(z)\}]$ results from the MSA equation of state evaluated via the energy route [@1; @2; @3; @7; @WL; @WL1]. In the above $ \Gamma(\{\bar\rho_i\})
=(\sqrt{1+2\kappa(\{\bar\rho_i\}) d }-1)/2d, $ where $\kappa(\{\bar\rho_i\}) $ denotes the inverse Debye screening length, $ \kappa ^{2}(\{\bar\rho_i\})=(4\pi e^{2}/T^{\ast
})[Z_{1}^{2}\bar\rho _{1}+Z_{2}^{2}\bar\rho _{2}]. $
The reference fluid densities $\{\bar\rho_{i}\}$ are evaluated by employing the approach of Gillespie et al. [@24; @25]. In the case of gas-liquid interface, the spatial symmetry of the RPM implies that the density profiles of the two ionic species should be the same. Therefore, in the presently considered situation, the approach of Gillespie et al. [@24; @25] is simplified and the reference fluid densities are given by $$\bar{\rho}_{i}(z)=\int \rho _{i}(z)W_{i}(|\mathbf{r}-\mathbf{r^{\prime }}%
|)\rd\mathbf{r^{\prime }},$$ where the weight function $W_{i}(r)$ is just a normalized step function $$W_{i}(|\mathbf{r}-\mathbf{r^{\prime }}|)=\frac{\theta (|\mathbf{r}-\mathbf{%
r^{\prime }}|-R_{f}(\mathbf{r^{\prime }}))}{(4\pi /3)R_{f}^{3}(\mathbf{%
r^{\prime }})}\,.$$ The radius of the sphere over which the averaging is performed, $R_f$, is approximated by the “capacitance” radius, i.e., by the ion radius plus the screening length $$R_{f}(\mathbf{r})=\frac{d}{2}+\frac{1}{2\Gamma (\{\bar{\rho}_{i}(\mathbf{r}%
)\})}\,.$$ The evaluation of $R_{f}$ requires an iteration procedure. This iteration loop should be carried out in addition to the main iteration procedure used to evaluate the density profiles, see for details [@24; @25].
Finally, the associative contribution, $f_{\mathrm{as}}$, is formulated at the level of the first-order thermodynamic perturbation theory [@pizio1; @we] $$f_{\mathrm{as}}/kT=\sum_{i=1,2} \bar{\rho}_{i}\left[\ln \alpha (\{\bar{\rho}_{i}\})+\frac{1%
}{2}-\frac{\alpha (\{\bar{\rho}_{i}\})}{2}\right],$$ where $\alpha $ is the degree of dissociation according to the mass action law, $
2{\alpha}={1-K{(\bar{\rho}_{1}+\bar{\rho}_{2})\alpha ^{2}}}.
$ The association constant $K\equiv K(\{\bar{\rho%
}_{i}\},T)$ is a product of solely temperature dependent term, $K^{0}$ and $K^{\gamma } $, $K=K^{0}K^{\gamma }$. The constant $K^{0}$, is [@13a] $$K^{0}=8\pi d^{3}\sum_{m=2}^{\infty} \frac{(T^{\ast })^{-2m}}{(2m)!(2m-3)}\,,$$ whereas $K^{\gamma }$ follows from the so-called simple interpolation scheme [@7; @pizio1; @f2] $$K^{\gamma }=\frac{(1-\bar{\eta}/2)}{(1-\bar{\eta})^{3}}\exp \left[ -\frac{%
\Gamma d}{T^{\ast }}\frac{(2+\Gamma d)}{(1+\Gamma d)^{2}}\right] .$$
The theory presented above uses different weighted densities to evaluate the hard sphere and electrostatic free energy contributions. Note that the idea of using different weighted densities to evaluate these contributions has also been employed by Patra et al. [@18; @19].
The density profile is obtained by minimizing the excess grand potential functional $\Delta \Omega =\Omega -\Omega _{b}$, $$\frac{\delta \Delta \Omega }{\delta \rho _{i}(z)}=0,{\ \ \mathrm{for=1,2}},$$ where $\Omega_b$ is the grand potential of a bulk uniform system of density $\{\rho_{b,i}\}$.
The evaluation of the density profiles requires the knowledge of the densities of both coexisting phases. Therefore, we have precisely evaluated the bulk phase diagram prior to the density profile calculations. Next, the local density calculations were carried out assuming that for $z\leqslant -L_z$ and for $\geqslant L_z$ the density of the fluid is equal to the bulk densities of gaseous and liquid phases, respectively. The value of $L_z$ was evaluated at each temperature by testing the convergence of the density profiles towards the final solutions. We have started with $L_z=30d$ and the consecutive runs were carried out doubling the value of $L_z$ for each new run. All the integrations were performed using Simpson method with the grid size of 0.01d. The convergence criterion definitly states that the maximum difference between two consecutive iterations must be smaller than 1E-8 percent. The knowledge of the density profiles of ions permits to calculate the excess grand thermodynamics potential per unit surface area, $A$, i.e., the surface tension $\gamma=\Delta\Omega/A$.
Results and discussion
======================
Figure 1 shows the phase diagram in the density-temperature plane. The phase diagram resulting from the MSA energy equation of state has been presented in several previous works, cf. [@7; @Groh]. Nevertheless, we have decided to include it here for completeness of the study. The solid line corresponds to the system without association (i.e. the associative free energy term is neglected in the grand thermodynamics potential functional), whereas the dashed line describes phase behavior of the model with the association effects included. Inclusion of chemical association into the theory provides a mechanism for the formation of ionic pairs. The fraction of pairs depends on density, on temperature and on the association constant. From physical point of view, the formation of pairs of oppositely charged ions, alters the effective interactions between all the particles. In effect, the critical temperature of liquid-vapor transition, $T^*_{\mathrm{c}}\approx 0.0745$, becomes lower compared to the critical temperature of the model without association, $T^*_{\mathrm{c}}\approx 0.0786$. The liquid-vapor phase diagrams are strongly asymmetric. At low temperatures the vapor phase is very dilute. For example, for the system without association at $T^*=0.05$ the vapor density is $\rho^*_b\approx 1.094E-5$, whereas the density of the coexisting liquid phase is $\rho^*_b\approx 0.2135$.
{width="45.00000%"}
![A comparison of the total density profiles, $\rho^*(z)=(\rho_1(z)+\rho_1(z))d^3$ resulting from the DFT without association theory (solid lines) and from MSA1 theory of Groh et al. [@Groh] (dashed lines). The temperatures $T/T_{\mathrm{c}}$ are given in the figure.](finf3.eps){width="45.00000%"}
First, we consider the system without association. The evaluation of the liquid-vapor density profiles was carried out assuming that the limiting values $\rho_i(-L_z)$ and $\rho_i(L_z)$ are equal to the MSA densities of coexisting phases. Of course, the agreement of our results with the results published previously [@Groh; @23] essentially depends on the agreement between the bulk phase diagrams. One should keep in mind the above remark while analyzing figure 2, where we show a comparison of our results with those of Groh et al.[@Groh]. The latter profiles were obtained from a different bulk theory (called by them “the MSA1 approach” [@Groh]), which yields the value of the critical temperature, $T_{\mathrm{c}}^*\approx 0.0846$, different from that obtained by MSA theory. Note that a comparison of the entire phase diagrams resulting from the MSA and MSA1 approaches in given in figure 1 of [@Groh].
![The convergence of the liquid parts of the single species density profiles towards the bulk liquid equilibrium density at three different temperatures given in the figure. Figure also contains the equations for straight lines (marked by solid lines) approximating the data, computed from density functional theory. Here $y=\ln[\rho){b,i}^{\mathrm{l}}-\rho_i(z)]$ and $ \rho_{b,i}^{\mathrm{l}}$ is the single species density of an ionic liquid at coexistence. The calculations are for the model without association.](finf4.eps){width="45.00000%"}
The way in which the density profile approaches the asymptotic bulk values has been discussed by Groh et al. [@Groh]. We have checked that the decay of the function $\rho_{b,i}^{\mathrm{l}}-\rho_i(z)$ (where $\rho_{b,i}^{\mathrm{l}}$ is the liquid-phase bulk density) with $z\to L_z$ is exponential, i.e. $\ln[\rho_i(z)-\rho_{b,i}^{\mathrm{l}}]\propto z/\xi$. This point is illustrated in figure 3, where we have plotted $\ln[\rho_{b,i}^{l}-\rho_i(z)]$ versus $z/d$. At larger distances this dependence is fairly well approximated by a straight line with the correlation coefficient, $R^2$, very close to unity.
![Part [*a*]{}. A comparison of the total density profiles for the models without (solid lines) and with association (symbols). Part [*b*]{}. Dissociation degrees across the vapor-liquid interface. The temperatures are given in the figure. ](rys4.eps "fig:"){width="43.00000%"} ![Part [*a*]{}. A comparison of the total density profiles for the models without (solid lines) and with association (symbols). Part [*b*]{}. Dissociation degrees across the vapor-liquid interface. The temperatures are given in the figure. ](rys4b.eps "fig:"){width="43.00000%"}
![The dependence of the surface tension on the temperature. Curves labelled MSA, MSA1 were evaluated by Groh et al. [@Groh], the curve labelled “square gradient theory” was evaluated by Telo da Gama et al. [@Gama], the curve abbreviated as DFT was obtained for the model without association, whereas black circles – for the model with the association effects included.](finf5.eps){width="45.00000%"}
We now consider the effect of the association, cf. figure 4. Part [*a*]{} shows a comparison of the total density profiles for the systems with (symbols) and without association (lines) at different temperatures. At low temperatures the liquid density for the system with the association included is higher than for the system with the association effects switched off, cf. figure 1. The profiles for the model with association are more “smeared out”, but this is the effect of lowering the critical temperature due to association effects. Similarly to the case of a system without association we have found exponential decay of the function $\rho_{b,i}^{\mathrm{l}}-\rho_i(z)$ with $z\to L_z$ (the corresponding plot was omitted for the sake of brevity). Figure 4b shows the profiles of $\alpha(z)$ (the phase diagram in the $\alpha$-$T$ plane has been already presented in [@pizio1]) We stress that the interfacial region of $\alpha(z)$ is shifted towards rarefied phase compared to the density profiles. It means that the position of the “pseudo-Gibbs” dividing surface, which can be introduced for the dissociation degree in analogy to the usual liquid-vapor dividing surface is also shifted compared to the position of the usual dividing surface (which is located at $z=0$). Note that similar shift of the dissociation degree was also observed in the case of liquid-vapor interfaces of non-ionic fluids [@bor].
=-1 The surface tension obtained from the density functional theory and from the MSA1 and MSA approaches, developed by Groh et al. [@Groh], as well as from the square gradient theory [@Gama] are shown in figure 5. We have plotted here the ratio $\gamma d^2/kT_{\mathrm{c}}$ versus the temperature reduced by the critical temperature, $T/T_{\mathrm{c}}$. Except for the temperatures very close to the critical temperature, the present theory predicts lower values of the surface tension than all the rest approaches. The associative free energy terms still lower the value of $\gamma d^2/kT_{\mathrm{c}}$. Note that the MSA theory grossly underestimates the densities of the coexisting liquid phase. This can suggest that the values of the surface tension are also underestimated. On the other hand, the value of the critical temperature is overestimated by the MSA approach, and hence can lead to too high values of the interfacial tension.
{width="45.00000%"}
Near critical temperature, both models (i.e., the model with and without association) yield the standard mean-field critical behavior, i.e., $\gamma d^2/kT_{\mathrm{c}}\propto
(1 -T/T_{\mathrm{c}})^{(3/2)}$ (see figure 6). As the temperature approaches the critical point, the theory yields a nearly linear variation of $\ln[\gamma d^2/kT_{\mathrm{c}}]$ with $\ln[(1 -T/T_{\mathrm{c}})]$ and the slope of the straight line approximating the obtained data is almost perfectly equal to (3/2).
{width="45.00000%"}
The width of the interfacial zone can be characterized by the parameter $W$, defined as [@Fischer] $$W=-[\rho(z=L_z)-\rho(z=-L_z)]
\left[{\frac{\rd\rho(z)}{\rd z}}\right]^{-1}_{z=z_0},$$ where $z_0$ is given by $\rho(z_0)=(1/2)[\rho_s(z=L_z)+\rho(z=-L_z)]$ and $\rho(z=L_z)$ and $\rho(z=-L_z)$ are the total densities of the coexisting liquid and vapor phases. Obviously, the interface becomes wider as the temperature increases. The effects of association slightly increases the interface width. As the temperature approaches the critical temperature, the interfacial width diverges. To investigate the character of this divergence, we have plotted the values of $\ln W/d$ versus $\ln(1-T/T_{\mathrm{c}})$, see figure 7. For $T$ approaching the critical temperature, the dependence of $\ln W/d$ versus $\ln(1-T/T_{\mathrm{c}})$ is linear. We have found the following scaling: $W^*\propto (1-T/T_{\mathrm{c}}(M))^{-0.5}$. The obtained value of the exponent (0.5) is characteristic of mean-field type theories [@Groh]. The scaling of $W$ is consistent with the scaling of the surface tension.
![A comparison of the total density profiles for the models without association (solid and dotted lines) and with association (dashed and dash-dotted lines) with computer simulation data. The simulation results for the profiles (see details in the text) are shown by symbols [@alejandre1; @alejandre2]. The temperatures are given in the figure.](compa.eps){width="45.00000%"}
Our discussion involved the results of different, but solely theoretical approaches. However, it must be clarified how accurate our approaches are (with and without association) compared to the available computer simulation data. In particular, Alejandre and co-workers investigated the liquid-vapor interface for the RPM using molecular dynamics simulation method supplemented by certain specific technical peculiarities dealing specifically with the RPM [@alejandre1; @alejandre2]. These authors presented the density profiles, $\rho^{*}(z)$ at four reduced temperatures, $T^*$=0.035, 0.038, 0.040 and 0.043 [@alejandre1]. On the other hand, a single density profile at $T^*=0.391$ relevant to our study was given in [@alejandre2] (this profile is for the soft primitive model but with parameters permitting comparison with the RPM). The authors have not performed estimates of the critical parameters coming from their method of study of the RPM. Thus, they somewhat arbitrarily used the critical temperature obtained in [@6], $T^*_{\mathrm{c}}=0.0492$, for their purposes. We have rescaled the temperature and density with respect to the critical parameters of both theoretical approaches (with and without association), respectively, and present here a comparison of the density profiles with computer simulation data in figure 8. Note that the critical density given in [@6] was used to rescale the simulation data. We observe that the *rescaled* liquid phase density at coexistence coming from the DFT for the model with ion pairing is in reasonable agreement with the simulation data. The DFT for the model without association greatly overestimates these liquid densities. However, the width of the interfacial region is not described well. Theoretical approaches yield narrower interface width for two reduced temperatures in question. It is difficult to attribute this specific inaccuracy to the particular term of the free energy functional. However, it seems that the mean field consideration requires improvement in order to better describe interparticle correlations and hopefully to obtain a more adequate interface width.
As concerns the relationship between the simulation data for the surface tension and the results of theoretical approaches we would like to comment on the following points. A set of computer simulation results for the surface tension were put together with the predictions of some theoretical approaches [@Gama; @Groh; @23] (that were discussed in the introductory part of our communication) on arbitrary temperature scale, see figure 2 of [@alejandre1]. Apparently, the theories yield the surface tension of the same magnitude as computer simulation data. This permitted to conclude that the surface tension from this set of theories [@Gama; @Groh] is overestimated because the effect of ion association is not taken into account and solely the theory that accounts for association [@23] is in good agreement with simulations. Then, it would seem that our approach taking ion pairing into account is the best, see figure 5. Moreover, a comparison of the theories, simulations and experimental data was performed [@alejandre2] using an arbitrary scale. Such an idyllic picture is, however, entirely destroyed, if one rescales the reduced temperature scale by the critical temperature of each theory and of simulations. This is actually an adequate comparison. According to [@alejandre1] the simulations yield, e.g. $\gamma d^2/kT_{\mathrm{c}}$=0.3 at $T/T_{\mathrm{c}}$=0.6. Such a value is very much higher than the results of any theory, c.f. figure 5, at this particular temperature. Moreover, this means that the inclusion of association effects makes things worse than from any other theory. This is difficult to accept, however. According to the more recent simulation work of the same authors [@alejandre2] it is difficult to draw a definite conclusion about the dependence of the surface tension on temperature for the model in question due to the statistical error associated with the value for surface tension. Moreover, it has been discussed [@alejandre2] that the inflection point at $T^*=0.04$ (again it is a reduced temperature of simulation) can be present on the dependence of surface tension on temperature as a result of chemical association of ions in the RPM in close similarity to hydrogen bonded fluids. In our opinion, additional work, both theoretical and simulational, is necessary to obtain definite answers about the behavior of surface tension on temperature for the model in question and the related models. From theoretical viewpoint one needs a theory that yields a more adequate shape of the bulk coexistence envelope. Then, the free energy expression coming from this approach, if available, would contribute to the development of better density functional approaches for inhomogeneous ionic fluids.
To summarize briefly, in this work we have applied density functional theory to the study of the liquid-vapor interface of a RPM fluid. Of course, the theory we use here is *ad hoc*. However, our previous calculations [@pizio; @reszko] have shown that it reproduces reasonably well the structure of a fluid at a charged and uncharged wall, predicts the dependence of the capacitance of the double layer on the temperature and yields “capillary evaporation” phase diagrams for confined ionic systems [@pizio; @reszko; @pizio1]. We have shown that the inclusion of ion pairing effect leads to a better agreement of the density profiles of ions across the liquid-vapor interface. A more sophisticated approaches [@7], compared to the present one, including the effects of ionic association can lead to even better agreement of the coexistence envelope with simulations and of the density profiles across the liquid-vapor interface. We plan to extend our study in this respect in the nearest future. However, the inaccuracies of the values for the liquid phase density at coexistence at different temeperatures prevent us from obtaining a reasonable agreement with the available simulation data for the surface tension. It seems, however, that a more ample set of simulation data using different techniques is necessary. Nevertheless, the theories of this study yield a decreasing surface tension with increasing temperature as one intuitively would expect. Still we are not able to conclude that the theoretical approach is entirely successful both for the fluid-solid and fluid-fluid interfaces and look forward to improving it in future work.
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[^1]: E-mail: pizio@servidor.unam.mx
|
---
abstract: 'A three-parameter (positive odd integer $s$, thickness factor $\lambda$, and asymmetry factor $a$) family of asymmetric thick brane solutions in five dimensions were constructed from a two-parameter ($s$ and $\lambda$) family of symmetric ones in \[R. Guerrero, R.O. Rodriguez, and R. Torrealba, Phys. Rev. D **72**, 124012 (2005).\]. The values $s=1$ and $s\geq3$ correspond to single branes and double branes, respectively. These branes have very rich inner structure. In this paper, by presenting the mass-independent potentials of Kaluza–Klein (KK) modes in the corresponding Schrödinger equations, we investigate the localization and mass spectra of fermions on the symmetric and asymmetric thick branes in an AdS background. In order to analyze the effect of gravity-fermion interaction (i.e., the effect of the inner structure of the branes) and scalar-fermion interaction to the spectrum of fermion KK modes, we consider three kinds of typical kink-fermion couplings. The spectra of left chiral fermions for these couplings are consisted of a bound zero mode and a series of gapless continuous massive KK modes, some discrete bound KK modes including zero mode (exist mass gaps) and a series of continuous massive KK modes, infinite discrete bound KK modes, respectively. The structures of the spectra are investigated in detail.'
author:
- 'Yu-Xiao Liu[^1][^2], Chun-E Fu[^3], Li Zhao[^4], Yi-Shi Duan[^5]'
title: Localization and Mass Spectra of Fermions on Symmetric and Asymmetric Thick Branes
---
Introduction
============
The suggestion that our observed four-dimensional world is a brane embedded in a higher-dimensional space-time [@Rubakov1983; @Akama1983; @ADD; @rs; @Lykken] can provide new insights for solving gauge hierarchy problem and cosmological constant problem, etc. In the framework of brane scenarios, gravity is free to propagate in all dimensions, while all the matter fields are confined to a 3–brane. By introducing large extra dimensions, Arkani-Hamed-Dimopoulos-Dvali (ADD) brane model [@ADD] drops the fundamental Planck scale to Tev. However, it introduces intermediate mass scales corresponding to the large extra dimensions between Planck and Tev scales. In Ref. [@rs], an alternative scenario, Randall-Sundrum (RS) warped brane model, had been proposed. In this scenario, the internal manifold does not need to be compactified to the Planck scale anymore and the exponential warp factor in the metric can generate a large hierarchy of scales, which are reasons why this new brane model has attracted so much attention.
Generalizations and extensions of RS brane model have been proposed for examples in Ref. [@ExtensionRS; @Starkman2001; @SenGuptaPRL2007]. In Ref. [@Starkman2001], the model with extra dimensions composed of a compact hyperbolic manifold is free of usual problems that plague the original ADD model and shares many common features with RS model. Recently, RS model is generalized to higher dimensions for a multiply space-time with negative cosmological constant [@SenGuptaPRL2007]. In this generalized scenario the observed hierarchy in the masses of standard model fermions can be explained geometrically without invoking any further hierarchy among the various moduli provided the warping is large in one direction and small in the other.
In RS warped scenario, however, the modulus, namely the brane separation, is not stable. Goldberger and Wise showed that it can be stabilized by introducing a scalar field in the bulk [@GW]. A bulk scalar also provides us with a way of generating the brane as a domain wall (thick brane) in five dimensions. Considering our four-dimensional Universe as an infinitely thin domain wall is an idealization. It is for this reason that an increasing interest has been focused on the study of thick brane scenarios based on gravity coupled to scalars in higher dimensional space-time [@dewolfe; @GremmPLB2000; @gremm; @Csaki; @CamposPRL2002; @varios; @Guerrero2002; @ThickBraneDzhunushaliev; @ThickBraneBazeia; @ShtanovJCAP2009]. A virtue of these models is that the branes can be obtained naturally rather than introduced by hand. In most thick brane scenarios, the scalar field is a standard topological kink interpolating between the minima of a potential with spontaneously broken symmetry. For a comprehensive review on thick brane solutions and related topics please see Ref. [@ThickBraneReview].
In brane world scenarios, an important problem is localization of various bulk fields on a brane by a natural mechanism. Especially, the localization of spin half fermions on thick branes is very interesting. Localizing fermions on branes or defects requires us to introduce other interactions besides gravity. Recently, localization mechanisms on a domain wall for fermions have been extensively analyzed in Ref. [@Volkas2007]. There are some other backgrounds, for example, gauge field [@Parameswaran0608074; @LiuJHEP2007], supergravity [@Mario] and vortex background [@LiuNPB2007; @LiuVortexFermion; @Rafael200803; @StojkovicPRD], could be considered. Localization of fermions in general space-times had been studied for example in [@RandjbarPLB2000]. In five dimensions, with the scalar–fermion coupling, there may exist a single bound state and a continuous gapless spectrum of massive fermion Kaluza–Klein (KK) states [@ThickBrane1; @ThickBrane2; @ThickBrane3; @Liu0708; @20082009], while for some other brane models, there exist finite discrete KK states (mass gaps) and a continuous gapless spectrum starting at a positive $m^2$ [@ThickBrane4; @Liu0803; @0803.1458]. In Ref. [@DubovskyPRD2000], it was found that fermions can escape into the bulk by tunnelling, and the rate depends on the parameters of the scalar potential. In Ref. [@KoleyCQG2005], the authors obtained trapped discrete massive fermion states on the brane, which in fact are quasibound and have a finite probability of escaping into the bulk.
In Ref. [@LiuJCAP2009], localization and mass spectra of various bulk matter fields including fermions on symmetric and asymmetric de Sitter thick single branes were investigated. It was shown that the massless modes of scalars and vectors are separated by a mass gap from the continuous modes. The asymmetry may increase the number of the bound KK modes of scalars but does not change that of vectors. The localization property of spin 1/2 fermions is dependent on the coupling of fermions and the background scalar $\eta\bar{\Psi}F(\phi)\Psi$. For the usual Yukawa coupling with $F(\phi(z))=\phi(z){\sim}\arctan(\sinh z)$ (a usual kink which is almost a constant at large $z$), the fermion zero mode can not be localized on the branes. For the scalar-fermion coupling with $F(\phi(z))$ a kink likes $\sinh z$, which increases quickly with $z$, there exist some discrete bound KK modes and a series of continuous ones, and one of the zero modes of left and right fermions is localized on the branes strongly. The asymmetry reduces the number of the bound fermion KK modes.
Fermions on symmetric and asymmetric double branes have been reported in Ref. [@Guerrero2006]. These double branes are stable Bogomol’nyi-Prasad-Sommerfeld (BPS) thick walls with two sub walls located at their edges. It was shown that, for the symmetric brane, the zero modes of fermions coupled to the scalar field through Yukawa interactions and gravitons are not peaked at the center of the brane, but instead a constant between the two sub branes. However, in the asymmetric scenario, as a consequence of the asymmetry, fermions are localized on one of the sub walls, while the gravitons are localized on another sub wall. Hence a large hierarchy between the Planck and the weak scales can be produced.
In Ref. [@asymdSBrane2], a three-parameter family of asymmetric thick brane solutions in five dimensions (including single branes and double branes) were constructed from a two-parameter family of symmetric ones given in Refs. [@MelfoPRD2003; @GregoryPRD2002; @BazeiaJCAP2004]. These branes have very rich inner structure. In this paper, we will investigate the localization problem and the mass spectra of fermions on the symmetric and asymmetric thick branes for three kinds of typical kink-fermion couplings in detail. It will be shown that the localization properties on asymmetric branes are very different from these given in Refs. [@LiuJCAP2009] and [@Guerrero2006]. The mass spectra of fermions are determined by the inner structures of the branes and the scalar-fermion couplings. The paper is organized as follows: In Sec. \[SecModel\], we first give a brief review of the symmetric and asymmetric double thick branes in an AdS background. Then, in Sec. \[SecLocalize\], we study localization of spin half fermions on the thick branes with different types of scalar-fermion interactions by presenting the shapes of the potentials of the corresponding Schrödinger problem. Finally, a brief discussion and conclusion are presented in Sec. \[secConclusion\].
Review of the symmetric and asymmetric thick branes {#SecModel}
===================================================
Let us consider thick branes arising from a real scalar field $\phi$ with a scalar potential $V(\phi)$. The action for such a system is given by $$S = \int d^5 x \sqrt{-g}\left [ \frac{1}{2\kappa_5^2} R-\frac{1}{2}
g^{MN}\partial_M \phi
\partial_N \phi - V(\phi) \right ],
\label{action}$$ where $R$ is the scalar curvature and $\kappa_5^2=8 \pi G_5$ with $G_5$ the five-dimensional Newton constant. Here we set $\kappa_5=1$. The line-element for a five-dimensional space-time is assumed as $$\begin{aligned}
ds^2&=&\text{e}^{2A(z)}\big(\eta_{\mu\nu}dx^\mu dx^\nu
+ dz^2\big), \label{linee}\end{aligned}$$ where $\text{e}^{2A(z)}$ is the warp factor and $z$ stands for the extra coordinate. The scalar field is considered to be a function of $z$ only, i.e., $\phi=\phi(z)$. In the model, the potential could provide a thick brane realization, and the soliton configuration of the scalar field dynamically generates the domain wall configuration with warped geometry. The field equations generated from the action (\[action\]) with the ansatz (\[linee\]) reduce to the following coupled nonlinear differential equations $$\begin{aligned}
\phi'^2 & = & 3(A'^2-A''), \\
V(\phi) & = & \frac{3}{2} (-3A'^2-A'') e^{-2A},\\
\frac{dV(\phi)}{d\phi} & = & (3A'\phi'+\phi'')e^{-2A},\end{aligned}$$ where the prime denotes derivative with respect to $z$. Now we consider static double thick branes in an AdS background. A two-parameter family of symmetric double thick branes in five dimensions for the potential $$\begin{aligned}
V(\phi)&=&\frac{3}{2} \lambda^2
\sin^{2-\frac{2}{s}} (\phi/\phi_{0})
\cos^2(\phi/\phi_{0}) \nonumber \\
&&\times\left[2s-1-4\tan^2(\phi/\phi_{0})\right],
\label{potencialdoble1}\end{aligned}$$ were presented and discussed in Refs. [@MelfoPRD2003; @GregoryPRD2002]: $$\begin{aligned}
e^{2A}&=&\frac{1}{\left[1+(\lambda z)^{2s}\right]^{1/s}},
\label{e2A3} \\
\phi~~&=&\phi_{0}\arctan(\lambda z)^s,
\label{phi3}\end{aligned}$$ where $\phi_{0}={\sqrt{3(2s-1)}}/{s}$, $\lambda$ is a positive real constant, and $s$ is a positive odd integer. This solution represents a family of plane symmetric static single ($s=1$) or double ($s>1$) domain wall space-times, being asymptotically AdS$_5$ with a cosmological constant $-6\lambda^2$. Similar solutions can be found in [@BazeiaJCAP2004].
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Based on the symmetric solution above, and in concordance with the approach presented in [@asymdSBrane2], a three-parameter family of asymmetric thick branes in five dimensions were constructed in Ref. [@asymdSBrane2]: $$\begin{aligned}
e^{2A}&=& \frac{1}{\left(1+(\lambda z)^{2s}\right)^{1/s}
{\cal F}(z)^{2} },\qquad
\label{e2A4} \\
\phi~~ &=&\phi_{0}\arctan(\lambda z)^s,
\label{phi4} \\
V(\phi)&=&
-\frac{3}{4}\sin^2(\phi/\phi_0)\tan^{-2/s}(\phi/\phi_0)
{\cal K}(\phi)\nonumber \\
&\times& \bigg\{16a\tan^{1/s}(\phi/\phi_0)
+\cos^{-2/s}(\phi/\phi_0)
\bigg[5 \nonumber \\
&& ~~-2s-(3+2s)\cos(2\phi/\phi_0)\bigg]
{\cal K}(\phi)\bigg\} \nonumber \\
&-& 6a^2\cos^{2/s}(\phi/\phi_0),~~~~~~ \label{Vphi4}\end{aligned}$$ where the asymmetric factor $a$ satisfies $$0<a <\frac{\Gamma(1/s)\;\lambda}{\Gamma(1/2s)\ \Gamma(1+1/2s)}
~\left(>\frac{\lambda}{2}\right), \label{constraintOna}$$ ${\cal F}(z)$ and ${\cal K}(\phi)$ are defined as $$\begin{aligned}
{\cal F}(z)&\equiv& 1+a z\
{}_2F_1\left(\frac{1}{2s},\frac{1}{s},1+\frac{1}{2s}
,-(\lambda z)^{2s}\right), \label{calFz}\\
{\cal K}(\phi)&\equiv&
\lambda+a\tan^{1/s}\big({\phi}/{\phi_0}\big)\nonumber \\
&&\times ~{}_2F_1\left(\frac{1}{2s},\frac{1}{s},1+\frac{1}{2s}
,-\tan^{2}\big(\frac{\phi}{\phi_0}\big)\right).\end{aligned}$$ The parameter $a$ describes the asymmetry of the solution. For $a\rightarrow 0$ and $s=1$ the regularized version of the Randall-Sundrum thin brane will be recovered [@GremmPLB2000; @Guerrero2002]. For $\alpha>0$ and $s>1$, this is a solution of an asymmetric static double domain wall space-time interpolating between different AdS$_{5}$ vacua. The scalar curvature $R$ and the energy density $\rho$ for the solution are read $$\begin{aligned}
R &=& -\frac{40 a (\lambda z)^{2 s}}
{z \left(1+(\lambda z)^{2s}\right)}{\cal F}(z)
-\frac{20 a^2 }{\left(1+(\lambda z)^{2 s}\right)^{1/s}}
\nonumber \\ &&
-\frac{4 (\lambda z)^{2 s}
\left(2-4 s+5 (\lambda z)^{2s}\right)}
{z^2\left(1+(\lambda z)^{2 s}\right)^{2-{1}/{s}}} {\cal F}^2(z),~~~
\label{R4} \\
\rho &=& -\frac{12 a (\lambda z)^{2 s}}
{z \left(1+(\lambda z)^{2 s}\right)}{\cal F}(z)
-\frac{6 a^2}{\left(1+(\lambda z)^{2s}\right)^{1/s}}
\nonumber \\ &&
-\frac{3 (\lambda z)^{2 s}
\left(1-2 s+2 (\lambda z)^{2 s}\right)}
{z^2\left(1+(\lambda z)^{2
s}\right)^{2-{1}/{s}}}{\cal F}^2(z).~~~~~~~
\label{EnergyDensity2}\end{aligned}$$ The shapes of the kink $\phi$, the potential $V(\phi)$, the warp factor $e^{2A}$, and the energy density $\rho$ are shown in Figs. \[fig\_Vphi\] and \[fig\_BPSBrane\]. It is clear that, the single brane is localized at $z=0$, while the two sub-branes are localized at $z=\pm 1/\lambda$ and the thickness of the double brane is $2/\lambda$. When $s\rightarrow\infty$, each sub-brane is a thin brane. More detailed discussions can be found in Ref. [@asymdSBrane2].
Localization and mass spectra of fermions on the thick branes {#SecLocalize}
=============================================================
In this section let us investigate the localization problem of spin 1/2 fermions on the family of symmetric and asymmetric thick branes given in Sec. \[SecModel\] by means of the gravitational interaction and sclar-fermion couplings. We will analyze the spectra of fermions on the thick branes by present the potential of the corresponding Schrödinger equation. It can be seen from the following calculations that the mass-independent potential can be obtained conveniently with the conformally flat metric (\[linee\]).
In five dimensions, fermions are four component spinors and their Dirac structure is described by $\Gamma^M= e^M _{\bar{M}}
\Gamma^{\bar{M}}$ with $\{\Gamma^M,\Gamma^N\}=2g^{MN}$, where $\bar{M}, \bar{N}, \cdots =0,1,2,3,5$ denote the five-dimensional local Lorentz indices, and $\Gamma^{\bar{M}}$ are the flat gamma matrices in five dimensions. In our set-up, $\Gamma^M=(\text{e}^{-A}\gamma^{\mu},\text{e}^{-A}\gamma^5)$, where $\gamma^{\mu}$ and $\gamma^5$ are the usual flat gamma matrices in the Dirac representation. The Dirac action of a massless spin 1/2 fermion coupled to the scalar is $$\begin{aligned}
S_{1/2} = \int d^5 x \sqrt{-g} \left(\bar{\Psi} \Gamma^M D_M
\Psi-\eta \bar{\Psi} F(\phi) \Psi\right), \label{DiracAction}\end{aligned}$$ where the covariant derivative $D_M$ is defined as $D_M\Psi =
(\partial_M + \omega_M) \Psi$ with the spin connection $\omega_M=
\frac{1}{4} \omega_M^{\bar{M} \bar{N}} \Gamma_{\bar{M}}
\Gamma_{\bar{N}}$. With the metric (\[linee\]), the nonvanishing components of the spin connection $\omega_M$ are $$\begin{aligned}
\omega_\mu =\frac{1}{2}(\partial_{z}A) \gamma_\mu \gamma_5. \label{eq4}\end{aligned}$$ Then the five-dimensional Dirac equation is read as $$\begin{aligned}
\left\{ \gamma^{\mu}\partial_{\mu}
+ \gamma^5 \left(\partial_z +2 \partial_{z} A \right)
-\eta\; \text{e}^A F(\phi)
\right \} \Psi =0, \label{DiracEq1}\end{aligned}$$ where $\gamma^{\mu} \partial_{\mu}$ is the Dirac operator on the brane. Note that the sign of the coupling $\eta$ of the spinor $\Psi$ to the scalar $\phi$ is arbitrary and represents a coupling either to kink or to antikink domain wall. For definiteness, we shall consider in what follows only the case of a kink coupling, and thus assume that $\eta>0$.
Now we study the above five-dimensional Dirac equation. Because of the Dirac structure of the fifth gamma matrix $\gamma^5$, we expect the left- and right-handed projections of the four-dimensional part to behave differently. From the equation of motion (\[DiracEq1\]), we will search for the solutions of the general chiral decomposition $$\Psi(x,z) = \text{e}^{-2A}\sum_n\bigg(\psi_{Ln}(x) f_{Ln}(z)
+\psi_{Rn}(x) f_{Rn}(z)\bigg),$$ where $\psi_{Ln}(x)=-\gamma^5 \psi_{Ln}(x)$ and $\psi_{Rn}(x)=\gamma^5 \psi_{Rn}(x)$ are the left-handed and right-handed components of a four-dimensional Dirac field, respectively, the sum over $n$ can be both discrete and continuous. Here, we assume that $\psi_{L}(x)$ and $\psi_{R}(x)$ satisfy the four-dimensional massive Dirac equations $\gamma^{\mu}\partial_{\mu}\psi_{Ln}(x)=m_n\psi_{R_n}(x)$ and $\gamma^{\mu}\partial_{\mu}\psi_{Rn}(x)=m_n\psi_{L_n}(x)$. Then $\alpha_{L}(z)$ and $\alpha_{R}(z)$ satisfy the following coupled equations
$$\begin{aligned}
\left[\partial_z + \eta\;\text{e}^A F(\phi) \right]f_{Ln}(z)
&=& ~~m_n f_{Rn}(z), \label{CoupleEq1a} \\
\left[\partial_z- \eta\;\text{e}^A F(\phi) \right]f_{Rn}(z)
&=& -m_n f_{Ln}(z). \label{CoupleEq1b}\end{aligned}$$
\[CoupleEq1\]
From the above coupled equations, we get the Schrödinger-like equations for the KK modes of the left and right chiral fermions $$\begin{aligned}
\big(-\partial^2_z + V_L(z) \big)f_{Ln}
&=&\mu_n^2 f_{Ln},
\label{SchEqLeftFermion} \\
\big(-\partial^2_z + V_R(z) \big)f_{Rn}
&=&\mu_n^2 f_{Rn},
\label{SchEqRightFermion}\end{aligned}$$ where the effective potentials are given by
$$\begin{aligned}
V_L(z)&=& \big(\eta\;\text{e}^{A} F(\phi)\big)^2
- \partial_z\big(\eta\;\text{e}^{A} F(\phi)\big), \label{VL}\\
V_R(z)&=& V_L(z)|_{\eta \rightarrow -\eta}. \label{VR}\end{aligned}$$
\[Vfermion\]
In order to obtain the standard four-dimensional action for the massive chiral fermions: $$\begin{aligned}
S_{1/2} &=& \int d^5 x \sqrt{-g} ~\bar{\Psi}
\left( \Gamma^M (\partial_M+\omega_M)
-\eta F(\phi)\right) \Psi {\nonumber}\\
&=& \sum_{n}\int d^4 x \left(~\bar{\psi}_{Rn}
\gamma^{\mu}\partial_{\mu}\psi_{Rn}
-~\bar{\psi}_{Rn}m_{n}\psi_{Ln} \right) {\nonumber}\\
&+&\sum_{n}\int d^4 x \left(~\bar{\psi}_{Ln}
\gamma^{\mu}\partial_{\mu}\psi_{Ln}
-~\bar{\psi}_{Ln}m_{n}\psi_{Rn} \right) {\nonumber}\\
&=&\sum_{n}\int d^4 x
~\bar{\psi}_{n}
(\gamma^{\mu}\partial_{\mu} -m_{n})\psi_{n},\end{aligned}$$ we need the following orthonormality conditions for $f_{L_{n}}$ and $f_{R_{n}}$: $$\begin{aligned}
&& \int_{-\infty}^{\infty} f_{Lm}(z) f_{Ln}(z)dz
=\delta_{mn},\nonumber\\
&&\int_{-\infty}^{\infty} f_{Rm}(z) f_{Rn}(z)dz
=\delta_{mn},\label{orthonormality}\\
&& \int_{-\infty}^{\infty} f_{Lm}(z) f_{Rn}(z)dz=0.
\nonumber\end{aligned}$$ Note that the differential equations (\[SchEqLeftFermion\]) and (\[SchEqRightFermion\]) can be factorized as $$\begin{aligned}
\left[-\partial_z+\eta\;\text{e}^A F(\phi)\right]
\left[\partial_z+\eta\;\text{e}^A F(\phi) \right]
f_{Ln}(z) &=& m_n^2 f_{Ln}(z), \label{SchEqLeftFermion2} \nonumber
\\ \\
\left[-\partial_z-\eta\;\text{e}^A F(\phi)\right]
\left[\partial_z-\eta\;\text{e}^A F(\phi) \right]
f_{Rn}(z) &=& m_n^2 f_{Rn}(z). \label{SchEqRightFermion2}\nonumber
\\\end{aligned}$$ It can be shown that $m_n^2$ is zero or positive since the resulting Hamiltonian can be factorized as the product of two operators which are adjoints of each other. Hence the system is stable against linear classical metric and scalar fluctuations.
It can be seen that, in order to localize left or right chiral fermions, there must be some kind of scalar-fermion coupling, and the effective potential $V_L(z)$ or $V_R(z)$ should have a minimum at the location of the brane. Furthermore, for the kink configuration of the scalar $\phi(z)$ (\[phi3\]), $F(\phi(z))$ should be an odd function of $\phi(z)$ when one demands that $V_{L,R}(z)$ are invariant under $Z_2$ reflection symmetry $z\rightarrow -z$. Thus we have $F(\phi(0))=0$ and $V_L(0)=-V_R(0)=-\eta\partial_z F(\phi(0))$, which results in the well-known conclusion: only one of the massless left and right chiral fermions could be localized on the brane. The spectra are determined by the behavior of the potentials at infinity. For $V_{L,R}\rightarrow 0$ as $|z|\rightarrow \infty$, one of the potentials would have a volcanolike shape and there exists only a bound massless mode followed by a continuous gapless spectrum of KK states, while another could not trap any bound states and the spectrum is also continuous and gapless. The simplest Yukawa coupling $F(\phi)=\phi$ and the generalized coupling $F(\phi)=\phi^k$ with positive odd integer $k\;(\geq3)$ belong to this type. For $V_{L,R}\rightarrow V_{\infty}=$ constant as $|z|\rightarrow \infty$, those modes with $m_n^2<V_{\infty}$ belong to discrete spectrum and modes with $m_n^2>V_{\infty}$ contribute to a continuous one. For this case, the simplest coupling is of the form $F(\phi)=\tan^{1/s}(\phi/\phi_0)$. If the potentials increase as $|z|\rightarrow \infty$, the spectrum is discrete. There are a lot of couplings for the case. The concrete behavior of the potentials is dependent on the function $F(\phi)$. In what following, we will discuss in detail three typical couplings for the above three cases as examples.
Case I: $F(\phi)=\phi^k$ {#sec3.1}
------------------------
We mainly consider the simplest case $F(\phi)=\phi$, for which the explicit forms of the potentials (\[Vfermion\]) are $$\begin{aligned}
V^S_L(z) &=& 3\eta^2\frac{(2s-1)}{s^2}
\frac{\arctan^2(\lambda^s z^s)}
{[1+({\lambda}z)^{2s}]^{\frac{1}{s}}}
\nonumber \\
&-& \eta\frac{\sqrt{6s-3}}{s}
\frac{({\lambda}z)^{s}\left[s -({\lambda}z)^{s}
\arctan(\lambda^s z^s)\right]}
{z[1+({\lambda}z)^{2s}]^{1+\frac{1}{2s}}}
, ~~~~~ \label{VSL_CaseI} \\
V^S_R(z) &=& V^S_L(z)|_{\eta \rightarrow -\eta},
\label{VSR_CaseI}\end{aligned}$$ and $$\begin{aligned}
V^A_L(z)
&=& \bigg\{3\eta^2\frac{(2s-1)}{s^2}
\frac{\arctan^2(\lambda^s z^s)}
{[1+({\lambda}z)^{2s}]^{\frac{1}{s}}}
\nonumber \\&&
~~ + a \eta \frac{\sqrt{6s-3}}{s}
\frac{\arctan(\lambda^s z^s)}
{\left[1+(z\lambda)^{2s}\right]^{\frac{3}{2s}}}
\bigg\} \frac{1}{{\cal F}^2(z)} \\
&&- \eta\frac{\sqrt{6s-3}}{s}
\frac{({\lambda}z)^{s}\left[s -({\lambda}z)^{s}
\arctan(\lambda^s z^s)\right]}
{z[1+({\lambda}z)^{2s}]^{1+\frac{1}{2s}}
{{\cal F}(z)} }
,~~~~\nonumber \\
V^A_R(z) &=& V^A_L(z)|_{\eta \rightarrow -\eta},\end{aligned}$$ for the symmetric and asymmetric brane solutions, respectively.
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All potentials have the asymptotic behavior: $V^{S,A}_{L,R}(z\rightarrow\pm \infty)\rightarrow0$. The values of the potentials for left and right chiral fermions at $z = 0$ are given by $$V^{S,A}_L(0) =-V^{S,A}_R(0)
= \left\{
\begin{array}{c}
-\sqrt{3}\;\eta\lambda \\
0
\end{array}
\begin{array}{l}
~~~\text{for} ~~~ s=1. \\
~~~\text{for} ~~~ s>1.
\end{array} \right.$$ So for a given coupling constant $\eta$ and $\lambda$, the values of the potentials for left and right chiral fermions at $z=0$ are opposite for $s=1$ and vanish for $s>1$. Note that there are a single brane and a double brane for $s=1$ and $s>1$, respectively. The shapes of the potentials are shown in Fig. \[fig\_VFermion\_Case1\] for given values of positive $\eta$ and $\lambda$. It can be seen that $V_L(z)$ is indeed a modified volcano type potential for the single brane scenario with $s=1$, and it has a well. While for the double brane case with $s>1$, the corresponding potential $V_L(z)$ has a double well, and the potential $V_R(z)$ of right chiral fermions has a single “well", which indicates that there may exist resonances (qusi-localized KK modes). The shape of the potentials is relative to the inner structure of the brane, or equivalently, it depends partly on the warp factor $e^{2A}$. Furthermore, the coupling type of scalar and fermion also affects the structure of the potentials. For example, for the case $F(\phi)=\phi^k$ with positive odd integer $k\geq 3$, we have $V^{S,A}_{L,R}(0)=0$ for both $s=1$ and $s>1$, and the potentials for left and right chiral fermions have a double well and a single well even for $s=1$, respectively (see Fig. \[fig\_VFermion\_Case1b\]).
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Since $V^{S,A}_L(z)\rightarrow 0$ when $z\rightarrow\pm\infty$, the potentials for left chiral fermions provides no mass gap to separate the fermion zero mode from the excited KK modes. Because the potentials $V^{S,A}_R(z)\geq 0$, there is no bound right chiral fermion zero mode. For both left and right chiral fermions, there exists a continuous gapless spectrum of the KK modes.
For positive $\eta$, only the potentials for left chiral fermions have a negative single well and a double well at the location of the branes for single brane and double brane, respectively, which could trap the left chiral fermion zero mode solved from (\[CoupleEq1a\]) by setting $m_0=0$: $$f_{L0}(z)
\propto \exp\left(-\eta\int^z d\bar{z}\text{e}^{A(\bar{z})}\phi(\bar{z})\right).
\label{zeroModefL0CaseI}$$ In order to check the normalization condition (\[orthonormality\]) for the zero mode (\[zeroModefL0CaseI\]), we need to check whether the inequality $$\int f_{L0}^2(z) dz \propto
\int \exp\left(-2\eta\int^z d\bar{z} \text{e}^{A(\bar{z})}\phi(\bar{z})\right) dz
< \infty \label{condition1}$$ is satisfied. For the integral $\int dz\text{e}^{A}\phi$, we only need to consider the asymptotic characteristic of the function $\eta\;\text{e}^{A}\phi$ for $z \rightarrow \infty$. For the asymmetric brane scenario, we have $$\begin{aligned}
2\;\text{e}^{A} \phi
&=& \frac{2{\sqrt{3(2s-1)}}\arctan(\lambda z)^s
\left(1+(\lambda z)^{2s}\right)^{-1/2s}}
{{s}\left[1+a z\
{}_2F_1\left(\frac{1}{2s},\frac{1}{s},1+\frac{1}{2s}
,-(\lambda z)^{2s}\right)\right]}
\nonumber \\
&\rightarrow& \frac{1}{{\eta_0}z} ~~~~~ \text{for} ~~~z \rightarrow \infty,\end{aligned}$$ where the constant ${\eta_0}$ is given by $$\begin{aligned}
{\eta_0} = \frac{s\lambda }{{\sqrt{3(2s-1)}}\;\pi}
\left( 1+ a\frac{\Gamma(1+1/2s)\Gamma(1/2s)}
{\lambda\;\Gamma(1/s)} \right).\end{aligned}$$ For the symmetric brane case we only need to take $a=0$. So, when $z
\rightarrow \infty$, we have $$\begin{aligned}
f_{L0}^2(z)\propto\exp\left(-2\eta\int^z
d\bar{z}\text{e}^{A(\bar{z})}\phi(\bar{z})\right)
\rightarrow z^{-\eta/{\eta_0}},\end{aligned}$$ which indicates that the normalization condition (\[condition1\]) is $$\eta>{\eta_0}. \label{conditionCaseI}$$ Provided the condition (\[conditionCaseI\]), the zero mode of left chiral fermions can be localized on the brane. In Refs. [@KoleyCQG2005; @LiuPRD2008], it was shown that the corresponding zero mode can also be localized on the brane in the background of Sine-Gordon kinks provided similar condition as (\[conditionCaseI\]). While the fermion zero mode can not be localized on the de Sitter brane with the same coupling $F(\phi)=\phi$ [@LiuJCAP2009]. Note that for large $s$, ${\eta_0}$ can be approximated as $$\begin{aligned}
{\eta_0} \approx \frac{(\lambda+2a)}{{\sqrt{6}}\;\pi}
\sqrt{s}.\end{aligned}$$ It is clear that in order for the potentials to localize the zero mode of left chiral fermions for larger $s$, $\lambda$ or asymmetric factor $a$, the stronger coupling of kink and fermions is required. That is to say, the massless mode of left chiral fermion is most easy to be localized on the symmetric single brane. The asymmetric factor $a$ may destroy the localization of massless fermions. This is different from the situation of the zero modes of scalars and vectors on symmetric and asymmetric de Sitter branes [@LiuJCAP2009] , where increasing the asymmetric factor $a$ does not change the number of the bound vector KK modes but would increase that of the bound scalar KK modes, and the zero modes of scalars and vectors are always localized on the de Sitter branes.
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In Fig. \[fig\_zeroModefL0CaseI\], we plot the left chiral fermion potentials $V^{S,A}_L(z)$ and the corresponding zero modes. We see that the zero modes are bound on the brane. They represents the lowest energy eigenfunction (ground state) of the Schrödinger equation (\[SchEqLeftFermion\]) since they have no zeros. Since the ground state has the lowest mass square $m_0^2=0$, there is no tachyonic left chiral fermion mode. The zero mode on both the symmetric and asymmetric double walls is essentially constant between the two interfaces. This is very different from the case of gravitons, scalars and vectors, where the massless modes on the asymmetric double wall are strongly localized only on the interface centered around the lower minimum of the potential. The massive modes will propagate along the extra dimension and those with lower energy would experience an attenuation due to the presence of the potential barriers near the location of the brane.
The potential $V_R$ is always positive near the brane location and vanishes when far away from the brane. This shows that it could not trap any bound fermions with right chirality and there is no zero mode of right chiral fermions. However, the shape of the potential is strongly dependent on the parameter $s$. For $s\geq
3$, a potential well around the brane location would appear and the well becomes deeper and deeper with increase of $\eta$. The appearance of the potential well could be related to resonances, i.e., massive fermions with a finite lifetime [@0901.3543; @Liu0904.1785]. In Ref. [@0901.3543], a similar potential and resonances for left and right chiral fermions were found in background of two-field generated thick branes with internal structure. We can investigate the massive modes of fermions by solving numerically Eqs. (\[SchEqLeftFermion\]) and (\[SchEqRightFermion\]).
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In Ref. [@0901.3543], the authors suggested that large peaks in the distribution of $f_{L,R}(0)$ as a function of $m$ would reveal the existence of resonant states. In Ref. [@Liu0904.1785], we extended this idea and proposed that large relative probabilities for finding massive KK modes within a narrow range $-z_b<z<z_b$ around the brane location, are called $P_{L,R}$, would indicate the existence of resonances. The relative probabilities are defined as follows: $$P_{L,R}(m)=\frac{\int_{-z_b}^{z_b} |f_{L,R}(z)|^2 dz}
{\int_{-z_{max}}^{z_{max}} |f_{L,R}(z)|^2 dz},
\label{Probability}$$ where we choose $z_b=0.1 z_{max}$. For the set of parameters: $\eta=4,\lambda=1$ and $s=7$, we find two resonances with mass square 2.4503 and 8.917 for both left and right chiral fermions (see Fig. \[fig\_fResonanceCase1\]). The configurations of Figs. \[fig\_fResonanceCase1\]c and \[fig\_fResonanceCase1\]d could present the $n=1$ and $n=2$ level KK resonance modes of left chiral fermions. The $n=0$ level mode with left chirality is in fact the only one bound state (the zero mode). While the configurations of Figs. \[fig\_fResonanceCase1\]a and \[fig\_fResonanceCase1\]b present the $n=0$ and $n=1$ level resonances of right chiral fermions. We note that the spectra of massive left-handed and right-handed fermionic resonances are the same, which demonstrates that a Dirac fermion could be composed from the left and right resonance KK modes [@Liu0904.1785]. The lifetime $\tau$ for a resonance can be estimated by the width in mass $\Gamma=\Delta{m}$ at half maximum of the corresponding peak in Fig. \[fig\_PLPRResonanceCase1\], which means that the fermion disappears into the extra dimension with time $\tau\sim\Gamma^{-1}$ [@RubakovPRL2000]. The lifetime of the resonances are listed in Table \[tab1\].
[|l||c|c|c|c|]{} $~$ & $m^2$ & $m$ & $\Gamma$ & $\tau$\
$n=0$(right) & 2.4503 & 1.5653 & 0.001177 & 849.6\
$n=1$(left) & 2.4503 & 1.5653 & 0.001184 & 844.5\
$n=1$(right) & 8.9179 & 2.9863 & 0.04606 & 21.71\
$n=2$(left) & 8.9179 & 2.9863 & 0.04580 & 21.83\
Case II: $F(\phi)=\tan^{1/s}(\phi/\phi_0)$ {#sec3.2}
------------------------------------------
Next, we consider the case $F(\phi)=\tan^{1/s}(\phi/\phi_0)$, for which the potentials take the forms of $$\begin{aligned}
V^S_L(z) &=& \frac{\eta^2({\lambda}z)^{2}}
{[1+({\lambda}z)^{2s}]^{\frac{1}{s}}}
- \frac{\eta\lambda}
{[1+({\lambda}z)^{2s}]^{1+\frac{1}{2s}}}
, \label{VSL_CaseII} \\
V^S_R(z) &=& \frac{\eta^2({\lambda}z)^{2}}
{[1+({\lambda}z)^{2s}]^{\frac{1}{s}}}
+ \frac{\eta\lambda}
{[1+({\lambda}z)^{2s}]^{1+\frac{1}{2s}}}
, \label{VSR_CaseII}\end{aligned}$$ and $$\begin{aligned}
V^A_L(z)
&=& \left\{
\frac{\eta^2 ({\lambda}z)^2 }
{ {\left[1+({\lambda}z)^{2s}\right]^{\frac{1}{s}}} }
+\frac{a \eta {\lambda}z }
{ {\left[1+({\lambda}z)^{2s}\right]^{\frac{3}{2s}}} }
\right\} \frac{1}{ {\cal{F}}^{2}(z) }
\nonumber \\ &&
-\frac{\eta\lambda}
{ [1+({\lambda}z)^{2s}]^{1+\frac{1}{2s}} }
\frac{1}{ {\cal{F}}(z) }, \label{VAL_CaseII} \\
V^A_R(z) &=& V^A_L(z)|_{\eta \rightarrow -\eta},\label{VAR_CaseII}\end{aligned}$$ for the symmetric and asymmetric brane solutions, respectively.
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\
**The symmetric potential**\
We first investigate the potential $V^S_L(z)$ (\[VSL\_CaseII\]) for the symmetric brane. It has a minimum (negative value) $-\eta\lambda$ at $z=0$ and a maximum (positive value) $\eta^2$ at $z=\pm\infty$. The shapes of the potential for various parameters are plotted in Fig. \[fig\_VSLR\_CaseII\], from which we can see that they are similar to that of a Pöschl-Teller (PT) potential for finite $s$. The massless KK mode can be solved as follows: $$\begin{aligned}
f_{L0}(z) \propto \exp\left\{-\frac{1}{2}\eta\lambda z^2
{~_2F_1}\left(\frac{1}{s},\frac{1}{2s},
1+\frac{1}{s},-({\lambda}z)^{2s}\right)\right\}.
\label{zeroModefSL0CaseII}\end{aligned}$$ Because $f^2_{L0}(z) \propto \exp(-2\eta z)$ when $z\rightarrow\infty$, the massless KK mode is normalizable without additional conditions, and it would be strongly localized on the brane with large coupling constant $\eta$ (see Fig. \[fig\_fL0\_CaseII\_s\]). We note that the potential $V^S_L(z)$ (\[VSL\_CaseII\]) and the zero mode $f_{L0}(z)$ (\[zeroModefSL0CaseII\]) are very different from those given in (\[VSL\_CaseI\]) and (\[zeroModefL0CaseI\]) for left chiral fermions:
\(1) The potential (\[VSL\_CaseII\]) has a single well but the potential (\[VSL\_CaseI\]) has a double well for $s\geq 3$, which results in that the zero mode (\[zeroModefSL0CaseII\]) is strongly localized at the center of the double brane while the zero mode (\[zeroModefL0CaseI\]) is localized between the two sub-branes of the double brane.
\(2) The potential (\[VSL\_CaseII\]) tends to a positive constant but the potential (\[VSL\_CaseI\]) runs to zero when far away from the brane, which results in that the localization of the zero modes here and (\[zeroModefL0CaseI\]) is unconditional and conditional (with condition (\[conditionCaseI\])), respectively.
\(3) The potential here provides mass gap to separate the zero mode from the excited KK modes.
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We have known that the massless KK mode is the lowest state. The massive bound KK modes would appear provided large $\eta$. Here we take $\lambda=1$ for convenience. The number of bound KK modes increases with the coupling constant $\eta$. For $\eta=0.1$, only zero modes are bound for all $s$. For $\eta=1$, there are two bound KK modes for $s=1$ and one bound KK mode (zero mode) for any $s\geq
3$, and the spectra of the KK modes are $$\begin{aligned}
\begin{array}{ll}
m_{Ln}^2 =\{0, 0.94\} \cup ~[1,\infty) & ~~~~~\text{for} ~~~~~s=1, \\
m_{Ln}^2 =\{0\} \cup ~[1,\infty) & ~~~~~\text{for} ~~~~~s\geq3.
\end{array}
\label{spectraMSLn1CaseII}\end{aligned}$$ For $\eta=6$, there are many bound KK modes for $s=1$ and four bound KK modes for any $s\geq 3$, and the spectra of the KK modes are $$\begin{aligned}
m_{Ln}^2 &=&\{0, 10.59, 18.57, 24.37, 28.44, 31.20, 33.01,
\nonumber \\
&& ~ 34.17,34.89, 35.33, 35.60, 35.76, \cdots\}\nonumber\\
&& \cup ~[36,\infty) ~~~ \text{for} ~~s=1, \label{spectraMSLn2CaseII}\\
m_{Ln}^2 &=&\{0, 11.89, 23.12, 32.21\} \cup ~[36,\infty)
~~~ \text{for} ~~s=3, \nonumber\\
m_{Ln}^2 &=&\{0, 11.99, 23.85, 34.49\} \cup ~[36,\infty)
~~~ \text{for} ~~s\rightarrow\infty.~\nonumber\end{aligned}$$ The spectra are plotted in Fig. \[fig\_fSL\_Spectra\_CaseII\]. The continuous spectrum starts at $m^2 = \eta^2$ and the KK modes asymptotically turn into plane waves when far away from the brane, which represent delocalized massive KK modes of fermions. It can be seen that the spectrum structure for the single brane scenario with $s=1$ is very different from that of the double brane scenario with $s\geq 3$. The single brane could trap more massive KK modes than the double brane. Noting that $$\begin{aligned}
V^S_L(z)=-\eta\lambda+ \eta^2 (\lambda z)^2
+\frac{2s+1}{2s}\eta\lambda (\lambda z)^{2s}
+ {\cal{O}} ((\lambda z)^{2s+2}),\nonumber\end{aligned}$$ we have the following simple potential for double thin brane scenario ($s\rightarrow \infty$): $$\begin{aligned}
V^S_L(z)=\left\{
\begin{array}{cc}
-\eta\lambda+ \eta^2 (\lambda z)^2, & ~~~|\lambda z|<1 \\
\eta^2, & ~~~|\lambda z|>1
\end{array}\right.\end{aligned}$$ which could be called “the harmonic oscillator potential well with finite depth". Similar to the square potential well with finite depth, the spectrum can be solved and there are finite number of bound KK modes.
The shape of the symmetric potential $V^S_{R}(z)$ (\[VSR\_CaseII\]) for right chiral fermions is more complex than that of left chiral fermions. It has a positive value $\eta\lambda$ at $z=0$ and trends to $\eta^2$ at $z=\pm\infty$. The shapes of the potential for various parameters are plotted in Fig. \[fig\_VSLR\_CaseII\]. For small $\eta$, the potential for any $s$ has no well to trap bound KK modes. With the increase of $\eta$, the potential for $s\geq 3$ will appear a single well, while the potential for $s=1$ will first appear a double well and then become a single well. For large $\eta$, they are similar to a PT potential only for small $s$ (see Fig. \[fig\_VSR\_CaseIIc\]). The massless KK mode for right chiral fermions is absent. The number of bound KK modes increases with the coupling constant $\eta$. For $\eta=0.1$, there is no any bound KK mode for all $s$. For $\eta=1$, there are one bound KK mode for $s=1$ and no bound KK mode for any $s\geq 3$, and the spectra of the KK modes are $$\begin{aligned}
\begin{array}{ll}
m_{Rn}^2 =\{0.94\} \cup ~[1,\infty) & ~~~~~\text{for} ~~~~~s=1, \\
m_{Rn}^2 =\{~~\} \cup ~[1,\infty) & ~~~~~\text{for} ~~~~~s\geq3.
\end{array}
\label{spectraMSRn1CaseII}\end{aligned}$$ For $\eta=6$, there are many bound KK modes for $s=1$ and three bound KK modes for any $s\geq 3$: $$\begin{aligned}
m_{Rn}^2 &=&\{10.59, 18.57, 24.37, 28.44, 31.20, 33.01, \nonumber \\
&& ~ 34.17, 34.89, 35.33, 35.60, 35.76, \cdots\}\nonumber \\
&& \cup ~[36,\infty)~~~~~~ \text{for} ~~~~~s=1, \label{spectraMSRn2CaseII}\\
m_{Rn}^2 &=&\{11.89, 23.12, 32.21\} \cup ~[36,\infty)
~~ \text{for} ~~s=3, \nonumber\\
m_{Rn}^2 &=&\{11.99, 23.85, 34.49\} \cup ~[36,\infty)
~~ \text{for} ~~s\rightarrow\infty,~~\nonumber\end{aligned}$$ The spectra are plotted in Fig. \[fig\_fSR\_Spectra\_CaseII\]. The continuous spectrum starts also at $m^2 = \eta^2$. It can be seen that the single brane could also trap more massive KK modes than the double brane. By comparing the mass spectra of right chiral fermions (\[spectraMSRn1CaseII\]) and (\[spectraMSRn2CaseII\]) with the ones of left chiral fermions (\[spectraMSLn1CaseII\]) and (\[spectraMSLn2CaseII\]) for $\eta=1$ and $\eta=6$, respectively, we come to the conclusion that the number of bound states of right chiral fermions $N_R$ is one less than that of left ones $N_L$, i.e., $N_R=N_L-1$. The mass spectra are almost the same, and the only one difference is the absent of the zero mode of right chiral fermions. Although a potential well around the brane location appears for $s\geq3$ in Fig. \[fig\_VSR\_CaseIIb\], we do not find any resonance.
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\
**The asymmetric potential**\
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Next, let us turn to the potentials (\[VAL\_CaseII\]) and (\[VAR\_CaseII\]) for the asymmetric brane. We note that, because of the appearance of the asymmetric factor $a$, the potentials have different limits at $z=\pm\infty$: $$\begin{aligned}
V^{A}_{L,R}(+\infty) = \frac{\eta^2}
{\left(1+ a \frac{\Gamma(1/2s) \Gamma(1+1/2s)}{\lambda\;\Gamma(1/s)}
\right)^2 }, \\
V^{A}_{L,R}(-\infty) = \frac{\eta^2}
{\left(1- a \frac{\Gamma(1/2s) \Gamma(1+1/2s)}{\lambda\;\Gamma(1/s)}
\right)^2 }.
\label{VAinftyCaseII}\end{aligned}$$ For $s=1$ and $s\rightarrow\infty$, we have $$V^{A}_{L,R}(\pm\infty) = \frac{\eta^2}
{\left(1\pm \frac{\pi}{2}\frac{a}{\lambda}\right)^2 }
\label{VAinftyCaseIIs1}$$ and $$V^{A}_{L,R}(\pm\infty) \rightarrow \frac{\eta^2}
{\left(1\pm 2\frac{a}{\lambda}\right)^2 },
\label{VAinftyCaseIIsinfty}$$ respectively. The constrain condition (\[constraintOna\]), i.e., $0< a \frac{\Gamma(1/2s) \Gamma(1+1/2s)}{\lambda\;\Gamma(1/s)} <1$, implies $$0<\frac{\eta^2}{4}<V^{A}_{L,R}(\infty) <\eta^2<V^{A}_{L,R}(-\infty)<\infty.
\label{VAinftyCaseII2}$$ Hence, comparing with the symmetric potentials $V^{S}_{L,R}(z)$ (\[VSL\_CaseII\]) and (\[VSR\_CaseII\]), the value of the asymmetric potentials $V^{A}_{L,R}(z)$ (\[VAL\_CaseII\]) and (\[VAR\_CaseII\]) are enlarged at $z\rightarrow +\infty$ and diminished at $z\rightarrow -\infty$, which would reduce the number of the bound KK modes of left and right fermions. The shapes of the potentials for various parameters are plotted in Figs. \[fig\_VAL\_CaseII\] and \[fig\_VAR\_CaseII\].
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The massless KK mode of left chiral fermions $$f_{L0}(z) \propto
\exp\left(\int^z d\bar{z}
\frac{-\eta\lambda \bar{z}\;{[1+({\lambda}\bar{z})^{2s}]^{-\frac{1}{2s}}}}
{1+a \bar{z} {~_2F_1}\left(\frac{1}{2 s},\frac{1}{s},
1+\frac{1}{2s},-({\lambda}\bar{z})^{2s}\right)}
\right)
\label{zeroModefAL0CaseII}$$ is also normalizable, and can be localized on the brane without additional conditions. The effect of the asymmetric factor $a$ and the coupling constant $\eta$ to the zero mode is shown in Fig. \[fig\_fL0\_CaseII\_a\]. It can be seen that, with increase of $\eta$, the effect of the asymmetric factor $a$ can be neglected. However, the effect of $a$ to the number of bound KK modes is remarkable. We also take $\lambda=1$ here. For $\eta=0.1$, only zero modes of left chiral fermions are bound for all $s$ and $a$. For $\eta=1$ and very small $a$, there is one bound massive KK mode of both the right and left chiral fermions for $s=1$, and no bound massive KK mode for $s\geq3$. For $\eta=1$ and large $a$, only massless modes of left chiral fermions are bound for all $s$. For $\eta=6$ and $a=0.2$, there are only three and two bound KK modes of left chiral fermions for $s=1$ and $s\geq 3$, respectively, and the spectra of the KK modes are $$\begin{aligned}
m_{Ln}^2 &=&\{0, 10.40, 17.42\} \cup [20.85,\infty)
~ \text{for} ~~s=1, \nonumber\\
m_{Ln}^2 &=&\{0, 11.74\} \cup ~[18.75,\infty)
~~~~~~~~ \text{for} ~~s=3,\label{spectraMALn2CaseII}\\
m_{Ln}^2 &=&\{0, 11.77\} \cup ~[18.37,\infty)
~~~~~~~~ \text{for} ~~s\rightarrow\infty,\nonumber\end{aligned}$$ and $$\begin{aligned}
m_{Rn}^2 &=&\{ 10.40, 17.42\} \cup [20.85,\infty)
~ \text{for} ~~s=1, \nonumber\\
m_{Rn}^2 &=&\{ 11.74\} \cup ~[18.75,\infty)
~~~~~~~~ \text{for} ~~s=3,\label{spectraMARn2CaseII}\\
m_{Rn}^2 &=&\{ 11.77\} \cup ~[18.37,\infty)
~~~~~~~~ \text{for} ~~s\rightarrow\infty,\nonumber\end{aligned}$$ for left and right chiral fermions, respectively. The spectra for left chiral fermions are shown in Fig. \[fig\_fAL\_Spectra\_CaseII\]. The continuous spectrum starts at different values for different $s$. It can be seen that the spectrum structure for the single brane case with $s=1$ is dramatically changed by the asymmetric factor $a$, i.e., the number of bound KK modes quickly decreases with the increase of $a$.
Case III: $F(\phi)=\tan^{k/s}(\phi/\phi_0)$ {#sec3.3}
-------------------------------------------
For the case $F(\phi)=\tan^{k/s}(\phi/\phi_0)$, considering the expression of $\phi$ (\[phi3\]), we have $F(\phi(z))=(\lambda
z)^k$. The potential (\[VL\]) for the asymmetric brane solution reads as $$\begin{aligned}
V^A_L(z) &=& \left\{\frac{\eta^2(\lambda z)^{2k}}
{[1+({\lambda}z)^{2s}]^{{1}/{s}} }
+ \frac{\eta\; a(\lambda z)^{k}}
{[1+({\lambda}z)^{2s}]^{{3}/{2s}} }
\right\}\frac{1}{{\cal F}^{2}(z)}
\nonumber \\&&
-\frac{\eta\lambda(\lambda z)^{k-1}[k+(k-1)(\lambda z)^{2s}]}
{[1+({\lambda}z)^{2s}]^{1+{1}/{2s}} }
\frac{1}{{\cal F}(z)}.\nonumber\\
\label{VAL_CaseIII}\end{aligned}$$ The special value $k=1$ belongs to case II considered in sub section \[sec3.2\]. Taking $a=0$, we will get the potential of left chiral fermions for the symmetric brane solution: $$\begin{aligned}
V^S_{L,R}(z) = \frac{\eta^2(\lambda z)^{2k}}
{[1+({\lambda}z)^{2s}]^{{1}/{s}} }
\mp\frac{\eta\lambda(\lambda z)^{k-1}[k+(k-1)(\lambda z)^{2s}]}
{[1+({\lambda}z)^{2s}]^{1+/{1}/{2s}} }.
\label{VSL_CaseIII}\end{aligned}$$
\
**The symmetric potential**\
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Let us first analyze the asymptotic property of the symmetric potential. When $z\rightarrow\infty$, $1 + (\lambda z)^{2
s}\rightarrow(\lambda z)^{2 s}$, we have $$\begin{aligned}
V^S_{L,R}(z\rightarrow\infty) &\rightarrow& \eta^2 (\lambda z)^{2k-2}
\mp \eta \lambda k (\lambda z)^{k-2-2s}\nonumber\\
&& \mp \eta \lambda (k-1) (\lambda z)^{k-2} \nonumber\\
&\rightarrow&\left\{
\begin{array}{ll}
\eta^2 (\lambda z)^{2k-2} \rightarrow\infty
& ~~ \text{for}~~ k>1 \\
\eta^2 >0 & ~~ \text{for}~~ k=1 \\
0 & ~~ \text{for}~~ k< 1
\end{array}\right..~~~~~\end{aligned}$$ When $z\rightarrow0$, $1 + (\lambda z)^{2 s}\rightarrow1$, we have $$\begin{aligned}
&&V^S_{L,R}(z\rightarrow0) \rightarrow
\eta^2 (\lambda z)^{2k} \left(1-\frac{1}{s}({\lambda}z)^{2s}\right)
\nonumber \\
&&\mp\eta\lambda(\lambda z)^{k-1}[k+(k-1)(\lambda z)^{2s}]
\left(1-\frac{2s+1}{2s}({\lambda}z)^{2s}\right) \nonumber\\
&&\rightarrow\left\{
\begin{array}{ll}
0 & ~~ \text{for}~~ k>1 \\
\mp \eta \lambda & ~~ \text{for}~~ k=1 \\
\frac{\eta(\eta\pm\lambda)}{(\lambda z)^{2}}
+ (-\eta^2 \pm \frac{1}{2}\eta\lambda)\delta_{s,1}
& ~~ \text{for}~~ k=-1\\
\infty & ~~ \text{for}~~ k<-1 \\
\end{array}\right. .\end{aligned}$$ For $\eta>0$, $\lambda>0$ and $k=-1$, $$\begin{aligned}
V^S_{L}(z\rightarrow0) &\rightarrow&
\infty \\
V^S_{R}(z\rightarrow0) &\rightarrow& \left\{
\begin{array}{cl}
\infty & ~~~~ \text{for}~~~~ \eta\neq\lambda \\
-\frac{3}{2}\eta^2 & ~~~~ \text{for}~~~~ \eta=\lambda, s=1 \\
0 & ~~~~ \text{for}~~~~ \eta=\lambda, s>1
\end{array}\right. .
\end{aligned}$$ The shapes of the symmetric potentials $V^S_L(z)$ and $V^S_R(z)$ for the case $F(\phi)=\tan^{k/s}(\phi/\phi_0)$ with odd $k<0$ and $k>1$ are plotted in Figs. \[fig\_VSLR\_CaseIII1\] and \[fig\_VSLR\_CaseIII2\], respectively. For the case $k<0$, the symmetric potential $V^S_L(z)$ of left chiral fermions has no well and can not trap any bound KK modes. For right chiral fermions, the symmetric potential $V^S_R(z)$ with $k<-1$ or $k=-1$, $\eta\neq\lambda$, has a double well and a infinite high bar, which can also not trap the massless mode. However, the case $k=-1$, $\eta=\lambda>0$ is very special, for which $V^S_R(z)$ has a single well (for $s=1$) or a double well (for $s\geq3$) (see Fig. \[fig\_VSR\_CaseIII1a\]). The bound KK modes corresponding to the potentials shown in Fig. \[fig\_VSR\_CaseIII1a\] for $s=1$, $s=3$ and $s\rightarrow\infty$ have mass square $-0.636$, $-0.1485$ and $-0.119$, respectively. Because all the potentials with $k<0$ can not trap zero modes, or even more, some of them could result in tachyonic KK modes with $m^2<0$, we do not consider the corresponding kink-fermion coupling of the type $\eta\overline{\Psi}\tan^{k/s}(\phi/\phi_0)\Psi$ with $k<0$.
For $k>1$, which is the case we are interesting in here, the potentials $V^S_{L,R}(z)$ trend to infinite when far away from the brane and vanish at $z=0$, which shows that there exist infinite discrete bound KK modes. We note from Fig. \[fig\_VSLR\_CaseIII2\] that the influence of $s$ is not important. For examples, the spectra for $k=3$ and $k=11$ are calculated as $$\begin{aligned}
m_{Ln}^2&=&\{0, 1.57, 4.80, 8.44, 12.61, 17.17, 22.08, \nonumber \\
&& 27.30, 32.78, 38.52, 44.49,
\cdots\} ~~~ \text{for} ~s=1, \nonumber\\
m_{Ln}^2 &=&\{0, 1.76, 5.39, 9.33, 13.73, 18.56, 23.70, \nonumber \\
&& 29.14, 34.85, 40.80, 46.97,
\cdots\} ~~~ \text{for} ~s=3,\\
m_{Ln}^2 &=&\{0, 1.81, 5.52, 9.44, 13.77, 18.65, 23.81, \nonumber \\
&& 29.19, 34.93, 40.88, 47.02,
\cdots\}~~~ \text{for} ~s\rightarrow\infty,\nonumber\end{aligned}$$ and $$\begin{aligned}
m_{Ln}^2 &=&\{0, 1.86, 7.30, 16.02, 27.68, 42.03, 58.93, \nonumber \\
&&78.30, 100.08, 124.22, \cdots\}
~~~ \text{for} ~s=1, \nonumber\\
m_{Ln}^2 &=&\{0, 1.93, 7.59, 16.66, 28.77, 43.67, 61.21, \nonumber \\
&& 81.30, 103.87, 128.88, \cdots\}
~~~ \text{for} ~s=3,\\
m_{Ln}^2 &=&\{0, 1.96, 7.69, 16.87, 29.12, 44.15, 61.81, \nonumber \\
&& 82.00, 104.68, 129.82, \cdots\}
~~~ \text{for} ~s\rightarrow\infty,\nonumber\end{aligned}$$ respectively, and shown in Fig. \[fig\_Mn2\_CaseIII1\].
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With the increase of $k$, the potentials vanish in a more wide range around $z=0$. Especially, for $k\rightarrow\infty$, we get an infinite deep square well for right hand fermions: $$\begin{aligned}
V^S_{R} = \left\{
\begin{array}{cc}
\infty, & ~~~~|z|>{1}/{\lambda} \\
0, & ~~~~|z|<{1}/{\lambda}
\end{array}
\right. ,\end{aligned}$$ The KK modes and the spectrum reads as $$\begin{aligned}
f_{Rn}=\left\{\begin{array}{cc}
\sqrt{\lambda}\cos({n\pi\lambda z}/{2}), & ~~n=1,3,5,\cdots \\
\sqrt{\lambda}\sin({n\pi\lambda z}/{2}), & ~~n=2,4,6,\cdots
\end{array}\right.
\label{fRn_CaseIIIa}\end{aligned}$$ $$\begin{aligned}
m_{Rn}^2=\left(\frac{\pi}{2}\lambda n \right)^2. ~~~(n=1,2,3,\cdots)
\label{mn2_CaseIIIa}\end{aligned}$$ The numeric result for $k=13111$, $s=\eta=\lambda=1$ is $$\begin{aligned}
m_{Rn}^2 =\{2.46, 9.85, 22.16, 39.39, 61.54,
88.62, \cdots\},\end{aligned}$$ from which we have $$\begin{aligned}
\frac{m_{Rn}}{m_{R1}} =\{1, 2.001, 3.001, 4.002, 5.002,
6.002, \cdots\}.\end{aligned}$$ For left hand fermions, the spectrum also takes the form (\[mn2\_CaseIIIa\]) but with $n=0,1,2,\cdots$, the KK modes can be calculated from Eqs. (\[CoupleEq1b\]) and (\[fRn\_CaseIIIa\]) as: $$\begin{aligned}
f_{Ln}=\left\{\begin{array}{ll}
-\sqrt{\lambda}\sin({n\pi\lambda z}/{2}), & ~~n=1,3,5,\cdots \\
\left\{
\begin{array}{cl}
\lambda/2, & ~|{z}|<1/\lambda \\
0, & ~|{z}|>1/\lambda
\end{array}\right.,
& ~~n=0 \\
-\sqrt{\lambda}\cos({n\pi\lambda z}/{2}), & ~~n=2,4,6,\cdots
\end{array}\right.
\label{fLn_CaseIIIa}\end{aligned}$$ The comparing of spectra of left chiral fermions for different $k$ is shown in Fig. \[fig\_Mn2\_CaseIII2\].
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\
**Asymmetric branes**\
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At last, we consider the asymmetric potential (\[VAL\_CaseIII\]) for left chiral fermions and the corresponding asymmetric potential (\[VR\]) for right chiral fermions. The asymptotic property of them is analyzed as follows. When $z\rightarrow\pm\infty$, $1 +
(\lambda z)^{2 s}\rightarrow(\lambda z)^{2 s}$, ${\cal{F}}(z)\rightarrow \left(1\pm a \frac{\Gamma(1/2s)
\Gamma(1+1/2s)}{\lambda\;\Gamma(1/s)}\right)$, we have $$\begin{aligned}
V^A_{L,R}(z) &\rightarrow&\left\{
\begin{array}{ll}
\frac{\eta^2 (\lambda z)^{2k-2}}
{\left(1\pm a \frac{\Gamma(1/2s) \Gamma(1+1/2s)}
{\lambda\;\Gamma(1/s)}\right)^2 }
\rightarrow\infty & \text{for}~ k>1 \\
\frac{\eta^2}
{\left(1\pm a \frac{\Gamma(1/2s) \Gamma(1+1/2s)}
{\lambda\;\Gamma(1/s)}\right)^2 }>0 & \text{for}~ k=1 \\
0 & \text{for}~ k< 1
\end{array}\right. .\end{aligned}$$ Since $V^S_{L,R}(z\rightarrow\pm\infty)$ are finite for $k\leq3$, we only consider the case $k\geq3$ here, for which $V^S_{L,R}(z\rightarrow0)\rightarrow0$. Comparison of the asymmetric potential $V^A_L(z)$ with the symmetric one $V^S_L(z)$ for the case $F(\phi)=\tan^{k/s}(\phi/\phi_0)$ with different $k$ and $s$ is shown in Fig. \[fig\_VAL\_CaseIII\]. We see that, for a fixed finite $k$, the difference of the two potentials would become large with increase of $s$. For $s\rightarrow\infty$, the difference is largest. While for a fixed $s$, the difference of the two potentials would become small with increase of $k$. The spectra for $k=3$ and $k=11$ are calculated as $$\begin{aligned}
m_{Ln}^2 &=&\{0, 1.59, 4.77, 8.50, 12.7, 17.4, 22.4, 27.7,
\nonumber\\ &&
33.3, 39.2, 45.4, 51.7, \cdots\},
~ (s=1) \\
m_{Ln}^2 &=&\{0, 1.83, 5.09, 9.27, 14.0, 19.4, 25.2, 31.5,
\nonumber\\ &&
38.2, 45.3, 52.7, 60.4, \cdots\},
~ (s\rightarrow\infty)\end{aligned}$$ and $$\begin{aligned}
m_{Ln}^2 &=&\{0, 1.88, 7.40, 16.2, 28.1, 42.6, 59.8, 79.5,
\nonumber\\ &&
102, 126, 153, 182, \cdots\},
~ (s=1) \\
m_{Ln}^2 &=&\{0, 2.09, 8.22, 18.1, 31.4, 47.8, 67.5, 90.3,
\nonumber\\ &&
116, 145, 177, 212, \cdots\},
~ (s\rightarrow\infty)\end{aligned}$$ respectively, where $a=0.5$, and the comparing with that of the symmetric potential is shown in Fig. \[fig\_Mn2\_CaseIII\]. For $k\rightarrow\infty$, the difference between $V^A_L(z)$ and $V^S_L(z)$ disappears, and the spectrum is $m_{Ln}=n\lambda{\pi}
/{2} ~(n=0,1,2,3,\cdots)$.
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The normalizable zero mode of left chiral fermions $$f_{L0}(z) \propto
\exp\left(\int^z d\bar{z}
\frac{-\eta(\lambda \bar{z})^k
[1+({\lambda}\bar{z})^{2s}]^{-\frac{1}{2s}}}
{1+a \bar{z} {~_2F_1}\left(\frac{1}{2 s},\frac{1}{s},
1+\frac{1}{2s},-({\lambda}\bar{z})^{2s}\right)}
\right) \label{zeroModefAL0CaseIII}$$ can also be localized on the brane without additional conditions. The effect of $\eta$, $k$, $s$ and the asymmetric factor $a$ to the zero mode is shown in Figs. \[fig\_fAL\_CaseIII\_eta\] and \[fig\_fAL\_CaseIII\]. It can be seen that, with increase of $\eta$ or $k$, the difference of zero modes with different $a$ would reduce, i.e., the effect of the asymmetric factor could be neglected. While, with increase of $a$, the effect of $s$ can not be neglected. For the case $a=0.5$ and $s\rightarrow\infty$, our four-dimensional massless left fermions can not appear in the range $z<-1/\lambda$ (see Fig. \[fig\_fAL\_CaseIIId\]), namely, the left sub-brane is the left boundary of the region that the four-dimensional massless left fermions could appear. The effect of $k$ to the zero mode is remarkable: with the increase of $k$ (i.e., the increase of the kink-fermion interaction), the region that the four-dimensional massless left fermions can appear would reduce. Especially, for the limit case $k\rightarrow\infty$, we have $$f_{L0}(z) \propto \left\{
\begin{array}{ll}
1, & ~|\lambda {z}|<1 \\
0, & ~|\lambda {z}|>1
\end{array}\right. , \label{zeroModefAL0CaseIII}$$ which shows that the four-dimensional massless left fermions can only exist in between the locations of two sub-branes and the probability they would appear is equal everywhere within the region.
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Discussions and conclusions {#secConclusion}
===========================
In this paper, we have investigated the localization problem and the spectrum of spin half fermions on a two-parameter ($s$ and $\lambda$) family of symmetric branes and on a three-parameter ($s$, $\lambda$ and $a$) family of asymmetric thick branes in an AdS background for three kinds of typical kink-fermion couplings. The parameter $a$, which is called the asymmetry factor in this paper, decides the asymmetry of the solution. The parameter $\lambda$ labels the thickness of the brane. For $s=1$, the solution denotes a usual thick one-brane, which is the regularized version of the Randall-Sundrum thin brane. For $a=0$ ($a\neq0$) and odd $s>1$, the solution stands for a symmetric (an asymmetric) static double brane interpolating between same (different) AdS$_{5}$ vacua. The thickness of the double brane is $2/\lambda$, the two sub-branes are localized at $z=\pm 1/\lambda$. The thickness of the sub-brane decreases with the increase of $s$. For the limit case $s\rightarrow\infty$, each thick sub-brane becomes a thin brane.
By presenting the mass-independent potentials (\[Vfermion\]) of KK modes in the corresponding Schrödinger equations, we investigate the localization and mass spectra of bulk fermions on the symmetric and asymmetric thick branes. The formation of the potentials (\[Vfermion\]) have two sources: the gravity-fermion coupling $\bar{\Psi} \Gamma^M \omega_M \Psi$ and the scalar-fermion coupling $-\eta \bar{\Psi} F(\phi) \Psi$. It can be seen that, without the gravity-fermion coupling, namely, only considering domain walls in a flat space-time, the potentials do not disappear and hence fermions could be localized on the domain walls (see e.g. [@Rubakov1983]). In fact, for a kink solution, it is clear that the potential for one of left and right chiral fermions would be a PT like potential, for which the massless mode of left or right chiral fermion can be localized on the domain wall without additional condition. However, without scalar-fermion coupling ($\eta=0$), there is no bound KK mode for both left and right chiral fermions. Hence, in order to localize the massless and massive left or right chiral fermions on the branes, some kind of Yukawa coupling should be introduced.
The spectra are determined by the behavior of the potentials at infinity. The potentials we are interesting in have three types:\
1) $V_{L,R}(|z|\rightarrow \infty)\rightarrow 0$,\
2) $V_{L,R}(|z|\rightarrow \infty)\rightarrow C$,\
3) $V_{L,R}(|z|\rightarrow \infty)\rightarrow \infty$,\
where $C$ is a positive constant. In order to realize the three type of potentials, we have considered three typical Yukawa couplings correspondingly in this paper:\
Case I: ‘weak’ interaction with $F(\phi)=\phi^k$ ($k\geq1$),\
Case II: ‘critical’ interaction with $F(\phi)=\tan^{1/s}(\phi/\phi_0)$,\
Case III: ‘strong’ interaction with $F(\phi)=\tan^{k/s}(\phi/\phi_0)$ ($k>1$).\
Note that, as discussed above, for a domain wall solution in a flat space-time, a weak kink-fermion interaction would become a strong interaction. This means that the interaction with gravity would destroy the localization of fermions on the brane, in a way. So, the localization of fermions on the brane is the result of the competition of two interactions.
For the simplest Yukawa coupling with $F(\phi)=\phi$ and $\eta>0$, the potentials for left chiral fermions provide no mass gap to separate the fermion zero mode from the excited KK modes. Provided the condition (\[conditionCaseI\]), the zero mode of left chiral fermions can be localized on the brane. The massless mode of left chiral fermion is most easy to be localized on the symmetric single brane (i.e., the $a=0$ case). The asymmetric factor $a$ may destroy the localization of massless fermions. For $s>1$ (the double brane case), the potential for left chiral fermions have a double well at the location of the branes. The corresponding zero mode on both the symmetric and asymmetric double walls is essentially constant between the two interfaces. This is very different from the case of gravitons, scalars and vectors, where the massless modes on the asymmetric double wall are strongly localized only on the interface centered around the lower minimum of the potential. The massive KK modes asymptotically turn into continuous plane waves when far away from the brane, which represent delocalized massive KK fermions. The massive modes with lower energy would experience an attenuation due to the presence of the potential barriers near the location of the brane. It is interesting to notice that, for $s\geq 3$, a potential well around the brane location for right chiral fermions would appear and the well becomes deeper and deeper with increase of $\eta$. We have shown that this potential would result in a series of massive fermions with a finite lifetime [@0901.3543; @Liu0904.1785]. The spectra of left-handed and right-handed fermionic resonances are the same, which demonstrates that a Dirac fermion could be composed from the left and right resonance KK modes [@Liu0904.1785].
For the critical interaction with $F(\phi)=\tan^{1/s}(\phi/\phi_0)$ and $\eta>0$, we get a PT-like potential for left chiral fermions, which provides mass gap to separate the zero mode from the excited KK modes. The mass spectra for left and right chiral fermions are almost the same, and the only one difference is the absent of the zero mode of right chiral fermions. The massless KK mode of left chiral fermions is normalizable without additional conditions, and it would be strongly localized on the brane with large coupling constant $\eta$. The massive bound KK modes would appear provided large $\eta$. The spectra for the single brane and the double brane are quite different. For large $\eta$, there are more bound massive KK modes on the single brane than on the double brane. For the double thin brane ($s\rightarrow\infty$), a harmonic oscillator potential well with finite depth will get for both left and right chiral fermions and there are finite bound KK modes. For the asymmetric brane case, the potentials $V^{S}_{L,R}(z)$ are enlarged at $z\rightarrow +\infty$ and diminished at $z\rightarrow -\infty$, which shows that the asymmetric factor would reduce the number of the bound KK modes of left and right fermions. The continuous spectrum starts at different values for different $s$. The spectrum structure for the single brane case ($s=1$) is dramatically changed by the asymmetric factor: the number of bound KK modes quickly decreases with the increase of $a$ and the difference with the double brane case is reduced.
For the strong interaction with $F(\phi)=\tan^{k/s}(\phi/\phi_0)$ ($k>1$), the potentials $V^{S,A}_{L,R}(z)$ trend to infinite when far away from the brane and vanish at $z=0$, and there exist infinite discrete bound KK modes. The influence of $s$ is not important for symmetric potentials $V^S_{L,R}(z)$, which indicates that the spectra on the single brane and the double brane are almost the same. While the increase of $k$ will dramatically changes the shape of the potentials. Especially, for $k\rightarrow\infty$, the potential for right hand fermions is an infinite deep square well. For a fixed finite $k$, the difference of the symmetric and asymmetric potentials would become large with the increase of $s$. For $s\rightarrow\infty$, the difference is largest. While for a fixed $s$, the difference of the two potentials would become small with the increase of $k$. The normalizable zero mode of left chiral fermions can also be localized on the brane without additional conditions. With the increase of $\eta$ or $k$, the effect of the asymmetric factor to the zero mode can be neglected. While, with the increase of $a$, the effect of $s$ is obvious. For the limit case $s\rightarrow\infty$, the left sub-brane is the left boundary of the region that the four-dimensional massless left fermions could appear. With the increase of $k$, the region that the four-dimensional massless left fermions can appear would reduces. Especially, for the limit case $k\rightarrow\infty$, the four-dimensional massless left fermions can only exist in between the locations of two sub-branes with equal probability.
Acknowledgement
===============
This work was supported by the Program for New Century Excellent Talents in University, the National Natural Science Foundation of China (No. 10705013), the Doctoral Program Foundation of Institutions of Higher Education of China (No. 20070730055), the Key Project of Chinese Ministry of Education (No. 109153).
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[^1]: liuyx@lzu.edu.cn
[^2]: Corresponding author.
[^3]: fuche08@lzu.cn
[^4]: lizhao@lzu.edu.cn
[^5]: ysduan@lzu.edu.cn
|
---
abstract: 'We study Lagrangian statistics of the magnitudes of velocity and pressure gradients in isotropic turbulence by quantifying their correlation functions and their characteristic time scales. In a recent work [@rf:Yu09a], it has been found that the Lagrangian time-correlations of the velocity and pressure gradient tensor and vector elements scale with the locally-defined Kolmogorov time scale, defined from the box-averaged dissipation-rate ($\epsilon_r$) and viscosity ($\nu$), according to $\tau_{K,r}=\sqrt{\nu/\epsilon_r}$. In this work, we study the Lagrangian time-correlations of the absolute values of velocity and pressure gradients. It has long been known that such correlations display longer memories into the inertial-range as well as possible intermittency effects. We explore the appropriate temporal scales with the aim to achieve collapse of the correlation functions. The data used in this study are sampled from the web-services accessible public turbulence database ([`http://turbulence.pha.jhu.edu`]{}). The database archives a $1024^4$ (space+time) pseudo-spectral direct numerical simulation of forced isotropic turbulence with Taylor-scale Reynolds number $Re_\lambda=433$, and supports spatial differentiation and spatial/temporal interpolation inside the database. The analysis shows that the temporal evolution of the auto-correlations of the absolute values are determined not by the local Kolmogorov time-scale but by the local eddy-turnover time scale defined as $\tau_{e,r}= r^{2/3}\epsilon_r^{-1/3}$. However, considerable scatter remains and appears to be reduced only after a further (intermittency) correction factor of the form of $(r/L)^\chi$ is introduced where $L$ is the turbulence integral scale. The exponent $\chi$ varies for different variables. The collapse of the correlation functions for absolute values is, however, less satisfactory than the collapse observed for the more rapidly decaying strain-rate tensor element correlation functions.'
author:
- Huidan Yu
- Charles Meneveau
date: 'Received: date / Accepted: date'
title: 'Scaling of conditional Lagrangian time correlation functions of velocity and pressure gradient magnitudes in isotropic turbulence [^1] '
---
Introduction {#sec:intro}
============
The study of turbulence from a Lagrangian viewpoint has a long history, with the earliest works of Taylor [@rf:Taylor21] and Richardson [@rf:Richardson26] both pre-dating Kolmogorov [@rf:Kolmogorov41]. The Kolmogorov 1941 theory used the constancy of the globally-averaged dissipation-rate $\langle \epsilon \rangle$ across scales to deduce, among others, the scaling properties of the wavenumber spectrum of kinetic energy. The 1941 theory was extended to account for intermittency by the introduction of the so-called refined Kolmogorov similarity hypothesis (RKSH) [@rf:Kolmogorov62] in 1962. In this K62 extension of the theory, conditional statistics, based on the dissipation rate averaged in some particular subregion of the flow, acquires a central role. The local dissipation rate, usually denoted by $\epsilon_r$, is defined according to $$\epsilon_r({\bf x}) = \frac{1}{V}\int \limits_{{\cal{R}}_r({\bf x})} 2 \nu \left[S_{ij}({\bf x}')\right]^2 d^3 {\bf x}',
\label{eq:defepsilonr}$$ where $V$ is the volume of the subregion ${\cal{R}}_r({\bf x})$ (e.g. a box or a sphere) of size $r$ centered at ${\bf x}$, $\nu$ is the kinematic viscosity of the fluid, and $S_{ij}$ is the strain-rate tensor. One of the main predictions of the RKSH relates to the longitudinal velocity increment at scale $r$, defined as $\delta_r u = [u_i({\bf x} + {\bf r}) - u_i({\bf x})](r_i/r)$. The RKSH states that in the inertial range of turbulence the statistics of $\delta_r u$ depend on $r$ and $\epsilon_r$. Therefore, from dimensional analysis, various moments of $\delta_r u$ conditioned upon a fixed value of $\epsilon_r$ will scale as $\langle \delta_r u^p \vert \epsilon_r \rangle = C_p (r \epsilon_r )^{p/3}$, essentially following Kolmogorov’s 1941 postulate, but locally. Anomalous scaling results from the additional global averaging and anomalous scaling behavior of moments of $\epsilon_r$. The existing literature to validate RKSH has focused mainly on velocity increments [@rf:Stolovitzky92; @rf:Thoroddsen; @rf:Chen93; @rf:Stolovitzky94; @rf:Chen95; @rf:Ching08] or acceleration [@rf:Yeung06] in which the analysis can be performed on single snapshot measurements of the turbulent flow, i.e. based on a relatively ‘static’ point of view of the flow.
In order to examine the RKSH in more depth, one would also like to inquire about its dynamical origin, specifically its role in the time evolution of the local structure of turbulence. It is well recognized that the dynamics of turbulence is best understood in a Lagrangian frame of reference, i.e. following fluid particles. Recent years have witnessed a strong revival of interest in Lagrangian statistics in turbulence. For reviews, see [@rf:Pope94; @rf:Yeung02]. The dynamics of turbulence following fluid particles also plays a central role in the PDF modeling framework developed over the past two decades by S. Pope, starting with his seminal 1985 paper [@rf:Pope85].
Besides which ‘frame’ to use in the description of the dynamics, it is also important to select variables of interest that convey rich information about the flow. In recent years, there has been growing attention placed in the dynamical evolution of the velocity gradient tensor ${\bf A}$ ($A_{ij}\equiv \partial u_i / \partial x_j$). This is due to the fact that ${\bf A}$ provides rich information about the topological and statistical properties of small-scale structure in turbulence. Pioneering studies of the Lagrangian structure and stochastic modeling of ${\bf A}$ are described in Refs. [@rf:Girimaji90a] and [@rf:Girimaji90], respectively. The Lagrangian time evolution of ${\bf A}$ can be obtained by taking gradient of the NS equation [@rf:Vieillefosse82]: $$\frac{dA_{ij}}{dt}=-A_{ik}A_{kj}-\frac{\partial ^2p}{\partial x_i \partial x_j}+\nu \frac{\partial ^2A_{ij}}{\partial x_k \partial x_k}.
\label{eq:Aij}$$ As usual, $d/dt$ stands for Lagrangian material derivative, $p$ is the pressure divided by the density of the fluid, and the second and third terms on the right-hand-side of this equation are the pressure Hessian tensor and viscous term respectively. Neglecting viscous effects and assuming the pressure Hessian isotropic lead to a closed formulation known as the Restricted-Euler (RE) dynamics [@rf:Vieillefosse82], [@rf:Cantwell92]. With analytically treatable solutions for the full tensor-level, the RE system provides a fruitful starting point for small structure modeling although there exist serious deficiencies in the RE dynamics, especially since it predicts nonphysical finite-time singularities [@rf:Cantwell92]. Models have been developed to mimic the regularization features of the neglected pressure Hessian and viscous terms. Efforts include a stochastic model in which the nonlinear term is modified to yield log-normal statistics of the dissipation [@rf:Girimaji90], a linear damping model for viscous term [@rf:Martin98], a tetrad model [@rf:Chertkov99] for pressure Hessian closure, a viscous diffusion closure [@rf:Jeong03], a new stochastic dynamic model, so-called Recent Fluid Deformation closure, for both viscous and pressure Hessian terms [@rf:Chevillard06], and a multi-scale model which includes energy exchange between scales [@rf:Biferale07]. The study of the temporal auto-correlation structure of various quantities associated with ${\bf A}$ assists in the further developments and improvements of such models.
It has recently been confirmed [@rf:Yu09a] that when averaging over the entire domain, auto-correlation functions of velocity gradient tensor elements decay on timescales on the order of the mean Kolmogorov turnover time scale. This time scale is computed from the globally averaged rate of dissipation and viscosity. However, when performing the analysis in different subregions of the flow, turbulence intermittency was found to lead to large spatial variability in the decay time scales. Remarkably, excellent collapse of the auto-correlation functions is recovered when using the ‘local Kolmogorov time-scale’ defined using the locally, rather than the globally, averaged dissipation-rate ($\tau_{K,r}\equiv \sqrt{\nu/\epsilon_r}$). This is an additional new evidence for the validity of Kolmogorov’s Refined Similarity Hypothesis, but from a Lagrangian viewpoint that provides a natural frame to describe the dynamical time evolution of turbulence.
In this paper, we study Lagrangian time-correlations of scalar measures (such as magnitudes) of velocity and pressure gradients and explore whether there is further evidence for the Lagrangian RKSH for these variables. Lagrangian correlation functions of square strain- and rotation-rate have already been studied in prior work [@rf:Guala07; @rf:Yeung07]. In examining the scaling of the magnitudes of the velocity gradient tensor and pressure gradient, the behavior of correlation functions will be shown here to be much more complex than for the tensor or vector elements themselves. Nontrivial dependencies on local-length scale $r$ and local dissipation-rate ($\epsilon_r$) are observed, and these require more detailed study. The present paper is devoted to such a study, based on analysis of Lagrangian data.
Lagrangian data can be extracted from direct numerical simulation (DNS) of NS equations with relative ease. The first such effort traces back to Riley and Patterson [@rf:Riley74]. The rapid development in computing power over the past few decades has spurred vast amount of such numerical investigations at increasing Reynolds numbers. For relevant reviews, see [@rf:She91; @rf:Pope94; @rf:Yeung02; @rf:Toschi09] and references therein. A new way to exploit large databases in turbulence has been recently proposed [@rf:Li08]. This approach is based on web-services that allow public access to turbulence DNS databases that store not only snapshots of 3D distributions but also the entire pre-computed time history. Using this public turbulence database, here we study the Lagrangian time evolution of velocity and pressure gradients and their magnitudes in isotropic turbulence.
The remainder of this paper is organized as follows. Section \[sec:database\] describes the public turbulence database and the numerical approach we use to perform the Lagrangian analysis. Results on the time evolution of auto-correlations of velocity and pressure gradient magnitudes are presented in Section \[sec:correlations\]. We conclude in Section \[sec:concl\] with a short discussion.
JHU public turbulence database and particle-tracking approach {#sec:database}
=============================================================
The DNS data of a forced isotropic turbulence archived in the JHU public database system are from a pseudo-spectral parallel computation of the forced NS equations in a $[0, 2\pi]^3$ domain, at a Taylor-microscale Reynolds number of $Re_\lambda \simeq 433$ [@rf:Li08]. The database contains of output on $1024^3$ spatial points and 1024 time samples (every tenth DNS time-step is stored) spanning about one large-scale eddy turnover time. The domain-wide averaged dissipation-rate ($<\epsilon>$) and the corresponding Kolmogorov time scale ($\tau_K$) are 0.092 and 0.045 respectively, in the units of the simulation. The turbulence integral scale is $L=1.376$. Some data processing functionalities such as spatial differentiation, and spatial and temporal interpolations are provided directly inside the database. This feature not only reduces data download cost but also allows users to obtain desired quantities at arbitrary locations and times. The whole database results in a 27 Terabyte storage size. The $1024^4$ space+time history of turbulence is publicly accessible through a web-service interface which serves as a bridge to connect user requests with the database nodes. Users may write and execute analysis programs using prevailing languages C, Fortran, or Matlab on their host computers such as desktops or laptops, while the programs request desired outputs from the database through GetFunctions (subroutine-like calls) over the Internet. Currently, eight GetFunctions listed in Table \[ta:getfunctions\] for velocity and pressure along with their derivatives and force are available. With these call functions, users can retrieve quantities simultaneously for large amounts of locations and time (within the stored time frames) without expensive memory and time costs. The details of the DNS data and JHU turbulence database can be found in a previous publication [@rf:Li08]. The instructions and sample codes in C, Fortran, and Matlab are available at `http://turbulence.pha.jhu.edu`.
Function name Spatial diff. Spatial int. Temporal int. Outputs
------------------------ --------------- ----------------- --------------- ---------------------------------------------------
GetVelocity – NoInt, Lag4,6,8 NoInt, PCHIP $u_i$
GetVelocityAndPressure – NoInt, Lag4,6,8 NoInt, PCHIP $u_i,p$
GetVelocityGradient FD4,6,8 NoInt, Lag4,6,8 NoInt, PCHIP $\frac{\partial u_i}{\partial x_j}$
GetPressureGradient FD4,6,8 NoInt, Lag4,6,8 NoInt, PCHIP $\frac{\partial p}{\partial x_i}$
GetVelocityHessian FD4,6,8 NoInt, Lag4,6,8 NoInt, PCHIP $\frac{\partial^2 u_k}{\partial x_i\partial x_j}$
GetPressureHessian FD4,6,8 NoInt, Lag4,6,8 NoInt, PCHIP $\frac{\partial^2 p}{\partial x_i\partial x_j}$
GetVelocityLaplacian FD4,6,8 NoInt, Lag4,6,8 NoInt, PCHIP $\frac{\partial^2 u_i}{\partial x_j\partial x_j}$
GetForce – NoInt, Lag4,6,8 NoInt, PCHIP $f_i$
: Subroutine-like call functions. diff: differetiation; int: interpolation; NoInt: no interpolation; FD: Centered finite difference, options for 4th-, 6th-, and 8th-order accuracies; Lag: Lagrangian polynomial interpolation, options for 4th-, 6th-, and 8th-order accuracies; PCHIP: Piecewise cubic Hermite interpolation. []{data-label="ta:getfunctions"}
We employ the particle-tracking algorithm of Ref. [@rf:Yeung88] to extract Lagrangian information along many particle trajectories simultaneously. Each particle is tagged and randomly assigned an initial position. Let ${\bf x}^+({\bf y},t)$ and ${\bf u}^+({\bf y},t)$ denote the position and velocity at time t of the fluid particle originating from position ${\bf y}$ at initial time $t_0$ with the superscript $+$ representing Lagrangian quantities following the fluid particle. Each particle is tracked by numerically integrating $$\frac{\partial {\bf x}^+({\bf y},t)}{\partial t}={\bf u}^+({\bf y},t)
\label{eq:particlemotionequation}$$ where the Lagrangian velocity ${\bf u}^+({\bf y},t)$ is replaced by the Eulerian velocity ${\bf u}({\bf x},t)$ where the particle is located, namely ${\bf u}^+({\bf y},t)={\bf u}({\bf x}^+({\bf y},t),t)$.
The particle displacement between two successive time instants $t_n$ and $t_{n+1}(=t_n+\delta t)$ is obtained through an integral of Eq. (\[eq:particlemotionequation\]) using a second-order Runge-Kutta method. At time $t_n$ for a particle located at ${\bf x}^+({\bf y},t_n)$, the predictor step yields an estimate ${\bf x}^*={\bf x}^+({\bf y},t_n) + \delta t ~{\bf u}^+({\bf y},t)$ for the destination position ${\bf x}^+({\bf y},t_{n+1})$. The corrector step then gives the particle position at $t_{n+1}$: ${\bf x}^+({\bf y},t_{n+1})={\bf x}^+({\bf y},t_n)+\delta t ~[{\bf u}^+({\bf y},t_n)+{\bf u}^+({\bf x}^*,t_{n+1})]/2$. It is proved that the time-stepping error is of order $(\delta t)^3$ over one time step [@rf:Yeung88]. In general, accurate spatial and time interpolations are crucial to obtain the fluid velocities while tracking particles along their trajectories. In JHU turbulence database, these operations have been built in with optional orders of accuracy. We use flags FD4Lag4, Lag8, and PHCIP (explained in the caption of Table \[ta:getfunctions\]) for the calls to specify spatial differentiation, spatial interpolation, and time interpolation.
Lagrangian time correlations of gradient magnitudes {#sec:correlations}
===================================================
As mentioned above, the velocity gradient tensor ${\bf A}$ has received considerable attention in recent years. It provides a rich characterization of the topological and statistical properties of the small-scale structure in turbulence in the viscous range. The antisymmetric part of the velocity gradient tensor, i.e., $\Omega_{ij} \equiv (A_{ij}-A_{ji})/2$, is the rate of rotation describing the vortex structure and dynamics. Whereas the symmetric part of ${\bf A}$, the strain-rate tensor, defined as $S_{ij} \equiv (A_{ij}+A_{ji})/2$, represents the strength and directions of fluid deformation rates. The dynamic evolution of ${\bf A}$ is given by Eq. (\[eq:Aij\]). Here we study Lagrangian auto-correlations for the absolute values of velocity-derivative tensor using the magnitudes of the strain-rate tensor ${\bf S}$ and rotation-rate tensor ${\bf \Omega}$. The magnitudes are defined here using the square invariant according to $|{\bf S}|\equiv \sqrt{S_{ij}S_{ij}}$ and $|{\bf \Omega}|\equiv \sqrt{\Omega_{ij}\Omega_{ij}}$ (we apply Einstein notation for repeating indexes unless indicated otherwise). We also study the magnitude of pressure gradient (approximately similar to the acceleration magnitude), defined as $|\bigtriangledown p|\equiv \sqrt{\nabla p \cdot \nabla p}$.
The Lagrangian time correlation of these scalar quantities is defined as usual: $$\rho_{f}(\tau) \equiv \frac{\langle{f(t_0)f(t_0+\tau)}\rangle}
{\sqrt{\langle{f(t_0)^2}\rangle \cdot \langle{f(t_0+\tau)^2}\rangle}},
\label{eq:scalarcorrelationfunction}$$ where $\tau$ is the time-lag along Lagrangian trajectories, $f$ can be $f=|{\bf S}|$, $f=|{\bf \Omega}|$, or $f=|\bigtriangledown p |$ as the case may be, and $\langle{\cdots}\rangle$ may represent ensemble or global volume averaging for homogeneous turbulence.
In order to study effects of intermittency, which in turbulence is characterized by local regions displaying different levels of turbulence activity, we also compute conditional correlation functions based on fluid particles that originate from various subregions of the flow domain. The subregions are characterized by the local dissipation-rate $\epsilon_r$ defined in Eq. (\[eq:defepsilonr\]) and the length-scale $r$. With this context, the global average in Eq.(\[eq:scalarcorrelationfunction\]) is replaced by the conditional average, i.e
$$\rho_{f}(\tau) \equiv \frac{\langle{f(t_0)f(t_0+\tau)\vert \epsilon_r}\rangle}
{\sqrt{\langle{f(t_0)^2\vert \epsilon_r}\rangle \cdot \langle{f(t_0+\tau)^2\vert \epsilon_r}\rangle}},
\label{eq:scalarconditional}$$
In the conditional average the initial position of particles contributing to the average at time $t_0$ are sampled from several local boxes of size $r$ that have a prescribed locally averaged dissipation-rate $\epsilon_r$. For practical reasons, a finite range of values of $\epsilon_r$ must be considered, i.e. we use sampling in bins of $\epsilon_r$ values. For each bin there are several such local cubes for which their $\epsilon_r$ falls in a prescribed range. Besides varying the bin location, we also consider four length scales, $r=34\eta_K, 68\eta_K,136\eta_K,272\eta_K$. They correspond to 16-, 32-, 64-, and 128 grid-point cubes, respectively. Each has four associated $\epsilon_r$ bins. An additional set of 64-cube cases ($r=136\eta_K$) with five bins has been studied. This set was also considered in [@rf:Yu09a] for the tensor element-based correlations. The specifications of scale and bin values are listed in Table \[ta:localcases\].
cube size $r$ bin index particle/cube cube/$\epsilon_r$ $\epsilon_r$ range nominal $\epsilon_r$
---------------- ----------- --------------- ------------------- -------------------------- ----------------------
34$\eta_K$ 1 50 120 0.0056 $\backsim$ 0.0093 0.0076
2 50 120 0.017 $\backsim$ 0.020 0.019
3 50 120 0.089 $\backsim$ 0.096 0.092
4 50 120 0.14 $\backsim$ 0.16 0.15
68$\eta_K$ 1 100 60 0.0074 $\backsim$ 0.011 0.0096
2 100 60 0.0241 $\backsim$ 0.028 0.019
3 100 60 0.089 $\backsim$ 0.096 0.093
4 100 60 0.14 $\backsim$ 0.16 0.15
136$\eta_K$-I 1 500 12 0.017 $\backsim$ 0.020 0.018
2 500 12 0.024 $\backsim$ 0.028 0.026
3 500 12 0.089 $\backsim$ 0.096 0.093
4 500 12 0.10 $\backsim$ 0.12 0.11
136$\eta_K$-II 1 4000 2 0.033 $\backsim$ 0.041 0.037
2 4000 3 0.045 $\backsim$ 0.046 0.046
3 4000 2 0.065 $\backsim$ 0.066 0.066
4 4000 2 0.083 $\backsim$ 0.088 0.086
5 4000 3 0.12 $\backsim$ 0.16 0.14
276$\eta_K$ 1 1000 6 0.030 $\backsim$ 0.037 0.034
2 1000 6 0.048 $\backsim$ 0.060 0.052
3 1000 6 0.081 $\backsim$ 0.10 0.094
4 1000 6 0.11 $\backsim$ 0.15 0.12
: Indication of different cases for calculation of conditional correlation functions. Shown for local subregion (cube) length size, bin index, number of particles in each cube, number of cubes in each bin, and range and nominal of local dissipation for each bin.[]{data-label="ta:localcases"}
![Left plot: PDFs of locally-averaged dissipation-rates $\epsilon_r$ with four different local length scales; Right plot: Sample particle trajectories starting from 12 randomly selected $64$-cubes characterized by local dissipation-rate $\epsilon_r$ at the initial time corresponding to case $136\eta_K-II$ in Table \[ta:localcases\]. []{data-label="Fig_PDFtrajectory"}](Fig_PDFtrajectory){width="5."}
In Fig. \[Fig\_PDFtrajectory\] we show PDFs of $\epsilon_r$ for the four length scales considered (left plot) and 12 representative $64$-cubes placed inside the $1024^3$ domain, with 50 sample fluid particle trajectories emanating from each and progressing during a time equal to 27$\tau_K$ (right plot). The required averages are taken over all the trajectories as well as over several cubes for which $\epsilon_r$ is in a bin’s prescribed range.
![Lagrangian time correlations of strain- and rotation-rate tensors and pressure gradient vector for all the cases in Table \[ta:localcases\]. Open squares are for global average over randomly located particles in the whole domain, whereas different lines correspond to subregions of the flow characterized by different $\epsilon_r$s. Time-lag is normalized using the global Kolmogorov time scale $\tau_{K}$ (left column) and the local time-scale $\tau_{K,r}$ (right column).[]{data-label="Fig_tensorcollapse"}](Fig_tensorcollapse){width="4.5"}
Lagrangian and conditional Lagrangian auto-correlation functions for tensors ${\bf A}$, ${\bf S}$, ${\bf \Omega}$, or vector $\bigtriangledown p$ have been studied in our previous work [@rf:Yu09a; @rf:Yu09b]. These tensor and vector time correlation functions are computed through expressions like $\langle{C_{ij}(t_0)C_{ij}(t_0+\tau)}\rangle$ and $\langle{C_{ij}(t_0)C_{ij}(t_0+\tau)\vert \epsilon_r}\rangle$ or $\langle{G_i(t_0)G_i(t_0+\tau)}\rangle$ and $\langle{G_i(t_0)G_i(t_0+\tau)\vert \epsilon_r}\rangle$ on tensor or vector element level for global and conditional correlation functions, respectively. Here we present these measurements in Fig. \[Fig\_tensorcollapse\] for three variables including all the cases listed in Table \[ta:localcases\] (in [@rf:Yu09a] only results for ${\bf S}$ were shown for the same cases). The evolution time is scaled by $\tau_K$ and $\tau_{K,r}$ where $$\tau_K = \sqrt{\frac{\nu}{\langle \epsilon \rangle}}, ~~~~ \tau_{K,r}= \sqrt{\frac{\nu}{\epsilon_r} }.$$ The main observations can be summarized as follows. First, the rotation-rate displays significantly longer time memory than the strain-rate. After about 6$\tau_{K,r}$, the strain-rate’s correlation is essentially zero, whereas it is still near 0.5 for the rotation-rate. We found that this trend holds true even if coherent vortex structures are excluded from the analysis. Second, the temporal auto-correlation functions scatter significantly when the time lag is scaled by $\tau_K$ (left column) but collapse well when scaled by $\tau_{K,r}$ (right column). This behavior demonstrates that the dynamics of flow variables such as velocity and pressure gradients following fluid particles depends upon the local dissipation-rate ($\epsilon_r$) rather than the global one ($\langle \epsilon \rangle$) which, as argued in [@rf:Yu09a], provides new evidence for the validity of Kolmogorov’s refined similarity hypothesis form a Lagrangian viewpoint.
In what follows, we study conditional Lagrangian time correlations for the absolute values of these tensors and vector.
![Auto-correlations of $|S|$ (top row), $|\Omega|$ (middle row), and $|\nabla p|$ (bottom row) vs. $\tau/\tau_K$(left column), $\tau/\tau_{K,r}$ (right column) for the cases listed in Table \[ta:localcases\]. Open square symbols correspond to global (unconditional) averaging over entire data volume.[]{data-label="Fig_scalar_tauK"}](Fig_scalar_tauK){width="4.5"}
Figure \[Fig\_scalar\_tauK\] shows the similar plots to Fig. \[Fig\_tensorcollapse\] but for the the absolute values of tensors ${\bf S}$ (top row) and ${\bf \Omega}$ (middle row) and vector $\nabla p$ (bottom row). It is quite clear that for all variables, especially for the strain-rate and pressure gradient, the correlations decay much more slowly for the magnitudes as compared to the tensor or vector elements. Similarly slow decay had been observed for the square of these variables in [@rf:Guala07; @rf:Yeung07]. Moreover, and unlike the tensor- or vector-based Lagrangian auto-correlations, poor collapse is seen when the time lag $\tau$ is scaled by the local Kolmogorov time $\tau_{K,r}$. Such scaling appears to work only for the viscous time-scale range near the origin of the curves ($\tau < \tau_{K,r}$). For the inertial range, the curves scatter significantly even when scaled by the local $\tau_{K,r}$.
Since there remain significant correlation scatters even after long time delays, we explore the use of other time-scales to express time. The characteristic time-scale that is believed to be relevant in the inertial range is the eddy-turnover scale appropriate for eddies of size $r$. Its global and local values are defined according to $$\tau_{e} = L^{2/3} \langle \epsilon\rangle^{-1/3}, ~~~~\tau_{e,r} = r^{2/3} \epsilon_r^{-1/3} .$$
![Auto-correlations of $|S|$ (top row), $|\Omega |$ (middle row), and $|\nabla p|$ (bottom row) vs. $\tau/\tau_e$ (left column) and $\tau/\tau_{e,r}$ (right column) for the case of 136$\eta_K$-II in Table \[ta:localcases\]. []{data-label="Fig_collapse_64"}](Fig_collapse_64){width="4.5"}
Fig. \[Fig\_collapse\_64\] shows the conditional Lagrangian time correlations of absolute values of ${\bf S}$ (top row), ${\bf \Omega}$ (middle row), and $\bigtriangledown p$ (bottom row) with time normalized by $\tau_e$ (left column) and $\tau_{e,r}$ (right column). These results are for a single length-scale corresponding to the case of $r=136\eta_K$ (case II) in Table \[ta:localcases\]. When the time lag $\tau$ is scaled by $\tau_e$, there are noticeable differences in the results depending on $\epsilon_r$. For the three variables, larger values of $\epsilon_r$ (i.e. in regions of more intense turbulence activity corresponding to smaller local eddy turn-over time) are associated with faster correlation decay. When the time lag is scaled by the local time-scale $\tau_{e,r}$, the curves collapse better than with the global value. This provides some evidence for a Lagrangian RKSH also at inertial-range ‘eddy-turnover’ scales.
The next question is whether good collapse also occurs for different length scales. Fig. \[Fig\_collapse\_er092\] plots the conditional auto-correlations of $|S|$, $|\Omega|$, and $|\nabla p|$ with approximately the same $\epsilon_r$ corresponding to bin No. 3 (see Table \[ta:localcases\]), but at different length scales. In the left column, the time lag is scaled by the local eddy time $\tau_{e,r}$. It is seen clearly that the correlation functions decay differently at different length scales, which implies that the normalization of time with $\tau_{e,r}$ does not account for the differences.
![Time correlations of $|S|$, $|\Omega|$, and $|\nabla p|$ for the $\epsilon_r$ value in bin No. 3 (see Table \[ta:localcases\]) vs. $\tau/\tau_{e,r}$ (left column) and $\tau/\tau_{e,r}\cdot \big(\frac{r}{L}\big)^{\chi}$ (right column, (b):$\chi=-0.3$ , (d):$\chi=-0.49$, and (f):$\chi=-0.55$). All are for for the cases in bin No. 3 with $\epsilon\approx 0.092$, which also close to value of global-averaged dissipation-rate.[]{data-label="Fig_collapse_er092"}](Fig_collapse_er092){width="4.5"}
Intermittency in turbulence is often known to connect the inertial range dynamics with the ratio of length-scale to the integral scale, i.e. the level of intermittency is related to “how far” the scale is from its original starting point at the large scales during the cascade. Often such effects are parameterized by factors of the form $(r/L)^{\chi}$, where ${\chi}$ is an appropriate intermittency exponent for the correction. We determine the exponents ${\chi}$ empirically (see below) to obtain improved collapse. The right column in Fig. \[Fig\_collapse\_er092\] shows the correlation functions with time now normalized by the intermittency corrected time scale $\tau_{e,r}(r/L)^{\chi}$. As can be observed, improved collapse can thus be obtained by using an intermittency correction.
In order to determine the exponents empirically, we set a threshold on the correlation function value. A value $\rho_f(\tau_{1/2}) = 1/2$ is used, which defines the time-scale $\tau_{1/2}$. For each case, the value of $\tau_{1/2}$ correspond to $\rho_f = 1/2$. A log-log plot of $\tau_{1/2}/\tau_{e,r}$ versus $r/L$ should have slope $\chi$, if a power-law intermittency correction is appropriate. In Fig. \[Fig\_exponent\] such plots are presented for each variable $|{\bf S}|$, $|{\bf \Omega}|$, $|{\bf \nabla p}|$, from top to bottom. Exponents are obtained by fitting straight lines through the data as shown by the dashed lines in the plots. The corresponding exponents of $\chi$ are $\chi_{|S|} = -0.3$, $\chi_{|\Omega|} = -0.49$, and $\chi_{|\nabla p|} = -0.55$. These are the values used to scale the results shown in right column of Fig. \[Fig\_collapse\_er092\]. We have not yet succeeded in relating the values of ${\chi}$ to the multifractal theory of turbulence. Note that these values are significantly larger than what is typically obtained from multifractal corrections.
![Plots of characteristic decay time $\tau_{1/2}$ (measured as the 1/2 point in the correlation function) versus length-scale. The scale $r$ is normalized by the integral length $L=1.367$. Different symbols are for different $\epsilon_r$ bins: $\circ$: $bin_1$; $\bigtriangleup$: $bin_2$; $\Box$: $bin_3$, $\diamond$: $bin_4$. The dashed lines are power-law fits yielding, in (a) for $|S|$, $\chi=-0.3$; in (b) for $|\Omega|$, $\chi=-0.49$; and in (c) for $|\nabla p|$, $\chi=-0.55$.[]{data-label="Fig_exponent"}](Fig_exponent){width="4."}
All the cases are plotted jointly in Fig. \[Fig\_collapse\_all\]. With global time scaling of $\tau_e$ (left column), the curves scatter significantly reflecting clear intermittency in the flow. When both $\tau_{e,r}$ and $(r/L)^{\chi}$ are considered in the scaling of time lag (right column), the curves collapse reasonably well. Some scatter remains, however.
![Time correlations of $|S|$ (top row), $|\Omega|$ (middle row), and $|\nabla p|$ (bottom row) vs. $\tau/\tau_e$(left column) and $\tau/\tau_{e,r}\cdot \big(\frac{r}{L}\big)^{\chi}$ (with (b):$\chi=-0.3$, (d):$\chi=-0.49$, and (f):$\chi=-0.55$) for all the cases in Table \[ta:localcases\]. The lines are solid lines: $16$-cube; $--$ : $32$-cube; $-- \cdot$: $64$-cube; $-- \cdot \cdot$ : $128$-cube.[]{data-label="Fig_collapse_all"}](Fig_collapse_all){width="4.5"}
Conclusions {#sec:concl}
===========
Using the JHU public turbulence database, we perform Lagrangian analysis of temporal time correlations for the absolute values of velocity and pressure gradients. Consistent with earlier results for the square strain-rate and rotation rates[@rf:Guala07; @rf:Yeung07], we find significantly longer decay times for the magnitudes as compared to the tensor or vector elements, especially for strain-rate and pressure gradients. It is demonstrated that Lagrangian dynamics of velocity gradient and pressure gradient (almost equivalent to acceleration since viscous effects are negligible) is determined mainly by scales provided by the locally averaged rate of dissipation, as predicated in the Kolmogorov Refined Similarity Hypothesis. We point out that a Lagrangian KRSH has also been shown to hold in the context of moments of two-time velocity increments[@rf:Benzi08]. That analysis is based on a fully Lagrangian rate of dissipation $\epsilon_{\tau}$ averaged over [*temporal*]{} domains of duration $\tau$ along the particle trajectory. By its nature, $\epsilon_{\tau}$ averages dissipation at various times. The present analysis is instead based on the more often used spatial average of dissipation at a single (initial-condition) time. Still, present results show that even after using local time-scales corrected for intermittency, there was remaining scatter observed in the correlation functions for absolute value variables. A better understanding of the origin of these deviations, as well as relating the relatively large intermittency corrections to various phenomenological models of turbulence, would be desirable developments.
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B.: Lagrangian PDF methods for turbulent flows, Annu. Rev. Fluid Mech.[**26**]{}, 23 (1994). Yeung, P. K.: Lagrangian investigations oF turbulence, Annu. Rev. Fluid Mech. [**34**]{},115 (2002). Pope, S.B.: PDF methods for turbulent reactive flows, Prog. Energy Combust. Sci. [**11**]{}, 119 (1985). Girimaji, S.S. and Pope, S.B.: Material element deformation in isotropic turbulence, J Fluid Mech. [**220**]{}, 427 (1990). Girimaji, S. S. and Pope, S. B.: A diffusion model for velocity gradients in turbulence, Phys. Fluids A [**2**]{}, 242 (1990). Vieillefosse, P.: Local interaction between vorticity and shear in a perfect incompressible fluid, J. Phys. (France) [**43**]{}, 837 (1982). Cantwell, B. J.: Exact solution of a restricted Euler equation, Phys. Fluids A [**4**]{}, 782(1992). Martin, J., Ooi, A., Chong, M. S., and Soria, J.: Dynamics of the velocity gradient tensor invariants in isotropic turbulence, Phys. Fluids [**10**]{}, 2336 (1998). Chertkov, M., Pumir, A., and Shraiman, B. I.: Lagrangian tetrad dynamics and the phenomenology of turbulence, Phys. Fluids [**11**]{}, 2394 (1999). Jeong E., and Girimaji, S. S.: Velocity-gradient dynamics in turbulence:Effect of viscosity and forcing, Theor. Comput. Fluid Dyn. [**16**]{}, 421 (2003). Chevillard, L., and Meneveau, C.: Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett. [**97**]{}, 174501(2006). Biferale, L., Chevillard, L., Meneveau, C. and Toschi, F.: Multiscale model of gradient evolution in turbulent flows, Phys. Rev. Lett. [**98**]{}, 214501 (2007). Guala, M., Liberzon, A., Tsinober, A., and Kinzelbach, W.: An experimental investigation on Lagrangian correlations of small-scale turbulence at low Reynolds number, J. Fluid Mech. [**574**]{}, 405 (2007). Yeung, P. K., Pope, S. B., Kurth, E. A., and Lamorgese, A. G.: Lagrangian conditional statistics, acceleration and local relative motion in numerically simulated isotropic turbulence, J. Fluid Mech. [**582**]{}, 399 (2007). Riley, J. J. and Patterson, G. S.: Diffusion experiments with numerically integrated isotopic turbulence, Phys. Fluids [**17**]{}, 292 (1974). She, Z-S., Jackson, E., and Sreenivasan, K. R.: Structure and dynamics of homogenous turbulence: models and simulations, Proc. R. Soc. Lord. A [**434**]{}, 101 (1991). Toschi F. and Bodenschatz, E.: Lagrangian properties of particles in turbulence, Annu. Rev. Fluid Mech.[**41**]{},375 (2009). Li, Y., Perlman, E., Wan, M., Yang, Y., Burns, R., Meneveau, C., Burns, R., Chen, S., Szalay, A., and Eyink, G.: A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence, J. Turbulence [**9**]{}, 31 (2008). Yeung, P. K. and Pope, S. B.: An algorithm for tracking fluid particles in numerical Simulations of homogeneous turbulence, J. Comp. Phys. [**79**]{}, 373 (1988). 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[^1]: The authors are delighted to present this paper in the context of a symposium held in celebration of Professor Stephen B. Pope’s pathbreaking contributions to turbulence and combustion research. They thankfully acknowledge the financial support from the Keck Foundation (L. C.) and the National Science Foundation (ITR-0428325 and CDI-0941530).
|
---
abstract: 'We introduce a novel kind of robustness in linear programming. A solution $x^*$ is called robust optimal if for all realizations of objective functions coefficients and constraint matrix entries from given interval domains there are appropriate choices of the right-hand side entries from their interval domains such that $x^*$ remains optimal. we propose a method to check for robustness of a given point, and also recommend how a suitable candidate can be found. We also discuss topological properties of the robust optimal solution set. We illustrate applicability of our concept in a transportation problem.'
author:
- 'Milan Hladík[^1]'
bibliography:
- 'ae\_ilp.bib'
title: 'Robust optimal solutions in interval linear programming with forall-exists quantifiers'
---
Introduction
============
Robustness in mathematical programming was intensively studied from diverse points of view [@BenBoy2006; @BenNem2009; @BenGor2004; @BenNem2002; @SoyMur2013]. Generally, robustness corresponds to stability of some key characteristics under limited input data change. In case of uncertainties in the objective function only, an optimal solution is usually called robust if the worst-case regret in the objective value is minimal.
One class of robustness is dealt with in the area of interval linear programming. Therein, we model uncertain parameters by intervals of admissible values and suppose that parameters can attain any value from their interval domains independently of other parameters. The effect of variations on the optimal value and interval solutions are the fundamental problems investigated [@AllNeh2013; @Hla2012a; @Hla2014:a]. Concerning to robustness, [@Hla2014a] was devoted to stability of an optimal basis in interval linear programming. In [@AveLeb2005; @GabMur2010b; @InuSak1995; @MauLag1998], the authors utilized maximum regret approach for finding robust solutions. In multiobjective case, [@HlaSit2013; @RivYag2013] studied robustness of a Parto optimal solution, and some specific nonlinear programming problems [@Hla2010c] were addressed in the context of interval robustness as well.
Recently, [@LiLuo2013; @LuoLi2013a; @LuoLi2014a] introduced a novel kind of interval robustness. They divided interval parameters into two sets, quantified respectively by universal and existential quantifiers. Roughly speaking, an optimal solution is robust in this sense if for each realization of universally quantified interval parameter there is some realization of the existentially quantified parameters such that the solutions remains optimal. Such forall-exists quantified problems are also studied in the context of interval linear equations [@Pop2012; @PopHla2013; @Sha2002]; imposing suitable quantifiers give us a more powerful technique in real-life problem modelling, and can more appropriately reflect various decision maker strategies.
This paper is a contribution to interval linear programming with quantified parameters. The robust optimal solutions considered must remain optimal for any admissible perturbation in the objective and matrix coefficients, compensated by suitable right-hand side change. We propose a method to check for this kind of robustness and present a cheap sufficient condition. We discuss properties of the set of all robust solutions, and propose a heuristic to find a robust solution. We apply the robustness concept to transportation problem in a small numerical study. The equality form of linear programming is then extended to a general form with mixed equations and inequalities (Section \[sGen\]).
#### Notation. {#notation. .unnumbered}
The $k$th row of a matrix $A$ is denoted as $A_{k*}$, and $\operatorname{diag}(s)$ stands for the diagonal matrix with entries given by $s$. The sign of a real $r$ is defined as $\operatorname{sgn}(r)=1$ if $r\geq0$ and $\operatorname{sgn}(r)=-1$ otherwise; for vectors the sign is meant entrywise.
An interval matrix is defined as $${\mbox{${\mbox{\mathversion{bold}$ A $}}$}}:=\{A\in{{\mathbb{R}}}^{m\times n}{;\,}{\mbox{$\underline{{{{A}}}}$}}\leq A\leq {\mbox{$\overline{{{{A}}}}$}}\},$$ where $ {\mbox{$\underline{{{{A}}}}$}}$ and ${\mbox{$\overline{{{{A}}}}$}}$, ${\mbox{$\underline{{{{A}}}}$}}\leq{\mbox{$\overline{{{{A}}}}$}}$, are given matrices. The midpoint and radius matrices are defined as $$\Mid{A}:=\frac{1}{2}({\mbox{$\underline{{{{A}}}}$}}+{\mbox{$\overline{{{{A}}}}$}}),\quad
\Mid{A}:=\frac{1}{2}({\mbox{$\overline{{{{A}}}}$}}-{\mbox{$\underline{{{{A}}}}$}}).$$ Naturally, intervals and interval vectors are consider as special cases of interval matrices. For interval arithmetic, we refer the readers to [@MooKea2009; @Neu1990], for instance.
#### Problem formulation. {#problem-formulation. .unnumbered}
Consider a linear programming problem in the equality form $$\begin{aligned}
\label{lp}
\min c^Tx{{\ \ \mbox{subject to}\ \ }}Ax=b,\ x\geq0.\end{aligned}$$ Let ${\mbox{${\mbox{\mathversion{bold}$ A $}}$}}\in{{\mathbb{IR}}}^{m\times n}$, ${\mbox{${\mbox{\mathversion{bold}$ b $}}$}}\in{{\mathbb{IR}}}^{m}$ and ${\mbox{${\mbox{\mathversion{bold}$ c $}}$}}\in{{\mathbb{IR}}}^{n}$ be given. Let $x^*\in{{\mathbb{R}}}^n$ be a candidate robustly optimal solution. The problem states as follows:
> For every $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$ and $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$, does there exists $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$ such that $x^*$ is optimal to ?
In other words, we ask whether $x^*$ is robustly optimal in the sense that any change in $c$ and $A$ within the prescribed bounds can be compensated by an adequate change in $b$. Thus, ${\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$ and ${\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$ play role of uncertain parameters all realizations of which must be taken into account. On the other hand, intervals in ${\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$ represent some reserves that we can utilize if necessary.
In [@Hla2012b], it was shown that checking whether $x^*$ is optimal for all evaluations $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$, with fixed $A$ and $b$, is a co-NP-complete problem. Since the class of problems studied in this manuscript covers this as a sub-class, we have as a consequence that our problem is co-NP-complete problem as well. This practically means that we hardly can hope for a polynomial time verification of robust optimality.
Checking robust optimality {#sEqForm}
==========================
Let $I:=\{i=1,\dots,n{;\,}x^*_i=0\}$ be the set of active indices of $x^*$. It is well known that $x^*$ is optimal if and only if $x^*$ is feasible, and there is no strictly better solution in the tangent cone at $x^*$ to the feasible set. In other words, the linear system $$\begin{aligned}
\label{optCond}
c^Tx=-1,\ \ Ax=0,\ \ x_i\geq0,\ i\in I,\end{aligned}$$ has no solution. We refer to this conditions as *feasibility* and *optimality*. In order that $x^*$ is robustly optimal, both conditions must hold with the given forall-exists quantifiers. Notice that only the entries of $A$ are situated in both conditions. Since there is the universal quantifier associated with $A$, we can check for feasibility and optimality separately.
#### Feasibility. {#feasibility. .unnumbered}
We have to check that for any $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$ there is $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$ such that $Ax^*=b$. This is well studied problem and $x^*$ satisfying this property is called tolerance (or tolerable) solution; see [@Fie2006; @Pop2013a; @Sha2002; @Sha2004]. By [@Fie2006 Thm. 2.28], $x^*$ is a tolerance solution if and only if it satisfies $$\begin{aligned}
\label{optFeas}
|\Mid{A}x^*-\Mid{b}|+\Rad{A}|x^*|\leq\Rad{b}.\end{aligned}$$ Thus, the feasibility question is easily answered.
#### Optimality. {#optimality. .unnumbered}
Denote by $A_I$ the restriction of $A$ to the columns indexed by $I$, and denote by $A_J$ the restriction to the columns indexed by $J:={{\{1, \ldots, {n}\}}}\setminus I$. In a similar manner we use $I$ and $J$ as sub-indices for other matrices and vectors.
We want to check whether is infeasible for any $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$ and $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$. By [@Hla2013b], this is equivalent to infeasibility of the system $$\begin{aligned}
({\mbox{$\underline{{{{c_I}}}}$}})^Tx_I+(\Mid{c_J})^Tx_J&\leq(\Rad{c_J})^T|x_J|-1,\\
-({\mbox{$\overline{{{{c_I}}}}$}})^Tx_I-(\Mid{c_J})^Tx_J&\leq(\Rad{c_J})^T|x_J|+1,\\
{\mbox{$\underline{{{{A_I}}}}$}}\,x_I+\Mid{A_J}x_J&\leq\Rad{A_J}|x_J|,\\
-{\mbox{$\overline{{{{A_I}}}}$}}\,x_I-\Mid{A_J}x_J&\leq\Rad{A_J}|x_J|,\\
x_I&\geq0.\end{aligned}$$ Due to the absolute values, the system is nonlinear in general, and it is the reason why checking robust optimality is co-NP-hard. Equivalently, this system is infeasible if and only if
\[optAeExp\] $$\begin{aligned}
({\mbox{$\underline{{{{c_I}}}}$}})^Tx_I+(\Mid{c_J}-\Rad{c_J}\operatorname{diag}(s))^Tx_J&\leq-1,\\
-({\mbox{$\overline{{{{c_I}}}}$}})^Tx_I-(\Mid{c_J}+\Rad{c_J}\operatorname{diag}(s))^Tx_J&\leq1,\\
{\mbox{$\underline{{{{A_I}}}}$}}\,x_I+(\Mid{A_J}-\Rad{A_J}\operatorname{diag}(s))x_J&\leq0,\\
-{\mbox{$\overline{{{{A_I}}}}$}}\,x_I+(\Mid{A_J}+\Rad{A_J}\operatorname{diag}(s))x_J&\leq0,\\
x_I&\geq0\end{aligned}$$
is infeasible for any sign vector $s\in\{\pm1\}^{|J|}$, where $|J|$ denotes the cardinality of $J$.
The system is linear, however, we have to verify infeasibility $2^{|J|}$ of instances. When $x^*$ is a basic feasible solution, then $|J|\leq m\leq n$. Thus, the number usually grows exponentially with respect to $m$, but not necessarily with respect to $n$. Therefore, we possibly can solve large problems provided the number of equations is low.
Sufficient condition {#ssSufCond}
--------------------
Since the number of systems can be very large, an easily computable sufficient condition for robust optimality is of interest.
Let us rewrite as $$\begin{aligned}
\label{optCond2}
c_I^Tx_I+c_J^Tx_J=-1,\ \ A_Ix_I+A_Jx_J=0,\ \ x_I\geq0.\end{aligned}$$ According to the Farkas lemma [@Fie2006; @Schr1998], this system is infeasible if and only if the dual system $$\begin{aligned}
\label{optCondDual}
A_I^Tu \leq c_I,\ \ A_J^Tu = c_J\end{aligned}$$ is feasible. Thus, in order that the optimality condition holds true, the linear system must be feasible for each $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$ and $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$.
If $x^*$ is a basic non-degenerate solution, then $A_J$ is square. If it is nonsingular in addition, then the system $A_J^Tu = c_J$ has a unique solution, and it suffices to check if the solution satisfies the remaining inequalities. Extending this idea to the interval case, consider the solution set defined as $$\{u\in{{\mathbb{R}}}^m{;\,}\exists A_J\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}_J\exists c_J\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}_J:
A_J^Tu = c_J\}.$$ There are plenty of methods to find an interval enclosure (superset) ${\mbox{${\mbox{\mathversion{bold}$ u $}}$}}$ of this solution set; see e.g. [@Fie2006; @Hla2014b; @MooKea2009; @Neu1990]. Now, if $${\mbox{$\overline{{{{{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}_I^T{\mbox{${\mbox{\mathversion{bold}$ u $}}$}}}}}}$}} \leq {\mbox{$\underline{{c}}$}}_I,$$ where the left-hand side is evaluated by interval arithmetic, then we are sure that has a solution in ${\mbox{${\mbox{\mathversion{bold}$ u $}}$}}$ for each realization of interval data, and therefore the optimality criterion is satisfied.
If $x^*$ is a basic degenerate solution, we can adopt a sufficient condition for checking similar kind of robust feasibility of mixed system of equations and inequalities proposed recently in [@Hla2013b]. We will briefly recall the method. First, solve the linear program $$\begin{aligned}
\max\alpha{{\ \ \mbox{subject to}\ \ }}(\Mid{A_I})^Tu+\alpha e \leq \Mid{c_I},\ \ (\Mid{A_J})^Tu = \Mid{c_J},\end{aligned}$$ where $e$ is the all-one vector. Let $u^*$ be its optimal solution. Let $B$ be an orthogonal basis of the null space of $(\Mid{A_J})^T$ and put $d:=Bu^*$. Now, compute an enclosure ${\mbox{${\mbox{\mathversion{bold}$ u $}}$}}\in{{\mathbb{IR}}}^m$ of the solutions set $$\{u\in{{\mathbb{R}}}^m{;\,}\exists A_J\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}_J\exists c_J\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}_J:
A_J^Tu = c_J,\ Bu=d\}.$$ Finally, if $${\mbox{$\overline{{{{{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}_I^T{\mbox{${\mbox{\mathversion{bold}$ u $}}$}}}}}}$}} \leq {\mbox{$\underline{{c}}$}}_I,$$ the the optimality criterion is satisfied.
Seeking for a candidate {#ssCand}
-----------------------
If we are not given a candidate vector $x^*$ for a robust optimal solution, then it may be a computationally difficult problem to find a robust optimal solution or to prove that there is no one. Below, we propose a simple heuristic for finding a promising candidate.
A candidate should be robustly feasible. The condition is can be rewritten in a linear form as $$\begin{aligned}
(\Mid{A}x^*-\Mid{b})+\Rad{A}x^*\leq\Rad{b},\ \
-(\Mid{A}x^*-\Mid{b})+\Rad{A}x^*\leq\Rad{b},\end{aligned}$$ or, equivalently, as $$\begin{aligned}
\label{ineqSetRob}
{\mbox{$\overline{{{{A}}}}$}}x^*\leq{\mbox{$\overline{{{{b}}}}$}},\ \
{\mbox{$\underline{{{{A}}}}$}}x^*\geq{\mbox{$\underline{{{{b}}}}$}}.\end{aligned}$$ This motivates us to find a good candidate $x^*$ as an optimal solution of the linear program $$\begin{aligned}
\min (\Mid{c})^Tx{{\ \ \mbox{subject to}\ \ }}x\in{{\mathcal{F}}},\end{aligned}$$ where $$\begin{aligned}
{{\mathcal{F}}}:=\{x\in{{\mathbb{R}}}^n{;\,}{\mbox{$\overline{{{{A}}}}$}}x\leq{\mbox{$\overline{{{{b}}}}$}},\ \
{\mbox{$\underline{{{{A}}}}$}}x\geq{\mbox{$\underline{{{{b}}}}$}},\ \
x\geq0\}.\end{aligned}$$
The set of robust solutions in more detail
------------------------------------------
Let us denote by ${\mbox{\large$\Sigma$}}$ the set of all robust optimal solutions.
\[propSsEqUniConv\] ${\mbox{\large$\Sigma$}}$ is formed by a union of at most $\binom{n}{\lfloor n/2\rfloor}$ convex polyhedral sets.
Each $x\in{\mbox{\large$\Sigma$}}$ must lie in ${{\mathcal{F}}}$ and must satisfy the optimality criterion. Since the optimality criterion does not depend directly on $x$, but only on the active set $I$ of $x$, we have that $$\begin{aligned}
{{\mathcal{F}}}_I:={{\mathcal{F}}}\cap
\{x\in{{\mathbb{R}}}^n{;\,}x_i=0,\ i\in I,\ x_i>0,\ i\not\in I\}\end{aligned}$$ either whole lies in ${\mbox{\large$\Sigma$}}$, or is disjoint with ${\mbox{\large$\Sigma$}}$. Hence ${\mbox{\large$\Sigma$}}$ is formed by a union of the sets ${{\mathcal{F}}}_I$ for several index sets $I\subseteq {{\{1, \ldots, {n}\}}}$. Since $$\begin{aligned}
{{\mathcal{F}}}_I\subseteq{\mbox{\large$\Sigma$}}\wedge I\subseteq I'\
\Rightarrow\ {{\mathcal{F}}}_{I'}\subseteq{\mbox{\large$\Sigma$}},\end{aligned}$$ we can replace the sets ${{\mathcal{F}}}_I$ by $$\begin{aligned}
\tilde{{{\mathcal{F}}}}_I:={{\mathcal{F}}}\cap
\{x\in{{\mathbb{R}}}^n{;\,}x_i=0,\ i\in I\}.\end{aligned}$$ Now, since $\tilde{{{\mathcal{F}}}}_I\supseteq\tilde{{{\mathcal{F}}}}_{I'}$ for $I\subseteq I'$, not all subsets of ${{\{1, \ldots, {n}\}}}$ have to be taken into account. By Sperner’s theorem (see, e.g., [@MatNes2008]), only $\binom{n}{\lfloor n/2\rfloor}$ of them it is sufficient to consider.
As illustrated by the following example, the robust solution set ${\mbox{\large$\Sigma$}}$ needn’t be topologically connected.
Consider the problem $$\begin{aligned}
\min x_1+x_2+c_3x_3
{{\ \ \mbox{subject to}\ \ }}x_1+x_2+x_3=1,\ x_1-x_2=b_2,\ x_1,x_2,x_3\geq0,\end{aligned}$$ where $c_3\in[0.5,1.5]$ and $b_2\in[-1,1]$. The robust feasible set ${{\mathcal{F}}}$ is formed by a triangle with vertices $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$. Concerning optimality, the system reads $$\begin{aligned}
\label{sysRobOptEx}
x_1+x_2+c_3x_3=-1,\ \
x_1+x_2+x_3=0,\ \
x_1-x_2=0,\ \
x_I\geq0.\end{aligned}$$ If $3\not\in I$, then has a solution $x=(1,1,-2)$ when $c_3=1.5$. Thus, it must be $3\in I$. If $I=\{3\}$, then has a solution $x=(-1,-1,2)$ when $c_3=0.5$. If $I=\{1,3\}$, then has no solution for any $c_3$, and the corresponding $\tilde{{{\mathcal{F}}}}_I=\{(0,1,0)\}$. Similarly, for $I=\{2,3\}$, the system has no solution for any $c_3$, and $\tilde{{{\mathcal{F}}}}_I=\{(1,0,0)\}$. In summary, the robust solution set ${\mbox{\large$\Sigma$}}$ consists of two isolated points $(1,0,0)$ and $(0,1,0)$.
Applications
============
Transportation problem
----------------------
Since linear programming is so widely used technique, the proposed concept of robust solution and the corresponding methodology is applicable in many practical problems. These problems include transportation problem and flows in networks, among others, in which the constraint matrix $A$ represents an incidence matrix of a (undirected or directed) graph. By imposing suitable intervals ($[0,1]$ or $[-1,1]$) as the matrix entries, we can model uncertainty in the knowledge of the edge existence.
More concretely, consider a transportation problem $$\begin{aligned}
\min\ &\sum_{i=1}^m\sum_{j=1}^nc_{ij}x_{ij}\\
{{\ \ \mbox{subject to}\ \ }}&\sum_{i=1}^m\alpha_{ij}x_{ij}=b_j,\quad j=1,\dots,n,\\
&\sum_{j=1}^n\alpha_{ij}x_{ij}=a_i,\quad i=1,\dots,m,\\
&x_{ij}\geq0,\quad i=1,\dots,m,\ j=1,\dots,n,\end{aligned}$$ where $c_{ij}\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}_{ij}$, $a_i\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}_i$ and $b_j\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}_j$. In contrast to the standard formulation $\alpha_{ij}\in\{0,1\}$ and in order to obtain interval parameters, we allow $\alpha_{ij}$ to attain values in the interval $[0,1]$.
Robustness here means that an optimal solution $x^*$ remains optimal for any $c_{ij}\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}_{ij}$. Moreover, $x^*$ should also remain optimal even when some selected edges are removed. The edge removal could be compensated by a suitable change of $a_i\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}_i$ and $b_j\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}_j$. Herein,the intervals ${\mbox{${\mbox{\mathversion{bold}$ a $}}$}}_i$ and ${\mbox{${\mbox{\mathversion{bold}$ b $}}$}}_j$ are interpreted as tolerances in supplies and demands.
For concreteness, let $$\begin{aligned}
C=\begin{pmatrix}20& 30& 10\\10& 20& 50\\40& 10& 20\end{pmatrix},\quad
a=(100, 160, 250),\quad
b=(150, 210, 150).\end{aligned}$$ Suppose that the objective coefficients $c_{ij}$ are known with $10\%$ precision only. Next suppose that the supplies and demands have $10\%$ tolerance in which they can be adjusted. Eventually, suppose that the connections from the second supplier to the second and third demanders, and from the third supplier to the first demander are all uncertain. Thus, we have interval data $$\begin{aligned}
{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}&=\begin{pmatrix}
[18,22]& [27,33]& [9,11]\\
[9,11]& [18,22]& [45,55]\\
[36,44]& [9,11]& [18,22]
\end{pmatrix},\\
{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}&=([90,110],\, [144,176],\, [225,275]),\\
{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}&=([135,165],\,[189,231],\,[135,165])\\
{\mbox{${\mbox{\mathversion{bold}$ \alpha $}}$}}_{22}&={\mbox{${\mbox{\mathversion{bold}$ \alpha $}}$}}_{23}={\mbox{${\mbox{\mathversion{bold}$ \alpha $}}$}}_{31}=[0,1],\ \
{\mbox{${\mbox{\mathversion{bold}$ \alpha $}}$}}_{ij}=1,\ (i,j)\not\in\{(2,2),(2,3),(3,1)\}.\end{aligned}$$
For the midpoint data, the optimal solution is $$\begin{aligned}
\begin{pmatrix}0& 0& 100\\150& 10& 0\\0& 200& 50\end{pmatrix}.\end{aligned}$$ It is robustly feasible, however, it is not robustly optimal.
Let us try our method from Section \[ssCand\]. It recommends to solve the problem $$\begin{aligned}
\min\ &\sum_{i=1}^m\sum_{j=1}^n\Mid{c}_{ij}x_{ij}\\
{{\ \ \mbox{subject to}\ \ }}&\sum_{i=1}^m{\mbox{$\overline{{\alpha}}$}}_{ij}x_{ij}\leq {\mbox{$\overline{{b}}$}}_j,\ \
\sum_{i=1}^m{\mbox{$\underline{{\alpha}}$}}_{ij}x_{ij}\geq {\mbox{$\underline{{b}}$}}_j,
\quad j=1,\dots,n,\\
&\sum_{j=1}^n{\mbox{$\overline{{\alpha}}$}}_{ij}x_{ij}\leq{\mbox{$\overline{{a}}$}}_i,\ \
\sum_{j=1}^n{\mbox{$\underline{{\alpha}}$}}_{ij}x_{ij}\geq{\mbox{$\underline{{a}}$}}_i,
\quad i=1,\dots,m,\\
&x_{ij}\geq0,\quad i=1,\dots,m,\ j=1,\dots,n.\end{aligned}$$ Its solution is $$\begin{aligned}
\begin{pmatrix}0& 0& 99\\144& 0& 0\\0& 189& 36\end{pmatrix}.\end{aligned}$$ It turns out that is is both robustly feasible and optimal, so it can serve as a robust solution in question. As the sufficient condition fails, optimality must have been verified by the exhausting feasibility checking of 16 systems of type . Nevertheless, if the edge $(2,2)$ becomes certain, and only the others are uncertain, then the sufficient condition succeeds.
\[exDpNum\] We carried out a limited numerical study about what is the efficiency of the sufficient condition and the heuristic for finding a candidate. In the transportation problem with given dimensions $m$ and $n$, we randomly chosen $C$, $a$ and $b$. In $C$, there were $10\%$ of randomly chosen entries subject to $10\%$ relative uncertainty. Tolerances for supplies and demands were also $10\%$. A given number randomly selected edges were considered as uncertain, i.e., the coefficients by $x_{ij}$ ranged in $[0,1]$.
$m$ $n$ $[0,1]$-edges candidate time (in $s$) robustness time (in $s$) success rate (in %)
----- ----- --------------- ------------------------- -------------------------- ---------------------
2 0.03138 0.01268 25.66
4 0.03155 0.00673 10.63
6 0.03178 0.00415 4.89
3 0.06980 0.02139 18.84
5 0.06904 0.01447 11.90
7 0.06873 0.01011 7.46
4 0.1364 0.02919 9.30
6 0.1370 0.02035 5.96
8 0.1336 0.01457 3.94
4 0.4737 0.11660 4.47
7 0.4520 0.08784 2.45
10 0.4265 0.06585 1.68
6 1.0825 0.2666 1.04
9 1.0119 0.2177 0.80
12 0.9657 0.1853 0.42
1 1.6982 0.04426 0
3 1.8670 0.04975 0
5 2.0025 0.08210 0
: (Example \[exDpNum\]) Computing time and percentual rate of finding robust optimal solution for different dimensions and numbers of uncertain edges in the transportation problem.\[tabDpNum\]
Table \[tabDpNum\] displays the results. Each row is a mean of 10000 runs, and shows the average running time in seconds and the success rate. The success rate measures for how many instances the heuristic found a candidate that was after that verified as a robust optimal solution by the sufficient condition. This means that the number of robust solution can be higher, but we were not able to check it because of its intractability. The displayed running time concerns both the heuristic for finding a suitable candidate and the sufficient condition for checking robustness.
The results show that in low computing time we found robust optimal solutions in 5% to 15% of the small dimensional cases. In large dimensions, the number of robust solutions is likely to be small. Even when we decreased the number of uncertain edges, the sufficient condition mostly failed. This may be due to 500 interval costs in the last data set.
Nutrition problem
-----------------
The diet problem is the classical linear programming problem, in which a combination of $n$ different types of food must be found such that $m$ nutritional demands are satisfied and the overall cost in minimal. The mathematical formulation has exactly the form of , where $x_j$ be the number of units of food $j$ to be consumed, $b_i$ is the required amount of nutrient $j$, $c_j$ is the price per unit of food $j$, and $a_{ij}$ is the the amount of nutrient $j$ contained in one unit of food $i$.
Since the amounts of nutricients is foods are not constant, it is reasonable to consider intervals of possible ranges instead of fixed values. The same considerations apply for the costs. The requirements on nutritional demands to be satisfied as equations are too strict. Usually, there are large tolerances on the amount of consumed nutricients (such as calories, proteins, vitamins, etc.), which leads to quite wide intervals of admissible tolerances for the entries of $b$. In this interval valued diet problem, we would like to find optimal solution $x$ that is robustly feasible in the sense that for each possible instance of $A$ there is an admissible $b$ such that $Ax=b$. This model is exactly the robustness model we are dealing with in this paper.
Consider Stigler’s nutrition model [@Dan1963], the GAMS model file containing the data is posted at <http://www.gams.com/modlib/libhtml/diet.htm>. The problem consists of $m=9$ nutrients and $n=20$ types of food. The data in $A$ are already normalized such that it gives nutritive values of foods per dollar spent. This means that the objective is $c=(1,\dots,1)^T$.
Suppose that the entries of $A$ can vary up to $5\%$ of their nominal values, and the tolerances in $b$ are $10\%$. Then the method from Section \[ssCand\] finds the solution $$\begin{aligned}
x^*
=(&0.0256, 0.0067, 0.0429, 0, 0, 0.0015, 0.0245,\\
&0.0108, 0, 0, 0, 0.0109, 0, 0, 0.0016, 0, 0, 0, 0, 0)^T.\end{aligned}$$ Even though our sufficient condition fails, it turns out by checking infeasibility of for each sign vector that this solution is robustly optimal.
General form of interval linear programming {#sGen}
===========================================
For the sake of simplicity of exposition, we considered the equality form of linear programming in the first part of this paper. It is well known in interval linear programming that different forms are not equivalent to each other [@Hla2012a] since transformations between the formulations lead to dependencies between interval coefficients. That is why we will consider a general form of interval linear programming in this section and extend the results developed so far.
The general form with $m$ equation, $m'$ inequalities and variables $x\in{{\mathbb{R}}}^n$, $y\in{{\mathbb{R}}}^{n'}$ reads $$\begin{aligned}
\label{lpGen}
\min c^Tx+d^Ty{{\ \ \mbox{subject to}\ \ }}Ax+By=b,\ Cx+Dy\leq a,\ x\geq0,\end{aligned}$$ where $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$, $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$, $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$, $d\in{\mbox{${\mbox{\mathversion{bold}$ d $}}$}}$, $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$, $B\in{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}$, $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$. Let $(x^*,y^*)$ be a candidate solution. The problem now states as follows.
> For every $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$, $d\in{\mbox{${\mbox{\mathversion{bold}$ d $}}$}}$, $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$, $B\in{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}$, $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$, $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, does there exist $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$ and $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$ such that $(x^*,y^*)$ is optimal to ?
As before, we will study feasibility and optimality conditions separately.
Feasibility
-----------
Here, we have to check whether for each $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$, $B\in{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}$, $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, there are $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$ and $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$ such that $Ax^*+By^*=b$ and $Cx^*+Dy^*\leq a$. We can check equations and inequalities independently. Equations are dealt with in a similar manner as in Section \[sEqForm\], and the sufficient and necessary condition is $$\begin{aligned}
\label{condGenFeasEq}
|\Mid{A}x^*+\Mid{B}y^*-\Mid{b}|+\Rad{A}x^*+\Rad{B}|y^*|\leq\Rad{b}.\end{aligned}$$ For inequalities, we have the following characterisation.
For each $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, there is $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$ such that $Cx^*+Dy^*\leq a$ if and only if $$\begin{aligned}
\label{condGenFeasIneq}
{\mbox{$\overline{{{{C}}}}$}}x^*+\Mid{D}y^*+\Rad{D}|y^*|\leq{\mbox{$\overline{{{{a}}}}$}}.\end{aligned}$$
For each $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$ we have $$\begin{aligned}
Cx^*+Dy^*
&=Cx^*+\Mid{D}y^*+(D-\Mid{D})y^*
\leq Cx^*+\Mid{D}y^*+|D-\Mid{D}||y^*|\\
&\leq {\mbox{$\overline{{{{C}}}}$}}x^*+\Mid{D}y^*+\Rad{D}|y^*|.\end{aligned}$$ This inequality chain holds as equation for $C:={\mbox{$\overline{{{{C}}}}$}}$ and $D:=\Mid{D}+\Rad{D}\operatorname{diag}(\operatorname{sgn}(y^*))$ since $|y^*|=\operatorname{diag}(\operatorname{sgn}(y^*))y^*$. That is, the largest value of the left-hand side is attained for this setting. Therefore, feasibility condition holds true if and only if the inequality is satisfied for this setting of $C$ and $D$, and for the largest possible right-hand side vector $a:={\mbox{$\overline{{{{a}}}}$}}$.
Optimality
----------
For checking optimality we have to define the active set first. For nonnegativity constraints, we can use the standard definition $I:=\{i=1,\dots,n{;\,}x^*_i=0\}$. However, we face a problem to define the active set for the other inequalities due to the variations in $C$ and $D$. Fortunately, we can define it as follows.
\[propGenOpt\] Each instance of the inequality system $Cx^*+Dy^*\leq a$, $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$, $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, with a suitable $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$ includes the index set $$\begin{aligned}
K:=\{k{;\,}{\mbox{$\underline{{{{C}}}}$}}_{k*}x^*+\Mid{D}_{k*}y^*-\Rad{D}_{k*}|y^*|\geq {\mbox{$\underline{{a}}$}}_k\}\end{aligned}$$ as a subset of its active set. Moreover, $K$ is attained as an active set for $C:={\mbox{$\underline{{{{C}}}}$}}$, $D:=\Mid{D}-\Rad{D}\operatorname{diag}(\operatorname{sgn}(y^*))$ and $a:=\max\{{\mbox{$\underline{{{{a}}}}$}},Cx^*+Dy^*\}$.
First, we show that $K$ is attained for $C:={\mbox{$\underline{{{{C}}}}$}}$, $D:=\Mid{D}-\Rad{D}\operatorname{diag}(\operatorname{sgn}(y^*))$, and $a:=\max\{{\mbox{$\underline{{{{a}}}}$}},Cx^*+Dy^*\}\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$. The condition $Cx^*+Dy^*\leq{\mbox{$\overline{{{{a}}}}$}}$, and hence also $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$, follows from feasibility of $(x^*,y^*)$, so $a$ is well defined. For $k\in K$, we have $$\begin{aligned}
{C}_{k*}x^*+{D}_{k*}y^*
={\mbox{$\underline{{{{C}}}}$}}_{k*}x^*+\Mid{D_{k*}}y^*-\Rad{D_{k*}}|y^*|
\geq {\mbox{$\underline{{a}}$}}_k,\end{aligned}$$ whence ${C}_{k*}x^*+{D}_{k*}y^*=a_k$. For $k\not\in K$, we have $$\begin{aligned}
{C}_{k*}x^*+{D}_{k*}y^*
={\mbox{$\underline{{{{C}}}}$}}_{k*}x^*+\Mid{D_{k*}}y^*-\Rad{D_{k*}}|y^*|
< {\mbox{$\underline{{a}}$}}_k=a_k.\end{aligned}$$
Now, let $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$, $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$ and $k\in K$ be arbitrary. From the feasibility of $(x^*,y^*)$ and $$\begin{aligned}
{\mbox{$\underline{{a}}$}}_k
\leq{\mbox{$\underline{{{{C}}}}$}}_{k*}x^*+\Mid{D_{k*}}y^*-\Rad{D_{k*}}|y^*|
\leq{C}_{k*}x^*+{D}_{k*}y^*\end{aligned}$$ we can put $a_k:={C}_{k*}x^*+{D}_{k*}y^*\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}_k$. Therefore, $k$ lies in the active set corresponding to $C$, $D$ and $a$.
Notice that the larger the active set the better since we have more constraints in the optimality criterion and the solution is more likely optimal. Proposition \[propGenOpt\] says that we can take $K$ as the active set to the interval inequalities. Since for each $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, this $K$ is the smallest active set, it is the worst case scenario that we can imagine. Similarly, the right-hand side vector $a$ from Proposition \[propGenOpt\] is the best response: If we decrease it, then $(x^*,y^*)$ or $a$ becomes infeasible, and if we increase it, then the active set becomes smaller.
To state the optimality criterion comprehensively, we have to introduce some notation first. Let $\tilde{A}:=A_I$, $\tilde{B}:=(A_J\mid B)$, $\tilde{c}:=c_I$, $\tilde{d}:=(c_J, d)$, $\tilde{x}:=x_I$, $\tilde{y}:=(x_J,y)$. Let $\tilde{C}$ be the restriction of $C_I$ to the rows indexed by $K$, and similarly $\tilde{D}$ be a restriction of $(C_J\mid D)$ to the rows indexed by $K$.
For a concrete setting $a\in{\mbox{${\mbox{\mathversion{bold}$ a $}}$}}$, $b\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$, $c\in{\mbox{${\mbox{\mathversion{bold}$ c $}}$}}$, $d\in{\mbox{${\mbox{\mathversion{bold}$ d $}}$}}$, $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$, $B\in{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}$, $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$, a feasible solution $(x^*,y^*)$ is optimal if and only if $$\begin{aligned}
\label{sysGenOpt}
\tilde{c}^T\tilde{x}+\tilde{d}^T\tilde{y}\leq-1,\ \
\tilde{A}\tilde{x}+\tilde{B}\tilde{y}=0,\ \
\tilde{C}\tilde{x}+\tilde{D}\tilde{y}\leq0,\ \
\tilde{x}\geq0\end{aligned}$$ has no solution. In order that $(x^*,y^*)$ is robustly optimal, this systems has to be infeasible for each realization from the given intervals. By [@Hla2013b], is infeasible for each realization if and only if the system $$\begin{aligned}
{\mbox{$\underline{{{{\tilde{c}}}}}$}}^T\tilde{x}+(\Mid{\tilde{d}})^T\tilde{y}
&\leq(\Rad{\tilde{d}})^T|\tilde{y}|-1,\\
{\mbox{$\underline{{{{\tilde{A}}}}}$}}\tilde{x}+\Mid{\tilde{B}}\tilde{y}
&\leq\Rad{\tilde{B}}|\tilde{y}|,\\
-{\mbox{$\overline{{{{\tilde{A}}}}}$}}\tilde{x}-\Mid{\tilde{B}}\tilde{y}
&\leq\Rad{\tilde{B}}|\tilde{y}|,\\
{\mbox{$\underline{{{{\tilde{C}}}}}$}}\tilde{x}+\Mid{\tilde{D}}\tilde{y}
&\leq\Rad{\tilde{D}}|\tilde{y}|,\\
\tilde{x}&\geq0\end{aligned}$$ is infeasible. Even though we reduced infeasibility checking from infinitely many systems to only one, the resulting system in nonlinear. As in Section \[sEqForm\], we can formulate it equivalently as infeasibility of $$\begin{aligned}
{\mbox{$\underline{{{{\tilde{c}}}}}$}}^T\tilde{x}
+(\Mid{\tilde{d}}-\Rad{\tilde{d}}\operatorname{diag}(s))^T\tilde{y}
&\leq-1,\\
{\mbox{$\underline{{{{\tilde{A}}}}}$}}\tilde{x}
+(\Mid{\tilde{B}}-\Rad{\tilde{B}}\operatorname{diag}(s))\tilde{y}
&\leq 0,\\
-{\mbox{$\overline{{{{\tilde{A}}}}}$}}\tilde{x}
-(\Mid{\tilde{B}}+\Rad{\tilde{B}}\operatorname{diag}(s))\tilde{y}
&\leq 0,\\
{\mbox{$\underline{{{{\tilde{C}}}}}$}}\tilde{x}
+(\Mid{\tilde{D}}-\Rad{\tilde{D}}\operatorname{diag}(s))\tilde{y}
&\leq 0,\\
\tilde{x}&\geq0\end{aligned}$$ for every $s\in\{\pm1\}^{n'+|J|}$. Now, we have to check infeasibility of $2^{n'+|J|}$ linear systems, which is large but finite. In case there are few sign-unrestricted variables and few positive components in $x^*$, the number fo systems can be acceptable for computation.
Sufficient condition {#sufficient-condition}
--------------------
Similarly as in Section \[ssSufCond\], we can derive a sufficient condition for optimality checking. We discuss it briefly and refer to [@Hla2013b] for more details. By Farkas lemma, the optimality criterion holds true if and only if the dual system $$\begin{aligned}
\label{optGenCondDual}
\tilde{A}^Tu-\tilde{C}^Tv\leq \tilde{c},\ \
\tilde{B}^Tu-\tilde{D}^Tv = \tilde{d},\ \
v \geq 0\end{aligned}$$ is feasible for each interval setting. First, solve the linear program $$\begin{aligned}
\max\alpha{{\ \ \mbox{subject to}\ \ }}(\Mid{\tilde{A}})^Tu-(\Mid{\tilde{C}})^Tv+\alpha e
\leq\Mid{\tilde{c}},\ \
(\Mid{\tilde{B}})^Tu-(\Mid{\tilde{D}})^Tv = \Mid{\tilde{d}},\ \
v \geq \alpha e.\end{aligned}$$ Let $(u^*,v^*)$ be its optimal solution, and let $(\hat{B}\mid-\hat{D})$ be an orthogonal basis of the null space of $((\Mid{\tilde{B}})^T\mid-(\Mid{\tilde{D}})^T)$ and put $\hat{d}:=\hat{B}u^*-\hat{D}v^*$. Compute an interval enclosure $({\mbox{${\mbox{\mathversion{bold}$ u $}}$}},{\mbox{${\mbox{\mathversion{bold}$ v $}}$}})$ to the solution set of the square interval system $$\begin{aligned}
\{(u,v){;\,}\exists \tilde{B}\in\tilde{{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}}
\exists \tilde{D}\in\tilde{{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}}
\exists \tilde{d}\in \tilde{{\mbox{${\mbox{\mathversion{bold}$ d $}}$}}}:
\tilde{B}^Tu-\tilde{D}^Tv = \tilde{d},\ \
\hat{B}u-\hat{D}v=\hat{d}\},\end{aligned}$$ and check whether ${\mbox{$\underline{{{{v}}}}$}}\geq0$ and $${\mbox{$\overline{{{{\tilde{{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}}^T{\mbox{${\mbox{\mathversion{bold}$ u $}}$}}-\tilde{{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}}^T{\mbox{${\mbox{\mathversion{bold}$ v $}}$}}}}}}$}}
\leq \tilde{{\mbox{$\underline{{{{c}}}}$}}}.$$ If they are satisfied, then has a solution in the set $({\mbox{${\mbox{\mathversion{bold}$ u $}}$}},{\mbox{${\mbox{\mathversion{bold}$ v $}}$}})$ for each interval realization, and we can claim that optimality criterion holds true.
Seeking for a candidate {#seeking-for-a-candidate}
-----------------------
Herein, we generalize the heuristic from Section \[ssCand\] to find a good candidate for robust optimal solution. Concerning the feasibility question, the conditions and are not convenient due to their nonlinearities. Thus, we state an equivalent, linear form of feasibility testing.
A vector $(x,y)$ is robustly feasible if and only if $y$ has the form of $y=y^1-y^2$ such that
\[condGenFeasProp\] $$\begin{aligned}
\label{condGenFeasProp1}
{\mbox{$\overline{{{{A}}}}$}}x+{\mbox{$\overline{{{{B}}}}$}}y^1-{\mbox{$\underline{{{{B}}}}$}}y^2&\leq{\mbox{$\overline{{{{b}}}}$}},\\
\label{condGenFeasProp2}
{\mbox{$\underline{{{{A}}}}$}}x+{\mbox{$\underline{{{{B}}}}$}}y^1-{\mbox{$\overline{{{{B}}}}$}}y^2&\geq{\mbox{$\underline{{{{b}}}}$}},\\
\label{condGenFeasProp3}
{\mbox{$\overline{{{{C}}}}$}}x+{\mbox{$\overline{{{{D}}}}$}}y^1-{\mbox{$\underline{{{{D}}}}$}}y^2&\leq{\mbox{$\overline{{{{a}}}}$}},\\
x,y^1,y^2&\geq0.\end{aligned}$$
Let $(x,y^1,y^2)$ be a solution to . For any $A\in{\mbox{${\mbox{\mathversion{bold}$ A $}}$}}$ and $B\in{\mbox{${\mbox{\mathversion{bold}$ B $}}$}}$ we have $$\begin{aligned}
Ax+B(y^1-y^2)\leq{\mbox{$\overline{{{{A}}}}$}}x+{\mbox{$\overline{{{{B}}}}$}}y^1-{\mbox{$\underline{{{{B}}}}$}}y^2&\leq{\mbox{$\overline{{{{b}}}}$}},\end{aligned}$$ and $$\begin{aligned}
Ax+B(y^1-y^2)\geq{\mbox{$\underline{{{{A}}}}$}}x+{\mbox{$\underline{{{{B}}}}$}}y^1-{\mbox{$\overline{{{{B}}}}$}}y^2&\geq{\mbox{$\underline{{{{b}}}}$}},\end{aligned}$$ whence $Ax+B(y^1-y^2)\in{\mbox{${\mbox{\mathversion{bold}$ b $}}$}}$. For any $C\in{\mbox{${\mbox{\mathversion{bold}$ C $}}$}}$ and $D\in{\mbox{${\mbox{\mathversion{bold}$ D $}}$}}$ we have $$\begin{aligned}
Cx+D(y^1-y^2)\leq{\mbox{$\overline{{{{C}}}}$}}x+{\mbox{$\overline{{{{D}}}}$}}y^1-{\mbox{$\underline{{{{D}}}}$}}y^2&\leq{\mbox{$\overline{{{{a}}}}$}},\end{aligned}$$ so $(x,y^1-y^2)$ is robustly feasible.
Conversely, let $(x,y)$ be robustly feasible. Put $y^1:=y^+$ and $y^2:=y^-$, the positive and negative parts of $y$. From , we derive $$\begin{aligned}
|\Mid{A}x+\Mid{B}(y^1-y^2)-\Mid{b}|+\Rad{A}x+\Rad{B}(y^1+y^2)
\leq\Rad{b}.\end{aligned}$$ This inequality gives rise to two linear inequalities: $$\begin{aligned}
\Mid{A}x+\Mid{B}(y^1-y^2)-\Mid{b}+\Rad{A}x+\Rad{B}(y^1+y^2)
&\leq\Rad{b},\\
-\Mid{A}x-\Mid{B}(y^1-y^2)+\Mid{b}+\Rad{A}x+\Rad{B}(y^1+y^2)
&\leq\Rad{b},\end{aligned}$$ which are equivalent to –. Similarly, implies $$\begin{aligned}
{\mbox{$\overline{{{{C}}}}$}}x+\Mid{D}(y^1-y^2)+\Rad{D}(y^1+y^2)\leq{\mbox{$\overline{{{{a}}}}$}},\end{aligned}$$ which is equivalent to .
Now, we recommend to take as a candidate solution the pair $(x^*,y^{*1}-y^{*2})$, where $(x^*,y^{*1},y^{*2})$ is an optimal solution of the linear program $$\begin{aligned}
\min (\Mid{c})^Tx+(\Mid{d})^Ty^1-(\Mid{d})^Ty^2
{{\ \ \mbox{subject to}\ \ }}(\ref{condGenFeasProp}).\end{aligned}$$
The set of robust solutions in more detail
------------------------------------------
As before, we denote by ${\mbox{\large$\Sigma$}}$ the set of all robust optimal solutions and state to following topological results on it.
\[propSsGenUniConv\] ${\mbox{\large$\Sigma$}}$ is formed by a union of at most $\binom{m'}{\lfloor m'/2\rfloor}\binom{n}{\lfloor n/2\rfloor}$ convex polyhedral sets.
Each $x\in{\mbox{\large$\Sigma$}}$ must satisfy both the feasibility and optimality criteria. The robust feasible set is a convex polyhedral set, so we focus on the optimality issue. The optimality depends only on the active sets $I$ and $K$, not on the concrete value of $x$. Given $I$ and $K$, the corresponding set $$\begin{aligned}
{{\mathcal{F}}}\cap
\{(x,y)\in{{\mathbb{R}}}^{n+n'}{;\,}&x_i=0,\ i\in I,\ x_i>0,\ i\not\in I,\\
&{\mbox{$\underline{{{{C}}}}$}}_{k*}x+\Mid{D}_{k*}y-\Rad{D}_{k*}|y|\geq{\mbox{$\underline{{a}}$}}_k,\
k\in K,\\
&{\mbox{$\underline{{{{C}}}}$}}_{k*}x+\Mid{D}_{k*}y-\Rad{D}_{k*}|y|<{\mbox{$\underline{{a}}$}}_k,\
k\not\in K\}\end{aligned}$$ either is a subset of ${\mbox{\large$\Sigma$}}$ or is disjoint. Since larger $I$ and $K$ preserve optimality, we can remove the strict inequalities, and ${\mbox{\large$\Sigma$}}$ is formed by a union of some of the sets $$\begin{aligned}
{{\mathcal{F}}}\cap
\{(x,y)\in{{\mathbb{R}}}^{n+n'}{;\,}&x_i=0,\ i\in I,\
{\mbox{$\underline{{{{C}}}}$}}_{k*}x+\Mid{D}_{k*}y-\Rad{D}_{k*}|y|\geq{\mbox{$\underline{{a}}$}}_k,\ k\in K\}.\end{aligned}$$ There are $2^{m'+n}$ possibilities to choose $I$ and $K$, but by Sperner’s theorem again, only at most $\binom{m'}{\lfloor m'/2\rfloor}\binom{n}{\lfloor n/2\rfloor}$ of them are sufficient to consider.
It remains to prove that the set of feasible solutions fulfilling the active set requirements is a convex polyhedral set. Concerning $I$, the condition $x_i=0$, $i\in I$, is obviously convex preserving. Concerning $K$, the condition $$\begin{aligned}
{\mbox{$\underline{{{{C}}}}$}}_{k*}x+\Mid{D}_{k*}y-\Rad{D}_{k*}|y|\geq {\mbox{$\underline{{a}}$}}_k\end{aligned}$$ can be reformulated as $$\begin{aligned}
{\mbox{$\underline{{{{C}}}}$}}_{k*}x+\Mid{D}_{k*}y-\Rad{D}_{k*}z\geq {\mbox{$\underline{{a}}$}}_k,\ \
z\geq y,\ \ z\geq -y.\end{aligned}$$ These inequalities describe a convex polyhedral set and its projection to the $x,y$-subspace is also convex polyhedral.
Conclusion
==========
We introduced a novel kind of robustness in linear programming. When showing some basic properties, some open questions raised. For example, the robust optimal solution set ${\mbox{\large$\Sigma$}}$ may be disconnected, but what can be the number of components at most? Similarly, how tight is the number of convex bodies (Propositions \[propSsEqUniConv\] and \[propSsGenUniConv\]) the set ${\mbox{\large$\Sigma$}}$ consists of?
### Acknowledgments. {#acknowledgments. .unnumbered}
The author was supported by the Czech Science Foundation Grant P402/13-10660S.
[^1]: Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 11800, Prague, Czech Republic, e-mail: `milan.hladik@matfyz.cz`
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abstract: 'Accretion-powered X-ray pulsars 4U 1626-67, GX 1+4, and OAO 1657-415 have recently shown puzzling torque reversals. These reversals are characterized by short time scales, on the order of days, nearly identical spin-up and spin-down rates, and very small changes in X-ray luminosity. We propose that this phenomenon is the result of sudden dynamical changes in the accretion disks triggered by a gradual variation of mass accretion rates. These sudden torque reversals may occur at a critical accretion rate $\sim$ $10^{15}-10^{16}g~s^{-1}$ when the system makes a transition from (to) a primarily Keplerian flow to (from) a substantially sub-Keplerian, radial advective flow in the inner disk. For systems near spin equilibrium, the spin-up torques in the Keplerian state are slightly larger than the spin-down torques in the advective state, in agreement with observation. The abrupt reversals could be a signature of pulsar systems near spin equilibrium with the mass accretion rates modulated on a time scale of a year or longer near the critical accretion rate. It is interesting that cataclysmic variables and black hole soft X-ray transients change their X-ray emission properties at accretion rates similar to the pulsars’ critical rate. We speculate that the dynamical change in pulsar systems shares a common physical origin with white dwarf and black hole accretion disk systems.'
author:
- 'Insu Yi$^1$, J. Craig Wheeler$^2$, and Ethan T. Vishniac$^2$'
title: 'Torque Reversal in Accretion-Powered X-ray Pulsars'
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Introduction
============
Detailed spin evolution on long time scales has been made available for several accretion-powered X-ray pulsars such as 4U 1626-67, GX 1+4, and OAO 1657-415 (for comprehensive reviews and data, see e.g. Chakrabarty et al. 1993, Chakrabarty 1995, and references therein). These systems have shown puzzling abrupt spin reversals. Before and after the observed torque reversals, their spin-up and spin-down torques were largely steady. It is intriguing that the spin-up and spin-down torques before and after reversals are nearly identical. These reversals are quite different from the random torque fluctuations seen in some pulsar systems believed to be fed by winds (e.g. Nagase 1989, Anzer & B[ö]{}rner 1995, and references therein). The nearly steady torques plausibly indicate the existence of ordered accretion disks. The observations indicate that the mass accretion rate is gradually modulated with small amplitudes on a time scale of at least a year, which is much longer than the typical reversal time scale.
In the disk-magnetosphere interaction models of the Ghosh-Lamb type (Ghosh & Lamb 1979ab, Campbell 1992, Yi 1995, Wang 1995), the magnetic torque is a function of the mass accretion rate, ${\dot M}$. The sign of the torque is reversed (spin-up/down) as ${\dot M}$ varies when the disk inner edge moves past the equilibrium radius at which the torque vanishes (e.g. Lipunov 1992). In this picture, however, the torque variation is expected to be smooth and continuous unless ${\dot M}$ varies discontinuously. Although this behavior may be relevant for some smooth torque reversals, the observed sudden reversals appear distinct (Chakrabarty 1995). Given the lack of any plausible mechanism for discontinuous change of ${\dot M}$, which must be tuned to occur near spin equilibrium, it is difficult for the existing magnetized disk models to provide an explanation. The observations indicate that the mass accretion rates vary little during transition (cf. Chakrabarty 1995), which makes the discontinuous change of ${\dot M}$ unlikely as an explanation.
In this Letter, we propose a possible explanation for the observed sudden torque reversal. We take the neutron star moment of inertia $I_*=10^{45} g~cm^2$, radius $R_*=10^{6} cm$, and mass $M_*=1.4M_{\odot}$. At a cylindrical radius $R$ from the star, the vertical component of the dipole magnetic field $B_z(R)=B_*(R_*/R)^3$ where $B_*$ is the stellar surface field strength. The spin period is $P_*$ or angular velocity $\Omega_*=2\pi/P_*$.
Keplerian Disk-Magnetosphere Interaction and Inner Region
=========================================================
In the conventional disk-magnetosphere interaction model of the Ghosh-Lamb type (Ghosh & Lamb 1979ab), the magnetic field of an accreting neutron star penetrates a geometrically thin accretion disk and exerts a magnetic torque. Except in a narrow region near the radius where the disk is disrupted, it is assumed that the accretion disk rotation is Keplerian, $\Omega_{K}(R)=(GM_*/R^3)^{1/2}$, the radial internal pressure gradient is small, the radial drift velocity is small, and the disk thickness becomes negligible (Campbell 1992, Yi 1995, Wang 1995 and references therein). In such a model, the $\phi$ component of the induction equation in steady state gives the azimuthal component of the field B\_(R)=[\_\*-\_[K]{}(R)\_[K]{}(R)]{}B\_z(R). where we have assumed that the internal viscosity and the magnetic diffusivity are due to a single turbulent process. Here we will assume that $\gamma$, defined as the ratio of $R$ to the vertical velocity shear length scale $\left|v_{\phi}/(\partial v_{\phi}/\partial z)\right|$, is $\sim 1$ (e.g. Campbell 1992, Yi 1995). The parameter $\alpha$ is the usual viscosity parameter (e.g. Frank et al. 1992) and we take $\alpha=0.3$. Although we adopt specific values of $\gamma$ and $\alpha$, the constraints on $B_*$ could always be rescaled in such a way that the exact individual values of the three parameters are not necessary (e.g. Kenyon et al. 1996). The inner edge of the disk, where the disk is magnetically disrupted, is at $R=R_o$ determined by the condition that the magnetic torque exceeds the internal torque in the disk (Campbell 1992, Yi 1995, Wang 1995) which can be expressed as (R\_oR\_c)\^[7/2]{}=[2N\_c(GM\_\*R\_c)\^[1/2]{}]{} where $N_c=(\gamma/\alpha)B_c^2R_c^3$, $B_c=B_z(R=R_c)$, and $R_c=(GM_*P_*^2/4\pi^2)^{1/3}$ is the Keplerian corotation radius. Integrating the magnetic torque over the disk, and allowing for the angular momentum carried by the gas which crosses the inner edge of the disk, the torque exerted on the star by the disk is N=[76]{}N\_0[1-(8/7)(R\_o/R\_c)\^[3/2]{}1-(R\_o/R\_c)\^[3/2]{}]{} where $N_0={\dot M}(GM_*R_o)^{1/2}$ (Campbell 1992, Yi 1995, Wang 1995) and $N\rightarrow 0$ as $R_o/R_c$ approaches the equilibrium point $x_{eq}=0.915$. This estimate is based on the assumption that there is a negligible flux of angular momentum from $R<R_o$. Most of the contribution to the torque in this type of model comes from field-disk interaction in a narrow region just outside of $R_o$. Several different phenomenological descriptions of disk-magnetosphere interaction cause little practical differences to our conclusions (Wang 1995).
Fig. 1(a) presents an example of the typical smooth torque reversal expected in the models of the Ghosh-Lamb type is shown. This example is based on a set of parameters similar to what we adopt below for 4U 1626-67. It is clear that any densely sampled spin evolution would reveal a gradual and continuous variation of the torque, a generic prediction of the Ghosh-Lamb type model. A sudden torque reversal with nearly constant $\left|N\right|$ is hard to explain unless there exists an unknown constraining mechanism or the ${\dot M}$ variation is discontinuous and fine-tuned. It is possible that some observed smooth torque reversals (Chakrabarty 1995) could be due to this type of reversal model.
Under suitable conditions the inner parts of the accretion disk may evolve to an optically thin, low density state, for example, inside the disk boundary layer around a white dwarf (e.g. Paczynski 1991, Narayan and Popham 1993). When $\dot M$ is sufficiently small this transition may occur over a broad range of radii within a disk, producing a state in which radiative losses are so inefficient that the disk retains a large fraction of the heat generated from the dissipation of orbital energy (the ‘advective state’, cf. Narayan and Yi 1995). The transition to this state is not well understood, and may be affected by various external factors, including X-ray irradiation (e.g. Meyer & Meyer-Hofmeister 1990) and coronal heating (Meyer & Meyer-Hofmeister 1994).
Here we will assume that, whatever the details of the transition, the disk will make the jump to a low density state when it becomes possible for it to do so. What is the critical ${\dot M}_{crit}$ below which the disk becomes hot and optically thin? Observationally, weakly magnetized cataclysmic variables generally show a trend in which the X-ray to optical flux ratio decreases as ${\dot M}$ increases. Above a critical rate of $\sim 10^{16} g~s^{-1}$, the X-ray emission becomes extremely weak. This has been attributed to the transition of the inner region to an optically thin (X-ray emitting) hot accretion disk (Patterson & Raymond 1985, Narayan & Popham 1993) at low ${\dot M}$. Interestingly, such a critical rate is largely consistent with the critical rate $\sim 2\times 10^{17}\alpha^2$ or $\sim 2\times 10^{16}
g~s^{-1}$ for $\alpha\sim 0.3$ based on the recently discussed advection-dominated hot accretion disks (Narayan & Yi 1995). For a given accretion rate, there exists a critical radius inside of which the disk makes a transition to sub-Keplerian while the disk at radii larger than the critical radius remains Keplerian (Narayan & Yi 1995). The exact location of this critical radius is not clearly understood yet. We assume that most of the magnetic torque contribution comes from the inner region (e.g. Wang 1995) which is inside the critical radius, e.g. $R_{crit}\gg R_c>R_o$. The relevance of the critical ${\dot M}_{crit}$ for the cataclysmic variables becomes more striking when we consider strongly magnetized neutron star systems. The inner edge of the accretion disk around a strongly magnetized neutron star lies roughly at $R_o\sim 5\times 10^{8}\left(B_*/10^{12}G\right)^{4/7}
\left({\dot M}/10^{16}g/s\right)^{-2/7} cm$ which is close to the typical white dwarf radius. Therefore, one may ask what would be the effects of the transition, at ${\dot M}_{crit}\sim 10^{16}g~s^{-1}$, to the hot optically thin accretion disk on the spin evolution of the pulsars?
Disk Transition and Torque Reversal
===================================
We propose an explanation for the torque reversal based on such a transition. The optically thin, hot accretion disk cannot be geometrically thin or Keplerian once its internal pressure $\sim\rho c_s^2$ becomes a significant fraction of its orbital energy. After the transition, the disk thickness $H\sim c_s/\Omega_K$ and radial drift velocity $v_R\sim\alpha c_s^2/\Omega_K$ increase. The rotation of the accretion disk $\Omega$ becomes sub-Keplerian $\Omega<\Omega_K$ (e.g. Narayan & Yi 1995). In the case of strongly magnetized pulsars, direct X-ray observation of such a disk transition is difficult because most of the X-ray luminosity, $L_x\sim GM_*{\dot M}/R_*$, comes from the surface of the star and the luminosity from the disk, which is truncated well above the stellar surface, is limited to $\sim GM_*{\dot M}/R_o\ll L_x$. The sub-Keplerian disk, however, may have observable dynamical consequences. Once the sub-Keplerian rotation is forced on the magnetosphere, the (sub-Keplerian) corotation radius is shifted inward and the position of the disk inner edge with respect to the new corotation radius is relocated. (In principle, the inner edge of the disk could end up beyond the new corotation radius but this would not account for the observed properties of the torque reversal systems.) As a result, the magnetic torque changes and there could be a visible change of spin-up/down torque on the disk transition time scale.
If such a transition does occur at a certain critical rate, the most likely time scales are the local thermal time scale $t_{th}\sim (\alpha\Omega_K)^{-1}$ or the disk viscous-thermal time scale $t_{dis}\sim R/(\alpha c_s)\sim (R/H)(\alpha\Omega_K)^{-1}\sim 10^3s$ for $\alpha\sim 0.3$, $R\sim 10^9cm$, and ${\dot M}\sim 10^{16} g~s^{-1}$ (e.g. Frank et al. 1992). This is plausible if the transition is mainly driven by the thermal instability of the local optically thin region of the accretion flow accompanied by the disk density change on the local viscous-thermal time scale (cf. Meyer & Meyer-Hofmeister 1994). In fact, the viscous time scale for the thin disk is unrealistically long, since the relevant infall time is the one for the hot, and thick disk, which will be only a few orbital times. In any case, these time scales indicate that the local disk surface density change could occur on a time scale short enough, much less than a day, to make the transition appear almost instantaneous. The long term gradual ${\dot M}$ modulation determines the overall evolutionary trend and possibly affects the residuals seen in some observations (Cutler et al. 1986, Chakrabarty 1995). The sudden torque reversal does not require any short term ($\sim day$) change of ${\dot M}$ but only that the condition ${\dot M}\sim {\dot M}_{crit}$ be satisfied around the time of reversal.
In order to model the reversal episode, we assume no dynamic vertical motion of the disk gas such as winds or outflows. The dominant effect of the transition is to reset the corotation radius and disk truncation radius $R_o$. We take the temperature of the sub-Keplerian hot disk to be a constant fraction $\xi$ of the local virial temperature, i.e. $c_s^2\sim\xi R^2\Omega_K^2$ (e.g. Narayan & Yi 1995). Then the ratio of the disk thickness to radius is $H/R\sim\xi^{1/2}$ and the radial drift velocity is $v_R\sim -\alpha\xi R\Omega_K$. Using the radial component of the momentum equation (e.g. Campbell 1992), we get the sub-Keplerian rotation frequency $\Omega/\Omega_K
\sim\left(1-5\xi/2-\alpha^2\xi^2/2\right)^{1/2}\equiv A.
$ where $\xi\rightarrow 0$ corresponds to the usual Keplerian limit. We note that $v_R/R\Omega_K\sim \alpha\xi <1$ and $H/R\sim \sqrt{\xi}\le 1$. Assuming a constant $\xi$ or $A$, after the transition to the sub-Keplerian rotation with $\Omega(R)=A\Omega_K(R)<\Omega_K(R)$, the corotation radius becomes $R_c^{\prime}=A^{2/3}R_c$ and the new inner disk edge is relocated to $R_o^{\prime}$ determined by (R\_o\^R\_c\^)\^[3]{}=[2N\_cN\_0\^ A]{} where $N_0^{\prime}=A{\dot M}(GM_*R_o^{\prime})^{1/2}$. The torque on the star after the transition is =[76]{} [1-(8/7)(R\_o\^/R\_c\^)\^[3/2]{}1-(R\_o\^/R\_c\^)\^[3/2]{}]{}. The torque vanishes when $R_o^{\prime}/R_c^{\prime}\rightarrow x_{eq}=0.915$ as in the Keplerian rotation (eq. 2-3). In our discussions, we take a constant $A=0.2$ (e.g. Narayan & Yi 1995). The parameters $\alpha=0.3$ and $\gamma=1$ are assumed to be constant before and after the transition. For the Keplerian disk, the equilibrium spin ($N=0$ in eq. (2-3)) period is P\_[eq]{}=\[4.9s\]()\^[3/7]{} (B\_\*10\^[12]{}G)\^[6/7]{}(R\_\*10\^6cm)\^[18/7]{} (M\_\*1.4M\_)\^[-5/7]{} ([M]{}10\^[16]{}g/s)\^[-3/7]{}. For $\Omega=A\Omega_K<\Omega_K$, the equilibrium spin period would become longer by a factor $1/A$ and the system begins to evolve toward the newly determined equilibrium after transition.
Sudden Torque Reversals
=======================
For the spin evolution calculation we integrate the torque equation =-[P\_\*\^2I\_\*]{}\[[Torque]{}\] where Torque $=N$ or $N^{\prime}$ depending on the physical state of the inner disk. We take a linear increase or decrease of ${\dot M}$ as an approximation to more complex ${\dot M}$ variations on longer time scales. The transition occurs on a time scale $\ll P_*/\left|dP_*/dt\right|$ before and after the transition. The transition at ${\dot M}_{crit}=10^{15}-10^{16} g/s$, which is determined by the fits to the observed spin evolution, is taken to be instantaneous (cf. $t_{th}$, $t_{dis}$). For each torque reversal event, we adjust $B_*$, ${\dot M}$, and the accretion rate time scale, ${\dot M}/\left|d{\dot M}/dt\right|$. We consider three X-ray pulsars for which abrupt torque reversals have been detected. For a given initial spin period $P_*$, a fit gives a set of the above parameters.
[*4U 1626-67:*]{} 4U1626-67 ($P_*\approx 7.7s$) was steadily spun-up on a time scale $\sim 10^4 yr$ (${\dot P_*}/P_*^2=-8.54(7)\times 10^{-13} s^{-2}$) during 1979-1989. The Keplerian corotation radius $R_c\approx
(GM_*P_*^2/4\pi^2)^{1/3}=6.5\times 10^8 cm$. The recent BATSE detection of a sudden torque reversal to spin-down is puzzling due to its very short time scale and the nearly equal spin-up/down rates. The steady spin-down torque suggests that there remains a dynamically stable (disk) structure after the sudden reversal (For details of observations, see Chakrabarty 1995). In Fig. 1(b), the observed torque reversal event is reproduced by $B_*=1.2\times 10^{11}G$, ${\dot M}=4\times 10^{15} g/s$, and $d{\dot M}/dt=-5\times 10^{13} g/s/yr$ which give $R_o/R_c=0.58$, $R_o^{\prime}/R_c^{\prime}=0.95$, and $R_o^{\prime}/R_o=0.56$. The derived accretion rate is slightly lower than the previously quoted values (Chakrabarty 1995), and accordingly the derived $B_*$ is also lower than the previous estimates (e.g. Pravado et al. 1979, Kii et al. 1986, Chakrabarty 1995). We note, however, that our estimated values ($B_*$ and ${\dot M}$) can always be rescaled by changing $\alpha$ and $\gamma$ (Kenyon et al. 1996). The values of $\alpha$ and $\gamma$ based on first principles are not available. The gradual decrease of ${\dot M}$ is consistent with the observed flux decrease (Mavromatakis 1994). The fit naturally achieves the spin-down torque which is slightly smaller than the spin-up torque. The fit requires for the transition to occur at ${\dot M}_{crit}=3.3\times 10^{15} g~s^{-1}$. The gradual decrease of the mass accretion rate on a time scale $\sim
20 yrs$ cannot be due to any viscous or thermal processes operating in the inner region ($t_{th}$, $t_{dis}$).
[*OAO 1657-415:*]{} OAO 1657-415 has an observed pulse period $P_*\approx 38s$. Recent observed spin-up/down torques are ${\dot P}_*/P_*^2\approx -7\times 10^{-12} s^{-2}$ and $\approx 2\times 10^{-12} s^{-2}$ respectively (Chakrabarty et al. 1993). For the torque reversal in Fig. 1(c), we get $B_*=10^{12}G$, ${\dot M}=2.0\times 10^{16} g/s$, and $d{\dot M}/dt=-5\times 10^{16} g/s/yr$. The characteristic ${\dot M}$ modulation time scale is $\sim 0.3yr$. The critical accretion rate ${\dot M}_{crit}=1.1\times 10^{16} g/s$ which is somewhat higher than the value required for 4U 1626-67.
[*GX 1+4:*]{} GX 1+4 has recently shown a sudden transition from spin-down to spin-up around the spin period $P_*\approx 122s$ (Chakrabarty 1995 and references therein). GX1+4 is peculiar in the sense that despite its very short spin time scale $\sim 40 yr$, the spin equilibrium has not been reached. It is likely that ${\dot M}$ fluctuates or oscillates on a time scale $\ll 40yr$ near spin equilibrium. The spin-down rate in the 1980’s, ${\dot P_*}/P_*^2\sim 3.7\times 10^{-12} s^{-2}$, is not far from the 1970’s spin-up rate, ${\dot P_*}/P_*^2\sim -6.0\times 10^{-12} s^{-2}$. The recent 1994 torque reversal from spin-down to spin-up lasted for $\sim 100d$. This system also showed very similar spin-down and spin-up rates. Although there is no significant spectral change in the hard X-ray emission spectra during the spin evolution, the flux appears to be increasing as spin-down torque increases (Chakrabarty 1995), which is in contradiction to the behavior expected in the Ghosh-Lamb type model (eqs. (2-2),(2-3)). The fit shown in Fig. 1(d) corresponds to $B_*=3.2\times 10^{12} G$, ${\dot M}=5.0\times 10^{15} g/s$, and $d{\dot M}/dt=1.0\times 10^{16} g/s/yr$. Once again, we note that these parameters could be rescaled by different choices of $\alpha$ and $\gamma$. It is interesting to observe that the ${\dot M}$ modulation time scale ${\dot M}/({\dot M}/dt)\sim 5yr$ is not far from the detected time scale in the coherent variation of pulse frequency residual (Cutler et al. 1986, Chakrabarty 1995). The critical mass accretion rate ${\dot M}_{crit}=6.5\times 10^{15} g/s$ lies between the two values derived above. The gradual decrease of the spin-up torque after reversal is not accurately fit in Fig. 1(d) with the linear ${\dot M}$ variation. This is not surprising given the reported unsteady behavior of the spin-down torque before the reversal (Chakrabarty 1995). The observed $\sim 300d$ coherent variation in pulse frequency residual is intriguing but such a time scale is not far from the estimated orbital time scale $\sim yr$ (Chakrabarty 1995).
Summary and Discussions
=======================
The proposed transition is most likely to occur at a critical accretion rate ${\dot M}_{crit}\sim 10^{15}-10^{16}g/s$. This suggests an interesting connection between the pulsar systems and other compact accretion systems such as cataclysmic variables and black hole soft X-ray transients. We speculate that the transitions seen in these systems may be due to a common physical mechanism, i.e. disk transition to optically thin hot flow. The model indicates that the sudden torque reversal could be a signature of a pulsar system near spin-equilibrium with ${\dot M}\sim {\dot M}_{crit}$.
There are some outstanding issues to look into. (i) The origin of the gradual ${\dot M}$ modulation on a time scale ranging from $\sim yr$ to a few decades remains unknown. The modulation by the orbital motion on a time scale $\sim yr$ is plausible in GX 1+4 (Chakrabarty 1995) but is unlikely in 4U 1626-67 (Rappaport et al. 1977, Joss et al. 1978, Shinoda et al. 1990, Chakrabarty 1995) and OAO 1657-415 (Chakrabarty et al. 1993). Several binary precession time scales (e.g. Thorne et al. 1986) and the mass flow oscillation time scale (induced by X-ray irradiation, Meyer & Meyer-Hofmeister 1990) could be relevant for long time scale ($>month$) modulations. In 4U 1626-67 the observed optical and X-ray pulsation frequencies are identical, which has been attributed to the reprocessing of X-rays by accretion disk (Ilovaisky et al. 1978, Chester 1979). The direct X-ray irradiation of the secondary star (Hameury et al. 1986) and the disk instability (e.g. Smak 1984) may also give rise to ${\dot M}$ modulations in neutron star systems. (ii) It is important to quantitatively understand ${\dot M}_{crit}$ and $R_{crit}$ (cf. Narayan & Yi 1995). (iii) Within our model, the observed smooth transitions back to spin-up (seen in OAO 1657-415 and GX 1+4) could result from the return from the advective to Keplerian flow. The characteristic time scale for such a back transition is likely to be the the viscous disk formation time scale. The observed UV-delay time scale, on the order of a day, in cataclysmic variables (Livio & Pringle 1992), may be similar to the postulated reverse transition time scale. (iv) It remains unexplained that in GX 1+4 the X-ray flux increased during the increase of the spin-down torque (Chakrabarty 1995). If X-rays come from the shocked polar accretion, as ${\dot M}$ decreases, the X-ray emission temperature and the apparent flux in a fixed X-ray band could decrease due to radiation drag (e.g. Yi & Vishniac 1994). A geometrically thick, but optically thin, inner disk is more apt to scatter X-ray emission, so that a partially obscured polar cap can actually become more conspicuous as $\dot M$ drops.
This research was supported in part by NSF Grant 95-28110 (JCW), by the SUAM Foundation (IY), and by NASA grant NAG5-2773 (ETV). We are happy to acknowledge related discussions on pulsars with Josh Grindlay and Ramesh Narayan.
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Figure Caption
Figure 1: (a) A typical smooth torque reversal event expected in the Ghosh-Lamb type model. This illustrative example assumes $B_*=10^{12} G$, $P_*=7.68s$ and ${\dot M}$ linearly decreasing from $7.2\times 10^{16} g/s$ to $5.0\times 10^{16} g/s$ over 10 yrs. The torque passes through $N=0$ (eq. (2-3)) and the continuous variation should show a wide range of torques before and after the torque reversal. Upper panel: time variation of torque. Lower panel: spin-up to spin-down transition. Examples of (b) 4U 1626-67, (c) OAO 1657-415, and (d) GX 1+4. The solid lines correspond to models described in the text and the dashed lines connect data points adopted from Chakrabarty et al. (1993) and Chakrabarty (1995). In OAO 1657-415 and 4U 1626-67, the observed spin periods are shown only schematically.
**ERRATUM**
In the Letter “Torque Reversal in Accretion-Powered X-ray Pulsars” by I. Yi, J. C. Wheeler, & E. T. Vishniac (ApJ, 481, L51 \[1997\]), there are errors in the values of the magnetic fields and the mass accretion rates. The correct values are as follows and Figure 1 is revised. [**4U 1626-67:**]{} $B_*=6\times 10^{11}G$, ${\dot M}=2.8\times 10^{16} g/s$, $d{\dot M}/dt=-4.5\times 10^{14} g/s/yr$, $R_o/R_c=0.760$, $R_o^{\prime}/R_c^{\prime}=0.987$, $R_o^{\prime}/R_o=0.444$, ${\dot M}_{crit}=2.2\times 10^{16} g/s$. [**OAO 1657-415:**]{} $B_*=5\times 10^{12}G$, ${\dot M}=1.2\times 10^{17} g/s$, $d{\dot M}/dt=-1.2\times 10^{17} g/s/yr$, ${\dot M}_{crit}=1.0\times 10^{17}g/s$. [**GX 1+4:**]{} $B_*=1.3\times 10^{13}G$, ${\dot M}=4\times 10^{16} g/s$, ${d{\dot M}/dt}=1.3\times 10^{17}g/s/yr$, ${\dot M}_{crit}=6\times 10^{16} g/s$.
0.5cm
Figure 1-Corrected
Torque reversal events in three X-ray pulsar systems. Short dashed lines correspond to the smooth torque transition (see text) and thick solid lines correspond to the sudden torque transition proposed in the present work. The points connected by long dashed lines are observed data points adopted from Chakrabarty et al. (1993) and Chakrabarty (1995). In OAO 1657-415 and 4U 1626-67, the observed spin periods are shown only schematically. Examples of (a) 4U 1626-67, (b) OAO 1657-415, and (c) GX 1+4. For the smooth transition events, the parameters ($B_*, {\dot M},
d{\dot M}/dt$) are ($10^{12}G$,$~4.2\times 10^{16}g/s$, $~-2\times 10^{15}g/s/yr$) for 4U 1626-67, ($10^{13}G$,$~2\times 10^{17}g/s$, $~-5\times10^{17}g/s/yr$) for OAO 1657-415, and ($4\times 10^{13}G$, $~10^{15}g/s$, $~1.8\times 10^{17}g/s/yr$) for GX 1+4.
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abstract: |
In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ is a finite blowup along smooth centers, here $X_0$ is a projective manifold of interest. Here we allow $X_0$ to be either one of the following manifolds: it has Picard number $1$, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of $X_0$ (may be upto $dim(X_0)-2$). Explicit constructions are given in Section \[SectionBlowupsAndNonVanishingConditions\], where we also show that the assumptions in the results in that section are necessary (see Example 6 in Section \[SectionBlowupsAndNonVanishingConditions\]).
As a consequence, we obtain new examples of manifolds $X$, whose any automorphism is either of zero topological entropy or is cohomologically hyperbolic.
address: 'Department of Mathematics, Syracuse University, Syracuse NY 13244'
author:
- Tuyen Trung Truong
title: On automorphisms of blowups of projective manifolds
---
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Introduction {#Introduction}
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The structure of the automorphism group of a compact Kähler manifold has been very extensively studied. Among many other things, the following question is very interesting: What compact Kähler manifolds have automorphisms of positive entropies? By Gromov-Yomdin’s theorem, this question reduces to the one about whether there is an automorphism with first dynamical degree $>1$. We recall that if $X$ is a compact Kähler manifold of dimension $k$ and $f:X\rightarrow X$ is a surjective holomorphic map, then the $p$-th dynamical degree $\lambda _p(f)$ of $f$ (here $0\leq p\leq k$) is the spectral radius of the pullback map $f^*:H^{p,p}(X)\rightarrow H^{p,p}(X)$. The theorem of Gromov and Yomdin states that the topological entropy is related to the dynamical degrees as follows $$\begin{aligned}
h_{top}(f)=\max _{p=1,\ldots ,k}\log \lambda _p(f). \end{aligned}$$ Since the dynamical degrees are log-concave (i.e. $\lambda_{p-1}(f)\lambda _{p+1}(f)\leq \lambda _p(f)^2$), we deduce that $h_{top}(f)>0$ if and only if $\lambda _1(f)>1$.
In dimension $k=2$, there are many constructions of automorphisms of positive entropies, starting as early as with the work of Coble (see Dolgachev-Ortland [@dolgachev-ortland]) who used Coxeter groups, see works of Cantat [@cantat], Bedford-Kim [@bedford-kim2][@bedford-kim1][@bedford-kim], McMullen [@mcmullen3][@mcmullen2][@mcmullen1][@mcmullen], Oguiso [@oguiso3][@oguiso2], Cantat-Dolgachev [@cantat-dolgachev], Zhang [@zhang], Diller [@diller], Déserti-Grivaux [@deserti-grivaux], Reschke [@reschke], Uehara [@uehara]...
In dimension $k\geq 3$, there are several general results on the structure of automorphism groups, see Bochner-Montgomery [@bochner-montgomery], Fujiki [@fujiki], Lieberman [@lieberman], Dinh-Sibony [@dinh-sibony], Oguiso [@oguiso0], Keum-Oguiso-Zhang [@keum-oguiso-zhang], Zhang [@zhang0],...However, the examples of compact Kähler manifolds having automorphisms of positive entropies are very rare (see Oguiso [@oguiso1][@oguiso], Oguiso-Perroni [@oguiso-perroni],...). On the other hand, for a class of maps very close to automorphisms, that is the class of pseudo-automorphisms or automorphisms in codimension $1$, there are systematic constructions of many examples of first dynamical degrees greater than $1$ by Bedford-Kim [@bedford-kim3], Perroni-Zhang [@perroni-zhang], Blanc [@blanc],... This leads to the natural question: How common are compact Kähler manifolds of dimension $\geq 3$ having automorphisms of positive entropies?
In our previous paper [@truong] on automorphisms of blowups of $\mathbb{P}^3$, we showed using a heuristic argument that for a “generic” compact Kähler manifold $X$, if $f\in Aut(X)$ then $h_{top}(f)=0$. Combined with results of Bayraktar and Cantat (see below), the same argument shows that for a “generic” compact Kähler manifold, the automorphism group $Aut(X)$ has only finitely many connected components. The idea is as follows. For an automorphism $f$, there is a non-zero nef cohomological class $\eta \in H^{1,1}(X)$ (we recall that nef classes are in the closure of Kähler classes) such that $f^*(\eta )=\lambda _1(f)\eta$. We observe that for a “generic” compact Kähler manifold $X$, the non-vanishing condition $B(0,0)$ below is satisfied, hence $\lambda _1(f)=1$ and $h_{top}(f)=0$. Here “generic” is used in the following sense:
[**Expectation of randomness for the intersection ring**]{}. We expect that when we choose randomly a compact Kähler manifold $X$ of dimension $k$ with a fixed value of $dim(H^{1,1}(X))$, because $dim (H^{1,1}(X))=dim (H^{k-1,k-1}(X))$ by the Poicare duality, the map $$(\zeta _1,\ldots ,\zeta _{k-1})\in H^{1,1}(X)^{k-1}\mapsto \zeta _1.\zeta _2.\ldots .\zeta _{k-1}\in H^{k-1,k-1}(X)$$ behaves randomly. In particular, the map $\zeta \in H^{1,1}(X)\mapsto \zeta ^{k-1}\in H^{k-1,k-1}(X)$ should be non-degenerate.
(This expectation of randomness is related to polar hypersurfaces, see the book Dolgachev [@dolgachev] for more detail on polar hypersurfaces.)
Note that for hyper-Kähler manifolds of dimension $k=2l$, the expectation of randomness with a smaller exponent was proved previously by Verbitsky [@verbitsky]: The map $(\zeta _1,\ldots ,\zeta _l)\mapsto \zeta _1.\ldots .\zeta _l$ is non-degenerate. In particular, if $\zeta$ is non-zero then $\zeta ^l$ is non-zero.
There are of course many manifolds for which this expectation of randomness is not satisfied. For example, if we start from any manifold $X_0$ and let $X\rightarrow X_0$ be a finite blowup of $X_0$ along smooth centers, the resulting manifold $X$ may not satisfy this expectation of randomness. Thus we may hope that $X$ contains some automorphisms of positive entropies, even if $X_0$ does not. A common approach in finding automorphisms of positive entropies, used very efficiently in the case $k=2$, is as follows: Start with a manifold $X_0$ of interest, find a birational meromorphic map $f:X_0\rightarrow X_0$, and then find a finite blowup $X\rightarrow X_0$ such that the lifting of $f$ to $X$ is an automorphism of positive entropy. Hence we can restate the question at the beginning of this section in the following form: Given a manifold of interest $X_0$, is there a finite blowup $X\rightarrow X_0$ carrying an automorphism of positive entropy? In the same paper [@truong], using the non-vanishing condition $A(0,0)$ below, we constructed systematically many finite blowup $X\rightarrow \mathbb{P}^3$ whose any automorphism has zero topological entropy. By the results of Bayraktar and Cantat (see below), it turns out that for many of these examples, the automorphism group has only finitely many connected components. This suggests that the answer to the above question is No for $X_0=\mathbb{P}^3$.
In recent works, Bayraktar-Cantat [@bayraktar][@bayraktar-cantat] used a generalized non-vanishing condition $B(r,0)$ (here $k>2r+2$) of the non-vanishing condition $B(0,0)$ to show that if $X_0$ is a projective manifold of dimension $k\geq 3$ and has Picard number $1$, and $X\rightarrow X_0$ is a finite blowup along smooth centers of dimension $\leq r$ then the automorphism group $Aut(X)$ has only finitely many connected components, in particular if $f\in Aut(X)$ then $h_{top}(f)=0$. In fact (see Theorem \[TheoremEk\]), for the projective examples given in [@bayraktar][@bayraktar-cantat], a stronger non-vanishing condition, which is very similar to the expectation of randomness for the intersection ring, is satisfied: If $\zeta \in H^{1,1}(X)$ (or $\zeta \in NS_{\mathbb{R}}(X)$ in the case of Picard number $1$) is non-zero ($\zeta$ needs not to be nef) then $\zeta ^{k-r-1}\not= 0$. Previously, for a hyper-Kähler manifold $X$ of dimension $k=2l$, Oguiso [@oguiso1] used the non-vanishing condition B(l-1,0) of Verbitsky [@verbitsky] (note that here the condition $k>2r+2$ is not satisfied) to show that if $f\in Aut(X)$ then either $h_{top}(f)=0$ or it is cohomologically hyperbolic, more precisely the middle dynamical degree $\lambda _{l}(f)$ is larger than other dynamical degrees.
In the current paper we combine the ideas in [@bayraktar][@bayraktar-cantat], and [@truong] to explore more general situations. More precise, while the results in [@bayraktar][@bayraktar-cantat] work in any dimension and yield better conclusion on the structure of the automorphism group, their arguments and conditions apply only for the case $X_0$ has Picard number $1$ and for blowups along centers of dimension $< (k-2)/2$ (e.g. when $k=3$ their arguments apply only for point blowups, while in the paper [@truong] we allow blowing up along curves satisfying certain conditions). Using some generalizations of the non-vanishing conditions $B(r,0)$ and $A(0,0)$, in this paper we can work with other manifolds $X_0$ (e.g. Fano manifolds, projective hyper-Kähler manifolds) and allow blowing ups along smooth centers of arbitrary dimensions under certain conditions.
We now introduce two non-vanishing conditions, including as special cases those referred to in the above. We recall that by Hodge decomposition, on a compact Kähler manifold $X$ we have $H^2(X,\mathbb{C})=H^{2,0}(X)\oplus H^{1,1}(X)\oplus H^{0,2}(X)$, and we define $H^{1,1}(X,\mathbb{Z})=H^2(X,\mathbb{Z})\cap H^{1,1}(X)$, $H^{1,1}(X,\mathbb{Q})=H^2(X,\mathbb{Q})\cap H^{1,1}(X)$ and $H^{1,1}(X,\mathbb{R})=H^2(X,\mathbb{R})\cap H^{1,1}(X)$. The cone of nef classes $H_{nef}^{1,1}(X)\subset H^{1,1}(X)$ is the closure of Kähler classes.
[**Non-vanishing condition $A(r,q)$**]{}: Let $r$, $q$ and $k$ be integers with $-1\leq r\leq k-1$ and $0\leq q\leq k-r-1$. We say that a compact Kähler manifold $X$ of dimension $k$ satisfies the non-vanishing condition $A(r,q)$ if for any nef class $\zeta$ on $X$, if $\zeta ^{k-r-1-q}.K_X^{q}=0$ then $\zeta$ is proportional to a rational cohomology class, i.e. a class in $H^{1,1}(X,\mathbb{Q})$. Here $K_X$ is the canonical divisor of $X$.
[**Non-vanishing condition $B(r,q)$**]{}: Let $r$, $q$ and $k$ be integers with $-1\leq r\leq k-1$ and $0\leq q\leq k-r-1$. We say that a compact Kähler manifold $X$ of dimension $k$ satisfies the non-vanishing condition $B(r,q)$ if for any non-zero nef class $\zeta$ on $X$ then $\zeta ^{k-r-1-q}.K_X^q\not= 0$.
These non-vanishing conditions also have algebraic analogs, which we will mainly use in the rest of this paper. We let $Pic(X)$ be the Picard group of $X$, and let $NS(X)=Pic(X)/$(algebraic equivalence), which can be regarded as a subset of $H^2(X,\mathbb{Z})$ via the Chern map $L$ a divisor $\mapsto c_1(L)\in H^{2}(X,\mathbb{Z})$. By Lefschetz $(1,1)$ theorem (see Chapter 0 in the book Griffiths-Harris [@griffiths-harris]), we have $NS(X)=H^2(X,\mathbb{Z})\cap H^{1,1}(X)$. We define $NS_{\mathbb{Q}}(X)=NS(X)\otimes _{\mathbb{Z}}\mathbb{Q}\subset H^{2}(X,\mathbb{Q})$ and $NS_{\mathbb{R}}(X)=NS(X)\otimes _{\mathbb{Z}}\mathbb{R}\subset H^2(X,\mathbb{Z})$. The nef cone $Nef(X)=H_{nef}^{1,1}(X)\cap NS_{\mathbb{R}}(X)\subset NS_{\mathbb{R}}(X)$ is the closure of ample divisors (with real coefficients). In fact, let $u\in Pic_{\mathbb{R}}(X)$ represent a class in $H_{nef}^{1,1}(X)\cap NS_{\mathbb{R}}(X)$. Theorem 4.5 in Demailly-Paun [@demailly-paun] shows that $u$ is then nef in the algebraic geometry sense. Then it follows by Kleiman’s result that $u$ is a limit of ample divisors (with real coefficients), see e.g. Corollary 1.4.9 in Lazarsfeld [@lazarsfeld].
[**Non-vanishing condition $NA(r,q)$**]{}: Let $r$, $q$ and $k$ be integers with $-1\leq r\leq k-1$ and $0\leq q\leq k-r-1$. We say that a complex projective manifold $X$ of dimension $k$ satisfies the non-vanishing condition $NA(r,q)$ if for any nef class $\zeta\in NS_{\mathbb{R}}(X)$, if $\zeta ^{k-r-1-q}.K_X^{q}=0$ then $\zeta$ is proportional to a rational class, i.e. a class in $NS_{\mathbb{Q}}(X)$. Here $K_X$ is the canonical divisor of $X$.
[**Non-vanishing condition $NB(r,q)$**]{}: Let $r$, $q$ and $k$ be integers with $-1\leq r\leq k-1$ and $0\leq q\leq k-r-1$. We say that a complex projective manifold $X$ of dimension $k$ satisfies the non-vanishing condition $NB(r,q)$ if for any non-zero nef class $\zeta\in NS_{\mathbb{R}}(X)$ then $\zeta ^{k-r-1-q}.K_X^q\not= 0$.
The use of these non-vanishing conditions to the question on the existence of automorphisms of positive entropies are given in the following two results.
Let $r$ be an integer such that $k>2r+2$, and let $q$ be such that $k-r-1-q>r+1$. If $X$ satisfies the non-vanishing condition $A(r,q)$ then for any $f\in Aut(X)$ we have $h_{top}(f)=0$. The same result holds if we replace $A(r,q)$ by $NA(r,q)$. \[TheoremArqCondition\]
Let $r$ be an integer such that $k>2r+2$, and let $q$ be such that $k-r-1-q>r+1$. If $X$ satisfies the non-vanishing condition $B(r,q)$ then the automorphism group $Aut(X)$ has only finitely many connected components. The same result holds if we replace $B(r,q)$ by $NB(r,q)$. \[TheoremBrqCondition\]
We list here some applications of Theorems \[TheoremArqCondition\] and \[TheoremBrqCondition\]. The first result concerns blowups of manifolds of Picard number $1$.
Let $X_0$ be a projective manifold of dimension $k$ and has Picard number $1$ then $X_0$ satisfies the non-vanishing condition $NB(r,0)$ for any $r\geq 0$.
1\) Let $V_1,V_2,\ldots ,V_m\subset X_0$ are pairwise disjoint submanifolds, where each $V_j$ is either of dimension $\leq 1$ or is a complete intersection of smooth hypersurfaces of $X_0$. Let $X_1\rightarrow X_0$ be the blowup along $V_1,V_2,\ldots ,V_m$. Then for a generic choice of $V_1,V_2,\ldots ,V_m$ the resulting space $X_1$ satisfies the non-vanishing condition $NB(1,0)$. Here generic is used in the sense of algebraic geometry, i.e. the claim is true out of a proper subvariety of the parameter space.
2\) Let $X_1$ be a blowup satisfying the generic condition in 1). If $\pi :X=X_{n}\rightarrow X_{n-1}\rightarrow \ldots\rightarrow X_1$ a finite blowup along smooth centers of dimensions $<(k-2)/2$, then any automorphism of $X$ has zero topological entropy. If moreover $k\geq 5$ or $X_1=X_0$ then the automorphism group $Aut(X)$ has only finitely many connected components.
3\) In 2) we can also allow the centers of blowups to have larger dimensions, by using Theorem \[TheoremEk\] and Corollary \[CorollaryExtremeCases\], see Examples 1-6 in Section \[SectionBlowupsAndNonVanishingConditions\]. \[TheoremPicardNumber1\]
[**Remarks.**]{} Part 1) is compatible with the expectation of randomness for intersection ring, here the exponent $k-2$ is only $1$ less than the exponent $k-1$ in the expectation of randomness. In part 1), when $3\leq dim(X_0)\leq 4$ we can also show that $X_1$ satisfies the non-vanishing condition $NA(0,0)$ (see Theorem 2 in [@truong] for the case $dim(X_0)=3$ and see Lemma \[LemmaP4\] for the case $dim(X_0)=4$.) The special case of part 2) when $X_1=X_0$ was proved in [@bayraktar][@bayraktar-cantat].
The next result concerns blowups of Fano manifolds. Recall that a manifold $X$ is Fano if its anti-canonical divisor is ample.
1\) Assume that $X_0$ is a projective manifold of dimension $k$ satisfying the non-vanishing condition $NB(l,0)$ and is Fano, i.e. $-K_X$ is ample. If $0\leq k-l-3$ and $0\leq 2l+3-k$, and $X\rightarrow X_0$ is a finite composition of blowups along smooth centers of dimensions $\leq k-3-l$, then the automorphism group $Aut(X)$ has only finitely many connected components.
Similar results hold if we replace $NB(l,0)$ by $A(l,0)$, $NA(l,0)$ and $B(l,0)$.
2\) Let $X_0=\mathbb{P}^{k_1}\times \mathbb{P}^{k_2}\times \ldots \times\mathbb{P}^{k_m}$ be a multi-projective space with $m\geq 2$, and we arrange that $1\leq k_1\leq k_2\leq \ldots k_m$. Denote by $k=k_1+\ldots +k_m$ the dimension of $X_0$.
i\) If $2\leq l\leq k_1+k_2$ and $2l\leq k+1$, and $X\rightarrow X_0$ is a finite blowup along smooth centers of dimensions $\leq l-2$ then any automorphism $f\in Aut(X)$ has zero topological entropy.
ii\) If $2\leq k_1$, and $X\rightarrow X_0$ is a finite blowup along smooth centers of dimensions $\leq k_1-2$ then the automorphism group $Aut(X)$ has only finitely many connected components.
3\) In 1) and 2) we can also allow the centers of blowups to have larger dimensions, by using Theorem \[TheoremEk\] and Corollary \[CorollaryExtremeCases\], see Examples 1-6 in Section \[SectionBlowupsAndNonVanishingConditions\]. \[TheoremAmpleCanonicalDivisor\]
For example, if $X_0=\mathbb{P}^{k_1}\times \mathbb{P}^{k_2}\times \ldots \times\mathbb{P}^{k_m}$ is a multi-projective space, where $m\geq 2$ and $dim(X_0)\geq 3$, and $X\rightarrow X_0$ is a finite composition of point-blowups then the automorphism group $Aut(X)$ has only finitely many connected components. Hence for any $f\in Aut(X)$ we have $h_{top}(f)=0$. In particular, the pseudo-automorphisms constructed in [@perroni-zhang] can never be automorphisms.
Besides helping to check whether a given manifold can only have automorphisms of zero topological entropy, the above non-vanishing conditions also help in checking whether a given manifold must have interesting automorphisms in case it has automorphisms of positive entropies, or to give constraints to dynamical degrees of holomorphic maps of a given manifold in general. We illustrate this with a result concerning automorphisms which are cohomologically hyperbolic. We recall that a surjective holomorphic map $f:X\rightarrow X$ is cohomologically hyperbolic if it has one dynamical degree larger than other dynamical degrees. In particular, such an automorphism has positive topological entropy.
Let $X_0$ be a projective hyper-Kähler manifold of dimension $k=2l$, and let $X\rightarrow X_0$ be a finite composition of blowups along smooth centers of dimension $\leq l-1$. Then any automorphism $f\in Aut(X)$ is either of zero topological entropy or is cohomologically hyperbolic with the dominant dynamical degree $\lambda _l(f)$.
Moreover, for any automorphism $f\in Aut(X)$ and $0\leq p\leq l$ we have $\lambda _{2l-p}(f)=\lambda _p(f)=\lambda _1(f)^p$.
We can also allow the centers of blowups to have larger dimensions by using Theorem \[TheoremEk\] and Corollary \[CorollaryExtremeCases\], see Examples 1-6 in Section \[SectionBlowupsAndNonVanishingConditions\].
In the above we can also start from other manifolds of even dimensions, such as $X_0$ has Picard number $1$. \[TheoremHyperKahler\]
Note that the case where $X=X_0=$ a compact hyper-Kähler manifold was proved in [@oguiso1], where some specific examples were also given. Cohomologically hyperbolic automorphisms have been shown to have good dynamical properties, see the papers Cantat [@cantat], Dinh-Sibony [@dinh-sibony3], [@dinh-sibony4], [@dinh-sibony5] and Dinh-deThelin [@dinh-deThelin] for more details.
This paper is arranged as follows. In Section \[SectionBlowupsAndNonVanishingConditions\], we explore the question when a blowup preserves the non-vanishing conditions $A(r,q)$, $B(r,q)$, $NA(r,q)$ and $NB(r,q)$, and give many explicit examples at the end of the section. The proofs of Theorems \[TheoremArqCondition\], \[TheoremBrqCondition\], \[TheoremPicardNumber1\], \[TheoremAmpleCanonicalDivisor\] and \[TheoremHyperKahler\] are given in Section \[SectionProofsOfTheorems\].
[**Acknowledgments.**]{} We thank Keiji Oguiso for suggesting extending the results in dimension $3$ to higher dimensions and for helpful correspondences. We thank Igor Dolgachev for explaining several aspects of automorphism groups, and thank Mattias Jonsson and Turgay Bayraktar for helpful comments and for pointing out the reference [@paun] which helped to improve the paper.
Blowups and the non-vanishing conditions $A(r,q)$, $NA(r,q)$, $B(r,q)$ and $NB(r,q)$ {#SectionBlowupsAndNonVanishingConditions}
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In this section we explore the question: if $Y$ is a projective manifold satisfying the non-vanishing condition $A(r,q)$ (correspondingly the non-vanishing conditions $NA(r,q)$, $B(r,q)$ and $NB(r,q)$), and $\pi :X\rightarrow Y$ is a blowup along a smooth submanifold $V\subset Y$, does $X$ also satisfy the non-vanishing condition $A(r,q)$ (respectively the non-vanishing conditions $NA(r,q)$, $B(r,q)$ and $NB(r,q)$)? We construct many examples of blowups when the answer to the question is Yes. The main idea is that if the normal vector bundle of $V$ in $Y$ is “[**negative enough**]{}” and $V$ is “movable” (for the precise conditions see Theorem \[TheoremEk\], Lemmas \[LemmaP4\] and \[LemmaManifoldDimension4\], and Corollary \[CorollaryExtremeCases\]) then $X$ will also satisfy the non-vanishing conditions. Example 6 at the end of the section show that these assumptions of “negative normal vector bundle” and “movability of the center of blowup” $V$ can not be removed. To motivate the constructions, in the first two results we consider blowups of projective manifolds of dimension $4$, and after that will consider blowups of manifolds of arbitrary dimensions. At the end of the section we will give some explicit examples.
For simplicity, for the results in this section we present the proofs only for the non-vanishing conditions $A(r,q)$ and $B(r,q)$. The proofs for algebraic non-vanishing conditions $NA(r,q)$ and $NB(r,q)$ are similar, using the following result: If $\pi :X\rightarrow Y$ is a birational morphism, then $\pi _*(NS(X))\subset NS(Y)$ and $\pi ^*(NS(Y))\subset NS(X)$ (see Example 19.1.6 in [@fulton]). Alternatively, we can see this by using $NS(X)=H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$ and $NS(Y)=H^{2}(Y,\mathbb{Z})\cap H^{1,1}(Y)$.
The first result of this section is for blowups of $\mathbb{P}^4$. See Theorem \[TheoremPicardNumber1\] 1) for an extension to higher dimensions.
Let $V_1,\ldots ,V_n\subset \mathbb{P}^4$ be irreducible pairwise disjoint smooth compact complex submanifolds of dimension $\leq 2$. Let $\pi :X\rightarrow \mathbb{P}^4$ be the blowup of $\mathbb{P}^4$ at $V_1,\ldots ,V_n$. If for any $j$ one of the following conditions are satisfied, then $X$ satisfies the non-vanishing condition $A(0,0)$
i\) $dim(V_j)\leq 1$.
ii\) $dim(V_j)=2$, and $V_j$ is a complete intersection of two smooth hypersurfaces $D_1$ and $D_2$ of degrees $d_1$ and $d_2$. For example, we may choose $V_j$ to be the intersection between a smooth hypersurface and a generic hyperplane. By Bertini’s theorem, such a $V_j$ is smooth.
(Note that since the manifolds $V_1,\ldots ,V_n$ are pairwise disjoint, there is at most one of them of dimension $2$.) \[LemmaP4\]
Let $H\subset \mathbb{P}^4$ be a hyperplane and let $E_1,\ldots ,E_n$ be the exceptional divisors.
Let $\zeta$ be a nef class on $X$ with $\zeta ^3=0$, we will show that $\zeta$ is proportional to a rational cohomology class. We can write $$\begin{aligned}
\zeta =aH+\sum _{j=1}^nb_jE_j,\end{aligned}$$ for real numbers $a$ and $b_i$. Since $\zeta$ is nef, either $a>0$ or $\zeta =0$. If $\zeta =0$ then we are done. Hence we now assume that $a\not= 0$, and upon dividing by $a$ can write $$\begin{aligned}
\zeta =H+\sum _{j=1}^nb_jE_j\end{aligned}$$ where $b_j\leq 0$. The conclusion of the lemma is equivalent to that $b_1,b_2,\ldots ,b_n$ are rational numbers.
By assumption we have $\zeta ^3.E_j=0$ for any $j=1,\ldots ,n$. Since $V_1,\ldots ,V_n$ are disjoint, it follows that $E_i.E_j=0$ for $i\not= j$. Hence $\zeta ^3.E_j=(H+b_jE_j)^3.E_j$ for any $j$.
Let $\mathcal{E}_j=N_{V_j/\mathbb{P}^4}$ be the normal vector bundle of $V_j$ in $\mathbb{P}^4$. Then $E_j=\mathbb{P}(\mathcal{E}_j)$. Let $c_1(\mathcal{E}_j)$ and $c_2(\mathcal{E}_j)$ be the first and second Chern classes of $\mathcal{E}_j$ (the higher Chern classes vanish since $\mathcal{E}_j$ is the vector bundle over a manifold of dimension $\leq 2$), and let $\pi _j:E_j\rightarrow V_j$ be the projections. Let $h_j=\pi ^*(H)|_{E_j}=\pi _j^*(H|_{V_j})$, and $e_j=E_j|_{E_j}$. Since $h_j^3=\pi _j^*(H^3|_{E_j})=0$, the equality $(H+b_jE_j)^3.E_j=0$ becomes $$b_j^3e_j^3+3b_j^2h_j.e_j^2+3b_jh_j^2.e_j=0,
\label{Equation1}$$ in $H^*(E_j)$.
We consider three cases:
Case 1: $V_j=$ a point. In this case $\mathcal{E}_j$ has rank $4$, $c_1(\mathcal{E}_j)=0$ and $c_2(\mathcal{E}_j)$, and we know that (see Remark 3.2.4 in Fulton’s book [@fulton]) $$\begin{aligned}
e_j^4=0.\end{aligned}$$ Moreover, we know that $H^*(E_j)$ is generated by $e_j$ as an algebra over $H^*(V_j)$, with the defining relation $e_j^4=0$. Therefore from Equation (\[Equation1\]) we see that $b_j=0$ hence is a rational number.
Case 2: $V_j=$ a curve. In this case $\mathcal{E}_j$ has rank $3$, and $c_2(\mathcal{E}_j)$, and we know that (see Remark 3.2.4 in Fulton’s book [@fulton]) $$\begin{aligned}
e_j^3-\pi _j^*c_1(\mathcal{E}_j)e_j^2=0.\end{aligned}$$ Moreover, we know that $H^*(E_j)$ is generated by $e_j$ as an algebra over $H^*(V_j)$, with the defining relation $e_j^3-\pi _j^*c_1(\mathcal{E}_j)e_j^2=0$. If $b_j=0$ then it is a rational number and we are done. If $b_j\not= 0$, dividing Equation (\[Equation1\]) by $b_j^3$ and compare to the defining equation of $H^*(E_j)$, we find $$\begin{aligned}
\frac{3}{b_j}h_j=-\pi _j^*c_1(\mathcal{E}_j). \end{aligned}$$ Since $h_j=\pi _j^*(H|_{V_j})=\pi _j^*(deg(V_j))$ and $\pi _j^*:H^*(V_j)\rightarrow H^*(E_j)$ is injective, we have $$\begin{aligned}
\frac{3}{b_j}deg(V_j)=-c_1(\mathcal{E}_j).\end{aligned}$$ Because $deg(V_j)$ is a positive integer, it follows that the integer $c_1(\mathcal{E}_j)$ is non-zero, and hence $$\begin{aligned}
b_j=-\frac{3deg(V_j)}{c_1(\mathcal{E}_j)}\in \mathbb{Q}\end{aligned}$$ as wanted.
Case 3: $V_j=$ a surface, and is a complete intersection of two hypersurfaces of degrees $d_1$ and $d_2$. In this case $\mathcal{E}_j$ has rank $2$, and we know that (see Remark 3.2.4 in Fulton’s book [@fulton]) $$\begin{aligned}
e_j^2-\pi _j^*c_1(\mathcal{E}_j)e_j+\pi _j^*c_2(\mathcal{E}_j)=0.\end{aligned}$$ Moreover, we know that $H^*(E_j)$ is generated by $e_j$ as an algebra over $H^*(V_j)$, with the defining relation $e_j^2-\pi _j^*c_1(\mathcal{E}_j)e_j+\pi _j^*c_2(\mathcal{E}_j)=0$. If $b_j=0$ then it is rational and we are done. Hence we can assume that $b_j\not= 0$. Dividing Equation \[Equation1\] by $b_j^3$ and defining $a_j=1/b_j$, we find $e_j^3+3a_jh_j.e_j^2+3a_j^2h_j^2.e_j=0$. Because $$\begin{aligned}
&&e_j^3+3a_jh_j.e_j^2+3a_j^2h_j^2.e_j\\
&=&[e_j^2-\pi _j^*c_1(\mathcal{E}_j)e_j+\pi _j^*c_2(\mathcal{E}_j)]\times [e_j+3a_jh_j+\pi _j^*c_1(\mathcal{E}_j)e_j]\\
&&+[3a_j^2h_j^2+3a_jh_j\pi _j^*(c_1(\mathcal{E}_j))+\pi _j^*(c_1(\mathcal{E}_j)^2)-\pi _j^*(c_2(\mathcal{E}_j))]\times e_j,\end{aligned}$$ we deduce $$\begin{aligned}
3a_j^2h_j^2+3a_jh_j\pi _j^*(c_1(\mathcal{E}_j))+\pi _j^*(c_1(\mathcal{E}_j)^2)-\pi _j^*(c_2(\mathcal{E}_j))=0.\end{aligned}$$ Because $\pi _j^*:H^j(V_j)\rightarrow H^*(E_j)$ is injective, we obtain $$3a_j^2(H^2|_{V_j})+3a_j(H|_{V_j})c_1(\mathcal{E}_j)+c_1(\mathcal{E}_j)^2-c_2(\mathcal{E}_j)=0.
\label{Equation2}$$ We now use the explicit values of the Chern classes. Example 3.2.12 in [@fulton] gives that $c(\mathcal{E}_j)=(1+d_1H|_{V_j})(1+d_2H|_{V_j})$, hence $c_1(\mathcal{E}_j)=(d_1+d_2)H|_{V_j}$ and $c_2(\mathcal{E}_j)=d_1d_2(H^2|_{V_j})$. Therefore $c_1(\mathcal{E}_j)^2=(d_1+d_2)^2(H^2|_{V_j})$. From this, upon dividing Equation (\[Equation2\]) by the integer number $(H^2|_{V_j})=deg(V_j)>0$, we obtain the equation $$\begin{aligned}
3a_j^2+3(d_1+d_2)a_j+(d_1+d_2)^2-d_1d_2=0.\end{aligned}$$ Using the quadratic formula, we see that the above has real solutions if and only if $d_1=d_2=d$ a positive integer, and in that case it has a repeated root $a_j=-d$ and hence $b_j=-1/d$ is a rational number, as wanted.
The next result considers blowups of general projective manifolds of dimension $4$.
Let $Y$ be a projective manifold of dimension $4$, and $V\subset Y$ be an irreducible compact complex submanifold. Let $\pi :X\rightarrow Y$ be the blowup of $Y$ at $V$, and let $E$ be the exceptional divisor. Assume that $Y$ satisfies the non-vanishing condition $A(0,0)$ (correspondingly the non-vanishing conditions $NA(0,0)$, $B(0,0)$ and $NB(0,0)$). If either one of the following three conditions is satisfied, then $X$ also satisfies the non-vanishing condition $A(0,0)$ (respectively the non-vanishing conditions $NA(0,0)$, $B(0,0)$ and $NB(0,0)$).
\(i) $V$ is a point.
\(ii) $V$ is a curve. In this case let $\mathcal{E}=N_{V/Y}$ be the normal vector bundle of $V$ in $Y$ and $c_1(\mathcal{E})$ be the first Chern class of $\mathcal{E}$. We then assume that $c_1(\mathcal{E})<0$ and $V$ is not the only effective cycle (with real coefficient) in its cohomology class.
\(iii) $V$ is a surface. In this case we assume that $\pi _*(E^3)$ can be represented by an effective curve (with real coefficients) whose intersection with $V$ has dimension $\leq 0$, $V$ can be represented by an effective cycle (with real coefficients) intersecting properly with $V$, and $E^4<0$.
Note that the conditions i), ii) and iii) can be stated in a uniform manner, see the proof of this lemma and see also Theorem \[TheoremEk\]. \[LemmaManifoldDimension4\]
(See also the proof of Theorem 2 in [@truong] for blowups of $\mathbb{P}^3$).
\(iii) Assume that the condition (iii) is satisfied and $Y$ satisfies the non-vanishing condition $A(0,0)$ (the case when $Y$ satisfies the non-vanishing condition $B(0,0)$ is similar; the algebraic analogs $NA(0,0)$ and $NB(0,0)$ are also similar by observing that Neron-Severi groups are preserved by pushing forward by $\pi$). Let $\zeta $ be a nef class on $X$ such that $\zeta ^3=0$. We then show that $\zeta$ is proportional to a rational cohomology class. We can write $\zeta =\pi ^*(\xi )-aE$ for some $a\geq 0$ and $\xi =\pi _*(\zeta )\in H^{1,1}(Y)$. Then $$0=\zeta ^3=\pi ^*(\xi ^3)-3a\pi ^*(\xi ^2).E+3a^2\pi ^*(\xi ).E^2-a^3E^3.
\label{Equation3}$$ Intersecting Equation (\[Equation3\]) with $E$, and using $\pi _*(E)=0$ and $\pi _*(E.E)=-V$ (see Section 4.3 in [@fulton]) we find that $$a^3E^4=-3a\pi ^*(\xi ^2).E^2+3a^2\pi ^*(\xi ).E^3.
\label{Equation4}$$ The assumptions imply that $-\pi ^*(\xi ^2).E^2,\pi ^*(\xi ).E^3\geq 0$ are psef. In fact, approximating $\zeta$ by Kähler classes, we may assume without loss of generality that $\zeta$ is represented by a positive closed smooth $(1,1)$ form. Then $\xi =\pi _*(\zeta )$ can be represented by a positive closed $(1,1)$ form smooth out of $V$. Since $codim(V)=2$, it follows (see e.g. Section 4 Chapter 3 in the book Demailly [@demailly]) that $\xi .\xi $ is represented by a positive closed $(2,2)$ form smooth out of $V$. Then $-\pi ^*(\xi .\xi ).E.E=\xi .\xi .V$ must be non-negative, since by assumptions we can find an effective cycle $V'$ of dimension $2$ in the cohomology class of $V$ and intersect properly with $V$. Then by the results in [@demailly] again, we have $$\begin{aligned}
\xi .\xi .V=\xi .\xi .V'\end{aligned}$$ can be represented by a non-negative measure, therefore is non-negative as wanted. Similarly we have $\pi ^*(\xi ).E^3\geq 0$.
From the above we must have $a=0$. Otherwise, $a>0$ and we obtain a contradiction $$\begin{aligned}
0>a^3E^4=-3a\pi ^*(\xi ^2).E^2+3a^2\pi ^*(\xi ).E^3\geq 0.\end{aligned}$$
Thus $\zeta =\pi ^*(\xi )$, and by a result of Paun (see [@paun]) $\xi$ is itself nef. Then since $\zeta ^3=0$, it follows that $\xi ^3=0$. Because $Y$ satisfies non-vanishing condition A and $\xi$ is nef on $Y$ as shown above, if follows that $\xi $ is proportional to a rational cohomology class and so is $\zeta =\pi ^*(\xi )$.
ii\) Assume that the assumption ii) is satisfied. We first observe that the condition $c_1(\mathcal{E})<0$ is equivalent to $E^4<0$. In fact, if $e=E|_E$ and $\pi _E:E\rightarrow V$ is the induced map, we know that $$\begin{aligned}
E^4=e^3=\pi _E^*(c_1(\mathcal{E}))e^2.\end{aligned}$$ Now $\pi _E^*(c_1(\mathcal{E}))=c_1(\mathcal{E})\mathbb{P}^2$, where $\mathbb{P}^2$ is a fiber of the map $\pi _E$. Since $E|_E$ is the tautological bundle, it follows that $e.\mathbb{P}^2=-\mathbb{P}^1$ and $e.e.\mathbb{P}^2=-e.\mathbb{P}^1=1$. Therefore $E^4=c_1(\mathcal{E})<0$.
Thus the assumptions in ii) can be restated as follows: Here $3=4-1$ is the codimension of $V$ in $Y$, $(-1)^{3-1}\pi _*(E^3)=V$ can be represented by an effective cycle (with real coefficients) whose intersection with $V$ has dimension $\leq 0$, and $(-1)^{4-1}\pi _*(E^4)$ is a positive number. Stated this way, we can see that the statements of ii) and iii) are similar. Given this, the proof of ii) is similar to that of iii), and hence is omitted.
i\) The proof is similar to those of iii) and ii) above.
The next result concerns blowups of projective manifolds of arbitrary dimension. Part (i) of Theorem \[TheoremEk\] below refines the results in the papers [@bayraktar] and [@bayraktar-cantat] (see the remark right after the statement of the theorem). Examples satisfying parts ii), ii’) and iii’) of Theorem \[TheoremEk\] are given at the end of this section. In Example 6 at the end of this section, we show that the assumptions in Theorem \[TheoremEk\] ii’) (and those of Lemmas \[LemmaP4\] and \[LemmaManifoldDimension4\] and Corollary \[CorollaryExtremeCases\]) can not be removed.
Let $Y$ be a projective manifold of dimension $k$, and let $V\subset Y$ be a compact complex submanifold. Let $\pi :X\rightarrow Y$ be the blowup of $Y$ at $V$, and let $E$ be the exceptional divisor. Let $r$ and $q$ be integers with $-1\leq r\leq k-1$ and $0\leq q\leq k-r-1$. Assume that $Y$ satisfies the non-vanishing condition $A(r,q)$ (correspondingly the non-vanishing conditions $NA(r,q)$, $B(r,q)$ and $NB(r,q)$). If one of the following four conditions is satisfied, then $X$ also satisfies the non-vanishing condition $A(r,q)$ (respectively the non-vanishing condition $NA(r,q)$, $B(r,q)$ and $NB(r,q)$).
i\) We assume that the dimension of $V$ is $dim(V)\leq r$.
ii\) Assume that $q=0$ and the dimension of $V$ is $>r$. Let $s=$codimension of $V$ in $Y$, then $s\leq k-r-1$. We assume that for any $j=s,s+1,\ldots ,k-r-1$, the cycle $(-1)^{j-1}\pi _*(E^j)$ can be represented by an effective cycle (with real coefficients) whose intersect with $V$ has dimension $\leq k-j-1$; and the cycle $(-1)^{k-r-1}\pi _*(E^{k-r})$ is strictly effective, i.e. it is effective and non-zero. Note that the cycles $(-1)^{j-1}\pi _*(E^j)$ can be represented in terms of the Chern classes of $\mathcal{E}$, see the Remark after the proof of the theorem.
When the center of blowup $V$ has Picard number $1$ (e.g. a curve or a projective space), the assumptions in ii) and iii) can be less restrictive, in that we do not require all the effective cycles $(-1)^{j-1}\pi _*(E^j)$ to have intersections of small enough dimensions with $V$.
ii’) Assume that $q=0$, the dimension of $V$ is $>r$, and $H^{1,1}(V)$ has dimension $1$ (or $V$ has Picard number $1$ for the algebraic non-vanishing conditions $NA(r,0)$ and $NB(r,0)$). Let $s=$codimension of $V$ in $Y$, then $s\leq k-r-1$. We assume that for any $j=s,s+1,\ldots ,k-r-1$, the cycle $(-1)^{j-1}\pi _*(E^j)$ is effective; and the cycle $(-1)^{k-r-1}\pi _*(E^{k-r})$ is strictly effective, i.e. it is effective and non-zero. We also assume that $V$ is not the only effective cycle in its cohomology class.
iii’) Assume that $q>0$, the dimension of $V$ is $>r$, and $H^{1,1}(V)$ has dimension $1$ (or $V$ has Picard number $1$ for the algebraic non-vanishing conditions $NA(r,q) $ and $NB(r,q)$). We assume that the other requirements in ii’) are satisfied. In addition we assume that $c_1(Y)|_V$ is ample in $V$.
\[TheoremEk\]
Remark: For the proof of i) when $q=0$ and $X\rightarrow X_0$ is a finite blowup along smooth centers of dimension $\leq r$ where $X_0$ has Picard number $1$, see also [@bayraktar] and [@bayraktar-cantat]. Our argument here is different from that used in those papers; in particular, here we can show that for the examples given in their papers, if $\zeta \in NS_{\mathbb{R}}(X)$ (not necessarily nef) is such that $\zeta ^{k-r-1}=0$ then $\zeta =0$.
i\) Assume that $Y$ satisfies the non-vanishing condition $B(r,q)$ (the case of non-vanishing condition $A(r,q)$ is similar; the algebraic analogs $NA(r,q)$ and $NB(r,q)$ are also similar, see the remark at the beginning of this section). Let $\zeta$ be a nef class on $X$ such that $\zeta ^{k-r-1-q}.K_X^q=0$. We need to show that $\zeta =0$. We can write $\zeta =\pi ^*(\xi )-aE$ where $a\geq 0$, and $\xi =\pi _*(\zeta )$. Let $t=dim(V)$ then we can write $K_X=\pi ^*(K_Y)+(k-1-t)E$. We first show that $a=0$. (Note that in the proof of i) we do not need that $a\geq 0$.)
By assumption we have $$\begin{aligned}
0&=&\zeta ^{k-r-1-q}.K_X^q\\
&=&(\pi ^*(\xi )-aE)^{k-r-1-q}.(\pi ^*(K_Y)+(k-1-t)E)^q\\
&=&[\sum _{j=0}^{k-r-1-q}(-a)^{k-r-1-j-q}C(j,k-r-1-q)\pi ^*(\xi ^j).E^{k-r-1-j-q}]\\
&&\times [\sum _{i=0}^q(k-1-t)^{q-i}C(i,q)\pi ^*(K_Y)^iE^{q-i}]\\
&=&\sum _{i,j}(-a)^{k-r-1-j-q}(k-1-t)^{q-i}C(j,k-r-1)C(i,q)\pi ^*(\xi ^j).\pi ^*(K_Y^i).E^{k-r-1-j-i}.\end{aligned}$$ Here $C(j,k-r-1-q)$ and $C(i,q)$ are the binomial numbers. Intersecting the above with $E^{r-t+1}$ and then pushing forward by the map $\pi$ we obtain by the projection formula $$\begin{aligned}
0=\sum _{i,j}(-a)^{k-r-1-j-q}(k-1-t)^{q-i}C(j,k-r-1)C(i,q)\xi ^j.K_Y^i.\pi _*(E^{k-t-j-i}).\end{aligned}$$ Observe that except for the first term $(-a)^{k-r-1-q}(k-1-t)^{q}\pi _*(E^{k-t})$, other terms are zero (in fact, if $i+j>0$ then $\pi _*(E^{k-t-j-i})$ must be zero because it has dimension $t+i+j>t$ and has support in $V=\pi (E)$ which is of dimension $t$); and the first term is $(-1)^{r-t+q}a^{k-r-1-q}(k-1-t)^qV$ (see the formula in Section 4.3 of [@fulton]). Therefore $a=0$ as wanted.
Thus $\zeta =\pi ^*(\xi )$, and from [@paun], it follows that $\xi $ is nef. Pushing forward the equality $\pi ^*(\xi ^{k-r-1-q}).(\pi ^*(K_Y)+(k-t-1)E)^q$ by the map $\pi$, from the assumption $k-q\geq r+1>r=dim(V)$ we see as in the above paragraph that $\xi ^{k-r-1-q} .K_Y^q=0$. Then the assumption on $Y$ implies that $\xi =0$ and hence $\zeta =\pi ^*(\xi )=0$.
ii\) Assume that $Y$ satisfies the non-vanishing condition $B(r,0)$ (the case of non-vanishing conditions $A(r,0)$, $NA(r,0)$ and $NB(r,0)$ are similar). Let $\zeta $ be a nef class on $X$ such that $\zeta ^{k-r-1}=0$. We will show that $\zeta =0$. We can write $\zeta =\pi ^*(\xi )-aE$ where $a\geq 0$, and $\xi =\pi _*(\xi )$. As in the proof of i), it suffices to show that $a=0$.
The proof proceeds similarly to that of i) and of Lemma \[LemmaManifoldDimension4\] iii). We have $$\begin{aligned}
0=\zeta ^{k-r-1}=\sum _{j=0}^{k-r-1}(-a)^{k-r-1-j}C(j,k-r-1)\pi ^*(\xi ^j).E^{k-r-1-j}. \end{aligned}$$ Intersecting with $E$ we obtain $$\begin{aligned}
0=\sum _{j=0}^{k-r-1}(-a)^{k-r-1-j}C(j,k-r-1)\pi ^*(\xi ^j).E^{k-r-j}. \end{aligned}$$ Pushing forward the above equality by $\pi$, we obtain $$0=\sum _{j=0}^{k-r-1}(-a)^{k-r-1-j}\xi ^j.\pi _*(E^{k-r-j}).
\label{Equation5}$$ In Equation (\[Equation5\]), the terms corresponding with $k-r-j<s$ (or equivalently $j>k-r-s$) are zero. Hence $$\begin{aligned}
0=\sum _{j=0}^{k-r-s}(-a)^{k-r-1-j}\xi ^j.\pi _*(E^{k-r-j}). \end{aligned}$$ By assumption and the argument in the proof of Lemma \[LemmaManifoldDimension4\] iii), each individual term in the above is effective, and in the first term $(-1)^{k-r-1}\pi _*(E^{k-r})$ is strictly effective. Since $a\geq 0$, this implies that $a=0$ as wanted.
ii’) We follow the proof of ii). Let $\omega$ be an ample class on $Y$. Since $V$ is not the only effective curve in its cohomology class, it follows that $\xi .V$ is effective. Let $\iota :V\subset Y$ be the inclusion. Because $V$ has Picard number $1$, it follows that $\xi |_V=\iota ^*(\xi )=b\omega |_{V}$ for some real number $b$. Since $b\omega .V=\iota _*(\xi |_V)=\xi .V$, and $\xi .V$ is effective and $\omega .V$ is strictly effective, it follows that $b\geq 0$. The equation $\zeta ^{k-r-1}=0$ implies $$\begin{aligned}
\zeta ^{k-r-1}.\pi ^*(\omega ^r).E=0,\end{aligned}$$ and the latter is the same as $$\begin{aligned}
(b\pi ^*(\omega )-aE)^{k-r-1}.\pi ^*(\omega ^r).E=0.\end{aligned}$$ Then we can proceed as in the proof of ii).
iii’) Assume that $Y$ satisfies the non-vanishing condition $B(r,q)$ (the case of non-vanishing conditions $A(r,q)$, $NA(r,q)$ and $NB(r,q)$ are similar). Let $\zeta $ be a nef class on $X$ such that $\zeta ^{k-r-1-q}.K_X^q=0$. We will show that $\zeta =0$. We can write $\zeta =\pi ^*(\xi )-aE$ where $a\geq 0$, and $\xi =\pi _*(\xi )$. First, we show that $a=0$.
Note that $\zeta ^{k-r-1-q}.K_X^q=0$ implies $\zeta ^{k-r-1-q}.K_X^q.\pi ^*(K_Y)^r.E=0$, and the latter is the same as $\zeta ^{k-r-1-q}.c_1(X)^q.\pi ^*(c_1(Y))^r.E=0$. The latter is the same as $\zeta ^{k-r-1-q}|_E.c_1(X)^q_E.\pi ^*(c_1(Y))^r|_E=0$. Since $V$ has Picard number $1$ and $c_1(Y)|_V$ is effective, as in the proof of ii’) we can write $c_1(Y)|_V=T|_V$, where $T$ is either zero or ample on $Y$. We have $$\pi ^*(c_1(Y))|_E=\pi _E^*(c_1(Y)|_V)=\pi _E^*(T |_V)=\pi ^*(T )|_E.$$ Using $c_1(X)=\pi ^*(c_1(Y))-aE$, we can then write $$\begin{aligned}
c_1(X)|_E=(\pi ^*(T )-(s-1)E)|_E.\end{aligned}$$ Let $\omega$ be an ample divisor on $Y$. Then the original equation $\zeta ^{k-r-1-q}.c_1(X)^q.\pi ^*(c_1(Y))^r.E=0$ becomes $$\begin{aligned}
(\pi ^*(\xi )-aE)^{k-r-1-q}(\pi ^*(T)-(s-1)E)^q.E.\pi ^*(\omega )^r=0. \end{aligned}$$ Pushforward this equation by $\pi$, we find by the projection formula $$\begin{aligned}
\pi _*[(\pi ^*(\xi )-aE)^{k-r-1-q}(\pi ^*(T )-(s-1)E)^q.E].\omega ^r=0.\end{aligned}$$ Arguing as in the proof of ii) we see that if $a>0$ then the term $\pi _*[(\pi ^*(\xi )-aE)^{k-r-1-q}(\pi ^*(T)-(s-1)E)^q.E]$ is psef and non-zero, and hence $\pi _*[(\pi ^*(\xi )-aE)^{k-r-1-q}(\pi ^*(T)-(s-1)E)^q.E].\omega ^r$ can not be zero since $\omega$ is ample on $Y$. Hence $a=0$ as wanted.
Hence, for the proof of iii’) it suffices to show that $\xi ^{k-r-1-q}.K_Y^q=0$. We consider two cases:
Case 1: $q<k-dim (V)$. Pushing the equation $\pi ^*(\xi ^{k-r-1-q}).(\pi ^*(c_1(Y))-(s-1)E)^q=0$ by the map $\pi$, we then find that $\xi ^{k-r-1-q}.K_Y^q=0$ as wanted.
Case 2: $q\geq k-dim(V)$. In this case we first pushforward the equation $\pi ^*(\xi ^{k-r-1-q}).(\pi ^*(c_1(Y))-(s-1)E)^q.E=0$ by the map $\pi$ and find that $$\begin{aligned}
\iota _*(\sum _{q\geq i\geq k-dim (V)}(\xi |_V)^{k-r-1-q}.(c_1(Y)|_V)^{q-i}(-1)^{i-1}(\pi _E)_*(E^{i-1}))=0.\end{aligned}$$ As argued above, each term insided the $\iota _*$ on the LHS of the above equation is psef, therefore each of them must be zero. In particular, the term with $i=k-dim(V)$, which is $(\xi |_V)^{k-r-1-q}.(c_1(Y)|_V)^{q-k+dim (V)}$, must be zero. Since $c_1(Y)|_V$ is ample by assumption, and since $\xi |_V$ is either zero or ample on $V$, it then follows that $\xi |_V=0$. Therefore $$\begin{aligned}
0=\pi ^*(\xi ^{k-r-1-q}).(\pi ^*(c_1(Y))-(s-1)E)^q=\pi ^*(\xi )^{k-r-1-q}.\pi ^*(c_1(Y)^q).\end{aligned}$$ Pushing this by the map $\pi$, we find that $\xi ^{k-r-1-q}.c_1(Y)^q=0$, as wanted.
In the border case $dim(V)=r+1$, we can make Theorem \[TheoremEk\] ii), ii’) and iii’) stronger. Part i) of the following result can be regarded as a generalization of Lemma \[LemmaManifoldDimension4\] ii). Examples satisfying Corollary \[CorollaryExtremeCases\] will be given at the end of this section (see in particular Examples 4 and 5).
Let $Y$ be a projective manifold of dimension $k$, and let $V\subset Y$ be a compact complex submanifold. Let $\pi :X\rightarrow Y$ be the blowup of $Y$ at $V$, and let $E$ be the exceptional divisor. Let $\mathcal{E}=N_{V/Y}$ be the normal vector bundle of $V$ in $Y$. Let $\iota :V\rightarrow Y$ be the inclusion, and let $\pi _E:E\rightarrow V$ be the projection.
i\) Assume that $Y$ satisfies the non-vanishing condition $A(r,0)$ (correspondingly the non-vanishing conditions $NA(r,0)$, $B(r,0)$ and $NB(r,0)$), and $dim(V)=r+1$. Assume moreover that $V$ is not the only effective variety (with real coefficients) in its cohomology class, and $\iota _*(c_1(\mathcal{E}))$ is not psef. Then $X$ also satisfies the non-vanishing condition $A(r,0)$ (respectively the non-vanishing conditions $NA(r,0)$, $B(r,0)$ and $NB(r,0)$).
ii\) (This is a generalization of i).) Assume that $Y$ satisfies the non-vanishing condition $A(r,0)$ (correspondingly the non-vanishing conditions $NA(r,0)$, $B(r,0)$ and $NB(r,0)$), and $dim(V)=r+1$. Assume moreover that there is an integer number $j\geq 1$, an effective cycle $V'$ having the same cohomology class as that of $V$ such that $dim(V'\cap V)\leq dim (V)-j$, and $\iota _*(c_j(\mathcal{E}))$ is not psef. Then $X$ also satisfies the non-vanishing condition $A(r,0)$ (respectively the non-vanishing conditions $NA(r,0)$, $B(r,0)$ and $NB(r,0)$).
\[CorollaryExtremeCases\]
i\) Let $\zeta $ be a nef class on $X$ with $\zeta ^{k-r-1}=0$. We write $\zeta =\pi ^*(\xi )-aE$ with $a\geq 0$. To prove ii) it suffices to show that $a=0$. Let $\iota _E:E\rightarrow X$ be the inclusion. Then we have $\iota _E^*(\zeta )^{k-r-1}=0$, which is the same as $$\begin{aligned}
(\pi _E^*(\xi |_V)-ae)^{k-r-1}=0.\end{aligned}$$
Because $dim(V)=r+1$, the defining equation for $H^*(E)$ is then $$\begin{aligned}
e^{k-r-1}-\pi _E^*(c_1(\mathcal{E}))e^{k-r-2}+\ldots =0.\end{aligned}$$ Comparing this equation with the equation $(\pi _E^*(\xi |_V)-ae)^{k-r-1}=0$, it follows that $\pi _E^*(\xi |_V)=a\pi _E^*(c_1(\mathcal{E}))$. Since the pullback maps $\pi _E^*:H^*(V)\rightarrow H^*(E)$ are injective, we obtain $\xi |_V=ac_1(\mathcal{E})$. From this, we must have $a=0$. Otherwise, pushing forward by $\iota$ we obtain $\xi .V=a\iota _*(c_1(\mathcal{E}))$. This is a contradiction, since the LHS is psef (as in the proof of Theorem \[TheoremEk\]) while the RHS is not psef by assumption.
ii\) The proof is similar to that of i): Rescaling, we may assume that $a=1$. We now use $\xi ^j|_V=C(k-r-1,j) c_j(\mathcal{E})$, and $\iota _*(\xi ^j|_V)=\xi ^j.V=\xi ^j.V'$ is psef.
[**Remark 1.**]{} In Theorem \[TheoremEk\] ii), we can represent $\pi _*(E^j)$ for $j=s,s+1,\ldots ,k-r$ in terms of the Chern classes of $\mathcal{E}$. Recall that $\pi :X\rightarrow Y$ is the blowup along a submanifold $V\subset Y$, $s=codim(V)$, $\mathcal{E}=N_{V/Y}$ the normal vector bundle of $V$ in $Y$, $E=\mathbb{P}(\mathcal{E})$ the exceptional divisor of $\pi$ with $\pi _E:E\rightarrow V$ the projection, and $e=E|_E$. First of all, we have $(-1)^{s-1}\pi _*(E^s)=V$ by the formula at the beginning of Section 4.3 in [@fulton]. To compute the pushforward of other $E^j$ we use the following formula (see Proposition 3.1 and the proofs of Lemma 3.3 and Proposition 6.7 in [@fulton]): $$(\pi _E)_*(\sum _{j=0}^{s-1}e^{j}.\pi _E^*(x_j))=(-1)^{s-1}x_{s-1}.
\label{Equation6}$$ It is then easy to compute the pushforward of $E^j$ for $j\geq s$. Let $\iota _E:E\subset X$ be the inclusion of $E$ in $X$. Then $$\begin{aligned}
\pi _*(E^{s+1})=\pi _*(\iota _E)_*(e^s)=\iota _*(\pi _E)_*(e^s).\end{aligned}$$ By the defining equation $$\begin{aligned}
\sum _{j=0}^{s}(-1)^je^{s-j}.\pi _E^*(c_j(\mathcal{E}))=0,\end{aligned}$$ we find that $$\begin{aligned}
e^s=e^{s-1}\pi _E^*(c_1(\mathcal{E}))-e^{s-2}\pi _E^{*}(c_2(\mathcal{E}))+\ldots .\end{aligned}$$ Using (\[Equation6\]), we find that $(\pi _E)_*(e^s)=(-1)^{s-1}c_1(\mathcal{E})$, and hence $\pi _*(E^{s+1})=(-1)^{s-1}\iota _*(c_1(\mathcal{E}))$. Similarly, we can compute $\pi _*(E^{s+2})$: We have $$\begin{aligned}
\pi _*(E^{s+2})=\pi _*(\iota _E)_*(e^{s+1})=\iota _*(\pi _E)_*(e^{s+1}).\end{aligned}$$ Now we have $$\begin{aligned}
e^{s+1}&=&e^s\pi _E^*(c_1(\mathcal{E}))-e^{s-1}\pi _E^*(c_2(\mathcal{E}))+\ldots \\
&=&(e^{s-1}\pi _E^*(c_1(\mathcal{E}))-e^{s-2}\pi _E^*(c_2(\mathcal{E}))+\ldots )\pi _E^*(c_1(\mathcal{E}))-e^{s-1}\pi _E^*(c_2(\mathcal{E}))+\ldots \\
&=&e^{s-1}\pi _E^*(c_1(\mathcal{E})^2-c_2(\mathcal{E}))+\ldots ,\end{aligned}$$ hence $(\pi _E)_*(e^{s+1})=(-1)^{s-1}[c_1(\mathcal{E})^2-c_2(\mathcal{E})]$. Therefore $$\begin{aligned}
\pi _*(E^{s+2})=(-1)^{s-1}\iota _*(c_1(\mathcal{E})^2-c_2(\mathcal{E})).\end{aligned}$$ Similarly we can compute the pushforward of other $E^j$’s in terms of Chern classes of $\mathcal{E}$.
[**Example 1.**]{} We now give a construction to provide many examples when the conditions of Theorem \[TheoremEk\] can be easily checked.
Assume that $\pi _1:Y_1\rightarrow Y$ is a blowup along a smooth submanifold $W_1\subset Y$ of dimension $d_1\geq 1$, and let $E_1$ be the exceptional divisor. Let $V\sim \mathbb{P}^{k-d_1-1}\subset E_1$ be a fiber of the restriction $\pi _1:E_1\rightarrow V_1$. Let $\pi :X\rightarrow Y_1$ be the blowup of $Y_1$ at $V$. If $Y_1$ satisfies the non-vanishing condition $A(r,q)$ (correspondingly $NA(r,q)$, $B(r,q)$ and $NB(r,q)$) then $X$ satisfies the non-vanishing condition $A(r,q)$ (respectively $NA(r,q)$, $B(r,q)$ and $NB(r,q)$).
First, if $dim(V)\leq r$ then we can apply Theorem \[TheoremEk\] i). Hence we can assume that $dim(V)\geq r+1$, and will show that Theorem \[TheoremEk\] ii), ii’) and iii’) apply.
In fact, let $\mathcal{E}=N_{V/Y_1}$ be the normal vector bundle of $V$ in $Y_1$. Then we have the following SES of vector bundles over $V$ (see Appendix B.7.4 in [@fulton]) $$\begin{aligned}
0\rightarrow N_{V/E_1}\rightarrow N_{V/Y_1}\rightarrow N_{E_1/Y_1}|_{V}\rightarrow 0.\end{aligned}$$ Therefore $c(N_{V/Y_1})=c(N_{V/E_1})c(N_{E_1/Y_1}|_V)$. Because $E_1$ is the projectivization of a vector bundle over $W_1$ and $V$ is a fiber of $E_1\rightarrow W_1$, it follows that the normal vector bundle $N_{V/E_1}$ is trivial. Hence $c(N_{V/E_1})=1$. If $h$ is the class of a hyperplane on $V$ then we have $$\begin{aligned}
N_{E_1/Y_1}|_V=(E_1|_{E_1})|_{V}=-h,\end{aligned}$$ since $E_1|_{E_1}$ is the tautological bundle. Therefore $c(N_{E_1/Y_1}|_V)=1-h$, and $c(N_{V/Y_1})=1-h$. Hence $c_0(N_{V/Y_1})=1$, $c_1(N_{V/Y_1})=-h$, and the other Chern classes are zero. Thus, if $E$ is the exceptional divisor of the blowup $\pi :X\rightarrow Y_1$ and $\pi _E:E\rightarrow V$ is the projection then the defining equation for $H^*(E)$ over $H^*(V)$ is $$\begin{aligned}
e^s=\pi _E^*(c_1(\mathcal{E}))e^{s-1}. \end{aligned}$$ Therefore $$(-1)^{s+j-1}\pi _*(E^{s+j})=(-1)^{j}\iota _*(c_1(\mathcal{E})^j)=(-1)^j\iota _*((-h)^j)=\iota _*(h^j),$$ are all strictly effective, for $j=0,\ldots ,k-s=dim(V)$. Since in Theorem \[TheoremEk\] ii) we assumed that $dim (V)\geq r+1$ we see that $$\begin{aligned}
(-1)^{k-r-1}\pi _*(E^{k-r})=(-1)^{k-r-1}\pi _*(E^{s+k-r-s})=\iota _*(h^{dim (V)-r})\end{aligned}$$ is strictly effective. Also, all of $\iota _*(h^j)$ can be represented by linear subspaces of the other fibers of $\pi _{E_1}$ disjoint from $V$. Hence if $q=0$ we see that all the assumptions of Theorem \[TheoremEk\] ii) are satisfied. In this case $V$ has Picard number $1$, and we can apply Theorem \[TheoremEk\] ii’). We now check that $c_1(Y_1)|_V$ is ample in order to be able to apply Theorem iii’) when $q>0$. In fact, from the SES of vector bundles on $V$ $$\begin{aligned}
0\rightarrow T_V\rightarrow T_{Y_1}|_V\rightarrow \mathcal{E}\rightarrow 0,\end{aligned}$$ we have $c_1(Y_1)|_V=c_1(V)+c_1(\mathcal{E})$. As computed above $c_1(\mathcal{E})=-h$, and because $V$ is a projective space we find $c_1(V)=(dim(V)+1)h$. Therefore $c_1(Y_1)|_V=dim(V)h$ is ample because $dim(V)>0$.
[**Example 2.**]{} In the situation of Example 1, we can also blowup a hyperplane $W$ of $V$ to produce an example satisfying the assumptions of Theorem \[TheoremEk\] ii) if either $dim (W)=dim (V)-1\leq r$ or the number $dim(W)-r$ is even. More precisely, assume that $\pi _1:Y_1\rightarrow Y$ is a blowup along a smooth submanifold $W_1\subset Y$ of dimension $d_1\geq 1$, and let $E_1$ be the exceptional divisor. Let $V\sim \mathbb{P}^{k-d_1-1}\subset E_1$ be a fiber of the restriction $\pi _1:E_1\rightarrow V_1$, and let $W\subset V$ be a hyperplane. Let $\pi :X\rightarrow Y_1$ be the blowup of $Y_1$ at $W$. If $Y_1$ satisfies the non-vanishing condition $A(r,q)$ (correspondingly $NA(r,q)$, $B(r,q)$ and $NB(r,q)$) and either
i\) $dim(W)\leq r$
or
ii\) $dim(W)-r$ is even,
or
iii\) $r\geq 1$,
then $X$ satisfies the non-vanishing condition $A(r,q)$ (respectively $NA(r,q)$, $B(r,q)$ and $NB(r,q)$).
In fact, in case i) we can apply Theorem \[TheoremEk\] i).
We consider case ii). The assumption in case ii) is the same as $(k-r)-s$ is even, where $s=k-dim(W)$ is the codimension of $W$. Let $E$ be the exceptional divisor of the blowup, and let $\pi _E:E\rightarrow W$ be the projection. Let $h$ be the hyperplane class in $W$. Then compute as in Example 1 we see that $$c(N_{W/Y_1})=c(N_{W/V}).c(N_{V/E_1}|_{W}).c(N_{E_1/Y_1}|_W)=(1+h)(1-h)=1-h^2.$$ Hence $c_0(N_{V/Y_1})=1$, $c_2(N_{V/Y_1})=-h^2$, and other Chern classes are zero. Thus the defining equation for $H^*(E)$ is $$\begin{aligned}
e^s=-\pi _E^*(c_2(N_{V/Y_1}))e^{s-2}=\pi _E^*(h^2)e^{s-2}. \end{aligned}$$ Then as in Example 1 we find $(-1)^{s+j-1}\pi _*(E^{s+j})=0$ if $j$ is odd, and $(-1)^{s+j-1}\pi _*(E^{s+j})=\iota _*(h^j)$ if $j$ is even. Hence all of them are effective, and if $j$ are even then they are strictly effective. Since we assume that $(k-r)-s$ is even, it follows that the term $(-1)^{k-r-1}\pi _*(E^{k-r})$ is strictly effective. Also, all of these classes can be represented by linear subspaces of the other fibers disjoint from $V$ and hence from $W$. Hence all the assumptions of Theorem \[TheoremEk\] ii) are satisfied. In this case $W$ has Picard number $1$, and we can apply Theorem \[TheoremEk\] ii’). We now check that $c_1(Y_1)|_W$ is ample so that Theorem \[TheoremEk\] iii’) also applies. As computed in Example 1, we have $c_1(Y_1)|_W=c_1(W)+c_1(\mathcal{E})$. Here we computed above that $c_1(\mathcal{E})=0$, and again have $c_1(W)=(dim(W)+1)h$. Hence $c_1(Y_1)|_W=(dim(W)+1)h$ is ample, as wanted.
In case iii) we apply the same argument as in ii) to the equation $\zeta ^{k-r-1-q}.K_X^q.E^2=0$, instead of the equation $\zeta ^{k-r-1-q}.K_X^q.E=0$.
[**Example 3.**]{} We give a specific application of Examples 1 and 2. Let $X_0$ be a projective manifold of even dimension $k=2l$ satisfying the non-vanishing condition $B(l-1,q)$ (or $A(l-1,q)$, $NA(l-1,q)$ and $NB(l-1,q)$). For example, we can take $X_0=$ a projective hyper-Kähler manifold or a manifold with Picard number $1$. Let $1\leq j\leq l-1$ and $V_0\subset X_0$ be a manifold of dimension $j$. We let $X_1\rightarrow X_0$ be the blowup of $X_0$ along $V_0$. By Theorem \[TheoremEk\] ii), we know that $X_1$ also satisfies the non-vanishing condition $B(l-1,q)$. Let $E_0\subset X_1$ be the exceptional divisor, then a fiber $V$ of $E_0\rightarrow V_0$ has dimension $l\leq k-1-j\leq k-2$. By Example 1, we see that if $X\rightarrow X_1$ is the blowup of $X_1$ along $V$, then $X$ also satisfies the non-vanishing condition $B(l-1,q)$. Hence by Theorem \[TheoremHyperKahler\], if $f\in Aut(X)$ is an automorphisms then either $h_{top}(f)=0$ or $f$ is cohomologically hyperbolic.
[**Example 4.**]{} We now give examples satisfying Corollary \[CorollaryExtremeCases\] i) and ii). This example allows blowing up higher codimension submanifolds in Examples 1 and 2. Assume that $\pi _1:Y_1\rightarrow Y$ is a blowup along a smooth submanifold $W_1\subset Y$ of dimension $d_1\geq 1$, and let $E_1$ be the exceptional divisor. Let $V\sim \mathbb{P}^{k-d_1-1}\subset E_1$ be a fiber of the restriction $\pi _1:E_1\rightarrow V_1$, and let $W\subset V$ be a complex submanifold of dimension $r+1$. Let $\pi :X\rightarrow Y_1$ be the blowup at $W$.
i\) Assume that $2r+1\geq dim(V)\geq r+1$. If $Y_1$ satisfies the non-vanishing condition $A(r,0)$ (correspondingly $NA(r,0)$, $B(r,0)$ and $NB(r,0)$) then $X$ satisfies the non-vanishing condition $A(r,0)$ (respectively $NA(r,0)$, $B(r,0)$ and $NB(r,0)$).
ii\) Assume that $[(dim (V)-r)/2]\leq r$, here $[x]$ is the largest integer less than or $x$. Assume moreover that $W\subset V$ is a linear subspace. If $Y_1$ satisfies the non-vanishing condition $A(r,q)$ (correspondingly $NA(r,q)$, $B(r,q)$ and $NB(r,q)$) then $X$ satisfies the non-vanishing condition $A(r,q)$ (respectively $NA(r,q)$, $B(r,q)$ and $NB(r,q)$).
Proof:
i\) Let $H$ be the hyperplane class in $V$, and $h=H|_W$. As in the proof of Example 2) we have $$\begin{aligned}
c(N_{W/Y_1})&=&c(N_{W/V})c(N_{V/E_1}|_W)c(N_{E_1/Y_1}|_W)\\
&=&c(N_{W/V})(1-h).\end{aligned}$$ Let $t=dim (V)-dim (W)$ be the codimension of $W$ in $V$. Then $$\begin{aligned}
c_{t+1}(N_{W/Y_1})=-hc_{t}(N_{W/V})=-deg(W)h^{t+1}, \end{aligned}$$ the last equality is a consequence of the self-intersection formula (see page 103 in [@fulton]). The assumption that $dim (V)\leq 2r+1$ implies that $c_{t+1}(N_{W/Y_1})$ is negative and is non-zero. Hence if $\iota :W\rightarrow Y_1$ is the inclusion map then $\iota _*(c_{t+1}(N_{W/Y_1}))$ is not psef. We can choose a submanifold $W'$ of another fiber disjoint from $V$ such that $W'$ has the same cohomology class as that of $W$. Hence Corollary \[CorollaryExtremeCases\] ii) can be applied to complete the proof of i).
ii\) If $dim(V)=r+1$ then $W=V$ and we can apply Example 1. Therefore we need to consider only the case when $dim(V)>r+1$. Define $t=dim (V)-dim (W)=dim (V)-r-1>0$ and $s=[(t+1)/2]=[(dim(V)-r)/2]$. Since $W$ is a complete intersection of hyperplanes, as computed above we find that $$\begin{aligned}
c(N_{W/Y_1})=(1-h)(1+h)^t=\sum _{j=0}^{t+1}(C(t,j)-C(t,j-1))h^{j}.\end{aligned}$$ By the properties of binomial numbers, $C(t,s)>C(t,s+1)$. This, together with the assumption that $s\leq r$ (hence $s+1\leq r+1$) implies that $c_{s+1}(N_{W/Y_1})=(C(t,s+1)-C(t,s))h^{s+1}$ is negative and is non-zero.
Also, $W$ has Picard number $1$. We computed above that $c_1(N_{W/Y_1})=c_1(N_{W/V})-h$. Since $V$ is a projective space, we have $c_1(V)=(dim(W)+1)H$. Therefore $$\begin{aligned}
c_1(Y_1)|_W&=&c_1(W)+c_1(N_{W/Y_1})=c_1(W)+c_1(N_{W/V})-h\\
&=&c_1(V)|_W-h=(dim (V)+1)H|_W-h=dim(V)h\end{aligned}$$ is ample. Thus we can apply the proofs of i) and Example 1.
[**Example 5.**]{} This example is to show that except for the requirement that $V$ is not the only effective cycle in its cohomology class, the condition in Corollary \[CorollaryExtremeCases\] is always satisfied if we blowup enough points in generic positions. More precisely, let $Y$ be a complex projective manifold of dimension $k$ and let $V\subset Y$ be a proper compact complex submanifold of dimension $\geq 1$. Let $x_1,\ldots ,x_n,\ldots $ be a sequence of distinct points in $V$ such that $\bigcup _{j=1}^{\infty}x_j$ is not contained in any subvariety of dimension $dim(V)-1$. For any $t$ , let $\pi _t:X_t\rightarrow Y$ be the blowup at $x_1,\ldots ,x_t$. Let $\widetilde{V}_t$ be the strict transform of $V$ in $X$, $\iota _t:\widetilde{V}_t\rightarrow X$ the inclusion map, and $\mathcal{E}_t=N_{\widetilde{V}_t/X}$ the normal vector bundle. If the number $t$ is large enough, then $(\iota _t)_*(c_1(\mathcal{E}_t))$ is not psef.
Proof: From the SES of vector bundles on $\widetilde{V}_t$: $$\begin{aligned}
0\rightarrow T_{\widetilde{V}_t}\rightarrow T_{X_t}|_{\widetilde{V}_t}\rightarrow \mathcal{E}_t\rightarrow 0,\end{aligned}$$ we have $c_1(\mathcal{E}_t)=\iota _t^*(c_1(X_t))-c_1(\widetilde{V}_t)$. Let $E_j=\pi ^{-1}(x_j)$ be the exceptional divisor over the point $x_j$. Then $$\begin{aligned}
c_1(X_t)=\pi _t^*(c_1(Y))-(k-1)\sum _{j=1}^tE_j.\end{aligned}$$ If $p_t=\pi _t|_{\widetilde{V}_t}:\widetilde{V}_t\rightarrow V$ is the restriction of $\pi _t$ to $\widetilde{V}_t$, then $p_t$ is the blowup of $V$ at $x_1,\ldots ,x_t$ (see Example 7.17 in the book Harris [@harris]). Moreover, $\alpha _j=E_j\cap \widetilde{V}_t=p_t^{-1}(x_j)$ is the exceptional divisor over $x_j$ of the map $p_t$. Hence $$\begin{aligned}
c_1(\widetilde{V}_t)=p_t^*(c_1(V))-(dim (V)-1)\sum _{j=1}^t\alpha _j,\end{aligned}$$ and $$\begin{aligned}
c_1(\mathcal{E}_t)&=&\iota _t^*\pi _t^*(c_1(Y))-p_t^*(c_1(V))-(k-dim (V))\sum _{j=1}^t\alpha _j\\
&=&p_t^*(c_1(Y)|_V)-p_t^*(c_1(V))-(k-dim (V))\sum _{j=1}^t\alpha _j.\end{aligned}$$
Now we show that $(\iota _t)_*(c_1(\mathcal{E}_t))$ is not psef for $t$ large enough.
We consider two cases:
Case 1: $V$ has dimension $1$. In this case, $c_1(Y)|_V$ and $c_1(V)$ are just numbers, and hence $c_1(\mathcal{E}_t)=c_1(Y)|_V-c_1(V)-t(k-dim(V))$ is negative when $t>c_1(Y)|_V-c_1(V)$.
Case 2: $V$ has dimension $\geq 2$. Assume otherwise that there are large values of $t$ (as large as desired) such that the cohomology class $(\iota _t)_*(c_1(\mathcal{E}))$ can be represented by positive closed currents $S_t$ of bidimension $(dim(V)-1,dim(V)-1)$ on $X_t$. We first check that for such currents $S_t$, the positive closed current $(\pi _t)_*(S_t)$ has Lelong number $\geq 1$ at $x_1,\ldots ,x_t$. We check this for example at the point $x_1$. Using the projection from $X_t$ to the blowup of $Y$ at $x_1$, it is enough to check the claim for the case $t=1$. We then need to check that $(\pi _1)_*(S_1)$ has Lelong number at least $1$ at $x_1$. Subtracting $S_1$ from its restriction to $E_1$ if needed, we may assume that $S_1$ has no mass on $E_1$ and in cohomology $\{S_1\}=(\iota _1)_*\{p_1^*(c_1(Y)|_V)-p_1^*(c_1(V))-(k-dim (V)+a)\alpha _1\}$ where $a\geq 0$. Let $b\geq 0$ be the Lelong number of $(\pi _1)_*(S_1)$ at $x_1$. By Siu’s theorem (see Siu [@siu]) on Lelong numbers combined with the approximation theorem for positive closed currents on compact Kähler manifolds of Dinh and Sibony (see Dinh-Sibony [@dinh-sibony1]), in cohomology $\{(\pi _1)^*(\pi _1)_*(S_1)\}=\{S_1\}+b\{\iota _*(\alpha _1)\}$. Then we can intersect with $E_1^{dim(V)-1}$ to obtain that $b=k-dim(V)+a\geq k-dim (V)\geq 1$.
We now finish the proof of Case 2. The positive closed currents $(\pi _t)_*(S_t)$ on $X$ have the same cohomology class: $$\begin{aligned}
(\pi _t)_*\{S_t\}&=&(\pi _t)_*(\iota _t)_*(c_1(\mathcal{E}))\\
&=&\iota _*(p_t)_*[p_t^*(c_1(Y)|_V)-p_t^*(c_1(V))-(k-dim (V))\sum _{j=1}^t\alpha _j]\\
&=&\iota _*(c_1(Y)|_V-c_1(V)).\end{aligned}$$ In particular, they have uniformly bounded masses, and we can extract a cluster point $S$, which is a positive closed current on $X$ of bidimension $(dim(V)-1,dim(V)-1)$. Since $(\pi _t)_*(S_t)$ has Lelong number at least $1$ at $x_{t_0}$ for $t\geq t_0$, it follows by the upper-semicontinuity of Lelong numbers (see e.g. Chapter 3 in [@demailly]), $S$ has Lelong number at least $1$ at the points $x_1,x
_2,\ldots $. But this contradicts to Siu’s theorem (see [@siu]) that the set of points where $S$ has Lelong number at least $1$ is a subvariety of $X$ of dimension $\leq dim (V)-1$ and our assumption that $\bigcup _{j=1}^{\infty}x_j$ is not contained in a variety of dimension $dim(V)-1$.
[**Example 6.**]{} In this example, we show that the assumptions in Theorem \[TheoremEk\] ii’) (and those of Lemmas \[LemmaP4\] and \[LemmaManifoldDimension4\] and Corollary \[CorollaryExtremeCases\]) can not be removed. We consider here $Y$ a complex projective manifold of dimension $3$ and $C\subset Y$ a smooth curve isomorphic to $\mathbb{P}^1$. Assume that $Y$ satisfies the non-vanishing condition $A(r,q)$ (where $r\geq 0$) (correspondingly the non-vanishing conditions $B(r,q)$, $NA(r,q)$ and $NB(r,q)$). In Corollary \[CorollaryExtremeCases\] and Example 4 we showed that if the following two conditions are satisfied:
i\) $c_1(Y).C-c_1(C)<0$,
and
ii\) $C$ is not the only effective cycle in its cohomology class,
then $X$ also satisfies the non-vanishing condition $A(r,q)$ (respectively the non-vanishing conditions $B(r,q)$, $NA(r,q)$ and $NB(r,q)$).
We now show that if either one of these two conditions is removed then the above result is no longer true.
Proof:
1\) (This example was given in Section 4.1 in [@truong]): It was proved by McMullen (see [@mcmullen1]) that there are distinct points $x_1,\ldots ,x_m\in \mathbb{P}^2$ such that the blowup $\pi :S\rightarrow \mathbb{P}^2$ at these points has an automorphism of positive entropy. Consider $X=S\times \mathbb{P}^1$, then $X$ also has an automorphism of positive entropy. Thus $X$ does not satisfy the non-vanishing condition $A(0,0)$. $X$ can also be represented as the blowup of $Y=\mathbb{P}^2\times \mathbb{P}^1$ at the disjoint curves $x_1\times \mathbb{P}^1,\ldots , x_m\times \mathbb{P}^1$, each of these curves is isomorphic to $\mathbb{P}^1$. Here $Y$ satisfies the non-vanishing condition $A(0,0)$. In this example we can see that $c_1(Y)|_{x_j\times \mathbb{P}^1}-c_1(x_j\times \mathbb{P}^1)=0$, hence condition i) above is not satisfied. Since the curves $x_j\times \mathbb{P}^1$ move in a family of dimension $2$ of curves, the condition ii) above is satisfied.
2\) Assume now that the following claim is true:
Claim 1: For any $Y$ of dimension $3$ satisfying the non-vanishing condition $A(0,0)$ and $C\subset Y$ a smooth curve isomorphic to $\mathbb{P}^1$ such that $c_1(Y)|_C-c_1(C)<0$ , the manifold $X$ always satisfies the non-vanishing condition $A(0,0)$. Then we will arrive at a contradiction. First, we show the following:
Claim 2: Assume that Claim 1 is true. Then for any $Y$ of dimension $3$ satisfying the non-vanishing condition $A(0,0)$ and $C\subset Y$ a smooth curve isomorphic to $\mathbb{P}^1$, the manifold $X$ always satisfies the non-vanishing condition $A(0,0)$. (In Claim 2 we do not need the assumption $c_1(Y)|_C-c_1(C)<0$.)
Proof of Claim 2: Let $t$ be a large number and $x_1,\ldots ,x_t$ be distinct points in $C$. Let $Z_1\rightarrow Y$ be the blowup of $Y$ at $x_1,\ldots ,x_t$, and let $Z\rightarrow Z_1$ be the blowup at the strict transform $\widetilde{C}$ of $C$. Since $Y$ satisfies $A(0,0)$ and $Z_1\rightarrow Y$ is a composition of point-blowups, $Z_1$ also satisfies $A(0,0)$. In Example 5 we showed that for $t$ large then $c_1(Z_1)|_{\widetilde{C}}-c_1(\widetilde{C})<0$. By Claim 1 applied to the blowup $Z\rightarrow Z_1$, $Z$ also satisfies $A(0,0)$.
Next we show that $X$, which is the blowup of $Y$ at the curve $C$, also satisfies the condition $A(0,0)$. Let $\tau :Z\rightarrow Y$ be the composition of the blowups $Z\rightarrow Z_1$ and $Z_1\rightarrow Y$. We first check that $\tau ^{-1}(C)$, as a subscheme of $Z$, is a hypersurface. This is easy to see on the level of sets. Now we check that the ideal of $\tau ^{-1}(C)$ is locally generated by an element. This question is local, hence we reduce to the case where $Y=\mathbb{C}^3$, $x_1=(0,0,0)$ and $C=\{x=y=0\}$. Then a local coordinate for $Z$ is given by (see e.g. Section 1 in [@bedford-kim3]): $\tau :Z\ni (t_0,\eta _1, \xi _2)\mapsto (t_0\eta _1\xi _2, \eta _1\xi _2,\xi _2)\in Y$. Hence $\tau ^{-1}(C)$ is generated by $t_0\eta _1\xi _2$ and $\eta _1\xi _2$, hence is generated by one element $\eta _1\xi _2$ as claimed.
Therefore, applying the universal property of blowups (see e.g. Proposition 7.14 in Chapter 2 in Hartshorne [@hartshorne] and Theorem 4.1 in Chapter 4 in Fischer [@fischer]), there is a birational holomorphic map $\sigma :Z\rightarrow X$. Then we can finish the proof of Claim 2 as follows: Let $\zeta$ be a nef class on $X$ such that $\zeta ^2=0$. Then $\xi =\sigma ^*(\zeta )$ is a nef class on $Z$ such that $\xi ^2=0$. Since $Z$ satisfies the non-vanishing condition $A(0,0)$, it follows that $\xi \in H^{1,1}(Z,\mathbb{Q})$. Then $$\zeta =\sigma _*(\xi )\in \sigma _*(H^{1,1}(Z,\mathbb{Q}))\subset H^{1,1}(X,\mathbb{Q}).$$ Thus $X$ satisfies the non-vanishing condition $A(0,0)$, and Claim 2 is proved.
Finally, we obtain a contradiction to Claim 1. Let $x_1,\ldots ,x_m\in \mathbb{P}^2$ be such that the blowup $S\rightarrow \mathbb{P}^2$ has an automorphism of positive entropy. Let $X=S\times \mathbb{P}^1$, then $X$ does not satisfy the non-vanishing condition $A(0,0)$. However, $X$ is a blowup of $Y=\mathbb{P}^2\times \mathbb{P}^1$ at curves $x_1\times \mathbb{P}^1,\ldots ,x_m\times \mathbb{P}^1$, and $Y$ satisfies the non-vanishing condition $A(0,0)$. Hence if Claim 1 were true, then by Claim 2 $X$ also satisfies the non-vanishing condition $A(0,0)$, which is impossible. Hence Claim 1 is not true.
Proofs of Theorems \[TheoremArqCondition\], \[TheoremBrqCondition\], \[TheoremPicardNumber1\], \[TheoremAmpleCanonicalDivisor\] and \[TheoremHyperKahler\] {#SectionProofsOfTheorems}
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Before giving the proofs of the results in Section \[Introduction\], we recall some facts about spectral radius of automorphisms. The readers may see e.g. [@dinh-sibony], [@bayraktar-cantat] or [@truong] and references therein for more on these facts.
Let $X$ be a compact Kähler manifold of dimension $k$ and let $f:X\rightarrow X$ be an automorphism. Then we define the dynamical degrees $\lambda _p(f)=$the spectral radius of the pullback map $f^*:H^{p,p}(X)\rightarrow H^{p,p}(X)$. Then $\lambda _p(f)$ are log-concave, in particular $\lambda _p(f)\leq \lambda _1(f)^p$ for all $p=0,\ldots ,k$. Also, by Poincare duality $\lambda _p(f)=\lambda _{k-p}(f^{-1})$. We can also compute dynamical degree in the following way: Let $\omega$ be a Kähler class and choose an arbitrary norm on $H^{p,p}(X)$. Then $$\begin{aligned}
\lambda _p(f)=\lim _{n\rightarrow\infty}||(f^n)^*(\omega ^p)||^{1/n}.\end{aligned}$$ The cone of nef cohomology classes $H^{1,1}_{nef}(X)$ is preserved by $f^*$. Then by a result of linear algebra, there is a non-zero $\zeta \in H^{1,1}_{nef}(X)$ such that $f^*(\zeta )=\lambda _1(f)\zeta$. Note that if $\lambda _1(f)>1$ then it is irrational. Since $f^*$ preserves $H^{2}(X,\mathbb{Z})$ it follows that when $\lambda _1(f)>1$ the eigen-class $\zeta$ can not be proportional to a rational cohomology class.
The algebraic analogs of the above facts are as follows. Let $X$ be a complex projective manifold of dimension $k$. Then $X$ is Kähler and we can define dynamical degrees as above. Moreover, $f^*$ preserves $NS(X)$, and hence also preserves $NS_{\mathbb{Q}}(X)$ and $NS_{\mathbb{R}}(X)$. Since $NS_{\mathbb{R}}(X)\subset H^{1,1}(X,\mathbb{R})$ and $NS_{\mathbb{R}}(X)$ contains ample divisors, it follows that $\lambda _1(f)=$the spectral radius of $f^*:NS_{\mathbb{R}}(X)\rightarrow NS_{\mathbb{R}}(X)$. Again, there is a non-zero nef class $\zeta\in NS_{\mathbb{R}}(X)$ such that $f^*(\zeta )=\lambda _1(f)\zeta$, and if $\lambda _1(f)>1$ then such a $\zeta$ can not be proportional to an element in $NS_{\mathbb{Q}}(X)$.
Now we give the proofs of Theorems \[TheoremArqCondition\], \[TheoremBrqCondition\], \[TheoremPicardNumber1\], \[TheoremAmpleCanonicalDivisor\] and \[TheoremHyperKahler\].
We prove for the case of non-vanishing condition $A(r,q)$, the case of non-vanishing condition $NA(r,q)$ is similar.
Let $f:X\rightarrow X$ be an automorphism. We will show that $\lambda_1(f)=0$. Assume otherwise, i.e. $\lambda _1(f)>1$, and we will reach a contradiction. Let $\zeta\in H^{1,1}(X)$ be a non-zero nef class such that $f^*(\zeta )=\lambda _1(f)\zeta$. Then $\zeta $ is not proportional to a rational cohomology class, hence by the non-vanishing condition $A(r,q)$ we have $\zeta ^{k-r-1-q}.K_X^q\not= 0$. In particular, $\zeta ^{k-r-1-q}\not= 0$ and $f^*(\zeta ^{k-r-1-q})=\lambda _1(f)^{k-r-1-q}\zeta ^{k-r-1-q}$. Hence we deduce $\lambda_{k-r-1-q}(f)\geq \lambda _1(f)^{k-r-1-q}$. The log-concavity of dynamical degrees implies $$\begin{aligned}
\lambda _j(f)=\lambda _1(f)^j\end{aligned}$$ for all $0\leq j\leq k-r-1-q$. The assumption that $k-r-1-q>r+1$ and $\lambda _1(f)>1$ implies $$\begin{aligned}
\lambda _{r+1}(f)=\lambda _1(f)^{r+1}<\lambda _1(f)^{k-r-1-q}. \end{aligned}$$ Because $f$ is an automorphism, we have $f^*(K_X)=K_X$. Therefore, since $\zeta ^{k-r-1-q}.K_X^q\not= 0$ and $f^*(\zeta ^{k-r-1-q}.K_X^q)=\lambda _1(f)^{k-r-1-q}\zeta ^{k-r-1-q}.K_X^q$, we deduce that $$\lambda _{k-r-1}(f)\geq \lambda _1(f)^{k-r-1-q}>\lambda _1(f)^{r+1}=\lambda _{r+1}(f).$$ Apply the same argument to the inverse map $f^{-1}$, we have $\lambda _{k-r-1}(f^{-1})>\lambda _{r+1}(f^{-1})$. Since $\lambda _{k-r-1}(f^{-1})=\lambda _{r+1}(f)$ and $\lambda _{r+1}(f^{-1})=\lambda _{k-r-1}(f)$ we obtain $\lambda _{r+1}(f)>\lambda _{k-r-1}(f)$. But the last inequality is contradict to the inequality $\lambda _{k-r-1}(f)>\lambda _{r+1}(f)$ which we obtained before, and we have a contradiction. Therefore $\lambda _1(f)=1$ as wanted.
We prove for the case of the non-vanishing condition $B(r,q)$, the case of $NB(r,q)$ is similar.
From Theorem \[TheoremArqCondition\], we already know that if $f\in Aut(X)$ then $\lambda _1(f)=1$. Replacing $f$ by an iterate of $f$ if needed, we can assume that all of the eigenvalues of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$ are $1$. To show that $Aut(X)$ has only finitely many connected components, it suffices to show that the size of the largest Jordan block of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$ is $1$.
We follow Steps 2-4 of the proof of Theorem 1.1 in [@bayraktar-cantat]. For each $p=1,\ldots ,k$, we choose a norm $||.||_p$ on $H^{p,p}(X)$, and define $$\begin{aligned}
m_p(f)=\lim _{n\rightarrow \infty}\frac{\log ||(f^*)^n||_p}{\log n}. \end{aligned}$$ Then to prove Theorem \[TheoremBrqCondition\], it suffices to show that $m_1(f)=0$.
First, we observe that the map $p\mapsto m_p(f)$ is concave, i.e. $m_{p-1}(f)+m_{p+1}(f)\leq 2m_p(f)$. This can be proved as proving the log-concavity of the dynamical degrees of $f$, see [@bayraktar-cantat] for more details.
Now we show that $m_1(f)=0$. Otherwise, then $m_1(f)\geq 1$ and we will obtain a contradiction. If we let $\omega$ be a Kähler class, then the sequence $n^{-m_1(f)}(f^*)^n(\omega )$ converges to a non-zero nef class $u$. By the non-vanishing condition $B(r,q)$, it follows that $u^{k-r-1-q}.K_X^q\not= 0$. In particular, $u^{k-r-1-q}\not= 0$. Hence $$\begin{aligned}
\lim _{n\rightarrow\infty}n^{-m_1(f)(k-r-1-q)}(f^*)^n(\omega ^{k-r-1-q})=u^{k-r-1-q}\not= 0.\end{aligned}$$ In particular $m_{k-r-1-q}(f)\geq (k-r-1-q)m_1(f)$, and the concavity of $p\mapsto m_p(f)$ implies that $m_j(f)=jm_1(f)$ for all $0\leq j\leq k-r-1-q$. Since $k-r-1-q>r+1$ and we assume that $m_1(f)>0$ we have $$\begin{aligned}
m_{r+1}(f)=(r+1)m_1(f)<(k-r-1-q)m_1(f).\end{aligned}$$
We also have $$\begin{aligned}
\lim _{n\rightarrow\infty}n^{-m_1(f)(k-r-1-q)}(f^*)^n(\omega ^{k-r-1-q}.K_X^q)=u^{k-r-1-q}.K_X^q\not= 0.\end{aligned}$$ Thus $m_{k-r-1}(f)\geq (k-r-1-q)m_1(f)>m_{r+1}(f)$. Apply the same argument to $f^{-1}$ we obtain $m_{k-r-1}(f^{-1})>m_{r+1}(f^{-1})$. But by Poincare duality, we have $m_{k-r-1}(f^{-1})=m_{r+1}(f)$ and $m_{r+1}(f^{-1})=m_{k-r-1}(f)$, and obtain a contradiction. Therefore we must have $m_1(f)=0$, as wanted.
Since $X_0$ has Picard number $1$, if $\zeta \in NS_{\mathbb{R}}(X_0)$ is nef and non-zero then it is ample, and hence $\zeta ^k\not= 0$. Hence $X_0$ satisfies the non-vanishing condition $B(r,0)$ for any $r\geq 0$.
1\) Since the submanifolds $V_1,\ldots ,V_m$ are pairwise disjoint, arguing as in proof of Lemma \[LemmaP4\], it suffices to consider the case where $\pi :X_1\rightarrow X_0$ is a single blowup along a smooth manifold $V$ of a fixed dimension $dim(V)$. We now need to show that for a generic choice of $V$, then $X_1$ satisfies the non-vanishing condition $NB(1,0)$, i.e. if $\zeta \in Nef(X)=H^{1,1}(X)\cap NS_{\mathbb{R}}(X)$ is non-zero then $\zeta ^{k-2}\not= 0$. Let $H\in NS_{\mathbb{R}}(X)$ is an ample divisor, normalized so that $H^{dim (V)}|V=1$. Then we may assume that $\zeta =\pi ^*(H)+bE$ for some real number $b$ (we do not need to use the fact the $b$ is non-positive here). Note that $b$ can not be zero, because then $\pi ^*(H^{k-2})=0$, which is absurd.
We consider two cases:
Case 1: $dim(V)\leq 1$. Pushing forward $\zeta ^{k-2}=0$ by $\pi $ we find that $H^{k-2}=0$ which is absurd since $H$ is ample.
Case 2: $dim(V)\geq 2$ and $V$ is a complete intersection of smooth hypersurfaces $D_1,\ldots ,D_t$ here $t=$ codimension of $D$ is fixed, where $D_j=d_jH$ in $NS_{\mathbb{R}}(X)$ for positive rational numbers $d_j$. We now show that for a generic choice of $d_1,\ldots ,d_t$ then there is no $b\in \mathbb{R}$ such that $(\pi ^*(H)+bE)^{k-2}=0$. In fact, if $(\pi ^*(H)+bE)^{k-2}=0$ then both $(\pi ^*(H)+bE)^{k-2}.E^2=0$ and $(\pi ^*(H)+bE)^{k-2}.\pi ^*(H)E=0$.
We expand these two equations explicitly in several first terms $$\begin{aligned}
0&=&(\pi ^*(H)+bE)^{k-2}.E^2=b^{k-2}E^k+C(1,k-2)b^{k-3}E^{k-1}.\pi ^*(H)\\
&&+C(2,k-2)b^{k-4}E^{k-2}.\pi ^*(H^2)+C(3,k-2)b^{k-5}E^{k-3}.\pi ^*(H^3)\\
&&+C(4,k-2)b^{k-6}E^{k-4}.\pi ^*(H^4)\ldots \\
0&=&(\pi ^*(H)+bE)^{k-2}.E.\pi ^*(H)=b^{k-2}E^{k-1}.\pi ^*(H)+C(1,k-2)b^{k-3}E^{k-2}.\pi ^*(H^2)\\
&&+C(2,k-2)b^{k-4}E^{k-3}.\pi ^*(H^3)+C(3,k-2)b^{k-5}E^{k-4}.\pi ^*(H^4)+\ldots \end{aligned}$$
Observation 1: Note also that in the polynomial $(\pi ^*(H)+bE)^{k-2}.E^2$, the coefficients of $b^{k-2-j}$ (where $j>dim(V)$) are zeros (because then $\pi _*(E^{k-j})=0$). Similarly, in the polynomial $(\pi ^*(H)+bE)^{k-2}.\pi ^*(H)E$ the coefficients of $b^{k-2-j}$ (where $j>dim (V)-1$) are zeros.
Using Observation 1 and that $b\not= 0$, we define two polynomials $$\begin{aligned}
f(b)&:=&\frac{1}{b^{k-2-dim(V)}}(\pi ^*(H)+bE)^{k-2}.E^2\\
g(b)&:=&\frac{1}{b^{k-1-dim(V)}}(\pi ^*(H)+bE)^{k-2}.E.\pi ^*(H).\end{aligned}$$
We deduce that any value $b$ for which $(\pi ^*(H)+bE)^{k-2}=0$ must be a common zero of $f(b)$ and $g(b)$. Let $\mathcal{E}=N_{V/X_0}$ is the normal vector bundle of $V$ in $X_0$. Then as in Remark 1 at the end of Section \[SectionBlowupsAndNonVanishingConditions\], we see that the coefficients of $f(b)$ and $g(b)$ can be described in terms of Chern classes $c_j(\mathcal{E})$. Since $V$ is the complete intersection of $D_1,\ldots ,D_t$ and $D_j=d_jH$ in $NS_{\mathbb{R}}(X)$, it follows from Example 3.2.12 that $c(\mathcal{E})=(1+d_1H)\ldots (1+d_tH)$. From this we see that $$c_j(\mathcal{E}).H^{dim (V)-j}|_V=s_j(d_1,\ldots ,d_t)H^{dim(V)}|_V,$$ where $s_j(d_1,\ldots ,d_t)$ is the $j$-th elementary symmetric function of $d_1,\ldots ,d_t$. Therefore the coefficients of $f(b)$ and $g(b)$ are polynomials in variables $d_1,\ldots ,d_t$.
Now these two polynomials $f(b)$ and $g(b)$ has at least one common solution if and only if their resultant $R(f,g)=0$. Since the coefficients of $f(b)$ and $g(b)$ are polynomials in variables $d_1,\ldots ,d_t$, this resultant $R(f,g)$ is also a polynomial in variables $d_1,\ldots ,d_t$. Hence either $R(f,g)$ is zero identically or it is non-zero for a generic choice of $d_1,\ldots ,d_t$. Thus if we can show that $R(f,g)\not= 0$ for a special choice of rational numbers $d_1,\ldots ,d_t$ then 1) is proved. Here we only use that $f(b)$ and $g(b)$ are polynomials of $b$ whose coefficients are fixed polynomials in variables $d_1,\ldots ,d_t$, and not the fact that they were constructed from some submanifolds $V$ of $X_0$. Hence we do not need to choose $d_1,\ldots ,d_t$ to be positive numbers, hence the choice may not correspond to any actual submanifold $V$. The special values we choose now is $d_1=1$ and $d_2=\ldots =d_t=0$. We will show that for this choice then $R(f,g)\not= 0$, and it is the same as showing the two polynomials $f(b)$ and $g(b)$ has no common zero. As stated before, this case does not correspond to any actual $V$, but it does not affect the argument below, $f(b)$ and $g(b)$ are formally constructed from the Chern classes of $V$ not from $V$ itself. Hence we assume that $d_1=1$ and $d_2=\ldots =d_t=0$ correspond to a (virtual) manifold $V$. This means that $V$ is a (virtual) manifold of codimension $t$ having $c_0(\mathcal{E})=1,$ $c_1(\mathcal{E})=H|_V$ and other Chern classes are zeros, and we ignore the fact that such a $V$ can not be a complete intersection when $t>1$ or can exist at all.
(This choice of $d_1=1$ and $d_2=\ldots =d_t=0$ can actually be made rigorous as follows. Assume that for any $V$ which is a complete intersection, then the two polynomials $f_V(b)$ and $g_V(b)$ has a common solution. We choose in particular $d_1=d$, and $d_2=\ldots =d_t=1$, here $d$ can be as large as we desire. The idea is to take the limit when $d$ goes to $\infty$. Note that when $d$ is large enough then $\pi _*(E^{t+j})$ is approximately $c_1(\mathcal{E})^j$ and hence is approximately $d^jH^j$, as can be seen from the computations in Example 1 in Section \[SectionBlowupsAndNonVanishingConditions\]. If we rescale $\widetilde{f}_V(b)=d^{dim(V)-2}f_V(b/d)$ and $\widetilde{g}_V(b)=d^{dim(V)-1}g_V(b/d)$, then we see that the polynomials $\widetilde{f}_V(b)$ and $\widetilde{g}_V(b)$ have bounded coefficients, of bounded degrees, and have top coefficients bounded away from zero. Therefore, since they have a common solution for any choice of $d_1=d$, the same is true for their limits. Their limits are exactly the polynomials $f(b)$ and $g(b)$ corresponding with the choice of $d_1=1$ and $d_2=\ldots =d_t=0$.)
Then as in Example 2 at the end of Section \[SectionBlowupsAndNonVanishingConditions\], we have the defining equation for $H^*(E)$ is $e^t=\pi _E^*(H|_V)e^{t-1}$. Then we can use the computations in Example 2 to show that $$\begin{aligned}
f(b)&=&\frac{1}{b^{k-2-dim(V)}}\times\\
&&[b^{k-2}+C(1,k-2)b^{k-3}+C(2,k-2)b^{k-4}+C(3,k-2)b^{k-5}+C(4,k-2)b^{k-6}\ldots ],\\
g(b)&=&\frac{1}{b^{k-1-dim(V)}}\times\\
&& [b^{k-2}+C(1,k-2)b^{k-3}+C(2,k-2)b^{k-4}+C(3,k-2)b^{k-5}+\ldots ].\end{aligned}$$ Here we use the convenience (see the definition of $f$ and $g$) that the coefficients for $b^{k-2-j}$ in the bracket for $f$ are zero when $j>dim (V)$, and the coefficients for $b^{k-1-j}$ in the bracket for $g$ are zero when $j>dim (V)$. Now it is easy to arrive at the proof of 1). We present in the below the proof for the cases $dim(V)=0,1,2,3$, the proofs for other cases are similar and hence are omitted.
Case $dim(V)=0$: In this case, the equations $f(b)=0$ and $g(b)=0$ becomes $$\begin{aligned}
0&=&f(b):=1\\
0&=&g(b):=0,\end{aligned}$$ and this system has no solution.
Case $dim(V)=1$: In this case, the equations $f(b)=0$ and $g(b)=0$ becomes $$\begin{aligned}
0&=&f(b):=b+C(1,k-2)\\
0&=&g(b):=1,\end{aligned}$$ and this system has no solution.
Case $dim(V)=2$: In this case, the equations $f(b)=0$ and $g(b)=0$ becomes $$\begin{aligned}
0&=&f(b):=b^2+C(1,k-2)b+C(2,k-2)\\
0&=&g(b):=b+C(1,k-2).\end{aligned}$$ Since $f(b)=bg(b)+C(2,k-2)$ it follows that this system has no solution.
Case $dim(V)=3$: In this case, the equations $f(b)=0$ and $g(b)=0$ becomes $$\begin{aligned}
0&=&f(b):=b^3+C(1,k-2)b^2+C(2,k-2)b+C(3,k-2)\\
0&=&g(b):=b^2+C(1,k-2)b+C(2,k-2).\end{aligned}$$ Since $f(b)=bg(b)+C(3,k-2)$ it follows that this system has no solution.
2\) and 3) follows from 1) and Theorems \[TheoremBrqCondition\] and \[TheoremEk\].
1\) Let $r=k-l-3$ and $q=2l+3-k$. By assumption we have $r,q\geq 0$. Observe also that $k-r-1-q=k-l-1=r+2>r+1$. By assumption, if $\zeta \in NS_{\mathbb{R}}(X_0)$ is nef and non-zero then $\zeta ^{k-r-1-q}=\zeta ^{k-l-1}\not= 0$. Because $K_{X_0}$ is anti-ample, we have $\zeta ^{k-r-1-q}.K_{X_0}^q$ is also non-zero. Hence we can apply Theorems \[TheoremBrqCondition\] and \[TheoremEk\] i).
2)
i\) We observe that $K_{X_0}$ is anti-ample. Hence applying 1), it suffices to show that $X_0$ satisfies the non-vanishing condition $A(k-l-1,0)$. Let $\zeta \in H^{1,1}_{nef}(X_0)$ be non-zero and such that $\zeta ^l=0$, we will show that $\zeta $ is proportional to a rational cohomology class. Since $l\leq k_1+k_2$ this would follow if we can show the claim for $l=k_1+k_2$. To this end, first we observe that $\zeta =\sum _{j=1}^m \pi _j^*(\zeta _j)$ where $\zeta _j\in H^{1,1}(\mathbb{P}^{k_j})$ are nef, and $\pi _j:X_0\rightarrow X_{0,j}$ are the projections. Since $$\begin{aligned}
0=\zeta ^{k_1+k_2}\geq \sum _{i<j}\pi _i^*(\zeta _i^{k_1}).\pi _j^*(\zeta _j^{k_2}),\end{aligned}$$ and $k_i\geq k_1$ and $k_j\geq k_2$ in the above, we have $$\begin{aligned}
\pi _i^*(\zeta _i^{k_1}).\pi _j^*(\zeta _j^{k_2})=0\end{aligned}$$ for any $i<j$. It follows that there is at most one index $i_0$ such that $\zeta _{i_0}\not= 0$. Then $\zeta =\pi _{i_0}^*(\zeta _{i_0})\in \pi _{i_0}^*(H^{1,1}(\mathbb{P}^{k_{i_0}}))$ is proportional to a rational cohomology class.
ii\) Similar to the proof of i).
For the first part of the theorem, it suffices to show that $X_0$ satisfies the non-vanishing condition $B(l-1,0)$. The latter is a result of Verbitsky, see [@verbitsky].
We end the proof showing $\lambda _{2l-p}(f)=\lambda _p(f)=\lambda _1(f)^p$ for any $0\leq p\leq l$. In [@oguiso1], this was proved by first showing that $\lambda _1(f)=\lambda _1(f^{-1})$ using special properties of automorphisms on hyper-Kähler manifolds. However, this can be proved directly for any compact Kähler manifold of even dimension $k=2l$ satisfying the condition $A(l-1,0)$ as follows. It suffices to show the claim for $\lambda _1(f)>1$. In this case, the eigenvector $\zeta$ for $\lambda _1(f)$ is not rational, hence $\zeta ^{l}\not= 0$. Therefore $\lambda _p(f)=\lambda _1(f)^p$ for $0\leq p\leq l$, by the log-concavity of dynamical degrees. Similarly, for $0\leq p\leq l$, we have $\lambda _{2l-p}(f)=\lambda _p(f^{-1})=\lambda _1(f^{-1})^p$. Because $l=2l-l$, we have $\lambda _l(f)=\lambda _{l}(f^{-1})$ by Poincare duality. Therefore, $$\lambda _1(f^{-1})^l=\lambda _l(f^{-1})=\lambda _l(f)=\lambda _1(f)^l,$$ which implies $\lambda _1(f^{-1})=\lambda _1(f)$, and the proof is completed.
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abstract: |
Proceeding from a homogeneous and isotropic Friedmann universe a conceptional problem concerning light propagation in an expanding universe is brought up. As a possible solution of this problem it is suggested that light waves do not scale with $R(t)$. With the aid of a Generalized Equivalence Principle a cosmologic model with variable “constants" $c$, $H$, and $G$ is constructed. It is shown that with an appropriate variation of the Boltzmann “constant" $k$ the thermal evolution of the universe is similar to the standard model. It is further shown that this model explains the cosmological redshift as well as certain problems of the standard model (horizon, flatness, accelerated expansion of the universe).
PACS numbers: 98.80.Bp, 98.80.Hw, 04.20.Cv.
Keywords: cosmology, velocity of light, expansion
author:
- |
Peter Huber\
Germanistisches Seminar\
University of Heidelberg\
Hauptstr. 207-209\
D-69117 Heidelberg\
Germany\
pethk@aol.com
title: |
A Cosmologic Model Based on the Equivalence of Expansion and Light Retardation\
Part 1: Large-Scale Aspects
---
Introduction
============
There is no doubt that the standard model of cosmology is the most successful approach in describing the universe as a whole while accounting for numerous empirical data provided by macro- and microphysical observations. Yet, the standard model causes some complications, such as an initial singularity, a flatness, horizon and density fluctuation problem, and some more. Although some of them could be avoided by re-introducing the cosmological constant $\Lambda$, which grants a variety of models like inflationary, extended or hyperextended expansion, the original simplicity and beauty of the model has disappeared.
In this paper it will be attempted to show that with the aid of a simple yet physically productive principle of the same category as the Equivalence or the Cosmological Principle it is possible to obtain an elementary description of the universe while avoiding the disadvantages of the standard model. The validity of the Cosmological Principle is assumed. Thus the scope is limited to Friedmann-Robertson-Lemaître-universes, which are based on the principles of spatial isotropy and homogeneity of the universe.
A Problem of Light Propagation in Friedmann universes
=====================================================
Among the various problems of the standard Friedmann universe there is one which had been mentioned occasionally in the past but which had not been paid full attention so far. It concerns the dependence of the dielectric and magnetic constants $\varepsilon_0$ and $\mu$ on the gravitational potential $\Phi$. The Maxwell equations in a constant gravitational field imply the relation $$\varepsilon = \mu = (1+2\Phi /c^2)^{-1/2} \label{0}$$ (M[ø]{}ller \[1969,1972\], Landau/Lifshitz \[1962,1975\]). In cosmology one is interested in large scale [*eigen*]{} gravitation of the universe rather than in local gravitational fields. It arises the question, in how far a universal $\Phi$ could vary. In Newtonian terms, the potential $\Phi_E$ of the proper gravitation of the universe can be written as $$\Phi_E = - M_u G / R \label{0a}$$ where $M_u$ ist the universal mass, $G$ the gravitational constant and $R$ the extension (radius) of the universe. In Friedmann models, $R$ is increasing as a function of time. This leads to a decreasing $\Phi_E(t)$, if one does not postulate [*ad hoc*]{} assumptions between $M_u$, $G$ and $R$ to keep the expression constant. According to eq. (\[0\]) we then obtain time-dependent “constants" $\varepsilon (t)$ and $\mu (t)$ with increasing values. (A detailed mathematical analysis on the variation of vacuum permittivity in Friedmann universes was given by Sumner \[1994\].) Because $(\varepsilon\mu)^{-1/2} = c$ the velocity of light should also decrease. We should become familiar with the idea, that a variable $c$ is not a heresy. Even Einstein (1911) was ready to give up the absolute constancy of light according to Special Relativity when he worked on the influence of gravitation on light propagation. In this paper he developed following equation which shows the velocity of light as a function of the gravitational potential $c = c_0 (1+\Phi / c^2)$. Years later he discovered, that it is more comfortable to keep $c$ constant and to interprete the elapsed time as a function of $\Phi$. This led to the present form of General Relativity. However, $\Phi$ in Einstein’s equation was restricted to local gravitational fields, which were regarded constant. Different values of $\Phi$ result in different lapses of time. Friedmann’s solutions of the gravitaional equations in accordance with Hubble’s observations of cosmological red shifts imply, as we have seen, a variable $\Phi_E(t)$. This is a completely different situation. Just like Einstein in the 1910’s we have the choice to interprete a globally decreasing $\Phi_E$ either as a global decrease of the speed of light or in the sense of varying universal time. Both alternatives, which base on mathematical (i.e. “absolute") scales, should be equivalent. However, the mere idea that reference scales could vary causes us to avoid the concept of absolute units and to rely on physical scales as it is practically done by defining space and time by electromagnetic (i.e. physical) processes. That means in particular that reference scales are not defined seperately but in relation to each other. The following equation (next section, eq.\[\[1\]\]) can be understood in this sense.
A New Mechanism of Light Propagation and its Relation to an Expanding Universe
==============================================================================
We define a relative decrease of the velocity of light with reference to an expanding universe. This implies the following mechanism of light propagation. Let there be a source $E$ emitting with a (mathematically) constant frequency $\nu$. Since $c$ decreases continuously, the wave-length $\lambda$ decreases proportionally to $c$ at emission. While traveling through the universe, however, $\lambda$ shall remain constant. Namely, it shall not expand with $R(t)$ as in the standard model. The assumption of constant $c$ and non-expanding light waves over cosmic distances means a retarded arrival of light in an expanding universe. To illustrate that point: Let the distance between an emission source $E$ and an observer $O$ at time $t_0$ be $m\lambda$ of a defined wave-length. At time $t_1 > t_0$ the distance $EO$ has expanded with $R(t)$, however, not the single $\lambda_r$. Therefore the distance at $t_1$ in terms of the light wave will be $n\lambda_r$ (with $n>m$). The runtime of a light beam with constant speed starting at $t_1$ will be longer than light that left $E$ at $t_0$. This problem casts a new light on the relation of expansion and the constancy of light: In the standard model a huge ominous force is needed to maintain expansion against gravity (vacuum energy, cosmological constant and quintessence are synonyms of possible explanations). What, if the nature of universal expansion were that in terms of decreasing wave-lengths $\lambda$ at emission? Then there would be no more necessity for a repulsive force, which maintains expansion. Usually length is defined by an invariable rigid rod. This is, however, a mathematical definition, because [*in praxi*]{} on has to utilize a physical reference scale like defined wave-lengths. There is no way for us to find out whether the universe is “really" expanding or whether the speed of light is “really" decreasing. Both phenomena are conditioning each other. This new principle of the equivalence of space expansion and light retardation can be formulated $$Rc = const. \label{1}$$ where $c$ is the seemingly retarded velocity of light in an expanding universe and $R$ is from now on regarded as the variable radius of the universe in the unit of meter (whereas the scale factor $a$ is dimensionless).
As the term “equivalence principle" is applied to inert and heavy masses, the meaning of eq.(\[1\]) will be referred to as “Indiscernibility Principle" (IP). It has to be mentioned that the IP is not an additional assumption; it just replaces $c=const.$ of standard cosmology. (By the way: This concept avoids the paradox situation of a light beam from a distant region of the universe entering a galaxy cluster: Its waves should suddenly cease expansion and carry on with it when leaving the cluster.)
To demonstrate the mathematical aspects of the IR (eq.\[1\]) we regard light propagation in Robertson-Walker-Metrics (RWM). Because of homogeneity and isotropy of space we can regard a light trajectory with $\chi (t), \theta = const., \phi = const.$ which reaches from $\chi(t_1) = 0$ to $\chi(t_0) = \chi$. Then the RWM reduces to $$ds^2 = c^2dt^2-R(t)^2d\chi^2 = 0 \iff d\chi = cdt/R(t)
\label{1a}$$ We now regard two subsequent wave peaks. Both have to travel the same distance from the source to the receiver (from $0$ to $\chi$): $$\chi = \int_{t_1}^{t_0} {c(t)dt\over R(t)} =\int_{t_1+\delta
t_1}^{t_0+\delta t_0}{c(t)dt\over R(t)} \label{1b}$$ The spatial distance of the two subsequent wave peaks is $\lambda = c/\nu$. The temporal distance $\delta t$ relates to the frequency $\nu$ via $$\delta t = 1/\nu \label{1c}$$ From eq.(\[1b\]) follows $$0 = \int_{t_0}^{t_0+\delta t_0}{c(t)dt\over R(t)} -
\int_{t_1}^{t_1+\delta t_1}{c(t)dt\over R(t)} = {c(t_0)\delta
t_0\over R(t_0)} - {c(t_1)\delta t_1\over R(t_1)} \label{1d}$$ (Because $\delta \ll \chi$ we can regard $R(t)$ in the integration interval as constant.) From (\[1c\]) and (\[1d\]) we have for the emitted frequency $\nu_1$ and the received frequency $\nu_0$ $$R(t_0)\nu_0 c(t_1) = R(t_1)\nu_1 c(t_0) \label{1e}$$ The standard model assumes $c=const.$ in (\[1d\]) and (\[1e\]) and concludes $R\nu = const.$ We see, however, that “stretching" waves are in general not an implication of the RWM but rather of the assumption of a constant $c$.
While in the standard model expansion and light propagation can be imagined within the szenario of an expanding balloon, the suggested mechanism of light propagation corresponds to the conveyor belt mechanism where a pen swinging rectangular to the belt’s movement, drawing waves upon it. The product of the pen’s frequency and the wave-length gives the speed of the belt identified with $c$. A diminishing speed of the belt reduces the wave-lengths at emission or the other way round: reduced wave-lengths caused by the expansion discrepancy of $R$ and $\lambda$ at constant swinging frequency diminishes the product $c$. The particle version is similar: A photon is placed with constant frequency on the decelerating belt.
This Retarded Light Model will in the following serve to treat the cosmological questions. (Of course the balloon picture can be further maintained if one is aware that in this model space expansion results in the fact that the coordinate system does not co-expand.)
Determination of Time Variabilities of Cosmological Quantities $H$, $c$ and $R$
===============================================================================
According to the Indiscernibility Principle (IP) it makes no sense to propagate either a “real" expanding universe or a “real" decreasing speed of light. The mathematical treatment combines both properties: Eq.(\[1\]) can be written $\dot R /R = -\dot c /c$. With the use of $\dot R/R = H$, where $H$ represents the Hubble parameter (rather than Hubble constant) we obtain the expression $${\dot R\over R} = H = -{\dot c\over c} \label{2}$$ We can write $R$ as a function of $c(t)$ and $H(t)$: $R=c/H$ and obtain $$Rc = c^2/H = const. \label{3}$$ (In the following time-dependent quantities like $R(t)$, $c(t)$ are abbreviated $R$, $c$, and so on. Certain value are characterized by indices like $R_0$.)
From (\[3\]) the time dependence of $H$ can be calculated: $${d\over dt}\left ({c^2\over H}\right ) = 0 \label{4}$$ and resolved to $\dot H$: $$\dot H = -2H^2 \label{5}$$
In regard of (\[2\]) and (\[5\]) the propagation of the universe is $$\dot R = c\label{6}$$ This was formerly deduced by Milne (1948) in the frame of his Kinematic Relativity. Differentiation of (\[6\]) gives according to (\[2\]) $$\ddot R = \dot c = -Hc \label{7}$$
From (\[5\]) we can derive $H(t)$ by seperating of variables and integration: $$H(t) = {1\over 1/H_0 +2t}={H_0\over 1+2H_0 t}\label{7a}$$
From (\[7\]) and (\[7a\]) the temporal variation of $c$ (in an expanding universe) is $$c(t) = {c_0\over\sqrt {1+2H_0t}} \label{7b}$$
Now we can specify the expansion of the universe. Its radius $R$ has a time dependence of $$R(t) = {c(t)\over H(t)} = {c_0\over H_0} \sqrt{1+2H_0t} =
R_0 \sqrt{1+2H_0t} \label{7c}$$
Cosmological Redshift in the Retarded Light Model
=================================================
Standard cosmology explains the cosmological redshift by expansion of $\lambda$ while the light beam travels a distance $r$. In the Retarded Light Model (RLM) a certain wave-length remains, once emitted, unchanged. However, $\lambda$ is a function of time. The earlier it was emitted, the faster $c$ and thus the larger $\lambda$ had been. When we observe a spectral redshift, this is because the wave was emitted much earlier in the history of the universe compared to the same electromagnetic process observed at present on our planet. In the following a simple deduction, based on conventional terms, is given.
Hubble’s law, as it is commonly referred to, is $$v = Hr \label{8}$$ Since the transmission speed of photons is finite we can express any distance $r$ by the runtime of light.
$$r = ct \label{9}$$
(Note that in standard cosmology $c$ is constant while in the RLM it is a function of time here. As it is calculated with functions and not with certain values, the expressions can be left unintegrated at the present state.)
Eqs. (\[9\]) and (\[8\]) give $$v = Hct \label{10}$$ Applying (\[10\]) to the expanding universe, $v$ means the receding velocity of cosmic objects and $t$ the time their light has taken to reach us. Division by $t$ gives the mean acceleration $\bar a$ of cosmic objects in dependence of their distance (standard interpretation) or, according to the RLM equivalence principle eq. $(\ref{1}$), the deceleration of $c$: $$-v/t = - \bar a = \dot c = -Hc \label{11}$$ Note that (\[11\]) is deduced from standard terms only, while the identical result in (\[7\]) uses a new concept of expansion based on the IP.
In the standard model one can describe the cosmological redshift in a first order approximation as a Doppler effect. This is also possible within the RLM, as first shown in Huber (1992). In the following we use the particle picture of the conveyor belt model: We regard photons which are emitted by a radiation source with a constant frequency $T_0^{-1}$ but with decreasing velocity of light. The frequency of radiation shall be the same everywhere in the universe and at all times. This condition guarantees the spatial and temporal invariance of physical (or chemical) processes and is just another formulation for the Cosmological Principle stating the homogeneity of the universe. The initial speed of the photon $n$ shall be $c_{0n}$. The index $n$ means that the initial speed $c_0$ decreases with time. We define a very first photon with the speed $c_{00}$. Then the varying initial speeds can be written $$c_{0n} = c_{00}-anT_0 \label{12}$$ where $a$ is the (negative) acceleration. At time $t>nT_0$ the $nth$ photon has the speed $$c_n(t)=c_{0n}-a(t-nT_0)=c_{00}-at \label{13}$$ At time $t$ photon $n$ has travelled a distance $$r(t) = \int_{nT_0}^t c_n(t')dt'=c_{00}t-{a\over 2}t^2 -
c_{00}nT_0 + {a\over 2}(nT_0)^2 \label{14}$$ This equation can be resolved for the time $t_r^{(n)}$ at which photon $n$ has travelled the distance $r$: $$\begin{aligned}
t_r^{(n)} & = & {1\over a}\left ( c_{00}-\sqrt{c_{00}^2 +
a^2(nT_0)^2-2ac_{00}nT_0-2ar}\right ) \nonumber\\
& = & {1\over a} \left ( c_{00}-(c_{00}-anT_0)
\sqrt{1-{2ar\over (c_{00}-anT_0)^2}}\right ) \label{15}\end{aligned}$$ where the positive root is excluded because one must have $t_r^{(n)} = nT_0$ for $r=0$. The time interval $T_r^{(n)}$ between the arrival of two photons $n$ and $n+1$ at the observer at distance $r$ is $$T_r^{(n)}=t_r^{(n+1)}-t_r^{(n)} \label{16}$$ Inserting (\[15\]) in (\[16\]) we get $$T_r^{(n)}=-{1\over a}\left \{\begin{array}{c}
(c_{00}-a[n+1]T_0) \sqrt{1-{2ar\over (c_{00}- a[n+1]T_0)^2}}
\\
-(c_{00}-anT_0) \sqrt{1-{2ar\over
(c_{00}-anT_0)^2}}\hfill\end{array}\right \} \label{17}$$
This time-distance-relation gives the absorbtion interval $T$ in dependence on the distance $r$ from the radiation source and the (absolute) time $nT_0$.
To obtain a Doppler interpretation we restrict (\[17\]) to relatively small distances $r$. (We know from standard cosmology, that the Doppler interpretation of the cosmological redshift is only valid for distances $r<0.5 R_0$.) Then we can expand (\[17\]) in powers of ${2ar\over (c_{00}-anT_0)^2} \ll 1$: $$T_r^{(n)}\approx -{1\over a}\left \{-aT_0-ar\left ( {aT_0\over
c_{00}^2\left[1-{a(2n+1)T_0\over c_{00}}+ {a^2n(n+1)T_0^2\over
c_{00}^2}\right ]}\right ) \right \} \label{18}$$ Under the additional assumption $c_{00}\gg anT_0$ this expansion leads to $$T_r^{(n)}\approx T_0\left( 1+{ar\over c_{00}^2}\right )
\label{19}$$ or written with frequency $\nu_r = T_r^{-1}$: $$\nu_r\approx {\nu_0\over 1+{ar\over c_{00}^2}} \label{20}$$
The classical Doppler effect for the frequency of light escaping from a source moving with velocity $v$ is approximately given by $$\nu = {\nu_0\over 1+{v\over c}} \label{21}$$ Interpreting the Hubble flow as Doppler redshift we have to replace $v$ by $Hr$ and get: $$\nu = {\nu_0\over 1+{Hr\over c}} \label{22}$$ Comparing (\[22\]) with (\[20\]) we immediately find that both expressions are equal for $c_{00} = c$ and $ar/c=Hr$ or $$a = Hc \label{23}$$ This calculation shows that within the Retarded Light Model the cosmological redshift can be interpreted as a Doppler effect just like in the standard model. The acceleration parameter $a$ has been introduced negatively in (\[12\]) so it bears no explicit sign. It can be associated with $-\dot c$. The result of (\[23\]) represents another independent determination of the results of (\[7\]) and (\[11\]).
There is yet another method to obtain this result. In the RWM we have for the distance D between an emitter and an observer the following relation, restricted to the first two powers of a Taylor expansion: $$D = D(t_0) = R(t_0)\chi \simeq c(t_0-t_1) + {Hc\over 2}
(t_0-t_1)^2 \label{23a}$$ where $\chi$ is the light trajectory from a remote light source ($\chi=0$) to the observer ($\chi$). This expression is within its limits identical with the formula of accelerated movement $d=v_0t + at^2/2$. Thus we can associate the acceleration $a$ with $Hc$. Because the increase of distance corresponds to the decrease of $c$ we have $a = -\dot c = Hc$. It seems that there is hardly a chance to get around the conclusion that the cosmological redshift is caused by a light deceleration of $\dot c = -Hc$. The last three of the four presented methods to determine $\dot c$ in order to compensate for the cosmological red shift do not make use of the IP, so this relation stated above can be regarded as a consequence of the new interpretation of the observed cosmological redshift.
Conclusions from Friedmann’s equations
======================================
Varying constants $c$ and $g$ applied in General Relativity usually lead to Brans-Dicke theories. Even there the Friedmann solution holds under certain preconditions, as was shown by Albrecht/Magueijo and Barrow. The situation in the RLM is different, however. It deals with the fact that Friedmann’s equations as a solution of Einstein’s gravitational equations imply varying magnitudes like wave-lengths and distances in the universe, while there still exist non-varying scales (at least by definition). As a consequence of the Friedmann scale variation the RLM eliminates mathematically defined “absolute" scales and refers to physical processes only. So the RLM is not a super theory replacing or changing General Relativity; it is applicable within its Friedmann solution [*only*]{} and it wouldn’t make sense elsewhere. The RLM is restricted to cosmological applications, where the universe as a whole and its [*eigen*]{} gravitation play a role. The description of local gravitational effects (star or galaxy interactions, for example) must follow the unaltered Einstein laws. Once one is aware of the hierarchy General Relativity – Friedmann solution – RLM, it is obvious that the Friedmann equations can be applied unaltered. There is just one exception: the cosmological constant $\Lambda$, which mediates in the standard model between gravitation and expansion, is superfluous, since the RLM itself is the theory of mediation. Friedmann’s equations without $\Lambda$ are: $$\left ({\dot R \over R}\right )^2 = - {kc^2\over R^2} +
{8\pi\over 3}G\varrho \label{24}$$ $$2 {\ddot R \over R} = - \left ({\dot R \over R}\right )^2 -
{kc^2\over R^2} - 8\pi Gp \label{25}$$
read as follows (with $R=c/H$, $\dot R = c$ and $\ddot
R=-Hc$):
$$H^2 = -kH^2+{8\pi\over 3}G\varrho \label{26}$$
$$-2H^2 = -H^2-kH^2-8\pi Gp \label{27}$$
In the RLM the space factor $k$ depends on the relation between the density $\varrho$ and the radiation pressure $p$, however, not on the universal mass $M$. For $p=\varrho/3$ (radiation era) $k$ must be $0$ to fit both (\[26\]) and (\[27\]). Accordingly, for $p=0$ (matter dominated era) we have necessarily $k=1$. This may indicate, that a change in space structure must have happened during the evolution of the universe. Hovever, because in the RLM the extension of the universe $\dot R$ occurs with the velocity of light, the photons, also moving with $c$, cannot maintain an internal pressure of radiation. For this reason it is more likely that the universe always had $p=0$ and $k=1$.
A relation between the Hubble radius $R_H$ and the Gravitational radius $R_G$:
------------------------------------------------------------------------------
From (\[24\]) follows in respect of $H=\dot R/R,\ \varrho :=
3M/4\pi R^3$ and $R_H := R=c/H$: $${c\over H} (k+1) = {2GM\over c^2} \label{28}$$ For $k=1$ and $R_G= GM/c^2$ this leads to $$R_H = R_G \label{29}$$
Within the RLM the identity of expansion radius $R_H$ (Hubble radius) and gravitational radius $R_G$ is not a coincidence. This will be pointed out in the next section.
A relation between $H$ and $q$:
-------------------------------
Differentiation of $H=\dot R/R$: $$\dot H = {\ddot RR - \dot R^2\over R^2} = {\ddot R\over R} - H^2
= H^2\left ({\ddot R R^2\over R\dot R^2}-1\right ) \label{30}$$ With regard to $q=-\ddot RR/\dot R^2$ we have $$\dot H = - H^2(q+1) \label{31}$$
According to (\[5\]) this equation is true for the deceleration parameter $$q=1 \label{32}$$
Gravitation, Universal Mass and “Distant Masses"
================================================
For the state $k=1$ equation (\[28\]) can be solved to the universal mass $M$: $$M = {c^3\over GH} \label{33}$$
Inserting $R$ for $c/H$ we obtain the so-called “Mach principle" $${GM\over c^2 R} = 1 \label{34}$$ as it was quantitatively formulated by Sciama (1959). This is remarkable, because it shows, that within the Retarded Light Model the Friedmann universe based on Einsteins theory of gravitation is in full concordance with the Mach principle. Eq.(\[33\]) leads to the energy equivalent of the universe $$E = {c^5\over GH}\label{35}$$ Equations (\[33\]) and (\[35\]) represent the sum of condensed matter and radiation. From the principle of energy conservation $\dot E= 0$ follows $${d\over dt}\left ({c^5\over GH}\right ) = 0 \label{36}$$ With (\[5\]) and (\[7\]) we find $$\dot G = -3GH \label{37}$$ and $$G(t) = G_0 (1+2H_0t)^{-3/2} \label{38}$$ A varying gravitational “constant" was assumed by Dirac (1937, 1938) propagating his and Eddington’s (1946) “large number observations" ($10^{40}$-relations). (The first remarks on the $10^{40}$-numbers were given 1923 by Weyl.) Gravitational experiments, though, seem to have almost excluded the alteration of $G$ stated above (Hellings et al., Damour et al.). One has to consider, however, that the values of $G$ and $M$ cannot be seperated in gravitational measurements (Canuto and Hsieh). Thus we have, regarding eqs. (\[37\]) and (\[40\]) for the product $MH$ a relative decrease of $-H$. If $c$ is needed for determinations of $G$, we even have $${GM\over c} = {c^2\over H} = const. \label{39}$$
With the deduced time variation of the universal “constants" $c$, $G$ and $H$ we can calculate the time dependence of the universal mass $M_H$. According to (\[5\]), (\[7\]), (\[33\]) and (\[37\]) (respectively to (\[36\]) and $E=
Mc^2$) we get $$\dot M = 2MH = {2c^3\over G} = const. \label{40}$$ and $$M(t) = M_0(1+2H_0t) \label{41}$$ An increase of universal mass has been proposed at first by Dirac to explain a large number relation, later by Narlikar and Arp to obtain a “tired light" mechanism for a non-expanding universe. They showed that when a nucleus increases in mass, the wave-length of emitted photons decrease. This process occurs similarly in the RLM.
We now define the [*eigen*]{} gravitation of the universe $F_G$ as $$F_G:= {M^2\over R^2} G \label{42}$$ where $M^2$ represents the self attraction of the universal matter at maximal distance, the universal radius $R$. With (\[33\]) and $R_H = c/H$ we obtain $$F_G = {c^4\over G} = {8\pi\over\kappa} \label{43}$$ where $\kappa$ is representing Einstein’s gravitational “constant" in the field equations of General Relativity. In a next step we describe the entire energy of the universe $E$ exclusively as the work of its [*eigen*]{} gravitation. $$E= F_G R \stackrel{(\ref{42})}{=} {M^2GH\over c} = Mc^2
\label{44}$$ Solved to $M$, the last two terms give $M = c^3/GH$. This expression has been deduced in a different way before. Here it results from the question: How large must a mass $m$ be, that its intrinsic energy is totally described by its proper gravitation?
So far only the self attraction of the universe was concerned. How does the universal “background" gravitation affect a test mass $m$? Inserting (\[33\]) and $R=c/H$ in (\[0a\]) we obtain the gravitational potential $$\Phi = -c^2 \label{44a}$$ That means, that the entire energy of a test mass $m$ is determined by the universal gravitational potential. In Special Relativity the famous formula $E=mc^2$ was obtained by kinematic reflections. Here it follows from the [*eigen*]{} gravitation of the universe. This demonstrates full concordance of inertia and gravity. One may also call it “identity", as will be shown with the following considerations.
On the other hand we can define an acceleration force $F_a$ of the universal mass caused by light retardation: $$F_a := M\dot c = {c^3\over GH} (-Hc) = - F_G \label{45}$$ This equation illustrates the principle of the equivalence of ponderable and inertial mass. While Einstein presumed the equivalence principle to proceed from Special to General Relativity he did not provide an explanation. Such an explanation is possible within the Retarded Light model: Let all matter of the universe be located on the ${\cal
R}_3$-“surface" of a 4-dimensional space ${\cal R}_4$. As stated above, light retardation is identical with isotropic space expansion in ${\cal R}_4$, which causes an inertial force on all mass particles on ${\cal R}_3$ towards the center of ${\cal R}_4$. This inertial force is being registered as gravitation in the ${\cal R}_3$-subsphere. This can be illustrated with an analogous [*gedanken*]{} experiment reduced by one dimension: Let us imagine an air balloon in gravitation-free space, which is half-ways blown up, and let us place a metal ball somewhere on its surface. We then blow up the balloon quickly. Due to inertia the metal ball will be pressed towards the balloon surface and cause a dent. Now we repeat the experiment with two metal balls being located close to each other on the surface. Then both balls will form a common dent in which they start moving against each other just as in a gravitational field. And indeed, a 2-dimensional observer on the balloon surface ${\cal R}_2$ not noticing space expansion will describe the phenomenon as gravitation, maybe even as hypothetical but imperceptible space curvature just like Einstein. Returning to the universal situation we can say that space expansion in ${\cal R}_4$ causes inertia, which is being percepted as gravitation in ${\cal R}_3$. Gravitation, on the other hand, causes energy degradation of electromagnetical processes, which can be understood as light retardation. Finally, light retardation is via gravity identical with expanding space. Because of that identity proposed by the RLM we emphasize the point that [*gravitation causes expansion*]{}, while the standard theory expects the expansion to be [*delayed*]{} by gravitation.
A historical remark: In Newtonian physics the equivalence of inert and ponderable mass was a pure coincidence. Einstein used this fact as an Equivalence Principle. Its explanation in the sense of their identity, however, is provided by the RLM.
In a further step we can write the inert mass $F_a$ as $M\ddot
R$ and obtain from (\[45\]) the following classical (Newtonian) differential equation for the equivalence principle: $$M\ddot R + {M^2 G\over R^2} = 0 \label{47}$$ Its formal solution $R(t)$ agrees with eq.(\[7c\]). With the values for $M$, $R$ and $\ddot R$ inserted, the identity of inertia and gravity is obvious.
With the identity of $|F_a| = Hcm$ and $|F_G| = MmG/R^2
= Hcm$ the energy $E_m$ of a test mass $m$ on the ${\cal R}_4$-surface with the radius $R$ is the product $$E_m = \int |F_a| dR = \int |F_G| dR = \int Hcm dR = mc^2
\label{47aa}$$ This result completes the above considerations concerning the interaction of the universal mass with a test mass $m$.
These examples show the high degree of self-consistence of the Retarded-Light-Universe. It reveals to be highly “Machian". In the course of its elaboration the equivalence of gravity and inertia was generalized with the aid of the IP and further to the equivalence of expansion and gravitation (resp. gravitation and light retardation). Referring to all of these aspects we use the term “Generalized Equivalence Principle" (GEP).
Friedmann Variables Versus Einstein Constants - An Alternative Approach
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In the last section we have found $|\phi|=c^2$. Inserting this result in M[ø]{}ller’s expression for $\varepsilon$ and $\mu$ we obtain time-independent electromagnetical constants. With these values the velocity of light would also be constant. Is there a contradiction to the varying $c$ as found above?The answer could be: yes, but only insofar as one is willing to admit a contradiction between General Relativity and Friedmann’s equations. With the universal expansion time, with the cosmic background radiation we have an absolute reference frame, which should not exist in the Theory of Relativity. On the other hand one can have the point of view, that quantities in the frame of GR do not necessarily have the same meaning as in the frame of a Friedmann universe. When $R$ varies in the Friedmann model while it is constant in GR and others of its solutions, why should not other quantities like $c$ and $G$ can be constant in GR and [*simultaneously*]{} varying in the Friedmann universe? If one accepts this argument one can ask how the light speed in the Friedmann model $c_F$ must vary so that Einsteins $c_E$ can be kept constant. From equation (\[0\]) follows $$c_E = c_0 (1+2\Phi/ c^2)^{-1/2} \label{82}$$ The gravitational potential is given by $$\Phi = MG/R \label{83}$$ We find that with increasing $R$, which is essential in the Friedmann model, $c_E$ can be kept constant in any case if $c_F^2$ varies proportionally to $\phi$. We then obtain in the Friedmann units $$\Phi = c^2 = {MG\over R}\times const. \label{84}$$ Solved to $R$ we immediatly obtain the satisfying result $$R(t) = MG/c^2 \times const. = R_G \label{85}$$ Only from adjusting $c_F$ in a way that $c_E$ remains absolutely constant, we achieve with $const. = 1$ the identity of gravitational and expansion radius! Replacing the universal mass $M$ by its energy we have $$EG/c^4 = R \label{86}$$ This is, beside the factor $8\pi$, the contracted, direction-independent form of Einsteins gravitational equations. With $R=c/H$ we have $E=c^5/GH$ for the total energy in the universe. Since this magnitude remains constant we obtain the following differential equation: $$5{\dot c\over c} - {\dot H\over H} - {\dot G\over G}
= 0 \label{87}$$ On the other hand we see from eq. (\[86\]) that $G/c^4$ must vary proportionally to $R$. Differentiation of $G/c^4\propto R$ gives with $GH\propto c^5$ the result $${\dot G\over G} - 4{\dot c\over c} \propto H \label{88}$$ Inserting (\[87\]) in (\[88\]) yields $$\dot R\propto c \label{89}$$ or $$R = c/H \propto \int cdt \label{90}$$ Starting out from the absolute constancy of $c$ in a varying gravitational field we obtain the same results as above. To achieve a complete solution of the two differential equations above, we still need the relative variation of one of the magnitudes $c_F$, $G_F$ or $H$. With the considerations of sections 4 and 5 we adopt the relation $\dot c = -Hc$ from observation (cosmological redshift) and have the Retarded Light Model.
To avoid two values $c_E$ and $c_F$ one could tentatively regard $c$ in (\[82\]) as constant. In this case $MG/R=MGH/c$ must be constant as well. The universal mass is then constant, too, because of $E/c^2 = const.
=M$. With this we would obtain $G\propto R$ and $G\propto 1/H$. That yields for both an increasing universal radius $R$ and universal time $1/H$ the very unlikely case of an increasing gravitational “constant" $G$. Beside this, there would arise another problem. Equation (\[82\]) is recursive, since $c^{-2}$ can be replaced by $\varepsilon\mu$, and both again by eq. (\[0\]), and so forth. With the distinction of $c_E$ and $c_F$ as suggested above this recursion can be avoided.
The intention of this chapter was to demonstrate that a constant $c_E$ in General Relativity implies a variable $c_F$ in a Friedmann model with varying $R$ respectively $\Phi$. We have adjusted $c_F$ in a way that $c_E$ remained constant. This led to the solution of the horizon problem. Restricting the varying constants $c$ and $G$ to the Friedmann model we do not have to alter the Einstein equations in the sense of Brans-Dicke-theories, since in this frame these magnitudes remain constant, as shown. The original approach was based on differing mathematical and physical units in a Friedmann universe. The alternative approach lined out in this section does not scrutinize the problem of measurement but assumes different behaviour of “constants" in Einstein and Friedmann frame, such as $c_E$ and $c_F$. Both approaches yield the same results and are obviously identical.
Emission of Light and Temporal Relations
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In a previous section we have described light propagation in the universe in terms of the conveyor belt model. We have assumed a light source with constant emission frequency $\nu$. This was in fact a mathematical setting. As pointed out before, we do not rely on “ideal" scales like an [*a priori*]{} constant time. We rather use a physical process like the electron’s change from one energetic level of the atom to another. The question is, whether the frequency changes in cosmic dimensions. If so, this would affect the theoretical explanation of the observed red shift. In the Bohr model (we use the Hydrogen atom) the emitted frequency $\nu$ from the $m$th to the $n$th level is given by $$\nu = {m_e e^4\over 8\varepsilon^2h^3}\left ({1\over n^2}
- {1\over m^2}\right ) \label{47a}$$ The Planck constant $h$ and the electric charge $e$ are absolute constants in this frame (for example M[ø]{}ller \[1972\], p. 416f). The dielectric parameter $\varepsilon$ has, as mentioned above, the temporal variation $$\varepsilon(t) = \varepsilon_0\sqrt{1+2H_0t}\label{47b}$$ (The magnetic field “constant" $\mu$ has the same time dependence. Both $\varepsilon$ and $\mu$ compute the function of $c(t)$ as given in eq. \[\[7b\]\].) It remains the mass $m_e$ of the electron. If it varied proportional to the universal mass $M$ (eq. (\[41\]), then the emission frequency of electromagnetic radiation would be constant, so that the mathematical time would be identical with the physical time. However, there is evidence that the electronic mass varies slightly (see part 2, section 5).
Regarding the conveyor belt model it is obvious that the observed frequency must decrease with runtime respectively distance from the source, since a remote observer receives longer wave-lengths being emitted in earlier times, while the speed of light is identic everywhere in three-space ${\cal R}_3$. We determine the variation of $\nu$ in dependence of the distance from the source. Inserting (\[1\]) in (\[1e\]) yields: $$R(t) \nu(t)/c(t) = const. \label{47c}$$ According to the values for $R(t)$ and $c(t)$ we have a variation of $\nu \propto 1/(1+2H_0t)$. This result can be evaluated more precisely. According to (\[1d\]) we have $$\int_{t_0}^{t_0+\delta t_0}{c(t)dt\over R(t)} -
\int_{t_1}^{t_1+\delta t_1}{c(t)dt\over R(t)} = 0 \label{47d}$$ Integration yields $$ln\left ({[1+2H_0(t_0+\delta t_0)](1+2H_0t)\over
(1+2H_0t_0)[1+2H_0(t_1+\delta t_1)]}\right ) = 0 \label{47e}$$ Because the counter must equal the denominator we obtain the relation $(1+2H_0t_0)/\delta t_0 = (1+
2H_0t_1)/\delta t_1$ or, according to (\[1c\]), $$\nu_0(1+2H_0t_0) = \nu_1(1+2H_0t_1) \label{47f}$$ Setting $\nu_0$ for $t_0=0$ and $\nu (t)$ for $\nu_1$ we obtain the expression $$\nu (t) = {\nu_0\over 1+2H_0t} \label{47g}$$ This equation corresponds to the situation of an observer receiving light from sources of various distances at the same time, as it is the case when looking at the nightly sky. Comparing this result with eq. (\[7a\]) we see that this delay in frequency as a function of runtime (or distance) represents the Hubble flow. Looking towards the past, cosmic time seems to elapse twice as fast as on earth. (This result may play a role for the determination of the Hubble parameter.) Accordingly, the age of the universe is $1/2H_0$. This seems to be a very short time. It has to be remarked, however, that this is a mathematical time with no physical relevance. If we use a physical clock and define the unit of time being equal to one electromagnetic oscillation of a certain frequency $\nu$, then we obtain for the number $N$ of oscillations, which represent the physical time elapsed between the present $t_0=0$ and $-1/2H_0$: $$N = \int_{-1/2H_0}^0 {\nu_0dt\over 1+2H_0t}
= -\infty \label{47h}$$ This physical time may not be “equidistant" in the mathematical sense, however, it provides at least the comforting argument that the universe has physically existed “forever".
Density of Matter and Radiation and its Temporal Variation
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The density of matter $\varrho_m$ and the density of radiation $\varrho_r=\varrho_m c^2$ are not distinguished in standard cosmology because $c=const.$ is assumed. This leads to a temporal variation of $\dot \varrho = (\varrho + P/c^2)(-3\dot
R)$ with the result that $\varrho_m(t) R(t)^3=const.$ and $\varrho_r(t) R(t)^4=const.$ The decrease of $\varrho_r$ with a power of 4 is explained by a redshift effect of wave-lengths in addition to the three spatial dimensions. The RLM has no expansion of wave-lengths. This fact must be clearly deduceable from the field equations of GR. They are $$R_{\mu\nu} = -{8\pi G\over c^4} \left( T_{\mu\nu}-{T\over
2}g_{\mu\nu}\right ) \label{48}$$ The matter distribution in the universe is described by the tensor $$T^{\mu\nu} = \left ( \varrho + {P\over c^2} \right ) u^\mu
u^\nu - g^{\mu\nu}P \label{49}$$ According to the Friedmann-Universe we have spatial homogeneity of density $\varrho$ and pressure $P$: $\varrho(r,t)=\varrho(t)$ and $P(r,t)=P(t)$. With these constraints the 00-component of eq. (\[48\]) is $$3\ddot R = -{4\pi G\over c^4} (\varrho c^2 + 3P)R \label{50}$$ The spatial components all lead to the equation $$R\ddot R + 2\dot R^2 + 2kc = {4\pi G\over c^4}(\varrho
c^2-P)R^2
\label{51}$$ For $P=\varrho c^2 /3$ both equations are identical only for $k=0$. ($R=c/H$, (\[6\]) and (\[7\]) have been used here.) This would result in a radiation density of $$\varrho_r = 3H^2 c^2/8\pi G =: \varrho_c\label{52}$$ As mentioned above, because of (\[6\]) it is doubtful, whether a radiation pressure had existed in a radiation-dominated era. For $P=0$, a scenario without remarkable radiation pressure, both equations are only identical for $k=1$. Here the radiation equivalent of the matter density $\varrho_r$ is $$\varrho_r = {3H^2 c^2\over 4\pi G} = 2\varrho_c \label{53}$$ Equations (\[52\]) and (\[53\]) show an important result: In both cases the temporal variation $\dot\varrho_r$ is $$\dot\varrho_r = - 3H\varrho_r \label{54}$$ Accordingly we have a temporal variation of matter density in the case $P=0$ and of the matter equivalent of radiation pressure in the case $P=\varrho c^2/3$ of $$\dot\varrho_m = - H\varrho_m \label{55}$$
The universal density can be derived from inserting (\[24\]) in (\[25\]) with $R=c/H$, (\[6\]), and (\[7\]). This gives the so-called equation of state $$\varrho c^2 + 3P = {3H^2c^2\over 4\pi G} \label{56}$$ Assuming $P=0$ for the whole history of the universe we always have the result of eq.(\[53\]). And indeed, Weinberg (1972) admits, that, “if we give credence to the values $q_0 \simeq 1$ and $H_0 \simeq 75$ km/sec/Mps \[…\], then we must conclude that the density of the universe is about $2\varrho_c$" (p. 476). In the RLM the critical density $\varrho_c$ is just a definition and does not have the meaning of the standard model, which has a strict opposition of gravitation and expansion (and wonders why their values seem to equal each other so perfectly).
As the spatial volume $V$ of the universe is $M/\varrho$, we obtain the result $$V = {4\pi R^3\over 3} \label{57}$$ This has been implicitly used for eq.(\[28\]). In standard cosmology this volume is a consequence of $k=0$, which indicates an equilibrium between gravity and expansion. This state of equilibrium is an integral part of the RLM, in which the meaning of $k$ differs from the standard model. Therefore the assumption (\[57\]) is subsequently justified. The next section will provide further evidence that the Retarded Light-universe is not “closed" as $k$ indicates.
On the Acceleration of Expansion
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The discovery in the recent years that the luminosity of supernovae is smaller than their redshifts suggest (Perlmutter et al., Riess et al.,), has brought much confusion into cosmological research. Anything could be expected but an accelerated expanding universe. On the contrary: in the standard model expansion is expected to be delayed by the [*eigen*]{} gravitation of the universe. The cosmological constant $\Lambda$ has been revived, models of antigravitation and even of a ominous “quintessence" (tracker field) have been suggested (Turner, Wang et al., Picon et al, Caldwell et al., Ostriker and Steinhardt). The RLM, however, predicts an acceleration.
First, it will be demonstrated that in the RLM a measured redshift $z$ results in a larger distance $D$ as in the standard model. This will explain the weaker luminosity observed at far supernovae. Then a mathematical indication for an accelerating universe will be given. As mentioned above, eq.(\[0\]) is valid in the standard model as well as in the RLM. This implies $$R(t_1)\nu_1 = R(t_0)\nu_0 \label{65}$$ where $\nu_1$ is the at time $t_1$ emitted and $\nu_0$ is the received frequency at time $t_0$. In the standard model with constant $c$ the frequency can be replaced by the wave-length as follows: $R(t_1)\lambda_0 = R(t_0)\lambda_1$. The spectral redshift $z$ is defined $$z={\lambda_0-\lambda_1\over \lambda_1} = {\lambda_0 \over
\lambda_1} -1 = {R(t_0)\over R(t_1)} -1 \simeq {H\over
c}D_{Standard} + \dots \label{66}$$ (The restriction to the linear term of distance $D$ does not alter the result in the sense of argumentation.) The RLM has at the point of emission: $$\nu = {c(t)\over \lambda(t)} = const. \label{67}$$ Insertion of (\[67\]) in (\[65\]) yields $${R(t_1)c_1\over\lambda_1} = {R(t_0)c_0\over\lambda_0}
\label{68}$$ This leads to an altered determination of $z$ in the RLM: $$z={R(t_0)\over R(t_1)} - 1 = {\lambda_0 c_1\over \lambda_1
c_0} - 1 = {H\over c_0} D_{RLM} + \dots \label{69}$$ Since $c$ decreases, the speed of light at emission time is always higher than at reception, thus we have $c_1 > c_0$. Comparing (\[69\]) with (\[66\]) we have $$D_{RLM} > D_{Standard} \label{70}$$ In words: For the same measured redshift $z$ the RLM yields a larger distance $D$ to cosmic objects like supernovae as the standard model. This important result explains the observed lack in luminosity.
The acceleration of expansion is kind of a natural process within the RLM without need of any external forces. Because the deceleration of $c$ the universal mass $M$ must increase for $E_{univ} = Mc^2 = const.$ In (\[40\]) and (\[41\]) the values of this process are given. Increasing mass of and in the universe means more gravitational delay of photons; they will slow down even more. The growing deceleration rate, however, is equivalent to a more and more accelerating universe, according to the GEP. The latest determinations of redshift and luminosity of supernovae can be understood as a confirmation of the Generalized Equivalence Principle of expansion, light retardation, and gravitation.
One more remark on extension, expansion, and accelerated expansion: The extension of the universe is given by $\dot R =
c$ (see eq.\[6\]). Its expansion is determined by the rate of light deceleration $\ddot R = \dot c = -Hc$ (see eq.\[7\]). We obtain the acceleration rate of expansion by differentiation of $\dot c$, $$\ddot c = 3H^2c \label{71}$$
Problems of the Standard Model: Horizon, Flatness, Density Fluctuation
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One of the classical problems of the standard model is the incompatibility of the event radius (extension) with expansion. The event radius ${\cal D}$ is given by ${\cal D}(t) =
ct$, constant $c$ provided. Thus the causal event horizon has a time dependence of ${\cal D} = const.\times t$. The time dependence of expansion, derived from Friedmann’s equations is $R(t)=
const. \times t^{2/3}$. That means, that different parts of the universe must have been less causally connected in the past – a contradiction to the demand of a homogeneous universe, which produced just those equations found by Friedmann.
In the RLM, both radii have the same temporal development. For ${\cal D}$ we have $$\begin{aligned}
{\cal D}(t) & = & \int c(t)dt = \int {c_0 dt\over \sqrt{1+2H_0t}}
\nonumber\\
& = & {c_0\over H_0}\sqrt{1+2H_0t} = R_0 \sqrt{1+2H_0t} = R(t)
\label{72}\end{aligned}$$ In particular we have from the origin of the universe ($t=-1/2H_0$) up to the presence ($t=0$) $${\cal D}_0 = \int_{-1/H_0}^{0} {c_0 dt\over
\sqrt{1+2H_0t}} = {c_0\over H_0}=R_0 \label{72b}$$ The identity of both horizons in the RLM is impressively confirmed; their time dependence is ${\cal D}(t) = R(t) = const.
\times t^{1/2}$. To verify this result we insert eq.(\[6\]), the functions of $c(t)$ (see eq.\[7b\]), $G(t)$ (see eq.\[38\]) and $$\varrho_m(t) = {\varrho_{m0} \over\sqrt{1+2H_0t}} \label{72a}$$ which follows from (\[55\]), in Friedmann’s first equation (\[24\]). $$\dot R^2 = -kc^2 + {8\over 3}\pi G\varrho_m R^2 \label{73}$$ Solved to $R$ and all of the above functions inserted yield $$R = \sqrt{{3(1+k)c_0^2(1+2H_0t)\over 8\pi G_0\varrho_{m0}}} =
const. \times t^{1/2} \label{74}$$ These calculations show that there is no horizon problem within the frame of the RLM.
The Retarded Light Model also provides an explanation for the present flatness of the universe. Dividing (\[24\]) by $H^2$ and setting $\varrho_c = 3H^2/8\pi G$ we obtain the following equation for the curvature of the universe: $${k\over R^2} = {H^2\over c^2}\times {\varrho (t) -
\varrho_c\over \varrho (t)} \label{75}$$ ($\varrho$ means the density of matter.) Multiplication of both sides with $\varrho_c/\varrho$ yields: $${\varrho - \varrho_c\over \varrho} = {3kc^2\over 8\pi
G\varrho R(t)^2} = const. \label{76}$$ This is valid for a matter dominated as well as for a radiation universe. The constancy in (\[76\]) is a consequence of the known functions of time $c$, $G$, $\varrho$, and $R$. Thus we get for the curvature after inserting the proportionality term of (\[76\]) into (\[75\]): $${k\over R^2} \propto {H^2\over c^2} \times const. = R(t)^{-2}
\times const. \label{77}$$ This result indicates that the curvature of the universe is proportional to $R(t)^{-2}$. Possible fluctuations around $\Omega = \varrho /\varrho_c = 1$ do not enlarge as in the standard model but remain in the same relation. And another remark: Since we know that in the RLM $k=1$, we can replace the proportionality symbol by equality in (\[77\]). Everything in the RLM fits perfectly, whereas the standard model produces sometimes – as in this case – weird results. To compensate the missing (or very small) space curvature observed today the standard model had to introduce the [*ad hoc*]{} process of an inflationary universe, reviving the cosmological constant, which causes even more trouble than it prevents (Abbott 1988).
Along these lines the remaining cosmological problems of the standard model, such as the increasing density fluctuation during expansion, can be solved.
The Temperature of the Universe
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Since in the presented model electromagnetic waves do not scale with $R(t)$, the relation between the scale factor or radius of the universe and its temperature $T$ should differ from the standard model. However, here arises a problem, which concerns the standard model just as well as the RLM. The question is, from which spectral distribution the temperature of the universe shall be determined. If the universe is regarded as a “black box", then the Planck distribution would be a good candidate. In an expanding universe, however, the Planck distribution needs a certain relation between $T$ and $R$ to maintain itself during expansion. This condition is $$T(t) \propto {1\over R(t)} \label{78}$$ This sounds surprising because this relation is in the standard model frequently derived from the Stefan-Boltzmann law $\varrho_r = aT^4$ with $a=\pi^2k^4/15\hbar^3c^2$ and a conclusion from one of the Friedmann equations $\varrho_r
\propto R^{-4}$. These two relations yield $T\propto 1/r$. However, the use of the Stefan-Boltzmann law implies the validity of the Planck distribution, whereas the Planck distribution needs the relation (\[78\]) to “survive" in an expanding universe. To break this vicious circle we must refer to experimental determinations of the cosmic background radiation (CBR), which show indeed a good approximation to a Planck distribution around 2.73 $K$. Because of this observational result the RLM will also rely on the relation described in (\[78\]), although the radiation density $\varrho_r$ relates in a different way to $R$ (see eq.\[54\]) as in the standard model. The Stefan-Boltzmann law solved to the universal temperature T(t) is $$T(t) = \root 4 \of {{15 \hbar^3c^5\varrho_r\over \pi^2k^4}}
\label{79}$$ where $\hbar$ is the Planck constant divided by $2\pi$, and $k$ is the Boltzmann constant. Most of the “constants" under the root symbol are functions of $t$. Since $R\propto \sqrt{t}$ the temperature $T$ must be proportional to $t^{-1/2}$ to satisfy (\[78\]). The time dependence of $c$ and $\varrho_r$ are given in (\[7b\]) and (\[54\]). The latter gives $$\varrho_r(t)= \varrho_{r0} (1+2H_0t)^{-3/2} \label{79a}$$ after integration. If the Planck constant $h$ is assumed to be constant, then the Boltzmann “constant" $k$ must obey the temporal variation $$k(t) = {k_0\over \sqrt{1+2H_0t}} \label{80}$$ or differentiated $$\dot k = -kH \label{81}$$ With the above conditions of $h$ and $k$ the relation (\[78\]) between $T$ and $R$ holds, so that the thermal evolution of the universe can be treated analogously to the standard model. It has to be emphasized that the variation of the Boltzmann “constant" $k$ is no additional assumption. It follows from the observation that the spectrum of CBR is distributed in a Planckian manner. As pointed out, the standard model also has to refer to that empirical fact, which is problematic in a certain way, because the assumption of a Planck distribution requires a radiation equilibrium, which should not exist in the standard model because of its horizon problems. The suggested model, however, is Mach connected and its radiation is much more in a state of equilibrium.
Summary
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The introduction of a new mechanism of light propagation in a Friedmann universe. avoids the problems of the standard model and provides a unifying description of various empirical facts. Its probably most important feature is that light waves do not scale with $R(t)$. The expanding universe is defined as expansion relative to a constant reference wave-length $\lambda_0$. Vice versa one could speak of a deceleration of $c$ relative to an expanding Friedmann universe. Both views are treated as mathematically and physically indiscernible. This principle of equivalence of expansion and light retardation (IP) is expressed by $Rc=const.$ The relation of these two cosmological quantities allows a classification of all the various cosmological models into four categories:
\(1) $c=const.$ and $R\ne const. \Rightarrow Rc\ne const.$: These conditions represent the Big Bang cosmologies including the standard model.
\(2) $c=const.$ and $R=const. \Rightarrow Rc = const.$: This describes the classical Steady State models (Bondi, Gold, Hoyle).
\(3) $c\ne const.$ and $R=const. \Rightarrow Rc \ne const.$: These conditions are preferred by modern Steady State theories including various Tired Light models.
\(4) $c\ne const.$ and $R\ne const. \Rightarrow Rc=const.$: This characterizes the Retarded Light Model (RLM).
Because the observed cosmological redshift is usually explained by scaling light waves, it first had to be provided another interpretation of the observed redshift. Its determined value $\dot c=-Hc$ has been found to be in accordance with theoretical considerations given before. Some important features of the RLM were developed, including a general condition of a universe: $M_{univ} = c^3/GH$. The universal density was found to be $2\varrho_c$, where the critical density $\varrho_c$ is just a definition taken from the standard model but otherwise meaningless in the RLM, because it was shown, that expansion and [*eigen*]{} gravitation of the universe correspond to each other in a way that a universal mass increase would lead to a delay in light propagation which is equivalent to accelerated expansion. This mechanism is referred to as the Generalized Equivalence Principle (GEP). It is obvious that there is no need of a cosmological constant $\Lambda$ in the RLM.
It was shown that the RLM avoids a variety of problems of the standard model, including the recently discovered mysterious acceleration of expansion. The RLM, based on General Relativity and the Cosmologic Principle of homogeneity and isotropy, includes Sciama’s version of the Mach Principle. It combines various isolated arguments which arose from dissatisfaction with some features of the standard model (Dirac, Narlikar, Milne) and puts them in a consistent frame. There is a good chance, that the Retarded Light Model can even more.
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abstract: 'We have determined new relations between $UBV$ colors and mass-to-light ratios ($M/L$) for dwarf irregular (dIrr) galaxies, as well as for transformed $g^\prime - r^\prime$. These $M/L$ to color relations (MLCRs) are based on stellar mass density profiles determined for 34 LITTLE THINGS dwarfs from spectral energy distribution fitting to multi-wavelength surface photometry in passbands from the FUV to the NIR. These relations can be used to determine stellar masses in dIrr galaxies for situations where other determinations of stellar mass are not possible. Our MLCRs are shallower than comparable MLCRs in the literature determined for spiral galaxies. We divided our dwarf data into four metallicity bins and found indications of a steepening of the MLCR with increased oxygen abundance, perhaps due to more line blanketing occurring at higher metallicity.'
author:
- 'Kimberly A. Herrmann, Deidre A. Hunter, Hong-Xin Zhang, and Bruce G. Elmegreen'
title: 'Mass-to-Light versus Color Relations for Dwarf Irregular Galaxies'
---
Introduction {#sec-intro}
============
Knowledge of the stellar mass in galaxies is important for a wide range of science problems. The best way to determine the stellar mass is through examination of the stellar populations, from either a star-by-star census or spectral energy distribution (SED) fitting of luminosities and colors based on stellar population synthesis (SPS) models. However, frequently, such data are not available, and we turn instead to a mass-to-light ratio, $M/L$, coupled with information on the luminosity or surface brightness, to derive a stellar mass or mass density.
Many studies have explored various calibrations of $M/L$ to color relations (MLCRs) for Johnson-Cousins optical bands (Bell & de Jong 2001; Portinari et al.2004, hereafter P+04; McGaugh & Schombert 2014, hereafter MS14), for Sloan Digital Sky Survey (SDSS) optical filters [@t+11; @rc15], and even some for both sets of filters (Bell et al.2003, hereafter B+03; Gallazzi & Bell 2009; Zibetti et al.2009, hereafter Z+09; Into & Portinari 2013, hereafter IP13). In particular, MS14 tested the self-consistency of MLCRs by applying relations from various studies (B+03, P+04, Z+09, IP13) and various bands (specifically $M/L$ in $V$, $I$, $K$, and 3.6 $\mu$m all as functions of $B-V$) to estimate masses for a sample of disk galaxies spanning over 10 mag in luminosity. They found reasonable agreement between the four studies in $V$, but determined revised MLCRs for $I$, $K$, and 3.6 $\mu$m to improve self-consistency.
For a large sample of galaxies from the Two Micron All Sky Survey (2MASS) and the SDSS, B+03 used models to construct linear fits of $\log_{10} (M/L)$ and SDSS colors as well as $B-V$ and $B-R$. The fits take the form: $$\log_{10} (M/L)_{\lambda} = a_{\lambda} + b_{\lambda} \times \rm{color}.$$ From there, one can go on, for example, to determine the stellar mass density profile, $\Sigma (r)$, from the surface brightness, $\mu$, and color profiles using the following from @btp08: $$\log_{10} \Sigma = \log_{10} (M/L)_{\lambda} - 0.4(\mu_{\lambda}-m_{\rm{abs},\odot,\lambda}) + 8.629,$$
where $m_{\rm{abs},\odot,\lambda}$ is the absolute magnitude of the Sun at wavelength $\lambda$ and $\Sigma$ is measured in $M_{\odot}$ pc$^{-2}$.
However, a reliable relation between some color and the $M/L$ is essential, and the MLCRs in the literature have largely been determined from models appropriate for spirals. It is questionable if any of these linear fits is suitable for dwarf galaxies, with lower metallicities and potentially different star formation histories (SFHs) than spirals. Therefore, we have determined relations between $M/L$ and colors for dwarf irregular (dIrr) galaxies. These relations, presented here, are based on empirical SED fitting of multi-wavelength surface photometry of a sample of 34 dIrrs [@z+12]. In §\[sec-data\] we describe the data on which our new MLCRs are based. In §\[sec-formula\] we present our MLCRs between stellar $M/L$ from the SED fitting and $UBVg^\prime r^\prime$ colors and compare them to several MLCRs from the literature. We explore metallicity effects in §\[sec-Zbins\] by breaking our data into four metallicity bins.
Data {#sec-data}
====
The sample of galaxies is taken from LITTLE THINGS [Local Irregulars That Trace Luminosity Extremes, The [H$\,$[i]{}]{} Nearby Galaxy Survey, @lt12]. This is a multi-wavelength survey of 41 nearby ($<10.3$ Mpc) dIrr galaxies and blue compact dwarfs, which builds on the THINGS project [@walter08], whose emphasis was on nearby spirals. The LITTLE THINGS galaxies were chosen to be gas-rich so they have the potential to form stars, although a few do not currently have [H$\alpha$]{} emission. They were also chosen to be fairly isolated, or at least not companions to a giant galaxy or obviously interacting with another system. The LITTLE THINGS data set includes [*Galaxy Evolution Explorer*]{} satellite [[*GALEX*]{}; @GALEX] images at FUV (1516 Å) and NUV (2267 Å) wavelengths, $UBV$ and some $JHK$ images from @he06, [H$\alpha$]{} images from @he04, and [*Spitzer*]{} Infrared Array Camera [@irac04 IRAC] 3.6 $\mu$m images. This yields 7-10 passbands for each galaxy from the FUV to the NIR.
@z+12 used azimuthally averaged surface photometry in these passbands and performed a SED analysis of each annulus for a subsample of 34 of the LITTLE THINGS dwarfs. They modeled the data with a library of four million different SFHs based on the unpublished 2007 version of GALAXEV (Bruzual & Charlot 2003) stellar population models and allowing dust extinction, metallicity, and relative star formation rate among different age bins to vary uniformly among physically reasonable ranges. They used the stellar initial mass function (IMF) of @c03 and took into account H$\alpha$ line emission, but not contamination from polycyclic aromatic hydrocarbon lines since nebular emission is arguably not important for the normal star-forming dIrr galaxies in the optical broadband images that were used.
From the fits to the data, they derived stellar mass surface density distributions for the 34 dIrr galaxies. @z+12 used two slightly different methods to determine the mass profile values at outer radii: (1) applying the $M/L$ from the most distant point where the SED analysis is applicable and (2) using the 3.6 $\mu$m light for some of the outermost points. We used the latter mass profiles for the 34 dwarfs in our averaging analysis.
The SED fitting analysis yielded robust mass profiles as a function of radius in a large sample of dwarf galaxies. @Paper1 and @Paper2 have determined surface brightness and color profiles for these same galaxies, as well as 107 other dwarfs. With these data, we have the information we need to determine $M/L$ trends with $U-B$ and $B-V$ colors, as well as the transformed color $g'-r'$ (using $g'-r' = 0.98(B-V) - 0.19$, $g' = V + 0.54(B-V) - 0.07$, and $r' = V - 0.44(B-V) + 0.12$ from Smith et al.2002), that are appropriate specifically for dIrrs.
[cccccccccccccc]{} \[tab-fits\] $U-B$ & $U$ & $-0.271$ & 0.081 & 1.147 & 0.260 & …& …& …& …& …& …& …& …\
$U-B$ & $B$ & $-0.231$ & 0.081 & 0.747 & 0.260 & …& …& …& …& …& …& …& …\
$B-V$ & $B$ & $-0.911$ & 0.157 & 1.452 & 0.393 & $-1.035$ & 1.737 & $-0.868$ & 1.690 & $-1.330$ & 2.237 & $-1.111$ & 2.027\
$B-V$ & $V$ & $-0.651$ & 0.157 & 1.053 & 0.393 & $-0.721$ & 1.305 & $-0.597$ & 1.290 & $-1.075$ & 1.837 & $-0.843$ & 1.627\
$g'-r'$ & $g'$ & $-0.601$ & 0.090 & 1.294 & 0.401 & $-0.592$ & 1.519 & …& …& $-1.030$ & 2.053 & $-0.794$ & 1.930\
$g'-r'$ & $r'$ & $-0.313$ & 0.090 & 0.894 & 0.401 & $-0.399$ & 1.097 & …& …& $-0.840$ & 1.654 & $-0.606$ & 1.530\
$V-3.6\mu$m & 3.6$\mu$m & $-0.175$ & 0.380 & $-0.149$ & 0.180 & …& …& …& …& …& …& …& …\
$B-V$ & 3.6$\mu$m & $-0.768$ & 0.148 & 0.776 & 0.367 & $-0.322$ & $-0.007$ & $-0.594$ & 0.467 & $-1.147$ & 1.289 & $-0.861$ & 0.849
Colors and mass-to-light ratios {#sec-formula}
===============================
The relations between $M/L$ and colors were determined via a minimizing $\chi^2$ fitting routine [@NumRec] iteratively reweighted to obtain a linear fit that was characteristic of as much of the data as possible, but not overly affected by outlying influential points. The weights were 1.0 for $|\Delta y| <= \sigma$ where $\Delta y$ was the residual and $\sigma$ was the weighted standard deviation, both from each previous fit, and $\sigma/|\Delta y|$ for more discrepant data, following an L1 procedure for non-Gaussian residuals. The $\chi^2$ fitting routine was iterated until the slope changed by $<10^{-8}$. Figures \[M\_L\] and \[new\] show the optical and 3.6 $\mu$m data and fits, respectively, as well as the parameters of the final fits, which are also listed in columns 3 and 5 of Table \[tab-fits\].
Stellar mass profiles are best calculated using the $B-V$ or $g'-r'$ relations; the $M/L$ ratio is not correlated as strongly with $U-B$ or $V-3.6 \mu$m as with $B-V$ or $g'-r'$, as seen by the larger scatter in the two $U-B$ panels of Figure \[M\_L\] and the $V-3.6 \mu$m panel of Figure \[new\]. This makes sense because redder passbands generally trace the mass better than bluer passbands. FUV$-$NUV colors would not have been useful at all for estimating $M/L$, though $M/L$ in 3.6 $\mu$m is not as strongly correlated with $B-V$ as the $M/L$ in $B$ or $V$.
Since our relations (and those of Z+09) are based on a @c03 stellar IMF, we have applied zero-point offsets to raise or lower MLCRs determined using other IMFs for a better comparison due to having one fewer variable between the different studies. @bdj01 (and B+03) used a “diet” Salpeter IMF from 0.1 to 125 M[\_$_{\sun}$]{} by modifying the $M/L$ by a factor of 0.7. This global factor was equivalent to reducing the contribution from low-mass stars, and was done to bring the absolute normalization of their $M/L$ in line with observational constraints of maximum disks determined from rotation curves of spiral galaxies. B+03 noted that $\log_{10} (M/L)$ must be reduced by 0.15 dex to convert from their “diet” Salpeter IMF to a @k01 IMF. @g+08 specified a similar correction (-0.093 dex) to convert from the “diet” Salpeter IMF to a @c03 IMF, which we applied to the MLCR of B+03. Since P+04 and IP13 both used Kroupa IMFs, we added 0.057 dex to adjust to a Chabrier IMF. To convert to a @salpeter55 IMF, add a constant of +0.243 to the right side of Equation 1. For a @k01 IMF, with a shallower IMF than Salpeter for low mass stars, subtract 0.057 dex.
With all the literature MLCRs adjusted (as needed) to a Chabrier IMF, the B+03 optical relations almost overlap our linear fits and the data, except our optical fits for the MLCRs of dIrrs are consistently slightly shallower. Actually, each cyan line (the dwarf best fit) in Figure \[M\_L\] has the [*shallowest*]{} slope of all the colored lines in each panel, and correspondingly column 5 (dwarf $b_{\lambda}$ slopes) of Table 1 contains the smallest slope in each of the first six rows. The Z+09 MLCRs provide a poor fit to our dIrr galaxy data for all five bands in the comparison. The P+04 MLCRs are slightly high for all three $B-V$ panels whereas the IP13 MLCRs fit the dwarf data in the $B-V$ panels fairly well (except for being too steep) but fall below the dwarf data in both transformed $g'-r'$ panels. The dwarf fit values for $\log_{10} (M/L_{r'})$ as a function of $g'-r'$ ($a_{r'} = -0.310$ and $b_{r'} = 0.890$) are similar to those used by @btp08 for their spiral analysis ($a_{r'} = -0.306$ and $b_{r'} = 1.097$) although the latter $a_{r'}$ becomes $-0.399$ after being adjusted from the “diet” Salpeter IMF to a Chabrier IMF.
The magenta asterisks in the lower central ($B-V$, $V$) panel of Figure \[M\_L\] show $\log_{10}(M/L)_V$ for a single starburst population of 0.1-100 $M_{\odot}$ stars [@bc03] reduced by 0.243 dex to adjust the data from their original Salpeter IMF to a Chabrier IMF. Interestingly, the dwarf data fall somewhere between the 1 Gyr ($\ast$9) and 10 Gyr ($\ast$10) points since the typical luminosity-weighted age of dIrrs is $\sim$1 Gyr whereas spirals are generally older than dwarfs in terms of their luminosity-weighted ages.
Figure \[fit\_params\] displays the fit parameters ($y$-intercepts, $a_{\lambda}$, and slopes, $b_{\lambda}$) of the MLCRs from this work as well as those from B+03, P+04, Z+09, and IP13. The left panel shows the fit parameters from each study with the B+03 data based on the “diet” Salpeter IMF, the P+04 and IP13 data based on a Kroupa IMF, and the data from this work and Z+09 based on a Chabrier IMF. The non-dwarf $V$, $I$, and $K$ data are directly from Table 2 of MS14. The right panel shows the $BVg'r'$ $y$-intercepts adjusted as needed to a Chabrier IMF (the cyan and green data points are unchanged) and the $I$ and 3.6 $\mu$m fits from Table 7 of MS14, revised to be self-consistent with the $V$ band masses and corrected for $V-I$ as a second color term in addition to a 3.6 $\mu$m data point for dwarfs. The $BVIK$ and 3.6 $\mu$m fits are all with respect to $B-V$; that is, they are of the form: $\log_{10} (M/L)_{\lambda} = a_{\lambda} + b_{\lambda} (B-V)$ whereas the $g'r'$ fits are with respect to $g'-r'$. In this work only, all $g'r'$ data have been transformed from $BV$ images. For completeness, we have included $IK$ MLCRs from the literature in Figure \[fit\_params\] even though the LITTLE THINGS survey does not have enough observations in these bands for us to determine $IK$ MLCRs for dwarfs.
The optical bands (but not the $K$ or 3.6 $\mu$m data) in both panels display a fairly smooth progression from steep slopes and most negative $y$-intercepts in $B$ (triangles, lower right) up to more shallow slopes and less negative $y$-intercepts in $r'$ (circles) and $I$ (open diamonds, both in the upper middle). However, in the right panel, the data points are more streamlined and the $I$-band points are all located in the upper middle instead of paralleling the $B$ to $r'$ points in the left panel. Furthermore, within each optical band, the shape obtained from connecting the data from lower right to upper left follows roughly the same pattern from Z+09 (green), IP13 (lavender), B+03 (pink), this work (cyan), to P+04 (gold). The differences between the various studies probably occur due to the variety of sample galaxy SFHs or assumed SFHs since SED masses and luminosities depend strongly on SFHs.
Why do the coefficients $a_{\lambda}$, the $y$-intercepts, and $b_{\lambda}$, the slopes, vary systematically with each other and with wavelength? The fact that $a_{\lambda}$ decreases (the $M/L$ gets smaller) when $b_{\lambda}$ increases (the color dependence gets stronger) for bluer passbands means that younger regions with small $M/L$s have a greater dependence between $M/L$ and $B-V$ color than older regions. This must occur since all the MLCR lines in Figure 1 have similar $M/L$ at the reddest colors. That is, if we think of $M/L$ versus $B-V$ as pinned at a particular $M/L$ for a particular large $B-V$, and all the other curves varying as straight lines around that pinning point, then naturally the $y$-intercept $a_{\lambda}$ will get smaller as the slope $b_{\lambda}$ gets larger. Physically this means that for red colors, the IMF and SFHs do not matter much because only old red stars remain and the $M/L$ and colors are universal for very old populations. The variety of SED models and galaxy fits is generally due to different histories or assumptions about [*young*]{} stars.
MLCRs and Metallicity {#sec-Zbins}
=====================
Since our MLCRs for dwarfs are shallower than the MLCRs in the literature and a lower metallicity is one characteristic that separates dwarfs from spirals, we broke our dwarf data into four bins based on oxygen abundance ($7.0 - 7.5$, $7.6 - 7.7$, $7.8 - 8.0$, and $8.1 - 8.7$) and fit the MLCR for each bin. Figure \[Z\_params\] shows the results, including an apparent steepening of the MLCRs with higher metallicity (see Table \[tab-Z\] for the fit parameters) but little metallicity dependence of the $y$-intercepts, especially in $g'$ and $r'$. Since the $g'$ and $r'$ analyses are not based on images but instead on transformations of $B$ and $V$, it is not surprising that the $g'$ and $r'$ fits have similar slopes to the $B$ and $V$ fits. Unfortunately, the metallicities of the samples of spiral galaxies explored to determine existing MLCRs are not available for comparison. However, for our slope trends from the left panel of Figure \[Z\_params\] to match the B+03 MLCR slopes, the metallicity for the spiral sample would need to be somewhere between 8.5 and 8.8, which might be reasonable for a galaxy-wide average considering that the solar value is 8.69 [@a+09]. Perhaps the shallower slope with lower metallicity is caused by less line blanketing occurring at lower metallicity [@h+15]. The color changes with changing line blanketing (bluer for less line blanketing) [*faster*]{} than the brightness (brighter for less line blanketing) of a population.
[cccccc]{} \[tab-Z\] $B-V$ & $B$ & 0.258 & $-0.537$ & 0.004 & 0.026\
$B-V$ & $V$ & 0.258 & $-0.937$ & 0.004 & 0.026\
$g'-r'$ & $g'$ & 0.264 & $-0.742$ & 0.004 & 0.029\
$g'-r'$ & $r'$ & 0.263 & $-1.139$ & 0.004 & 0.028
Summary {#sec-summary}
=======
We have used the stellar mass density radial profiles of @z+12 produced from SED fitting to 7-10 passbands from the FUV to the NIR and the surface brightness and color profiles of @Paper1 [@Paper2] to determine relations between $\log_{10}(M/L)$ and $U-B$ and $B-V$ colors as well as the transformed color $g'-r'$ given by @SDSSfilters. These MLCRs are specifically for dIrr type galaxies. The $B-V$ relationship in particular can be used to determine stellar masses in dIrrs in situations where SED fitting of multi-wavelength photometry or a census of individual stars is not available. Our MLCRs are consistently shallower than those reported in the literature and the differences could be due at least partly to a steepening of MLCRs with metallicity, perhaps from more line blanketing occurring at higher metallicity.
We would like to thank the anonymous reviewer for helpful suggestions. KAH acknowledges funding from Pennsylvania State-Mont Alto that enabled extended visits to Lowell Observatory where some of this work was carried out. DAH also appreciates funding from the Lowell Research Fund and is grateful to John and Meg Menke for funding for page charges. HZ acknowledges support from FONDECYT Postdoctoral Fellowship Project No. 3160538 and earlier from the Chinese Academy of Sciences (CAS) through a CAS-CONICYT Postdoctoral Fellowship administered by the CAS South America Centre for Astronomy (CASSACA) in Santiago, Chile.
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abstract: 'We use molecular dynamics simulations to test integral equation theory predictions for the structure of fluids of spherical particles with eight different piecewise-constant pair interaction forms comprising a hard core and a combination of two shoulders and/or wells. Since model pair potentials like these are of interest for discretized or coarse-grained representations of effective interactions in complex fluids (e.g., for computationally intensive inverse optimization problems), we focus here on assessing how accurately their properties can be predicted by analytical or simple numerical closures including Percus-Yevick, hypernetted chain, reference hypernetted chain, first-order mean spherical approximation, and a modified first-order mean spherical approximation. To make quantitative comparisons between the predicted and simulated radial distribution functions, we introduce a cumulative structural error metric. For equilibrium fluid state points of these models, we find that the reference hypernetted chain closure is the most accurate of the tested approximations as characterized by this metric or related thermodynamic quantities.'
author:
- 'Kyle B. Hollingshead'
- 'Thomas M. Truskett'
title: 'Predicting the structure of fluids with piecewise constant interactions: Comparing the accuracy of five efficient integral equation theories'
---
Introduction
============
A common challenge in materials science is the “inverse design problem” [@torquato:invstatmech; @jain:invstatmechrev], wherein one seeks to use theoretical models to discover the microscopic characteristics (e.g., the effective pair interactions) of a new system which, if fabricated or synthesized, would yield a targeted material property. Recent applications include designing materials that self-assemble into specific crystalline lattices [@jain:invstatmech; @marcotte:invstatmech2; @cohn:invstatmech; @edlund:invstatmech], fluids that display optimized structural correlations and related transport properties [@stillinger:iso-g; @carmer:tracer; @goel:tuneddensity], or solids that exhibit specific optical characteristics [@andkjaer:photonics]. Inverse design problems are commonly addressed by stochastic optimization strategies like simulated annealing. Such approaches have the advantage of being general and easy to apply, and they can also be effective as long as material properties required for evaluating the objective function can be accurately and efficiently computed for large numbers of trial interactions during the optimization. This requirement typically means that “exact” yet computationally intensive methods for property determination (e.g., molecular simulations) are impractical for use within such calculations. Approximate theories with analytical or simple numerical solutions are attractive alternatives to molecular simulation in these contexts, provided that they can make sufficiently accurate predictions for a wide range of microscopic interaction types.
For bulk fluids, a key aim for property prediction is to discover the one-to-one link [@henderson:uniqueg] between $g(r)$, the radial distribution function (RDF) of a system at a given set of conditions, and $\varphi(r)$, the interparticle pair potential. Knowledge of these functions of interparticle separation $r$ allows for the direct calculation of the static structure factor, the energy, the pressure, and the isothermal compressibility [@thysimpliq]. Estimations of other properties can be directly obtained from knowledge of the RDF as well. One example is the two-body excess entropy, which is often a good approximation of the total excess entropy [@nettleton:s2dominant] for simple liquids. Another is the information-theoretic estimate for the probability $p_n (\Omega)$ of observing $n$ particle centers in a molecular-scale subvolume $\Omega$, a quantity which characterizes the fluid’s density fluctuations [@hummer:insertprob]. Excess entropy, its two-body approximation, and $p_0$ have been shown to correlate with various dynamic properties of equilibrium fluids, e.g. diffusivity or viscosity [@rosenfeld:sexscaling2; @dzugutov:sexscaling; @rosenfeld:sexscaling1; @chakravarty:waterlikesexscaling; @abramson:h2osexscaling; @krekelberg:s2; @abramson:n2sexscaling; @mittal:p0scaling; @abramson:co2sexscaling; @dyre:isomorphs; @goel:confinedstructure; @krekelberg:rosenfeld; @pond:s2gauss; @errington:dumbbellsexscaling; @chakravarty:ionmeltsexscaling; @pond:rosenfeldbrownian; @carmer:tracer; @bollinger:sexscaling]. Mode-coupling theory also predicts that dynamic phenomena can be directly estimated from knowledge of the static structure factor [@reichman:mctreview].
With these considerations in mind, herein we use molecular simulations to test the accuracy of RDF predictions for five approximate integral-equation theory closures: Percus-Yevick, hypernetted chain and reference hypernetted chain [@thysimpliq], first-order mean spherical approximation (FMSA) [@tang:fmsa2], and a modified exponential version of FMSA [@hlushak:sexpfmsasquareshoulder]. Other more resource-intensive theories, like the Rogers-Young and hybrid mean-spherical approximations [@rogers:ry; @zerah:hmsa], self-consistent Ornstein-Zernike approaches [@thysimpliq], and thermodynamic perturbation theories [@barkerhenderson:swtpt; @zhou:3rdordertpt; @zhou:5thordertpt] are not considered here. We apply the simpler five theories listed above to a diverse suite of eight pair potentials previously introduced by Santos et al [@santos:2steps], each composed of a hard core at $r=\sigma$ plus two piece-wise constant sections at larger $r$ (i.e. wells or shoulders), that qualitatively mimic some of the features observed in the effective interactions of complex fluid systems. For each interaction, we investigate four thermodynamic state points with various combinations of low and high density and low and high temperature, and we compare the theoretical predictions for the RDF, the energy, and the two-body excess entropy to data from event-driven molecular dynamics simulations. To facilitate the RDF comparisons we introduce a “cumulative squared error” metric, which provides a quantitative characterization of the overall quality of each theoretical prediction. We also assess the accuracy of predictions for the potential energy and the two-body excess entropy.
Methods
=======
Integral Equation Theory
------------------------
Integral equation theories for uniform, isotropic fluids typically involve solving a system of two equations: the Ornstein-Zernike relation, $$\label{eq:oz}
h(r)=c(r)+\rho\int c\left(|\mathbf{r'}-\mathbf{r}|\right)h(r')d\mathbf{r'},$$ which defines the direct correlation function $c(r)$ in terms of the number density $\rho$ and the total correlation function $h(r) = g(r)-1 $, and a closure, e.g., $$\label{eq:closure}
h(r)+1=\exp\left[-\beta\varphi(r)+h(r)-c(r)+B(r)\right],$$ which introduces the link to the pair potential $\varphi(r)$, where $\beta = (k_\text{B}T)^{-1}$, $T$ is temperature, $k_\text{B}$ is Boltzmann’s constant, and $B(r)$ is the so-called bridge function.
Two common approximations for $B(r)$ are the Percus-Yevick (PY) closure, $$B_{\text{PY}}(r) = \ln\left[h(r)-c(r)+1\right]-h(r)+c(r),$$ and the hypernetted chain (HNC) closure, $$B_{\text{HNC}}(r) = 0.$$
Another is the so-called reference hypernetted chain approximation (RHNC), which assumes that the bridge function can be accurately approximated by that of a reference fluid, typically one of hard spheres at the same density: $$B_{\text{RHNC}}(r) = B_\text{HS}(r) .$$ The hard-sphere fluid’s bridge function $B_\text{HS}(r)$ has been calculated through careful molecular simulations, and multiple parameterizations for its density dependence exist [@malijevsky:bridgefunc; @verletweis:hsrdf; @grunkehenderson:hsdcf]. For this work, we employ the analytical parameterization proposed by Malijevský and Labík [@malijevsky:bridgefunc] for the RHNC closure.
With $B(r)$ specified by these closures, we solve the coupled equations (\[eq:oz\]) and (\[eq:closure\]) using a rapidly-converging combination of Newton-Raphson and Picard root-finding methods developed by Labík et al. [@labik:fastthy].
An alternative strategy is to replace the closure of Eq. \[eq:closure\] with separate expressions. For example, the mean spherical approximation (MSA) assumes the following relations hold, $$\begin{aligned}
g_{\text{MSA}}(r) &= 0 & \quad r < \sigma, \\
c_{\text{MSA}}(r) &= 0 & \quad r \geq \sigma. \\
\end{aligned}$$ By further assuming first-order expansions in the characteristic dimensionless energy of the potential $\beta\varepsilon$ for both $g(r)$ and $c(r)$–e.g., $g_{\text{FMSA}}(r) = g_\text{HS}(r) + \beta\varepsilon g_1(r)$, where $g_\text{HS}(r)$ is the pair correlation function for a hard sphere system at the same density–Tang and Lu closed the equations analytically for several common pair interactions, including square wells [@tang:fmsa2]. We refer to this solution as the first-order mean spherical approximation (FMSA). In principle, FMSA can be applied to potentials with square shoulders as well. But for strong interactions, FMSA is known to incorrectly predict RDFs with negative values for some interparticle separations [@hlushak:sexpfmsasquareshoulder]. To resolve this, Hlushak et al. modified the FMSA to make it equally applicable to wells and shoulders by rearranging the terms in the series expansion, so that $g_{\text{EFMSA}}(r)=g_\text{HS}(r)\exp[-\beta\varepsilon g_1(r)]$ [@hlushak:sexpfmsasquareshoulder]. In this work, we refer to this analytical solution as the exponential first-order mean spherical approximation (EFMSA).
Suite of Two-Step Potentials
----------------------------
Motivated by Santos et al. [@santos:2steps], we examine predictions for fluids from a set of pair interactions comprising a hard core and two piecewise-constant steps, $$\label{eq-potential}
\varphi(r) = \left\{ \begin{array}{ll}
\infty & \quad r < \sigma, \\
\varepsilon_1 & \quad \sigma \leq r < \lambda_1, \\
\varepsilon_2 & \quad \lambda_1 \leq r < \lambda_2, \\
0 & ~~r \ge \lambda_2, \\
\end{array} \right.$$ where $\varepsilon_1$ and $\varepsilon_2$ are the energies of the first and second steps, respectively, and $\lambda_1$ and $\lambda_2$ are the outer edges of the first and second steps, respectively.
![\[fig:interactions\] The suite of eight pair interactions considered in this study, inspired by Santos et al. [@santos:2steps], is topologically exhaustive (e.g., there are no other qualitative arrangements of two constant pairwise pieces that are not more appropriately labeled single wells or shoulders).](00_potentials){width="8cm"}
Furthermore, as in Santos et al., we restrict the values of $\varepsilon_i$ to the set $\{-\varepsilon, -\varepsilon/2, 0, \varepsilon/2, \varepsilon\}$, where $\varepsilon$ is a characteristic energy scale. Cases where $\varepsilon_1=\varepsilon_2$ or $\varepsilon_2 = 0$ reduce to either single square wells or shoulders, or hard spheres, which have all been studied extensively elsewhere (see, e.g., refs. 1-41 in [@yuste:rfass]) and are not considered here. We also exclude cases where $\max\{|\varepsilon_1|, |\varepsilon_2|\} = \varepsilon/2$. Of the cases where $\varepsilon_1$ and $\varepsilon_2$ have opposite sign, we consider only combinations where $\varepsilon_2=-\varepsilon_1=\pm\varepsilon$. We choose $\lambda_1=1.5\sigma$ and $\lambda_2=2\sigma$ in order to provide challenging perturbations to the bare hard sphere system that are still amenable to molecular simulation and theoretical treatment. After imposing these restrictions, the remaining eight pairwise interactions shown in Fig. \[fig:interactions\], which we refer to as “Type A–H,” form our test suite. To explore how the accuracy of the various theories varies with density and temperature, we investigate each interaction at the four state points comprising combinations of packing fraction $\eta=\rho\pi\sigma^3/6=0.15$ or $0.45$ and dimensionless temperature $T^*=k_\text{B}T/\varepsilon=0.67$ or $2.0$.
Molecular Simulations
---------------------
We compare the theoretical predictions for the RDF, the energy, and the two-body excess entropy to the results of event-driven molecular dynamics simulations performed with the DynamO simulation engine [@bannerman:dynamo]. Periodic boundary conditions were used, and the simulated systems were sized such that adequate RDF statistics could be collected for separations up to at least $r=10\sigma$. In practice, this required $N=4000$ particles when $\eta=0.15$, and $N=8788$ particles when $\eta=0.45$. The “bins” for particle counts were $0.005\sigma$ wide. Temperatures were set and maintained using an Andersen thermostat [@andersen:thermostat].
Each simulation was initialized as an FCC lattice of the desired density at a high temperature, with randomly assigned particle velocities. After equilibrating for ten million events, the simulations were cooled to the desired temperature and re-equilibrated for a further ten million events. Then, the thermostat was removed, and the RDF was measured over the final five million events.
Quantifying Error in Predictions
--------------------------------
To compare the various RDF theoretical predictions to simulations at a given state point, we define a metric we call the cumulative squared error, $\text{CSE}(r)$:
$$\label{eq:CSE}
\text{CSE}(r)=\frac{\int_{\sigma}^r \left[h_\text{sim}(r')-h_\text{thy}(r')\right]^2r'^2\,dr'}
{\int_{\sigma}^\infty h_\text{sim}^2(r')r'^2\,dr'}.$$
The integrand in the numerator characterize the squared deviation in the total correlation function between the prediction of a given theory $h_\text{thy} (r)$ and the result of the ‘exact’ simulation $h_\text{sim} (r)$; the power of two eliminates any possible cancellation of error, e.g. for cases where a theory both underpredicts and overpredicts the value of $h(r)$ at different values of $r$. The denominator accumulates the total squared correlations in the simulated system, and thus normalizes the overall function to facilitate comparison between systems with different degrees of correlation (e.g., between low-density and high-density systems).
As $r$ approaches infinity, all $h(r)$ curves converge to zero and the $\text{CSE}$ converges to a finite value, $\text{CSE}_\infty$: $$\text{CSE}_\infty = \lim_{r \to \infty}{\text{CSE}(r)},$$ which is a measure of the summed squared correlations as a fraction of the total squared correlations in the system; thus, a larger value of $\text{CSE}_\infty$ indicates that a theoretical prediction deviates more significantly from the “exact” simulation results. By construction, $\text{CSE}_\infty$ has a defined minimum of $0$ and, while it does not have a rigorous maximum, its value is typically less than $1$ except in cases where the theoretical predictions are qualitatively very poor.
We also calculate the potential energy per particle $U/\varepsilon$, $$\label{eq:potenergy}
\frac{U}{\varepsilon} = \frac{\rho}{2}\int_0^\infty \frac{\varphi(r)}{\varepsilon} g(r) \, d{\bf r},$$ and the two-body contribution to excess entropy $s^{(2)}/k_\text{B}$, $$\label{eq:s2}
\frac{s^{(2)}}{k_\text{B}} = -\frac{\rho}{2}\int_0^\infty \left[g(r)\ln g(r) - g(r) + 1 \right] \, d{\bf r},$$ from simulations and theoretical predictions. Both quantities can also be directly computed from $g(r)$ and thus, the normalized absolute deviation of the predicted versus simulated values can be used as an indication of the success of theoretical predictions. However, note that different RDFs can, in principle, give rise to the same value of $U/\varepsilon$ or $s^{(2)}/k_\text{B}$. Moreover, $U/\varepsilon$ only depends on correlations within the range of the pair interaction. As a result, we argue here that since the RDF is weighted differently for each thermodynamic quantity, the CSE metric we introduce–which tests the overall similarity between predicted and simulated RDFs–represents a more sensitive measure for the overall predictive quality of particular theory.
![\[fig:adata\] Radial distribution functions $g(r)=h(r)+1$ and the associated cumulative squared errors (CSE, see Eq. (\[eq:CSE\])) predicted by the reference hypernetted chain (RHNC), hypernetted chain (HNC), and Percus-Yevick (PY) Ornstein-Zernike closures [@thysimpliq; @malijevsky:bridgefunc]; the first-order mean spherical approximation solution (FMSA) [@tang:fmsa2]; and the simple exponential first-order mean spherical approximation (EFMSA) [@hlushak:sexpfmsasquareshoulder], for the “type A” pair interaction. Shaded regions adjacent to each $g(r)$ indicate the difference between the theory and simulation results.](00_fulldata_typeA){width="8cm"}
Results and Discussion
======================
Structural predictions for the Type A pair interaction are compared to simulation results in Fig. \[fig:adata\], along with the corresponding cumulative squared errors as calculated via Eq. (\[eq:CSE\]). For this interaction, the analytic solutions (FMSA and EFMSA) perform better at higher rather than at lower equilibrium fluid densities. As density increases, the effect of the excluded volume captured by the well-modeled hard-sphere RDF, $g_\text{HS}(r)$, overwhelm the energetic perturbations from the repulsive steps and dominate the resulting structure. Of the tested integral-equation theories with simple numerical closures, the PY closure tends to perform least well near contact, and for interaction Type A, the RHNC offers the best predictions at all four state points investigated. Analogous figures for each of the other interactions are presented for the interested reader in Appendix \[app:plots\].
It is tempting to conclude from a visual comparison of theoretical and simulated radial distribution functions that all of the theories perform similarly well, especially at the higher temperature (Figs. \[fig:adata\]d and \[fig:adata\]h). However, the resulting CSEs differ [*by nearly two orders of magnitude*]{} from most to least accurate (Figs. \[fig:adata\]c and \[fig:adata\]g), which underscores the utility and sensitivity of the CSE metric. As discussed below, these differences in the CSE become important when computing other quantities that depend on the RDF, especially when one considers that each thermodynamic quantity weights the RDF in a different way.
The total cumulative squared errors $\text{CSE}_\infty$ for all interactions, state points, and theories are listed in Table \[table:cse\]. Six of the total thirty-two combinations of interaction type and state point considered did not produce single-phase, uniform fluids when simulated. Of the remaining twenty-six systems, the RHNC offered the most accurate structural predictions for all but four; however, at three of these four points, the $\text{CSE}_\infty$ of the RHNC is still within ca. 65% of the most accurate theory (HNC). All four points are at low temperature ($T^*=0.67$) and high packing fraction ($\eta=0.45$), and each of the pair interactions include attractions (types D, F, G, and H).
![\[fig:corr\] Correlations between total cumulative squared error $\text{CSE}_\infty$ and either (a) absolute normalized two-body excess entropy error or (b) absolute normalized potential energy error for all data collected.](00_correlations_horiz){width="8cm"}
We also compare $\text{CSE}_\infty$ against the absolute normalized errors for predictions of two example thermodynamic quantities, two-body excess entropy $s^{(2)}/k_\text{B}$ and potential energy $U/\varepsilon$, in Fig. \[fig:corr\]. Fig. \[fig:corr\]a shows that $\text{CSE}_\infty$ is generally a good predictor of $s^{(2)}/k_\text{B}$ accuracy, although there are a handful of instances where the fractional error in the excess entropy is low while $\text{CSE}_\infty$ is higher.The correlation between $\text{CSE}_\infty$ and the potential energy is weaker, but still present; this is likely due to opportunities for fortuitous cancellation of error when pair interactions contain both positive and negative contributions (e.g., types D and H), when portions of the interactions are zero (types C and G), or when significant contributions to $\text{CSE}_\infty$ occur beyond the range of the pair interaction. Overall, however, it is clear that the accuracies of both example thermodynamic quantity predictions correlate well with the cumulative squared error. For the interested reader, the values of $\left|\left(s^{(2)}_\text{thy}/s^{(2)}_\text{sim}\right)-1\right|$ and $\left|\left(U_\text{thy}/U_\text{sim}\right)-1\right|$ are tabulated in Appendix \[app:tables\]. If other thermodynamic quantities that depend on the RDF in a different way (e.g., the pressure or the isothermal compressibility) are also of interest, then the necessity to have an independent structural metric like $\text{CSE}_\infty$ to assess the quality of the structural predictions is even more critical.
Conclusion
==========
In order to quantify the overall accuracy of theoretical predictions for fluid structure, we have introduced the total cumulative squared error ($\text{CSE}_\infty$) metric, which accumulates squared discrepancies between a theoretical prediction and a reference “exact” result at all separation distances along the total correlation function and avoids any possible cancellation of error. We find that this $\text{CSE}_\infty$ metric is very sensitive and tends to forecast the overall accuracy of structure-dependent thermodynamic calculations. As a result, it is an excellent tool for comparing accuracy between multiple theories, particularly when differences are difficult to discern by visual inspection.
We have used this metric to test the performance of five integral equation theory-based approaches for predicting equilibrium fluid structure in systems with pair interactions comprising a hard core plus two piecewise constant interactions, and we find that the reference hypernetted chain (RHNC) integral equation closure offers accurate and efficient predictions across a broad range of interactions and thermodynamic state points. This kind of analysis, i.e., considering the accuracy of various efficient theoretical methods for predicting the structure consistent with a broad range of possible interactions, will be particularly important for inverse design problems where the goal is to rather accurately predict which interaction is consistent with a targeted structure (or structurally-related property).
The authors thank Anatol Malijevský for sharing his rapidly-converging integral equation theory code. T.M.T. acknowledges support of the Welch Foundation (F-1696) and the National Science Foundation (CBET-1403768). We also acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research findings reported within this paper.
Extended Type B-H Structure Plots {#app:plots}
=================================
The predicted radial distribution functions $g(r)$ compared against simulation results, and the resulting cumulative squared errors $\text{CSE}(r)$, are shown for interaction types B through H in Figs. \[fig:bdata\]–\[fig:hdata\], respectively.
![\[fig:bdata\] Radial distribution functions $g(r)=h(r)+1$ and the associated cumulative squared errors (CSE, see Eq. (\[eq:CSE\])) predicted by the reference hypernetted chain (RHNC), hypernetted chain (HNC), and Percus-Yevick (PY) Ornstein-Zernike closures [@thysimpliq; @malijevsky:bridgefunc]; the first-order mean spherical approximation solution (FMSA) [@tang:fmsa2]; and the simple exponential first-order mean spherical approximation (EFMSA) [@hlushak:sexpfmsasquareshoulder], for the “type A” pair interaction. Shaded regions adjacent to each $g(r)$ indicate the difference between the theory and simulation results.](00_fulldata_typeB){width="8cm"}
![\[fig:cdata\] Radial distribution functions and cumulative squared errors for the “type C” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeC){width="8cm"}
![\[fig:ddata\] Radial distribution functions and cumulative squared errors for the “type D” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeD){width="8cm"}
![\[fig:edata\] Radial distribution functions and cumulative squared errors for the “type E” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeE){width="4.5cm"}
![\[fig:fdata\] Radial distribution functions and cumulative squared errors for the “type F” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeF){width="8cm"}
![\[fig:gdata\] Radial distribution functions and cumulative squared errors for the “type G” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeG){width="8cm"}
![\[fig:hdata\] Radial distribution functions and cumulative squared errors for the “type H” interaction. Series and labeling are as in Fig. \[fig:bdata\].](00_fulldata_typeH){width="8cm"}
Complete Thermodynamic Error Tables {#app:tables}
===================================
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abstract: 'We perform molecular dynamics simulations on an interacting electron gas confined to a cylindrical surface and subject to a radial magnetic field and the field of the positive background. In order to study the system at lowest energy states that still carry a current, initial configurations are obtained by a special quenching procedure. We observe the formation of a steady state in which the entire electron-lattice cycles with a common uniform velocity. Certain runs show an intermediate instability leading to lattice rearrangements. A Hall resistance can be defined and depends linearly on the magnetic field with an anomalous coefficient reflecting the manybody contributions peculiar to two dimensions.'
address: |
Departamento de Física, Universidade Federal de São Carlos,\
Rodovia Washington Luiz km 235, 13565-905, Caixa Postal 676 , São\
Carlos,SP,Brazil
author:
- Vishal Mehra and Jayme De Luca
title: Perfectly Translating Lattices on a Cylinder
---
Introduction
============
The advent of modern computers has made it possible to study many-dimensional dynamical systems in detail and to ferret out previously unsuspected characteristics. Phenomena such as relaxation to equilibrium [@deluca], lack of ergodicity [@thirring], and connection of dynamical to statistical properties [@ruffo] have been explored numerically. A category of models that continue to enjoy great attention is that of interacting-electron systems. In this paper we consider a particular member of this class: the classical dynamics of the confined two-dimensional electron gas (2DEG). While various aspects of 2DEG have been well-studied, it remains a convenient and relatively tractable system to understand dynamical properties of complex systems. It is well-known that at low temperatures the electron system freezes into an hexagonal Wigner lattice [@peeters]. Extensive studies have been made of the possible conformations that result when this simple system is perturbed [@d93; @tkt97]. A simple example is provided by the inter-layer coupling between vertically separated 2DEGs which enhances the stability of the square lattice relative to the hexagonal [@SSP60]. The melting and structural transitions have also been well-studied [@transition]. The confinement is also a possible perturbation—ideal hexagonal lattice is obtained only for infinite system.
However, commonly used confining potentials preserve the hexagonal character with only slight edge distortions. Perturbed 2DEG models have relevance beyond electron systems: experiments on dusty plasmas [@dusty] and ion plasmas [@ion.plasma] have been usefully explained in terms of formation of few layered and bi-layered Coulomb lattices.
The classical behavior of a confined 2DEG subject to an external magnetic field has been of some recent theoretical use [@Magnetic1; @magnetic2; @magnetic3]. In particular the magnetotransport phenomena in 2DEG have lead to important physical insights. New systems have been proposed in which the electron motion in 2DEG is nontrivially altered by a (possibly non-homogeneous) magnetic field [@PMI96]. Some possibilities have been realized by advances in the semiconductor technology [@skww98]. Such systems are studied quantum mechanically for a full understanding but many features can already be appreciated within the classical theory. For example, the classical chaos was found to control the low-field transport in systems with competing magnetic and electrostatic modulation [@HS99; @classical; @fgk92]. In this paper we analyze the structure and dynamics of 2DEG on a novel geometry. The electrons are constrained to the surface of a cylinder of radius $R_{d}$ and confined along the $z$ direction to a strip of width $W=2\pi R_{d}$ by the potential of the uniform static positive background. We chose this geometry because it allows the electrons to carry a current along the $\phi $ direction. We apply the external electric and magnetic fields and observe the time-dependent response.
We have been motivated to this problem by certain reflections on usual textbook derivations of the classical 3-dimensional Hall effect [@textbook]. These derivations are based on free-electron Drude theory and consider independent electrons drifting with a common and constant drift velocity $V$. It is then easy to show that a constant transverse electric field, generated by excess boundary charge, can balance the magnetic force and bring a steady state. In the context of a classical 2-dimensional many degree-of-freedom interacting system this simple picture could be dynamically unstable. Here we conduct numerical experiments to see if a regime of interacting electrons drifting with a common uniform velocity can be attained in a 2D many-body system. A peculiarity of 2DEG is that the Coulomb field escapes from the surface; in 3D a constant transverse Hall field can be nicely produced by the boundary electrons only, while in 2D, because of Gauss’s law, these boundary electrons can only produce a $1/r$ field (the field of a charged wire). We see then that a global charge redistribution is called upon for to produce a constant field.
Simulation and results
======================
The Lagrangian for $N$ interacting electrons confined by the background potential $V_{CON}$ and subject to an external radial magnetic field $B$ is $${\cal L}=\sum_{i}\frac{1}{2}m\dot{\vec{x}_{i}}^{2}-\sum_{i>j}\frac{e^{2}}{%
r_{ij}}-V_{CON}(z_{i})-BR_{d}\sum_{i}z_{i}\dot{\phi _{i}},$$ where $\vec{x}_{i}$ is the coordinate of the $i$th electron and $r_{ij}$ is the distance between $i$th and $j$th electrons and $c$ is the speed of light. The electronic mass and charge are $m$ and $e$ respectively. To remove any ambiguity we clarify that distances are not calculated along the surface but are ordinary three-dimensional distances: $%
r_{ij}^{2}=(z_{i}-z_{j})^{2}+2R_{d}(1-\cos (\phi _{i}-\phi _{j}))$. The rotation symmetry possessed by ${\cal {L}}$ implies the Noether’s constant: $$l_{z}=\sum_{i}mR_{d}^{2}\dot{\phi _{i}}-BR_{d}\sum_{i}z_{i}.$$ The resulting equations of motion are $$m\frac{d^{2}\vec{x}_{i}}{dt^{2}}=-\frac{\partial }{\partial \vec{x}_{i}}%
\left( V_{CON}+\sum_{i\neq j}^{N}\frac{e^{2}}{r_{ij}}\right) +\frac{e}{c}%
\vec{B}\times \frac{d\vec{x}_{i}}{dt}, \label{Unscaled}$$ where the confining potential $V_{CON}$ of the positive background is taken to be either
$$V_{CON}=-(\ln (z+W/2)+\ln (-z+W/2)), \label{confine1}$$
or
$$V_{CON}=((2z+W)\ln (2z+W)+(-2z+W)\ln (-2z+W)), \label{confine2}$$
the stripe being symmetrical about $z=0$ and extending from $-W/2$ to $W/2$. The first potential is flat but rises steeply near the edges. The second potential has more parabolic character and is an approximation to the exact potential of a positive background: we found analytical solution for the potential of a positively charged flat rectangle with the same width and length of our cylindrical background, and the above second potential approximates this analytic form. The use of potentials with differing characters helps in distinguishing any peculiar effect of confinement from overall observations. These potentials are sketched in Fig.(\[confine\]).
The form of equation (1) is simplified by using scaled units: lengths may be scaled by the average interelectronic distance $R$: $\vec{x}\rightarrow R%
\vec{x}$ and time is scaled as $dt\rightarrow \omega ^{-1}d\tau $ where $%
\omega ^{2}\equiv e^{2}/mR^{3}$. In these units the equations of motion are
$$\frac{d^{2}\vec{x}_{i}}{d\tau ^{2}}=-\frac{\partial }{\partial \vec{x}_{i}}%
\left( \frac{R}{e^{2}}V_{CON}(R\vec{x}_{i})+\sum_{i\neq j}\frac{1}{r_{ij}}%
\right) +\hat{B}\times \frac{d\vec{x}_{i}}{d\tau }, \label{Scaled}$$
where $\hat{B}=n^{-3/4}B/\sqrt{(}mc^{2})$ and $n$ is the two-dimensional number density of the electron gas. For example at the experimentally attainable $n=10^{10}$ cm$^{-2}$ the above formula gives $B=0.284~\hat{B}$ Tesla. Because of the scale-invariance of the Coulomb interaction, $\frac{R}{%
e^{2}}V_{CON}(R\vec{x}_{i})$ is independent of $R.$ Equation (4) has two components for each particle corresponding to $z$ and $\phi $ motions. We deliberately add an additionally force along the $\phi $ direction represented by a small electric field $E_{\phi }<<1$ and a weak damping so that the final equation for $\phi $ is
$$R_{d}^{2}\ddot{\phi}_{i}=\hat{B}\dot{z}R_{d}+E_{\phi }(1-\frac{R_{d}\dot{\phi%
}_{i}}{V})+R_{d}^{2}\sum_{j\neq i}\frac{\sin (\phi _{i}-\phi _{j})}{%
r_{ij}^{3}}.$$
We chose this form of damping such that if all electrons cycle with a constant common velocity $V$, the damping force is balanced by $E_{\phi }.$ The last term in the above equation is just the Coulomb repulsion between electrons. The equations of motion are integrated numerically using a 7/8 order embedded Runge-Kutta pair with self-adjusted step.
The initial states for the numerical integrations are generated in the following way: the energy of the confined Coulomb system is $$E=\frac{1}{2}\sum_{i}v_{i}^{2}+e^{2}\sum_{i<j}\frac{1}{r_{ij}}%
+\sum_{i}V_{CON}(z_{i}).$$ A naive candidate for a minimal-energy initial condition would be one which minimizes the above energy and with all electrons cycling with the same velocity $V.$ It is straightforward to find this condition by steepest descent quenching and one obtains a translating lattice which approximates an ideal triangular lattice with some edge distortions. The problem with this condition is that in the presence of the external magnetic field $B$ (which does not appear explicitly in the energy), the translating electrons deflect upwards causing a lattice deformation. We observe, by integrating the equations of motion numerically from this condition with inclusion of the above defined weak dissipation, that the lattice shifts to a different configuration consistent with the presence of magnetic field and with a slightly higher steady state energy. (Notice that we are driving the system, so energy does not have to be constant). An equivalent way to accomplish this same final state is to seek minimum energy configurations with a certain property: they must describe a uniform motion of the entire system along the $\phi $ direction under a radial magnetic field $B$ ($v_{i}=V\hat{%
\phi}).$ What we are looking for is that the total force acting on each electron be zero
$$\nabla \Phi +BV=0,\qquad \Phi =\sum_{i\neq j}\frac{1}{r_{ij}}+V_{CON}.$$
This configuration can be obtained by a steepest descent procedure: we integrate the (modified) quenching equation
$$\frac{d\vec{x}_{i}}{ds}=-\frac{\partial }{\partial x_{i}}(\Phi
+BV\sum_{i}z_{i}/c),$$
$s$ being the parameter along the quenching path.
The quenched configurations are obtained for various values of the parameter $BV$ and then used as initial conditions for subsequent molecular dynamical runs. Fig.(2) shows a representative configuration obtained with the confinement of Eq.(2). The electrons arrange themselves in an hexagonal lattice slightly distorted by the confinement and magnetic field (notice that this is a global distortion, not a simple boundary perturbation of a perfect lattice). Electrons are projected out with an initial $\phi $ velocity $V/R_{d}$ plus small random components along $\phi $ and $z$ directions. Various quantities are calculated along the trajectory: the instantaneous rotation rate $\sum_{i}\dot{\phi}_{i}/N$ which is related to the current $I=\sum_{i}\dot{\phi}_{i}/2\pi $; the $\phi $ averaged potential difference between the top and bottom edges : $V_{H}$ (Hall voltage); and the Hall resistance $R_{H}=V_{H}/I$. All these quantities are function of time and hence their time-development is likely to be informative. We also observe snapshots of the system at regular intervals. These snapshots could be folded to $0-2\pi $ or left unfolded to preserve information about angular motion.
We report the results of the simulations of $N=216$ and 484 particles performed with the confining potential of Eqs (2) and (3). We distinguish between runs carried out with $E_{\phi }$ zero and nonzero. Simulations with nonzero $E_{\phi }$ reach a steady state after a brief transient which, is not attained by zero $E_{\phi }$ runs. We henceforth call this state a Perfectly Translating Lattice (PTL). In the PTL state the electrons cycle with a common constant $\phi $ velocity $V/R_{d}$; motion along $z$ being rapidly damped out. The resulting Hall resistance $R_{H}$ should be a constant in such circumstances but we observe small fluctuations in $%
R_{H}(t) $. The amplitude of these fluctuations is an irregular function of $%
B$ generally varying between 0.1-1.0 percent. The presence of fluctuating $%
R_{H} $ is not disquieting however and can be understood as an artifact of the method employed to calculate Hall resistance of small finite number of electrons: Because we used a finite number of points to average the potential difference, small instantaneous fluctuations are generated if the number of electrons is small and we verify that the fluctuations become smaller for larger $N$ (number of electrons).
Two cases can be further differentiated with regard to the initial phase of the dynamics. Initial configuration for certain values of the external magnetic field $B$ turns out to be dynamically unstable. The system makes a transition to a new configuration through coordinated row-jumping of many electrons simultaneously. Such a transition is made possible by the existence of numerous local minima in the energy surface whose presence has been confirmed numerically by extensive quenching runs starting from distinct initial conditions. The instability only appears for scaled magnetic fields greater than 7.0 for $N=216$ and 8.5 for $N=484$ (for electron density of $10^{10}$ cm$^{-2}$ these fields correspond to 1.9 T and 2.3 T respectively ). In a way, this result shows that the initial state found by minimization ceased to be stable and another extremum (not the minimum anymore) of the functional became an stable fixed point, a bifurcation. This instability is accompanied by slower relaxation to a PTL state and can be seen in Fig.(3) which plots the spread in the instantaneous rotation rate vs. time. Even more dramatically, velocity inhomogeneities are developed if all electrons are released with a common velocity $V$ with no random components. These inhomogeneities are not long lasting and ultimately a PTL is attained.
The resultant of this temporarily existing velocity profile can be visualized most easily in Fig.(4) where the $\phi $ coordinates have not been folded to $(0-2\pi )$. This striking profile is not apparent after folding and the finally established PTLs do not differ from PTLs in cases where this instability is absent. We call this instability a [*shearing*]{} instability since it is visible as a shear in the velocity profile of the electrons. This shear is seen as a dispersion in unfolded lattice along $%
\phi $ coordinate.
We have carried out runs without forcing and dissipation i.e. $E_{\phi }=0$ and these do not achieve a PTL state. They may or may not display the initial shearing instability but in all cases the initial randomness in velocity distribution is magnified and the $z$ dynamics is not damped. An irregular velocity distribution develops even from a perfectly homogeneous initial velocity distribution. The instantaneous Hall resistance $R_{H}(t)$ is unsteady with large amplitude fluctuations and the system can not be said to be in a Hall regime.
From these simulations a plot of the Hall resistance $R_{H}$ vs. $B$ can be drawn. Only converged values of $R_{H}$ are used which rules out our undriven simulations ($E_{\phi }=0$). In Fig.(5) results from unsheared and sheared states are displayed for $N=216$ and 484. Data from the simulations employing confining potential of equation (3) has been plotted for $N=216$ also. All are PTL configurations but the sheared states have experienced the shearing instability referred to earlier. We observe that $R_{H}$ from the unsheared states lie on straight lines though different slopes are obtained for the two confining potentials used: 1.07 and 1.05 for potential (1) and (2) respectively. Data for $N=216$ and 484 overlap. Points from the sheared states are scattered haphazardly about this straight line.
How far do these data match our expectations? A plausible argument can be made for reasonableness of our simulation results: The Hall voltage $V_{H}$ is the difference between the top and the bottom edges of the stripe and may be expressed as
$$V_{H}=V(W/2)-V(-W/2)=\int_{-W/2}^{W/2}\nabla _{0}\left( \sum_{i}\frac{1}{%
r_{i0}}+V_{CON}\right) .dl_{0},$$
where $V=\sum_{i}1/r_{i0}+V_{CON}$ and the integral is taken along a straight line from bottom to the top edge; $r_{i0}$ is the distance of the $%
i $th electron from integration element. Now the integrand may be split as
$$\nabla _{0}\left( \sum_{i}\frac{1}{r_{i0}}+V_{CON}\right) \approx BV\hat{z}%
+\nabla \left( \frac{1}{r_{n0}}\right) ,$$
where $n$ labels the electron nearest to the integration element and $\hat{z}
$ is the unit vector along $z$ direction. This follows from the assuming that the force-balance condition (Eq. 9) holds in a neighborhood of the electron nearest to 0:
$$\nabla \left( \sum_{i\neq j}\frac{1}{r_{ij}}+V_{CON}\right) =BV,$$
where $i,j$ label electrons. This equation holds only at the position of an electron and we use as an approximate equality in a neighborhood of the nearest point $0$. From the above it follows
$$V_{H}=\int BV\hat{z}.dl_{0}+\int \nabla \left( \frac{1}{r_{n0}}\right)
.dl_{0}.$$
The first term on the right yields the straight line dependence of $R_{H}$ vs. $B$ with a unit slope but the second term provides a correction. To evaluate this correction, we take an integration path passing through a column of electrons, avoiding each electrons by making a small semi-circle around it. If the distance between the $i$th and $(i+1)$th electrons along this path is $d_{i}$, this correction evaluates to $$2\sum_{i}\left( \frac{d_{i+1}-d_{i}}{d_{i}}\right) .\frac{1}{d_{i}}$$ Hence the correction to the unit slope depends on the degree of compression that the lattice undergoes under the external magnetic field. A 3D lattice does not suffer this kind of bulk squishing but only edge distortions. If the lattice rearrangement is global in the sense that lattice distances are affected throughout the bulk and not just at the edges then we can fairly expect a significant alteration of the slope of Hall resistance plot. This expectation is realized in our simulations as we have seen that the slopes of $R_{H}$ vs $B$ plot indeed differ from unity by a few percent. Lattices which are dynamically unstable and undergo rearrangement via the shearing instability would seem to require more correction according to this picture and in fact provide a needed check for the theory.
Discussion and Conclusion
=========================
In conclusion we have performed dynamical simulations on a 2DEG constrained to a cylindrical stripe and subject to crossed electric and magnetic fields. The classical confined electron gas has a natural non-trivial minimal energy state and provides a convenient test bed to study many-body dynamics and long-range effects. In this work we included many-body effects on the dynamical picture used in the classical derivations of the Hall effect in two dimensional systems. We have analyzed the formation of the steady state presupposed in classical derivations. This state, which we refer to as a Perfectly Translating Lattice (PTL), in which all electrons cycle with a common constant velocity, is formed by a relaxation process. The initial configurations are obtained by a generalized quenching procedure. This initial configuration is liable to be dynamically unstable for magnetic fields above a threshold. The quenching procedure sometimes yields rather shallow local minima which readily allow further rearrangements to nearby wide basins. The inter-basin like motion is manifested in simultaneous jumping of many electrons and a slower relaxation to PTL state. The Hall resistance $R_{H}$ can be calculated and plotted as a function of external magnetic field $B.$ Appropriately for PTL states $R_{H}$ is a linear function of $B$ except for sheared states. An explanation has been put forward based upon force-balance condition as it obtains for PTL states. If one repeated the same study for a confined 3D electron-gas, the minimal energy state would be a tridimensional Wigner lattice and inclusion of a drift velocity in the presence of an external magnetic field would produce only boundary charge rearrangement. The obtained PTL would be the same ideal Wigner lattice and there would be no many-body correction to the Hall coefficient. Our results are specific to 2D classical systems.
The present work evolved from our earlier attempts to study the same problem but with the electrons interacting via the Darwin Lagrangian, which is the first relativistic correction to the Coulomb interaction. We find that the relativistic corrections break the scale-invariance of the Coulomb interaction, even in the absence of a magnetic field, only by requiring a critical density of the electron gas[@Vishal]. Even though these is a much richer dynamical system, we did not continue the studies because the equations of motion are algebraic-differential and become impossible to integrate above the critical density even by use of the modern specialized integrators RADAU [@RADAU]and DASSL [@DASSL].
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|
---
abstract: 'We present an experimental study of stochastic resonance in an electronic Chua circuit operating in the chaotic regime. We study in detail the switch-phase distribution and the phase-shift between sinusoidal forcing for two responses of the circuit: one depending on both inter-well and intra-well dynamics and the other depending only on inter-well dynamics. We describe the two relevant de-synchronizatrion mechanisms for high and low frequencies of the forcing and present a method to detect the optimal noise intensity from switch phases which coincides with the one derived from the observation of the signal-to-noise ratio or residence times.'
author:
- Wojciech Korneta
- 'Iacyel Gomes, Claudio R. Mirasso, Raúl Toral'
title: 'Phase-shifts in stochastic resonance in a Chua circuit'
---
[^1]
Introduction
============
Stochastic resonance, by which a weak signal acting upon a nonlinear system can be amplified by the addition of noise of optimal intensity, has attracted considerable attention in recent years. While most of the research has focused on bistable and excitable systems [@1; @2], studies have also been performed in chaotic systems with two attractors, in what can be considered as a generalized form of a bistable system [@3; @4]. Several measures have been introduced to quantify the strength of the response to the forcing: the response amplitude originally used by Benzi et al.[@5], the signal-to-noise ratio [@1; @2], as well as other measures determined from the residence and switching times probability distributions [@6; @7; @8] or the work done by the external force [@9]. Although all these different quantifiers display a maximum as a function of the noise intensity, the values of the optimal noise intensity obtained in each case do not necessarily coincide.
In both theoretical and experimental studies the phase-shift between the force and the response has been much less considered, although it is interesting from the viewpoint of relating stochastic resonance to standard resonance phenomena. Moreover, the magnitude of the phase-shift and its variation with noise intensity has been the subject of some controversy [@10; @11; @12]. The phase-shift has been analytically calculated in bistable system within the linear response theory taking into account both intra-well and inter-well dynamics [@10; @13]. It displays a bell-shape dependence on the noise intensity with a extremum at a smaller value of the noise intensity than the optimal noise intensity obtained from the signal-to-noise ratio. This result was confirmed by measurements on an analog electronic circuit [@14] and the same dependence has also been obtained numerically [@9; @15]. However, results in a one-dimensional Ising model show a monotonous change of the phase-shift with the noise intensity[@16]. This controversy [@11; @12] was considered in Refs.[@16; @17; @18] based on numerical studies of a bistable system and an assembly of superparamagnetic particles. One obtains the monotonous change of the phase-shift with the noise intensity if only inter-well dynamics is taken into account. The bell-shape dependence thus reflects the competition between inter-well and intra-well dynamics. It was then concluded that the extrema in the dependence of the phase-shift and signal-to-noise ratio have a different origin. The phase-shift has also been studied in terms of switch-phase distributions in Ref.[@19]. The authors presented a numerical investigation in the symmetric Schmitt trigger and proposed the de-synchronization mechanism responsible for the disappearance of stochastic resonance.
The aim of this paper is to present experimental results of our thorough studies of the phase-shift in stochastic resonance in the Chua electronic circuit [@20] operating in a chaotic regime where two single-scroll attractors coexist. In this case the circuit can be thought of as bistable and the observed phenomena correspond to conventional stochastic resonance within two stable wells[@14]. We have determined the phase-shift between a sinusoidal forcing $E(t)=E_0\sin(\omega t)$ and two responses of the circuit: one depending on both inter-well and intra-well dynamics and the other depending only on inter-well dynamics. This clarifies the dependence of the phase-shift on the noise intensity and parameters of external forcing. Considering switching-phase distributions one can distinguish two de-synchronization mechanisms for low and high frequencies of external forcing. This leads us to propose different methods to determine the optimal noise intensity. We show that the same optimal noise intensity can be obtained from the signal-to-noise ratio and the switching, residence time or switching-phase distributions. Stochastic resonance is a widespread phenomenon and we hope that our methods will provide an easy and appropriate way to determine numerically or experimentally the optimal noise intensity.
Experimental setup and results
==============================
The Chua circuit and parameters of its components used in our experiments are given in Fig.\[fig:1\]. Stochastic resonance in this circuit was observed and described in Ref.[@4]. In the absence of the forcing, $E(t)$, and the noise source, $\xi(t)$, the dynamics displays two single-scroll attractors and trajectories flow to one or another depending on initial conditions. Jumps between the attractors can be observed if the amplitude $E_0$ of the forcing is above a threshold value that depends on the frequency $\omega$ of the forcing [@4]. In all experiments we first set the amplitude of forcing just below that threshold value and then add the noise signal to the forcing so to induce jumps of the system between single-scroll attractors. During the experiments we recorded both $E(t)$ and the voltage $V_1(t)$ on the capacitor $C_1$. This voltage is selected because it clearly shows the position of the dynamical trajectory on either attractor and it represents the response of the circuit depending on both inter-well and intra-well dynamics. The step function representation, $S(t)$, of this voltage has been used in many studies [@1] as it represents the response of the circuit depending only on inter-well dynamics.
We have computed phase-shifts both for $V_1(t)$ and $S(t)$. An example of temporal evolution of $E(t)$, $V_1(t)$ and $S(t)$ is shown in Fig.\[fig:2\]. At forcing frequencies below $100$Hz one can determine the phase-shift between the forcing $E(t)$ and the response, e.g. $V_1(t)$, as the value $\Phi_{V_1}$ of the phase $\phi$ such that the cross-correlation function $\langle E(t)V_1(t+\phi/\omega)\rangle$ takes the maximum value (where $\langle\cdot\cdot\cdot\rangle$ represents a time average). In Fig. \[fig:3\] we present the dependence of the phase-shift $\Phi_{V_1}$ on the noise intensity (measured as the standard deviation of its probabiity distribution) for different amplitudes $E_0$ and frequencies $\omega$. This is the bell-shape dependence observed and explained by Dykman et al. [@13]. For small noise levels, inter-well hoppings can be neglected and the circuit dynamics depends almost totally on the interaction between $E(t)$ and the intra-well motion. The characteristic frequency of the dynamics of our circuit at the single-scroll attractor is around $2740$Hz, so the phase-shift remains small. The abrupt decrease in the phase is associated with the onset of inter-well jumps.
One can notice that the minimum of the phase-shift depends both on $E_0$ and $\omega$. This is illustrated by the data presented in Fig.\[fig:3\] in the case $\omega=1$Hz. The threshold amplitude value in this case is about $555$mV. For lower forcing amplitudes higher noise intensities are necessary to induce jumps between attractors and the minimum of the bell-shape curve moves to higher noise intensities. At low $\omega$ the phase-shift $\Phi_S$ between $E(t)$ and the step function representation of the voltage $S(t)$ can also be determined from the maximum of the corresponding correlation function. An example of the dependence of this phase-shift on the noise intensity is shown in Fig.\[fig:4\] in the case $\omega=10$Hz. This dependence is monotonic and similar to what has been obtained in a one-dimensional Ising model [@16]. By superimposing in this figure also the phase-shift $\Phi_{V_1}$ one can notice that as soon as inter-well jumps are activated they immediately become dominant and cause the decrease of the phase-shift between $E(t)$ and $V_1(t)$. The inter-well hoppings also make the phase-shift tend to zero for higher noise intensities.
Phase-shifts can also be determined from the switch-phase distribution. This method has been used in the symmetric Schmitt trigger [@19] and it is specially important for theoretical and experimental studies of stochastic resonance in neurons [@21]. In our case we looked at the distribution $f_+(\phi_+)$ of the oriented switch phases $\phi_+$ defined as the phases of the sinusoidal forcing signal (modulo $2\pi$) corresponding to switches of the step function $S(t)$ from negative to positive value. The distribution $f_-(\phi_-)$ of the switch phases $\phi_-$ in the opposite direction can be obtained by a translation by $\pi$ of the distribution of the switch phases $f_+(\phi_+)$. The average of the switch phases $\Phi_+$ is defined by the following equation: $$\label{eq:1}
\rho \exp{(i\Phi_+)}=\frac{1}{N}\sum_{k=1}^{N}\exp{(i\phi_+^k)}$$ where $\phi_+^k$, with $k=1, \dots, N$, are the different phase switches observed during the time evolution and the amplitude $\rho$ of the complex quantity serves as an order parameter characterizing the degree of phase synchronization in neurons [@21]. The dependence of $\Phi_+$ on the noise intensity coincides with that of the phase-shift $\Phi_S$ between $E(t)$ and $S(t)$, as shown in Fig.\[fig:4\]. Both measures can thus be equivalently used to determine the phase-shift of a system whose response depends only on inter-well dynamics. In Fig.\[fig:5\] we present distributions of switch phases $f_+(\phi_+)$ obtained at forcing frequency $\omega=7.5$Hz and different noise intensities. At low noise intensities there is only one peak in the distribution. As the noise intensity increases, this peak shifts towards zero and flattens. For a standard deviation of noise around $300$mV a second peak appears at distance $\pi$ from the dominating peak. The onset of the second peak signals the onset of de-synchronization in stochastic resonance. As the noise intensity increases the second peak grows until both peaks become comparable. This de-synchronization mechanism was observed and described in the numerical studies of the symmetric Schmitt trigger [@19] and signals the degradation of the stochastic resonance with the noise intensity.
These observations allow us to propose a quantity which can be used to determine the optimal noise level from switch-phase distributions. Let us denote by $P_{\phi_+}$ the probability that the switch-phase $\phi_+$ belongs to the interval $[\phi_+^D-\pi/2,\phi_+^D+\pi/2]$, where $\phi_+^D$ is the position of the dominating peak in the switch-phase distribution. The dependence of this probability on the noise intensity for different forcing frequencies is presented in Fig.\[fig:6\]. For frequencies higher than $7.5$Hz all curves in this figure have the inflection point at a noise intensity of $300$mV. In a previous paper [@4] we considered the residence time distributions and defined an appropriate quantity which is suitable to detect the optimal noise level from these distributions. This quantity is the probability $P_{T_r}$ that the residence time is in one of intervals $[(i-3/4)\frac{2\pi}{\omega},(i-1/4)\frac{2\pi}{\omega}]$, where $i=1,2,\dots$. The dependence of this probability on the noise intensity for the same forcing frequencies as in Fig.\[fig:6\] is shown in Fig.\[fig:7\]. All curves at this figure also have the inflection point for the same noise level as curves in Fig.\[fig:6\]. We observed [@4] that this noise level is the same as the optimal noise level obtained experimentally from the dependence of signal-to-noise ratio on noise intensity. We thus propose to use alternatively the dependence of the signal-to-noise ratio on the noise intensity or the dependence of probabilities $P_{\Phi_+}$ or $P_{T_r}$ on the noise intensity. In the first case the optimal noise level corresponds to the maximum whereas in the other cases it corresponds to the inflection point.
The methods described above to determine the phase-shift and de-synchronization mechanism change for forcing signals with frequencies above $100$Hz. In this case, the correlation function between the response of the system (either $V_1(t)$ or $S(t)$) and the forcing $E(t)$ vanishes and the synchronization between the forcing and the response can only be determined from the switch-phase distributions, as the ones presented in Fig.\[fig:8\] for different noise intensities and $\omega=540$Hz. One can note that there is only one peak in the distribution for all noise intensities. Increasing the noise intensity the position of the peak does not change but the distribution flattens. Finally for very large noise intensities the distributions of switch phases $\phi_+$ and $\phi_-$ start to overlap. This is the second mechanism of de-synchronization that ensues stochastic resonance on increasing further the noise level. The position of the peak in the switch-phase distribution changes with the forcing frequency. We present in Fig.\[fig:9\] the dependence on the forcing frequency of the average switch-phase $\Phi_+$, as obtained from Eq. (\[eq:1\]). The values $\Phi_+=-\pi/8,-\pi/4,-\pi/2$ correspond, respectively, to frequencies $\omega=590$Hz, $1080$Hz, $2160$Hz. This dependence does not depend on the value of the sub-threshold amplitude of the forcing signal. It is evident from Figs. \[fig:5\] and \[fig:6\] that also for high frequencies the optimal noise intensity is determined by the inflection point of the dependence of the probability $P_{\phi_+}$ or $P_{T_r}$ on the noise intensity. As we have shown in Ref.[@4] this corresponds to the optimal noise intensity as determined by the maximum of the signal-to-noise ratio.
Conclusions
===========
In this paper we have addressed the study of the the phase-shift in stochastic resonance. We have analyzed two mechanism of de-synchronization observed at high and low forcing frequencies. Our results point out the attention to switch-phases which are able quantify the synchronization between the forcing and response in the whole frequency range of the forcing signal. This paper also suggests methods to detect the optimal noise intensity by observing the signal-to-noise ratio, the residence times, the switching times or switch-phases. The optimal noise intensity determined from any of these observations is the same and they can be used alternatively. We have derived our main results by studying experimentally a Chua circuit operating in a chaotic regime, but we believe that our main conclusions are rather general and could be useful in many investigations and applications of the phenomenon of stochastic resonance.
We acknowledge financial support by the MEC (Spain) and FEDER (EU) through projects FIS2006-09966 and FIS2007-60327.
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[^1]: wkorneta@op.pl
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abstract: 'Young stars in the disks of galaxies produce HI from their parent H$_2$ clouds by photodissociation. This paper describes the observational evidence for and the morphology of such HI. Simple estimates of the amount of dissociated gas lead to the startling conclusion that much, and perhaps even all, of the HI in galaxy disks can be produced in this way.'
author:
- 'Ronald J. Allen'
title: |
On The Origin of HI in Galaxies:\
The Sizes and Masses of HI Photodissociation Regions
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Most observers view the clouds of HI we see in galaxies as the raw material out of which the stars were formed. These clouds are thought to form higher-density complexes of gas and dust, turn molecular (H$_2$), and then form stars. In this context, the observed correlation between the star formation rate in galaxy disks and the HI content (often called the Schmidt Law) is generally viewed as being at the basis of an understanding of the global star formation process in galaxies. However, after many years of work to elucidate the specific physics of this process, we have a complicated and still rather rudimentary picture of just how all this might happen.
I want to propose a different view for a part of this “star-formation” story. This view is contrary to the conventional wisdom, but it has the virtue of being physically much simpler. I hope to convince you that this view is supported both qualitatively, by the detailed morphology of the HI observed in galaxy disks, as well as quantitatively, by the theory of photodissociation of H$_2$. In this view, the basic star construction material in galaxies is gas which is already mostly molecular, and out of which the stars form directly. HI appears in the region when the leftover H$_2$ is illuminated with UV photons from nearby young stars. The physics of photodissociation regions provides a natural and quantitative explanation for the appearance of HI envelopes around the clouds, and for the CO(1-0) emission which is sometimes seen emanating from the warmer, higher-density parts of their surfaces. The rather surprising new result is that the total amount of Far-UV emission produced in galaxy disks by run-of-the-mill, non-ionizing B stars is actually sufficient to account for most, and perhaps even all, of the HI present.
Background
==========
For me, the first example of failure of the conventional wisdom about HI being the precursor to star formation appeared in the early VLA-HI synthesis images of the relatively nearby galaxy M83, which I first saw at an afternoon seminar given by Mark Ondrechen at the Kapteyn Laboratory in Groningen in 1985. The morphology of the HI in the spiral arms of this galaxy did not fit the expected picture; rather than coinciding with the dust lanes as markers of the highest gas column density, the HI ridges appeared to be shifted to larger galactocentric distances and to correspond better with the line of HII regions marking the locus of the youngest stars in the spiral arm. The density wave picture for spiral structure introduced by Lin & Shu (1964) led to a simple explanation (Roberts 1969) for the separation of the arm of maximum gas column density (located upstream in the flow) and the HII regions (downstream) in terms of the flow velocities and the time to convert the H$_2$ in GMCs into stars. But if the HI was a precursor to the H$_2$ in GMCs, this picture did not explain why the HI ridge also appeared downstream with the HII regions. The explanation we offered (Allen, Atherton, & Tilanus 1986) was that the ISM was changing its physical state as it moved along, and doing so in a big way. Primarily molecular to start with, the residual gas after star formation becomes atomic as it is bathed in UV photons from the most massive of the nearby newly-formed stars. It later returns to the molecular form after the young stars die off and the UV flux decreases.
Other studies have followed on M83 and on other nearby spirals (M51, M100) and have generally reached the same conclusions; a list of relevant references is given in the introduction to the paper on photodissociation in M101 by Smith et al. (2000); that introduction also summarizes the basic physics and provides many additional references to the literature on the physics of Photo-Dissociation Regions (PDRs).
With the exception of the most recent work on M101, the studies referenced above have been carried out on relatively coarse linear scales, typically 500 - 1000 pc. This is too coarse to even begin to resolve the morphology of individual PDRs, which form “blankets” and “blisters” on the surfaces of GMCs with the largest scales of $\sim 100$ pc. Such structures were first identified in M81 with a resolution of $\sim 150$ pc (Allen et al. 1997) and in M101 with a resolution of $\sim 220$ pc (Smith et al.2000).
In this paper I want to start from the “other” end, at length scales of $\sim 1$ pc, and explore the nature of the HI structures associated with GMCs in the Galaxy. These structures can also be explained with the photodissociation model, and provide new information on how to quantify the relation between UV-producing B stars and HI. To give the story some structure I will describe the examples on scales of a factor of 10 from 1 to 1000 pc, providing overlap with the studies described above and showing the continuity and ubiquity of the photodissociation process in the ISM of galaxies.
HI associated with H$_2$ on the $\approx 1$ pc scale
====================================================
On the scale of $\approx 1$ pc the radio – HI resolution is too poor to look for HI – H$_2$ associations anywhere except in the Galaxy, and there, confusion along the line of sight usually makes the identification of HI features with specific stars and GMCs very difficult. Nevertheless, new all-sky HI surveys at high resolution have revealed several cases of discrete HI features associated with young stars deeply embedded in dense molecular clouds.
Deeply-embedded young stars
---------------------------
The prototype of this class is IRAS 23545+6508 (Dewdney et al. 1991). This object is a strong IR source and a weak radio continuum source, and also has a faint reflection nebula apparently associated with it. The HI source is compact, $\approx 0.6$ pc in diameter, with M(HI) $\approx 1.4$ M$_{\odot}$. The star is B3 or B4 but behind $A_V \approx 11$ mag of extinction, embedded in a dense GMC at $\approx 1$ kpc distance.
Several other examples like IRAS 23545+6508 have been found (Purton, private communication), including an object associated with the star BD +65 1638 in the star-forming region around NGC 7129 (Matthews et al., in preparation), IRAS 01312+6545 found in the Canadian Galactic Plane Survey, and IRAS 06084-0611 associated with BD +30 549. These are all examples of “dissociating stars” with several M$_{\odot}$ of HI and a very small amount of ionized gas (1/1000 of the HI) surrounding B stars of types ranging from 0.5 to 5. Charles Kerton (private communication) has recently provided a list of additional candidates (Table \[table:IRAS-B\]). These objects all have HI masses of a few M$_{\odot}$.
[ccc]{} IRAS Number & D$_{kin}$(pc) & Sp. Type\
& &\
01431+6232 & 6.0 & B3\
01448+6239 & 4.8 & B3\
01546+6319 & 6.0 & B3\
01524+6332 & 4.5 & B5\
The identification of these objects on HI maps of the Galactic plane is facilitated by the very compact character of their HI emission. Presently the search criterion involves a correspondence of an HI concentration with a compact source of IR emission, but the best way to find these objects in the Galaxy is still under discussion. Figure \[fig:IRAS01431+6232\] shows the HI and CO(1-0) emission identified with IRAS 01431+6232.
Table \[table:IRAS-B\] is likely to be affected by observational selection. For instance, the HI emission from such objects which are closer to us will be larger in angular size and more diffuse, and therefore increasingly difficult to discern in the HI maps; on the other hand, at distances of a few kpc, these objects are barely resolved, so more distant HI objects will be fainter and offer less contrast with the ambient HI. The concentration to mid-B spectral types may also be a selection effect; later-type stars produce too little dissociating radiation, so not enough HI is produced, whereas the ionizing flux from earlier-type stars destroys the HI, creating a substantial HII region instead.
The dissociated HI mass; a simple calculation
---------------------------------------------
The central star e.g. in IRAS 23545+6508 has T$_{\rm eff} \approx$ 17,000 - 18,000 K, so the rate of production of ionizing photons is ${\mathcal
R_{\rm LYC}} \approx 10^{43.3}$ photons/sec. But the rate of production of [*dissociating*]{} photons is much higher, ${\mathcal R_{\rm FUV}} \approx
10^{46.5}$ photons/sec (e.g. Puxley et al. 1990) in the range $912 < \lambda < 1108$ Å.
According to Dewdney et al. (1991) the GMC density $n_2 \approx 900$ molecules cm$^{-3}$, so the 2 $\times$ HI $\rightarrow$ H$_2$ reformation time on grains $ \tau_{\rm rec} \approx 3.3 \times 10^8/n_H$ years is $\approx 1.8 \times 10^5$ yr for $n_H = n_1 + 2n_2$, significantly longer than the estimated present age of the B star of $ \tau_{\rm B3V}
\approx 10^4$ yr. In this case a simple calculation can be made of the total mass of dissociated HI around the star. Assuming that 100% of the dissociating photons produced are used inside the GMC, and taking the efficiency $\eta \approx 0.15$ for the dissociation of an H$_2$ molecule into two HI atoms, the number $\mathcal{N}$ of HI atoms produced by the star up to the present time is: $$\begin{aligned}
% this eqnarray environment has no equation numbers.
\mathcal{N}({\rm HI}) & = & 2 \times \eta \times \mathcal{R}_{\rm FUV}
\times \tau_{\rm B3V}, \\
& \approx & 3 \times 10^{57}\; {\rm HI \; atoms}, \\
{\rm M} & \approx & 2.5\; {\rm M}_\odot {\rm \; of \; HI.}\end{aligned}$$ The excellent agreement with the observed value of 1.4 M$_{\odot}$ must be fortuitous, considering the approximations. Dewdney et al. (1991) used a more complete, time-dependent model of HI production they developed themselves (Roger & Dewdney 1992) to arrive at a value of 2 – 3 M$_{\odot}$ of HI, again assuming a lifetime of $10^4$ yr for the star.
Note that if the volume density of the molecular ISM in which these stars are embedded was lower, say $n_2 \approx 100$ cm$^{-3}$ (the typical GMC), the HI sources would be $\approx 10$ pc in size (with the same HI mass of a few M$_{\odot}$), and would begin to merge into the generally-lumpy distribution of Galactic HI, making them even more difficult to identify.
HI associated with H$_2$ on the $\approx$ 10 pc scale
=====================================================
On the $\approx 10$ pc scale in the Galaxy, confusion is a major problem, and specific associations of HI and H$_2$ have been made only in cases where the geometry is particularly simple, or the region is not confused by other features along the line of sight (e.g. at high Galactic latitude).
HI and H$_2$ in interstellar cirrus clouds
------------------------------------------
A study of the HI, CO, and IR emission from a sample of 26 isolated, high-Galactic-latitude cirrus clouds (Reach, Koo, & Heiles 1994) with typical sizes 1 - 10 pc shows that the H$_2$ content of these clouds is of the same order as the HI mass, in spite of their location above the Galactic plane where we might expect the UV flux from the Galaxy to have destroyed all the H$_2$. In one well-studied case (G236+39) the inferred mass of H$_2$ is 70 M$_{\odot}$, compared to an HI mass of 90 M$_{\odot}$.
Warm HI “envelopes” around Galactic GMCs
----------------------------------------
An analysis of observations of several Galactic GMCs (Andersson & Wannier 1993 and references there) reveals warmed HI surface layers with characteristic depths of $\approx 2$ pc and maximal extents of $\approx 10$ pc. The HI production is ascribed to photodissociation. In one well-studied case (Andersson, Roger, & Wannier 1992) the envelope around the GMC called “B5” in the Per OB2 association has 350 M$_{\odot}$ of HI in an envelope which is moderately dense, $\approx 35$ cm$^{-3}$, and warm, $\approx 70$ K, and is expanding away from the cloud at approximately the escape velocity. Blitz & Terndrup (quoted in Blitz 1993) have reported on about a dozen cases of HI envelopes surrounding Galactic GMCs in or near well-known star-forming regions (NGC 7023, S140, Per OB2, Orion, ...). The HI masses range from $\approx 500$ to 500,000 M$_{\odot}$.
Another simple calculation
--------------------------
How much HI could be produced by photodissociation around a typical GMC embedded in the mean interstellar radiation field (ISRF)? For this estimate our initial approach assuming a very young PDR and neglecting re-formation of H$_2$ will not be adequate. We need a steady-state calculation, where the HI production rate by photodissociation of H$_2$ is balanced by the 2 $\times$ HI $\rightarrow$ H$_2$ reformation rate on grain surfaces. We use the model developed by Sternberg (1988) to determine the steady-state column density of HI. For standard values of ISM parameters in the solar neighborhood we have: $${\rm N}(HI) \approx 5 \times 10^{20} \times
\ln (90 \chi/n_{\rm H} + 1)$$ where:\
------------- --- ---------------------------------------------------
N(HI) = the HI column density in atoms cm$^{-2}$,
$\chi$ = the FUV intensity relative to the local ISRF, and
$n_{\rm H}$ = the total proton volume density of the gas.
------------- --- ---------------------------------------------------
\
The standard GMC has a diameter of 50 pc, an average H$_2$ volume density $\langle n_2 \rangle \approx 100$ and a total mass of $10^{5-6}$ M$_{\odot}$ (Blitz 1993). The ISRF therefore creates a “skin” of HI on the surface of this cloud with N(HI) $\approx 3.5 \times 10^{20}$ atoms cm$^{-2}$. Assuming the cloud is spherical, its surface area is $7 \times
10^{40}$ cm$^2$, so the warm HI envelope around the GMC will contain:\
------------------- ----------- -----------------------------------------------------------------
$\mathcal{N}$(HI) $\approx$ $3.5 \times 10^{20} \times 7 \times 10^{40}$ HI atoms, which is
M(HI) $\approx$ $2.1 \times 10^4$ M$_{\odot}$ of HI,
------------------- ----------- -----------------------------------------------------------------
\
...close to the average observed HI mass in the GMC sample of Blitz & Terndrup. Such GMCs are apparently about 10% HI and 90 % H$_2$.
HI associated with H$_2$ on the $\approx$ 100 pc scale
======================================================
On the $\approx 100$ pc scale in the Galaxy, confusion remains a big problem. Unusual morphologies, correspondences in several tracers, and careful isolation of features along the line of sight are necessary.
HI “tails” and “cones”
----------------------
The Canadian Galactic Plane Survey in HI and radio continuum, coupled with a similar coverage in CO emission with the FCRAO radio telescope, has revealed many features in the Galaxy which bear the signature of the destruction of GMCs by photodissociation (e.g. Wallace & Knee 1999). The object WK-7 is a typical example (Figure \[fig:WK-7\]), with CO emission at the apex of an HI structure that fans out away from the CO. For many of these associations B stars have been found nearby.
Maddelena’s cloud
-----------------
A careful separation of features in radial velocity has permitted the identification of a large ($\approx 50 \times 200$ pc), “blanket” of HI located in the Galaxy and associated with CO emission from the GMC G216-2.5 (Williams & Maddelena 1996) and two nearby young stars (at least one of which is surely a B star). This structure is especially important to the discussion now because it is large enough to be identified in high-resolution HI studies of nearby galaxies, and shows us the morphology we ought to be seeking in that case. An interesting point with this object is that both the HI and the CO emission from this structure appear roughly to fit into the PDR picture. The observations give the “excess” HI associated with the PDR as $\approx 2 \times 10^{20}$ atoms cm$^{-2}$ and the average CO(1-0) intensity as $\approx 6$ K km s$^{-1}$. These two points are in rough agreement with the combined predicted CO(1-0) (Kaufman et al. 1999) and HI intensity (equation given above) for an H$_2$ density of $\approx 100$ cm$^{-3}$, typical for GMCs in the Galaxy (Blitz 1993), and for an incident FUV flux of $\chi \approx 1$, which is just the estimated “excess” value coming from the nearby B stars according to Williams & Maddelena. A more refined model is under construction (Allen, Heaton, & Kaufman, in preparation).
The step to nearby galaxies
---------------------------
With its $\approx 100$ pc size scale, Maddalena’s Cloud provides a vital link between the structures we have so far been discussing in the Galaxy and the structures we can discern in the nearby galaxies with current instrumentation. HI “blankets” and “blisters” similar in structure to Maddalena’s Cloud and characteristic of large-scale PDR morphology have now been found all over the disks of two nearby galaxies, M81 and M101, thanks to the availability of FUV imaging from the ASTRO/UIT missions and high-resolution HI images from the VLA. The combination permits a comparison of the FUV and HI morphologies on a $\approx 100 - 200$ pc scale. Although even better linear resolution would be highly desirable, there are three important advantages of working on nearby galaxies:
- The line-of-sight confusion problem which plagues the Galactic work is now virtually absent, so the features are more easily identified.
- The distances to different structures are all very nearly the same, so we do not have to contend with widely varying angular scales on the sky for what are basically the same physical structures just seen at very different distances (e.g. a given physical structure in the Local arm of the Galaxy can be 10 or more times the angular size of the same structure in the Perseus arm). The observational selection effects are therefore more nearly the same over the whole of the galaxy disk.
- Observational selection effects act to enhance the utility of HI for detecting the PDRs arising when low-density GMCs are illuminated by modest FUV radiation fields...the HI features in this case are generally larger, so the telescope beam filling factors are larger, and the HI is therefore easier to detect on the 21-cm VLA maps.
### FUV and HI in the Sb I-II spiral M81
HI “blisters” and “shells” are common in M81 (Allen et al. 1997) and are clearly associated with nearby clusters of young stars. These clusters sometimes also emit H$\alpha$, but not always, whereas the association with HI is common; B stars dominate the FUV morphology at $\lambda 150$ nm. In addition to the association of discrete HI and FUV features in M81, the general FUV brightness correlates well with the HI brightness in spiral arms (more on this later), indicating that HI production by a general, scattered distribution of dissociating stars is important.
### FUV and HI in the Sc I spiral M101
In the first really detailed, quantitative study of the HI – FUV association, Smith et al. (2000) have developed a method using the equations given earlier in this paper to discover a new probe for the molecular gas in galaxies. They conclude that GMCs in M101 have volume densities in the range of 30 – 1000 cm$^{-3}$ with no clear trend from the inner to the outer parts of the galaxy. The large-scale and well-known decrease in N(HI) in the inner parts of M101 is explained in the context of their PDR model as a result of the increasing dust/gas ratio there.
HI associated with H$_2$ on the $\approx$ 1 kpc scale
=====================================================
On the $\approx 1$ kpc scale in nearby galaxies the individual PDRs can not be resolved, and we must make do with trying to interpret the general surface brightness distributions of FUV and HI. A recent result in the outer parts of M31 bridges the gap, providing a clear association between a general distribution of identified B stars and the associated (low-resolution) HI surface brightness.
Gas, dust, and young stars in the outer disk of M31
---------------------------------------------------
Cuillandre et al. (2001) have studied a field in the far outer parts of M31, beyond the De Vaucouleurs radius along the major axis of the galaxy to the south-west. Besides other results in that paper we note the following points of interest here:
- There is dust mixed in with the HI on the large scale, so we may expect that H$_2$ is also present.
- Young stars are distributed generally with the HI. These stars have the signatures (V mag, V-I colors) of B stars in the outer parts of M31.
- These B stars are found at radial distances of 23 to 33 kpc (4 to 5.7 times the disk scale length in B) in areas where the galaxy optical surface brightness is $\mu_B > 27$ mag arcsec$^{-2}$. This is approaching the range of a low-surface-brightness galaxy.
- Faint H$\alpha$ emission has very recently been detected from some of the richest concentrations of bright MS stars, confirming the interpretation that these are mostly B stars (Cuillandre, private communication).
- Computations of the FUV output from the census of observed B stars and a fit of the data to PDR models is under way; preliminary results are encouraging and give reasonable values for the densities of the GMCs which must be the antecedents of the B stars.
FUV and HI surface brightness correlations in M81
-------------------------------------------------
A general correlation of VLA-HI and UIT-FUV surface brightness on the $\approx 1$ kpc scale has very recently been shown in M81 (Emonts et al., in preparation). Figure \[fig:M81-HI-FUV\] shows this correlation when the data is smoothed to $\approx 1'$. The results are consistent with a photodissociation model, although the precise parameters (e.g. the H$_2$ volume density) are uncertain owing to the averaging procedure on the model. The correlation is closely similar to that determined for a sample of nearby galaxies from the FAUST Far-UV survey (Deharveng et al. 1994), although the interpretation is quite different; this particular result and its relation to the subject of the global star formation rate in galaxies is discussed in more detail by Allen (2002).
Yet another simple calculation
------------------------------
What fraction of the total HI in a galaxy is maintained by the photodissociation process we have been discussing here? As an example, let’s take the case of NGC 4152, an Sc galaxy at D $\approx 19.5$ Mpc in Virgo. The FAUST Far-UV flux is $6.3 \times 10^{-14}$ ergs/cm$^2$/sec/Å (Deharveng et al. 1994) at $\lambda \approx 1500$ Å. This corresponds to about 1 photon/cm$^2$/sec at earth, or $\approx 4 \times 10^{52}$ Far-UV photons/sec at the galaxy. Let us take the fraction trapped in the galaxy to be $f_t$. What is the appropriate time scale? This depends of course on the reformation time for the process 2 $\times$ HI $\rightarrow$ H$_2$ on grains, which we mentioned briefly earlier in this talk. If we take a typical GMC volume density to be $n_2 \approx 5 - 50$ H$_2$ molecules cm$^{-3}$, the reformation time is typically $3 \times 10^{6 - 7}$ yr for standard dust parameters. The Far-UV photon production rate in NGC 4152 then accounts for:
$$2 \times 0.15 \times f_t \times 4 \times 10^{52} \times 3 \times
10^{6 - 7} \times 3 \times 10^7 \approx f_t \times 10^{66 - 67}\:
{\rm HI~atoms.}$$
A typical value for $f_t \sim 0.5$, and this galaxy contains $2.08 \times
10^9$ M$_{\odot}$ of HI, or $2.5 \times 10^{66}$ HI atoms. So from 20% – 200% of the HI present could be accounted for by photodissociation! This result is startling, and perhaps even a bit outrageous; a more precise calculation is now clearly required.
Conclusions
===========
- The [*morphology*]{} of HI features near Far-UV sources in disk galaxies is consistent with that expected for (low-density) PDRs.
- The [*quantity*]{} of HI in these features can be calculated using simple photodissociation physics.
- The [*distribution*]{} of HI, even at faint levels, can be explained for the most part as a steady-state photodissociation – reformation process of H$_2 \Leftrightarrow 2 \times $ HI, where the Far-UV photons come mainly from non- or only weakly-ionizing B stars, and the reformation occurs on the surfaces of dust grains.
These conclusions raise two very important questions, still unanswered:
1. Just how much H$_2$ does a typical galaxy contain?
2. How much of the HI in these galaxies is primordial?
The hints at answers provided by the work reviewed in this talk suggest that significantly more H$_2$ is present than we currently suspect, and that the HI clouds in the ISM of galaxies may be mostly “processed” gas.
I am grateful to my colleagues at STScI for discussions on the topics covered in this paper, and for the invigorating research atmosphere they have helped to create at the Institute. Through his annual visits to STScI, Ken has contributed very directly to that atmosphere. It is a special pleasure to acknowledge the intellectual stimulation and the personal friendship which Ken has provided to me over the years.
Allen, R.J. 2002, in [*Seeing Through The Dust – The Detection of HI and the Exploration of the ISM of Galaxies*]{}, eds.A.R. Taylor, T.L. Landecker, & A. Willis (ASP Conference Series \# TBD), in press. Allen, R.J., Atherton, P.D., & Tilanus, R.P.J. 1986, Nature, 319, 296 Allen, R.J., Knapen, J.H., Bohlin, R., & Stecher, T.P. 1997, , 487, 171 Andersson, B.-G., & Wannier, P.G. 1993, , 402, 585 Andersson, B.-G., Roger, R.S., & Wannier, P.G. 1992, , 260, 355 Blitz, L. 1993, in [*Protostars & Planets III*]{}, eds. E.H. Levy & J.I. Lunine (Tucson; Univ. Arizona Press), 125 Cuillandre, J.-C., Lequeux, J., Allen, R.J., Mellier, Y., & Bertin, E. 2001, , 554, 190 Deharveng, J.-M., Sasseen, T.P., Buat, V., Bowyer, S., Lampton, M., & Wu, X. 1994, , 289, 715 Dewdney, P.E., Roger, R.S., Purton, C.R., & McCutcheon, W.H. 1991, , 370, 243 Kaufman, M.J., Wolfire, M.G., Hollenbach, D.J., & Luhman, M.L. 1999, , 527, 795 Lin, C.C., & Shu, F.H. 1964, , 140, 646 Puxley, P.J., Hawarden, T.G., & Mountain, C.M. 1990, , 364, 77 Reach, W.T., Koo, B.-C., & Heiles, C. 1994, , 429, 672 Roberts, W.W. 1969, , 158, 123 Roger, R.S., & Dewdney, P.E. 1992, , 385, 536 Sternberg, A. 1988, , 332, 400 Smith, D.A., Allen, R.J., Bohlin, R.C., Nicholson, N., & Stecher, T.P. 2000, , 538, 608 Wallace, B.J., & Knee, L.B.G. 1999, in [*New Perspectives on the Interstellar Medium*]{}, eds. A.R. Taylor, T.L. Landecker, & G. Joncas (ASP Conference Series \# 168), 261 Williams, J.P., & Maddalena, R.J. 1996, , 464, 247
Discussion {#discussion .unnumbered}
==========
[*Silk:*]{} Your account of the origin of the HI by photodissociation should scale to Low Surface Brightness (LSB) galaxies. What is the status of searching for the evidence for your ideas in these systems?
[*Allen:*]{} The basic idea here is that B stars ought to [*always*]{} accompany HI, so they ought to be present even in the LSB disks. In general, one needs HST to detect upper-main-sequence stars, and only a small number of LSB galaxies are nearby enough to make this feasible. In fact Ken and I have tried to obtain HST time for such work, but we have not been successful. However, I understand that the classic LSB galaxy NGC 2915 is on the program for HST observations in guaranteed time with the new Advanced Camera; this galaxy is near enough, and there is enough known about the detailed distribution of HI, that the results ought to be clear.
[*Harding:*]{} What is the source of the primordial H$_2$ if it is the source of gas for star formation and for HI production?
[*Allen:*]{} This question is in the realm of cosmology and galaxy formation, and I don’t really know much about those subjects. I hear that some of the latest work on galaxy formation shows that H$_2$ could be formed at an early epoch from primordial HI and H$^+$ by reactions that do not require dust. The reaction rates are slow, but on the other hand there is lots of time.
[*Bosma:*]{} There is a lot of HI around the galaxy NGC 3077 in the M81 group. According to your ideas, there should be UV emission associated with it. Did you look for this, and did you detect anything?
[*Allen:*]{} The UIT field containing M81 does not extend out to NGC 3077, so we can not do the check you describe. However, the field does indeed include some of the HI “streamers” located in the far outer parts of M81 to the east, in the general direction of NGC3077. My student Mr. Emonts has recently produced 1$\arcmin$ smoothed versions of the HI and FUV images, and there is indeed detectible FUV emission associated with this “bridge” HI. We are looking more closely at this result, but at first sight the correlation between HI and FUV emission seems to agree with the general trend shown in Figure 3 of my talk.
[*King:*]{} There’s an implication here that you didn’t state outright. There’s got to be a reservoir from which future stars will be made; are you saying this is H$_2$ rather than HI?
[*Allen:*]{} I think the conventional wisdom is that star formation, at least for stars of masses less than a few M$_{\odot}$, requires “processed” gas containing heavy elements and dust as the raw material. Such “stuff” turns molecular relatively quickly compared, for example, to the rotation time of a galaxy. So indeed it is likely that the reservoir of gas required for future generations of star formation will be molecular, and the HI we see is a photodissociation product of the star formation process itself. This suggests that galaxies have a larger reservoir of H$_2$ than we think they do, but unfortunately I don’t yet know how to calculate how much!
|
---
author:
- |
[ ]{}, Dimitri Colferai$^{2}$, Joey Huston$^{3}$ and [ ]{}\
\
1 - Deutsches Elektronen-Synchrotron DESY,\
Platanenallee 6, D-15738 Zeuthen, Germany &\
Institute for High Energy Physics IHEP,\
Pobeda 1, 142281 Protvino, Russia.\
E-mail:\
\
2 - University of Florence,\
Department of Physics,\
Via G. Sansone 1, I-50019 Sesto Fiorentino (Firenze), Italy.\
E-mail:\
\
3 - Michigan State University,\
Department of Physics and Astronomy,\
3218 Biomedical Physical Science Building,\
East Lansing, MI 48824, USA.\
E-mail:\
\
4 - Deutsches Elektronen Synchrotron DESY,\
Notkestrasse 85, D-22607 Hamburg, Germany.\
E-mail:\
title: 'Parton densities from DIS and hadron colliders to LHC: WG summary'
---
Parton distribution functions (PDFs) are determined from global analyses of a wide range of data resulting from a variety of hard-scattering processes; these processes include deep inelastic scattering (DIS) of leptons off of a nucleon, lepton pair production in hadron-hadron collisions (the Drell-Yan process), and jet production in proton-antiproton and lepton-proton collisions (see Table \[tab:pdfs\]). The data used in these global analyses are continually updated, taking advantage both of improvements to the data sets and of progress in the theory. At this workshop, new experimental results from HERA, Fermilab, and JLAB experiments were presented and analyzed in the context of PDF determination [@sumtalk].\
Name Data used QCD order Scheme Reference
--------- ------------- ----------- -------- -------------
ABKM DIS+DY NLO/NNLO FFN [@ABKM]
CTEQ DIS+DY+jets NLO GMVFN [@CTEQ]
HERAPDF DIS NLO GMVFN [@HERACOMB]
JR DIS+DY NLO/NNLO FFN [@JR]
MSTW DIS+DY+jets NLO/NNLO GMVFN [@MSTW]
NNPDF DIS+DY+jets NLO ZMVFN [@NNPDF]
\
The large sample of data accumulated in the fifteen years of successful HERA operation provides a comprehensive view of the proton, particularly at small values of the Bjorken variable $x$. The latest HERA results presented in this DIS2010 workshop include (1) the combination of DIS cross sections measured at various proton-beam energies, (2) the extraction of the longitudinal structure function $F_L$, and (3) updated neutral current (NC) and charged current (CC) measurements. Recent HERA data measurements have resulted in improved determination of both PDFs and of the weak coupling constants.
A new combined data set on inclusive $e^{+}p$ DIS cross sections measured at different proton-beam energies ($E_p = 920$ GeV, $E_p = 460$ GeV and $E_p$= $575$ GeV) was recently produced by the H1 and ZEUS Collaborations [@loweprelim]. The variation of the centre-of-mass energy $\sqrt s$ allows for the discrimination of the structure functions $F_L$ and $F_2$. Since DIS cross sections are proportional to $F_2 -y^2 F_L/ (1+(1-y)^2)$, $F_L$ and $F_2$ can be extracted from measurements at two or more different values of inelasticity $y=Q^2/xs$. The combination of the H1 and ZEUS data includes the cross-calibration of two experiments, and results in an improved precision for the data. The proton structure function $F_L$ is extracted from the combined cross sections in the range of $2.5 \le Q^2 \le 800$ GeV$^2$ (see Figure \[combFl\]).
In addition to the combined HERA data of Ref. [@loweprelim], a new extended NC measurement of the $e^+p$ inclusive cross sections at different proton-beam energies has been obtained by the ZEUS Collaboration [@loweprelim]
![The combined H1 and ZEUS measurement of the structure function $F_L$, averaged in x at a given value of $Q^2$, compared to a QCD prediction based on the PDFs of Ref. [@HERACOMB] ([**left**]{}) and different variants of the fits of Ref. [@radescu1] [**(right)**]{}. []{data-label="combFl"}](H1prelim-10-043.fig5.eps){width="7.4cm"}
![The combined H1 and ZEUS measurement of the structure function $F_L$, averaged in x at a given value of $Q^2$, compared to a QCD prediction based on the PDFs of Ref. [@HERACOMB] ([**left**]{}) and different variants of the fits of Ref. [@radescu1] [**(right)**]{}. []{data-label="combFl"}](H1prelim-10-044.fig22.eps){width="7.4cm"}
For this analysis, satellite vertex events were used to access $Q^2$ values down to 5 GeV$^2$ for the reduced proton beam energies. This measurement should further improve constraints on $F_L$ provided by HERA. The combined data of Ref. [@loweprelim] were employed in a QCD fit taking into account QCD corrections up to NNLO [@radescu1]. Two variants of the fit with two different cuts on $Q^2$ were tried. In this way, the sensitivity of the PDFs to the low $Q^2$ portion of the data was determined. The fit is sensitive to the treatment of the heavy quark contribution to the inclusive DIS cross sections. In particular, the predictions based on the optimized Thorne prescription of Ref. [@Thorne] undershoot the HERA $F_L$ measurement (see Figure \[combFl\]). The ACOT prescription [@ACOTfull; @ACOTchi] and the (3-flavor) fixed-flavor number (FFN) scheme are in better agreement with the data.
New NC and CC DIS cross section measurements at high $Q^2$ were obtained by the H1 and ZEUS Collaborations using the HERA-II sample obtained with longitudinally polarised leptons. These data provide additional constraints on PDFs through the NC polarisation asymmetries and test the chiral structure of the weak interactions.
The ZEUS Collaboration has measured the CC cross sections based on a sample with an integrated luminosity of 132 $pb^{-1}$ and different polarizations of the positron beam [@zeuscc]. The H1 Collaboration has determined the CC cross sections using the complete HERA-II data sample with polarized $e^{+}$ and $e^{-}$ beams [@h1cc]. In order to achieve better precision in the CC data, the HERA-II CC polarised measurements were combined with the unpolarised HERA-I data. The combined sample corresponds to the total luminosity of 280.8 $pb^{-1}$ and 165.5 $pb^{-1}$ for $e^+p$ and $e^-p$ scattering, respectively.
The NC double-differential cross sections for $e^{\pm}p$ scattering with longitudinally polarised lepton beams were measured by the H1 Collaboration [@h1nc]. The charge dependent polarisation asymmetry in the NC cross sections is sensitive to the quark vector and axial couplings to the Z boson. Similar to the case of CC interactions, the NC cross sections were combined with the earlier H1 measurements. The structure functions $x \tilde F_3$ and $xF_3^{\gamma Z}$ were extracted from the combined unpolarised cross section measurements.
A new method to measure the NC cross sections up to values of $x$ close to 1 was employed by the ZEUS Collaboration [@zeusnc]. In this region, PDFs are poorly known and the data of Ref. [@zeusnc] provide valuable additional constraints for the global PDF fits. The NC cross sections were extracted at $Q^2 \ge 575$ GeV$^2$ using the $e^-p$ collision data sample with an integrated luminosity of 187 $pb^{-1}$. This is a factor of ten larger than the one used for the earlier ZEUS measurement [@zeusnchera1]. The systematic uncertainties are also reduced as compared to Ref. [@zeusnchera1] due to improved kinematic reconstruction methods. The data of Ref. [@zeusnc] were found to be in good agreement with the predictions based on the CTEQ6D PDFs.
A new combined electroweak and QCD analysis was performed by the H1 Collaboration using the full low and high $Q^2$ HERA data [@zhang]. Weak vector and axial couplings of light up- and down-type quarks to the $Z$ boson ($v_q, a_q$) were extracted from the combined fit simultaneously with the PDFs. Due to the additional sensitivity of the polarised NC measurement to the quark vector coupling, the accuracy of the vector coupling $v_q$ was improved with respect to earlier results based on the unpolarised HERA data only [@ewqcdhera1]. The constraint obtained on the up-type quark coupling $v_{u}$ from HERA is better than those obtained from LEP [@lepcouplings] and from the TEVATRON [@cdfcouplings] (see Figure \[qcdewfits\]). Moreover, the HERA data are also sensitive to the sign of weak couplings.
![ The 68$\%$ confidence level for the determination of the weak couplings of up-([**left**]{}) and down-type([**right**]{}) quarks to the $Z$ boson determined in a combined electroweak-QCD analysis using the full HERA H1 data [@h1nc]; the results are compared with the corresponding results published previously using the HERA-I data alone [@ewqcdhera1]; couplings determined by the LEP [@lepcouplings] and TEVATRON [@cdfcouplings] are given for the comparison.[]{data-label="qcdewfits"}](H1prelim-10-042.fig1a.eps){width="7cm"}
![ The 68$\%$ confidence level for the determination of the weak couplings of up-([**left**]{}) and down-type([**right**]{}) quarks to the $Z$ boson determined in a combined electroweak-QCD analysis using the full HERA H1 data [@h1nc]; the results are compared with the corresponding results published previously using the HERA-I data alone [@ewqcdhera1]; couplings determined by the LEP [@lepcouplings] and TEVATRON [@cdfcouplings] are given for the comparison.[]{data-label="qcdewfits"}](H1prelim-10-042.fig1b.eps){width="7cm"}
The H1 Collaboration presented proton structure function measurements at low and medium $Q^2$ determined from the first period of HERA operation. The inclusive double differential cross sections for $e^{+}p$ scattering were obtained at 12 $\le~Q^2~\le~150$ GeV$^2$ for an integrated luminosity of 22 pb$^{-1}$ [@klein]. The data were collected at the beam energies of E$_e$ = 27.6 GeV and E$_p$ = 920 GeV and were combined with earlier data obtained with E$_p$ = 820 GeV. The typical precision of the combined measurements is $1.3-2\%$. This is a record level of accuracy for DIS measurements. The NLO QCD fit to the H1 data based on the new measurements provides an improved determination of the gluon and quark densities in the proton, particularly at small x [@medQ2paper]. The analysis of Ref. [@medQ2paper] includes the study of the experimental, model and parameterisation uncertainties in PDFs.
The inclusive NC double differential cross sections of $e^{+}p$ scattering at HERA were obtained by the H1 Collaboration at small $x$ and low $Q^2$ ($0.2 \le Q^2 \le 12 $ GeV$^2$) [@glazov; @lowQ2paper]. These data were collected in two dedicated periods with nominal and shifted interaction vertex, where a shifted vertex provided better acceptance at low $Q^2$. The two samples overlap at $0.5~\le~Q^{2}~\le~3.5$ GeV$^2$, and the proton structure functions $F_2$ and $F_L$ were extracted from the combination of these measurements. The data at $Q^2 \lesssim 2$ GeV$^2$ correspond to the transition between DIS and photoproduction regimes. With the improved data accuracy, one can discriminate between different theoretical approaches used to model $F_{2}$ and $F_{L}$ in this region [@lowQ2paper].
The inclusive $e^{\pm}p$ NC and CC cross sections measured by the H1 and ZEUS Collaborations in the first period of HERA operation were combined into a common data set [@habib]. The combined data cover the range of $6 \cdot 10^{-7} < x < 0.65$, $0.045 \le Q^2 \le 30000$ GeV$^2$ for the NC and $0.013 < x < 0.4$, $300 \le Q^2 \le 30000$ GeV$^2$ for the CC scattering, respectively. The H1 and ZEUS input data used in the combination were found to be consistent with each other with $\chi^2/DOF = 636.5/656$. The total uncertainty reached with this combination is 1$\%$ for NC interactions in the most accurately measured region ($20 < Q^2 <100$ GeV$^2$) [@HERACOMB]. A NLO QCD analysis of the combined $e^{\pm}p$ scattering cross-section data was performed and a new set of parton distribution functions, HERAPDF 1.0, was obtained from the analysis [@HERACOMB]. The experimental uncertainties in the PDFs were reduced with respect to the earlier H1 and ZEUS PDF sets, due to the improved accuracy in the combined data. The theoretical uncertainties obtained by varying the input assumptions of the fit, e.g. the charm quark mass, were also studied.\
[r]{}[0.5]{}
\[fig:cteq10\]
Several groups [@MSTW-DIS; @CTEQ10; @NNPDF; @ABKM-DIS] have recently replaced the inclusive H1 and ZEUS DIS data of Refs. [@H1ZEUS] used in their PDF fits with the combined HERA data of Ref. [@HERACOMB]. The new HERA data are in reasonable agreement with the other data sets used in the fits. Moreover, this replacement leads to an improvement in the small-$x$ gluon and sea uncertainties, typically by the order of 20-30% (see Figure \[fig:cteq10\]). At scales of $O({\rm GeV}^2)$, the accuracy in the HERA data is better than 2%. Due to such precision, NNLO QCD corrections are needed for the most accurate interpretation of these data. The NNLO (3-loop) corrections to the PDF evolution have been calculated in Ref. [@Moch:2004pa]. In addition, the 3-loop corrections to the massless DIS coefficient functions are also known [@Vermaseren:2005qc]. In the factorization scheme with 3 light flavors in the initial state (the FFN scheme), the corrections to the heavy-quark contribution to DIS are only known up to NLO [@Laenen:1992zk]. This makes an NNLO analysis of the DIS data somewhat inconsistent. The higher-order QCD corrections are partially taken into account through the massless evolution of the heavy quarks, which is employed in the zero-mass variable-flavor-number (ZMVFN) scheme. However, the ZMVFN scheme is applicable only at asymptotically large transfers $Q\gg m_{\rm h}$, where $m_{\rm h}$ denotes the heavy quark mass. For realistic kinematics, the exact results are generally overestimated, due to the missed power corrections to the heavy-quark production coefficient functions. This shortcoming is overcome in the general-mass variable-flavor-number (GMVFN) extensions of the ZMVFN scheme by a special modeling of the Wilson coefficients at low $Q$. In this way, the GMVFN scheme prescriptions of ACOT, Roberts-Thorne, and Thorne were obtained (cf. review of Ref. [@Jung:2009eq] and Refs. [@Nadolsky:2009ge; @Thorne] for the recent update of the ACOT and Thorne’s prescriptions).
[r]{}[0.5]{} {width="19pc" height="17pc"}
The GMVFN schemes facilitate combination of the DIS and the hadron collider data in the global PDF fits. However, the GMVFN PDFs are sensitive to the specific choice of the prescription. This results to an additional uncertainty in the hard cross section predictions based on these PDFs [@Tung:2006tb]. The FFN scheme and the different prescriptions of GMVFN schemes provide an equally good description of the combined HERA data on the inclusive and semi-inclusive structure functions $F_2^c$ [@mandy]. Meanwhile, the FFN small-$x$ gluon distribution obtained in the fit of Ref. [@mandy] is substantially higher than gluon distribution determined in the GMVFN variants of the fit (see Figure \[fig:mandy\]), and is clearly positive at small $x$. In part, this is explained by the definition of the GMVFN PDFs; the remaining discrepancy can be considered as the PDF uncertainty due to the scheme choice. It is worth noting that the PDFs obtained in the FFN fits correspond to the $\overline{\rm{MS}}$ scheme. This is also the case for the GMVFN prescription suggested by Buza-Matiounine-Smith-van Neerven (BMSN) [@BMSN] and for the FONLL prescription of Ref. [@FONLL], which contains the main BMSN features. This is not generally guaranteed for PDFs obtained within the GMVFN fit formalism [@chuvakin]. However, the higher-order corrections to heavy-quark DIS production suppress the difference between the FFN and ZMVFN schemes at large $Q$ (see Ref. [@GLUCK]). For the NLO case, this difference can barely be resolved by the existing data [@ABKM] and must be even smaller taking into account the NNLO corrections to the matching conditions for the 4- and 5-flavor distributions [@BBK]. The interpretation of the heavy-quark electro-production and inclusive DIS data also depends on the heavy-quark masses values and their definition. The HERA data alone are not sensitive to the heavy-quark masses since the effect of the mass variation is compensated by changes in the PDFs [@radescu1]. The heavy quark masses can be more precisely determined in a global fit including data from other processes. Such a determination was carried out by MSTW and a set of PDFs corresponding to different values of the heavy quark masses is provided in Ref. [@MSTWHQ].
Hadron collider data on $W/Z$ and jet production provide more information for PDF determination to one available in the DIS data alone. They provide constraints on PDFs at large factorization scales and at large parton $x$ values and help to disentangle the distributions of different PDF species. Moreover, $W/Z$ production is considered as one of the primary LHC standard candle processes [@dittmar], due to the large cross section and small experimental and theoretical uncertainties. The $W/Z$ distributions have been calculated up to NNLO accuracy [@DYDIST]. The higher-order corrections of Refs. [@DYDIST] are usually employed in the global PDF fits in the form of fixed $K$-factors [@MSTW; @CTEQ] applied to each data point, and updated periodically through the fitting process. In the NNPDF fit [@NNPDF], the higher order corrections are applied in each fitting iteration with the use of FastKernel tool. A similar technique was employed for the first time in the FastNLO code for the fast calculation of the jet production cross sections [@Kluge:2006xs] and has been developed recently within the APPLGRID project for the NLO QCD analysis of the LHC data on the jet and $W/Z$ production [@Carli:2010rw].
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Recently, precise Run II $W$ lepton asymmetry data have become available [@RUNII]. The CTEQ and MSTW collaborations found that the lepton asymmetry data disagree with other data sets used in the global fitting, particularly those from the NMC [@nmc] and BCDMS [@bcdms1],[@bcdms2] experiments. The disagreement leads to a non-negligible increase in global $\chi^2$ for the global fits, and a poor agreement of the resultant predictions with the lepton asymmetry data. As a result, both CTEQ and MSTW have left the Run II lepton asymmetry data out of their latest global fits. The CTEQ and MSTW predictions, as well as those from ABKM, overshoot the data in general. CTEQ has a variant fit (CT10W) [@CTEQ10] in which the asymmetry data is included in the fit, but given a large weight. This is the only manner in which a reasonable $\chi^2$ can be obtained for the asymmetry data sets. The Run II lepton asymmetry analyses have been carried out by both CDF (for electron) and D0 (for both electron and muon). The CDF electron asymmetry data agree with the results from the D0 electron analysis. To make matters more complex, there is some tension between the D0 electron and muon asymmetry data sets. Thus, there is no final conclusion yet as to the impact of the Run II lepton asymmetry data. The best agreement with the data was obtained for the case of the JR PDFs (see Figure \[fig:wasym\]).
The Tevatron charge asymmetry data are sensitive to the $d$-quark distribution, and to the $d/u$ quark ratio. Therefore, the agreement might be improved due to modification of the correction for nuclear effects in deuterium, which affects the $d$-quark distribution extracted from the deuteron fixed target DIS data that have the conflict with the Run II lepton asymmetry data. A model-independent form of deuteron correction was attempted in the MSTW fit of Ref. [@MSTW-DIS]. While this correction somewhat improves the description of the Tevatron charge asymmetry data, the shape of the deuteron correction preferred by this fit cannot be justified by a reasonable nuclear model.
In Ref. [@Schienbein:2007fs], nuclear corrections were also fitted to the charged-lepton and neutrino DIS data in the spirit of the nuclear PDF concept [@Eskola:2009uj; @Hirai:2007sx; @deFlorian:2003qf]. In this manner, different PDF shapes were found for neutral-current and charged-current DIS off of an iron target. Therefore the application of the resulting nuclear PDFs to collider predictions is somewhat problematic. Meanwhile, the observation of Ref. [@Schienbein:2007fs] is based on the analysis of data from one experiment only; it therefore requires independent confirmation. In the model of Ref.[@Kulagin:2004ie], the various nuclear effects are considered separately, in contrast to the nuclear PDF approach. This model describes a wide set of nuclear DIS data, including the recent JLAB data for helium-3 [@JLAB]. Thus, it can be conjectured that this approach can be also reliably extrapolated to the case of deuteron targets for the benefit of interpretation of the hadron collider lepton asymmetry data.
The production of jets at high transverse momentum in hadron collisions is sensitive to the large-$x$ gluon distribution at both the Tevatron and LHC colliders. The NNLO corrections to the hadronic jet production have not yet been calculated; therefore the jet data can be consistently used only in the NLO version of the PDF fit. For some time, the Run I jet Tevatron data have provided the main constraints on the high $x$ gluon distribution [@MRST; @CTEQ4]. The Run I data, and in particular the Run I data from D0, prefer a higher large-$x$ gluon distribution as compared to the gluon determined from fits to DIS data alone. The Run II inclusive jet data, especially those from D0, are relatively lower at high jet transverse momentum $p_T$ and jet rapidity $Y$, and the resultant global fits using this data alone have a lower gluon distribution at high $x$. Thus, there is some tension between the the Run I and the Run II Tevatron jet data. This tension was examined by CTEQ [@Pumplin:2009nk]; although some degree of tension does exist, the data sets from Run I and Run II were found to be statistically compatible with each other, with the tensions similar to that between other data sets used in the global fit. Thus, both generations (Run I and Run II) of Tevatron jet data have been kept in recent CTEQ PDF fits [@CTEQ10], although only the Run II data have been used in the MSTW and NNPDF global fits. The resultant high $x$ gluon distribution for CTEQ is thus larger than either MSTW or NNPDF. MSTW has also found the impact of the Run II jet data on the uncertainty in the large-$x$ gluon distribution obtained in the global PDF fits to be relatively minor (see Figure \[fig:gluon\]), while CTEQ finds the constraints to still be appreciable.
[r]{}[0.5]{} {width="21pc"}
A substantial fraction of jets produced in the forward direction results from the simultaneous scattering of two pairs of partons. The rate for such processes is defined by double-parton distribution functions (dPDFs), i.e. the probability to find two partons with certain momenta inside the nucleon. Due to experimental constraints that are currently insufficient, the dPDFs are usually derived as a product of conventional collinear PDFs. However, the dPDFs obtained in such a way do not enjoy the fermion and momentum sum rules fulfilled for the dPDFs evolution equations. Addressing this shortcoming, a novel set of dPDFs was generated with the MSTW08 PDFs taken as an input [@Gaunt:2009re].
For many important processes the fixed-order perturbative QCD calculations demonstrate excellent convergence [@Gehrmann:2010rj] and in most cases the NNLO approximation is sufficient to meet the data accuracy. However, at small $x$ the perturbative corrections are unstable. In particular, the 3-loop terms in the massless DIS coefficient functions are quite large [@Vermaseren:2005qc]. On the other hand, the small-$x$ resummation of the splitting functions and the DIS Wilson coefficients softens the small-$x$ terms and compensate to some extent the NNLO corrections [@Ciafaloni:2007gf; @Altarelli:2008aj]. The impact of small-$x$ resummation in DIS was examined with a variant of the NNPDF1.0 PDFs [@NNPDF1] fitted to the high-$Q$ part of the DIS data, and then extrapolated to the low-$Q$ region with the heavy-quark contribution calculated in the ZMVFN scheme [@Caola:2009iy]. In this way, possible deviations from the standard NLO evolution of the inclusive HERA data were found. The effect observed is explained in Ref. [@Caola:2009iy] by the impact of small-$x$ resummation or by parton saturation [@Gribov:1984tu]. In line with this observation, the GMVFN variant of the HERAPDF fit of Ref. [@radescu1] is sensitive to the low-$Q$ part of the HERA data; however the FFN variant of the fit is much more stable to the cut on $Q$. Resummation effects in DIS thus still need to be systematically explored. This is an important issue for collider phenomenology since resummation contributes to many important collider channels such as heavy-quark and the lepton pair production (cf. Ref. [@Diana:2010ef] for a recent study of the effects of resummation in direct photon production). Small-$x$ dynamics is often considered within the framework of non-collinear PDF evolution, using $k_T$ or angular ordering. The unintegrated parton distributions appear in the parton evolution with the angular ordering depending on both longitudinal and transverse variables; therefore, the calculation of final state transverse momentum distributions is better suited for this formalism. With the appearance of the combined HERA I data, a determination of the unintegrated gluon distribution was updated and found enhanced as compared to the previous determination [@knut].
PDF shapes cannot be calculated in perturbative QCD. Instead, usually they are parameterized in a model-independent way with loose constraints imposed on the high-$x$ and the low-$x$ PDF exponents, coming from quark counting rules and Regge phenomenology, respectively [@MSTW; @CTEQ; @JR; @ABKM; @HERACOMB]. If the PDFs are evolved starting from a scale $\mu_{\rm F}^0<1~{\rm GeV}$, like in the case of JR PDFs, at the hard process scales they enjoy the asymptotic behavior defined by the leading evolution kernel singularities [@derujula]. In a similar way, the non-singlet PDF combinations can be constrained by the infrared QCD evolution kernel [@ermolaev]. An additional constraint may come from unitarity [@tiwon] and the universalities observed for the proton, photon, and diffractive structure functions [@klim].
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In contrast, the NNPDF collaboration uses a very flexible form of PDF parameterisation, with 259 free parameters [@NNPDF]. The gradient minimization methods are inapplicable for that large a number of parameters, and instead the NNPDF PDFs were fitted to the data using the neural network technique. Within the neural network approach, the PDF uncertainties are calculated from the probability distribution of the neural network replicas. Since no tensions between the different data sets used in the fit were observed (see Figure \[fig:chi2\]) the standard statistical methods are employed for the NNPDF PDFs. The uncertainties in the HERAPDF, ABKM, and JR PDFs are also calculated using the standard statistical methods. For comparison, the CTEQ and MSTW collaborations apply larger tolerance factors to take into account discrepancies between the data sets used in the fit and theoretical uncertainties, such as parameterisation choices. Despite the different statistical treatments, the PDF errors provided by all groups are in qualitative agreement with each other.
Outside the kinematic region covered by the existing data, the NNPDF PDF uncertainties are larger than the other PDF uncertainties, due to the lack of prior theoretical/parameterisation constraints on the PDFs. On the other hand, the uncertainties for the JR PDFs are reduced, compared to other fits due to additional theoretical constraints. A study of the small-$x$ sea and gluon distribution flexibility allowed by the data was also performed with the PDFs parameterized by the Chebyshev polynomials [@radescu2; @Pumplin:2009bb]. The gluon distribution obtained in Ref. [@radescu2] is stable with respect to the polynomial powers used at $x\gtrsim 0.001$; at smaller $x$ the gluon distribution is unstable, due to the lack of the experimental constraints. The PDFs given in Table \[tab:pdfs\] are similar, but do not completely overlap within $1\sigma$ uncertainty bands. This discrepancy appears to be due to differences in the theoretical formalisms used in the global fits and in details of the data treatment; however, no definitive understanding has yet been reached. In addition, the treatment of the strong coupling constant $\alpha_{\rm s}$ is different for the PDFs listed in Table \[tab:pdfs\].
[hr]{}[0.5]{} {width="19pc" height="14pc"}
In Refs. [@MSTW; @JR; @ABKM] the value of $\alpha_{\rm s}$ is fitted to the data simultaneously with the PDFs, while in the fits of Refs.[@CTEQ; @NNPDF; @HERACOMB] it is fixed at a value close to the world average [@Bethke:2009jm]. For many cross sections, such as Higgs and top quark pair production, the rates are quite sensitive to the value of $\alpha_{\rm s}$ used (cf. Refs. [@Martin:2009bu; @Lai:2010nw; @Demartin:2010er]). This sensitivity, and the variations in the values of $\alpha_s$ used, results in additional uncertainties on the hard-scattering cross section predictions. For the NLO Higgs cross section benchmark, the spread of the predictions is as large as 20% (see Figure \[fig:higgs\]). Further consolidation of the predictions would make LHC predictions for the Higgs cross section more definitive. The corresponding spread for predictions at the Tevatron can also be important for the interpretation of the Fermilab collider data [@Baglio:2010um].
The PDF4LHC working group is carrying out a benchmarking exercise [@huston] where each PDF group has been invited to provide NLO predictions for benchmark processes (such as the Higgs production cross sections for the Higgs masses of 120, 180 and 240 GeV, as well as for $W,Z$ and $t\bar{t}$ production. The predictions are to be made for the default value of $\alpha_s$, as well as for a range of values from 0.116 to 0.118. Comparisons are available at the website [@pdf4lhc], along with a prescription for the calculation of the PDF uncertainties at the LHC.\
\
The last year has seen tremendous progress on both the theoretical and the experimental fronts. In particular, the release of the combined H1+ZEUS HERA I data has led to increased precision and thus better constraints on PDFs, especially the low-$x$ gluon distribution. The Tevatron Run II W-lepton asymmetry measurements have the potential to allow for improvements in the description of high-$x$ quark distributions, but conflict with some of the data sets currently in use in the global fits. A resolution of the conflict, perhaps with better understanding of some of the nuclear corrections for the fixed target data that conflict with the lepton asymmetry data, is needed.
The LHC is presenting its first results for standard model cross sections. Predictions for these cross sections are available using PDFs from a number of fitting groups. It is useful and important at this stage of LHC running to have benchmark comparisons of the predictions from the various PDF groups. Such an exercise has been carried out by the PDF4LHC working group. Further standardization would be extremely useful, in particular common estimate of the uncertainty in the value of $\alpha_s$.
At the next DIS workshop, we expect our first inputs from LHC data to PDF fitting. To significantly affect the existing PDF fits, the systematic errors must be reasonably small and well-known for the measurements to be included. The most likely cross sections to be used in such a way are Drell-Yan measurements, especially of $W$ and $Z$ boson production, with event yields of the order of a million events per experiment to be expected for 1 $fb^{-1}$, and with the possibility of accessing new kinematic regions in $x$ and $Q^2$[@keaveney; @lorenzi; @dahmes].\
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---
abstract: 'Multi- or many-objective evolutionary algorithms(MOEAs), especially the decomposition-based MOEAs have been widely concerned in recent years. The decomposition-based MOEAs emphasize convergence and diversity in a simple model and have made a great success in dealing with theoretical and practical multi- or many-objective optimization problems. In this paper, we focus on update strategies of the decomposition-based MOEAs, and their criteria for comparing solutions. Three disadvantages of the decomposition-based MOEAs with local update strategies and several existing criteria for comparing solutions are analyzed and discussed. And a global loop update strategy and two hybrid criteria are suggested. Subsequently, an evolutionary algorithm with the global loop update is implemented and compared to several of the best multi- or many-objective optimization algorithms on two famous unconstraint test suites with up to 15 objectives. Experimental results demonstrate that unlike evolutionary algorithms with local update strategies, the population of our algorithm does not degenerate at any generation of its evolution, which guarantees the diversity of the resulting population. In addition, our algorithm wins in most instances of the two test suites, indicating that it is very competitive in terms of convergence and diversity. Running results of our algorithm with different criteria for comparing solutions are also compared. Their differences are very significant, indicating that the performance of our algorithm is affected by the criterion it adopts.'
author:
- 'Yingyu Zhang [^1] [^2]Bing Zeng [^3] Yuanzhen Li [^4] Junqing Li [^5]'
bibliography:
- 'moea.bib'
title: 'A Multi- or Many-Objective Evolutionary Algorithm with Global Loop Update'
---
evolutionary algorithms, many-objective optimization, global update strategy, Pareto optimality, decomposition.
Introduction
============
lot of real-world problems such as electric power system reconfiguration problems[@Panda2009937], water distribution system design or rehabilitation problems[@WBS2013], automotive engine calibration problems[@AEC2013], land use management problems[@LUM2012], optimal design problems [@Ganesan2015293; @Ganesan2013; @DomingoPerez201695], and problems of balancing between performance and cost in energy systems[@Najafi201446], etc., can be formulated into multi- or many-objective optimization problems(MOPs) involving more than one objective function. MOPs have attracted extensive attention in recent years and different kinds of algorithms for solving them have been proposed. Although algorithms based on particle swarm optimization[@PSO] and simulated annealing[@Suman2006] developed to solve MOPs are not ignorable, multi- or many-objective evolutionary algorithms(MOEAs) are more popular and representative in solving MOPs, such as the non-dominated sorting genetic algorithm-II (NSGA-II)[@NSGAII], the strength pareto evolutionary algorithm 2(SPEA-2)[@SPEA2], and the multi-objective evolutionary algorithm based on decomposition(MOEA/D)[@MOEAD],etc. In General, MOEAs can be divided into three categories[@Survey2016]. The first category is known as the indicator-based MOEAs. In an indication-based MOEA, the fitness of an individual is usually evaluated by a performance indicator such as hypervolume[@Emmerich2005]. Such a performance indicator is designed to measure the convergence and diversity of the MOEA, and hence expected to drive the population of the MOEA to converge to the Pareto Front(PF) quickly with good distribution. The second category is the domination-based MOEAs, in which the domination principle plays a key role. However, in the domination-based MOEAs, other measures have to be adopted to maintain the population diversity. In NSGA-II, crowding distances of all the individuals are calculated at each generation and used to keep the population diversity , while reference points are used in NSGA-III[@NSGAIII]. The third category is the decomposition-based MOEAs. In a decomposition based MOEA, a MOP is decomposed into a set of subproblems and then optimized simultaneously. A uniformly generated set of weight vectors associated with a fitness assignment method such as the weighted sum approach, the Tchebycheff approach and the penalty-based boundary intersection(PBI) approach, is usually used to decompose a given MOP. Generally, a weight vector determines a subproblem and defines a neighborhood. Subproblems in a neighborhood are expected to own similar solutions and might be updated by a newly generated solution. The decomposition-based MOEA framework emphasizes the convergence and diversity of the population in a simple model. Therefore, it was studied extensively and improved from different points of view [@Carvalho2012; @Ray2013; @Tam2016; @RVEA; @MOEADD; @Chen2017; @Survey2017] since it was first proposed by Zhang and Li in 2007[@MOEAD].
Recently, some efforts have been made to blend different ideas appeared in the domination-based MOEAs and the decomposition-based MOEAs. For examples, an evolutionary many-objective optimization algorithm based on dominance and decomposition(MOEA/DD) is proposed in [@MOEADD], and a reference vector guided evolutionary algorithm is proposed in [@RVEA]. In MOEA/DD, each individual is associated with a subregion uniquely determined by a weight vector, and each weight vector (or subregion) is assigned to a neighborhood. In an iterative step, mating parents is chosen from the neighboring subregions of the current weight vector with a given probability $\delta$, or the whole population with a low probability $1-\delta$. In case that no associated individual exists in the selected subregions, mating parents are randomly chosen from the whole population. And then serval classical genetic operators such as the simulated binary crossover(SBX)[@SBX] and the polynomial mutation[@PM],etc., are applied on the chosen parents to generate an offspring. Subsequently, the offspring is used to update the current population according to a complicated but well-designed rule based on decomposition and dominance.
In this paper, we focus on update strategies of the decomposition-based evolutionary algorithms and the criteria for comparing solutions. Three disadvantages of the decomposition-based MOEAs with local update strategies and several existing criteria for comparing solutions are analyzed and discussed. And a global loop update (GLU) strategy and two hybrid criteria are suggested. Also, we propose an evolutionary algorithm with the GLU strategy for solving multi- or many-objective optimization problems(MOEA/GLU). The GLU strategy is designed to try to avoid the shortcomings of the decomposition-based MOEAs with local update strategies and eliminate bad solutions in the initial stage of the evolution, which is expected to force the population to converge faster to the PF.
The rest of the paper is organized as follows. In section II, we provide some preliminaries used in MOEA/GLU and review serval existing criteria for comparing solutions, i.e., PBI criterion, dominance criterion and distance criterion. And then two hybrid criteria for judging the quality of two given solutions are suggested. The disadvantages of the decomposition-based MOEAs with local update strategies are also analyzed in this section. In section III, the algorithm MOEA/GLU is proposed. A general framework of it is first presented. Subsequently, the initialization procedure, the reproduction procedure, and the GLU procedure are elaborated. Some discussions about advantages and disadvantages of the algorithm are also made. In section IV, empirical results of MOEA/GLU on DTLZ1 to DTLZ4 and WFG1 to WFG9 are compared to those of several other MOEAs, i.e., NSGA-III, MOEA/D, MOEA/DD and GrEA. Running results of MOEA/GLU with different criteria are also compared in this section. The paper is concluded in section V.
Preliminaries and Motivations
=============================
MOP
---
Without loss of generality, a MOP can be formulated as a minimization problem as follows: $$\label{MOP}
\begin{split}
Minimize \quad &F(x)=(f_1(x),f_2(x),...,f_M(x))^T \\
&Subject \quad to \quad x\in\Omega,
\end{split}$$ where $M\geq 2$ is the number of objective functions, x is a decision vector, $\Omega$ is the feasible set of decision vectors, and $F(x)$ is composed of M conflicting objective functions. $F(x)$ is usually considered as a many-objective optimization problems when M is greater than or equal to 4.
A solution $x$ of Eq.(\[MOP\]) is said to dominate the other one $y$ ($x\preccurlyeq y$), if and only if $f_i(x)\leq f_i(y)$ for $i\in(1,...,M)$ and $f_j(x)<f_j(y)$ for at least one index $j\in(1,...,M)$. It is clear that x and y are non-dominated with each other, when both $x\preccurlyeq y$ and $y \preccurlyeq x $ are not satisfied. A solution x is Pareto-optimal to Eq.(\[MOP\]) if there is no solution $y\in\Omega$ such that $y\preccurlyeq x$. F(x) is then called a Pareto-optimal objective vector. The set of all the Pareto optimal objective vectors is the PF[@PF]. The goal of a MOEA is to find a set of solutions, the corresponding objective vectors of which are approximate to the PF.
Criteria for Comparing Solutions
--------------------------------
### Dominance criterion
Dominance is usually used to judge whether or not one solution is better than the other in the dominance-based MOEAs. As a criterion for comparing two given solutions, dominance can be described as follows.
***Dominance criterion:A solution x is considered to be better than the other one y when $x\preccurlyeq y$.***
As it is discussed in [@Hisao2008], the selection pressure exerted by the dominance criterion is weak in a dominance-based MOEA, and becomes weaker as the number of the objective functions increases. It indicates that such a criterion is too stringent for MOEAs to choose the better one from two given solutions. Therefore, in practice, the dominance criterion is usually used together with other measures.
### PBI criterion
In a decomposition-based MOEA, approaches used to decompose a MOP into subproblems can be considered as criteria for comparing two solutions, such as the weighted sum approach, the Tchebycheff approach and the PBI approach[@MOEAD]. Here, we describe the PBI approach as a criterion for comparing two given solutions.
***PBI criterion:A solution x is considered to be better than the other one y when $PBI(x)<PBI(y)$ , where $PBI(\bullet)$ is defined as $PBI(x)=g^{PBI}(x|w,z^{*})$, $\omega$ is a given weight vector, and $z^*$ is the ideal point.***
The PBI function can be elaborated as[@MOEAD]: $$\label{PBI}
\begin{split}
Minimize \quad &g^{PBI}(x|w,z^{*})=d_1+\theta d_2 \\
&Subject \quad to \quad x\in \Omega
\end{split}$$ where $$\label{TwoDists}
\begin{split}
&d_1=\frac{\left\|(F(x)-z^{*})^{T}w\right\|}{\|w\|}\\
&d_2=\left\|F(x)-\left(z^{*}+d_1\frac{w}{\|w\|}\right)\right\|,
\end{split}$$ and $\theta$ is a used-defined constant penalty parameter. In a decomposition-based MOEA with the PBI criterion, the set of the weight vectors is usually generated at the initialization stage by the systematic sampling approach and remains unchanged in the running process of the algorithm. The ideal point is also set at the initialization stage, but can be updated by every newly generated offspring.
### Distance criterion
In [@IDBEA], a criterion with respect to the two Euclidean distances $d_1$ and $d_2$ defined by Eq.(\[TwoDists\]) are used to judge whether or not a solution is better than the other. Denote the two Euclidean distances of x and y as $\{d_{1x},d_{2x}\}$ and $\{d_{1y},d_{2y}\}$ ,respectively. A criterion for comparing two given solutions with respect to the two distances can be written as follows.
***Distance criterion:A solution x is considered to be better than the other one y when $d_{2x}<d_{2y}$. In the case that $d_{2x}=d_{2y}$, x is considered to be better than y when $d_{1x}<d_{1y}$.***
### Two Hybrid Criteria
It has been shown that the dominance criterion can be a good criterion for choosing better solutions in conjunction with other measures[@NSGAII; @NSGAIII] . And likely, the PBI criterion has achieved great success in MOEAs[@MOEAD; @MOEADD]. However, there are two facts with respect to these two criteria respectively can not be ignored. The first one is that using dominance comparison alone can not exert too much selection pressure on the current population, and hence, can not drive the population to converge to the PF of a given MOP quickly. The second one is that it is not necessarily $PBI(x)<PBI(y)$ when $x\preccurlyeq y$, and vice versa.
Therefore, it might be natural to combine these two criteria in consideration of the two facts. Here, we suggest two hybrid criteria.
***H1 criterion: One solution x is considered to be better than the other one y when $x\preccurlyeq y$. In the case that the two solutions do not dominate with each other, x is considered to be better than y when $PBI(x)<PBI(y)$.***
***H2 criterion: One solution x is considered to be better than the other one y when $x\preccurlyeq y$. In the case that the two solutions do not dominate with each other, x is considered to be better than y when $d_{2x}<d_{2y}$.***
It is clear that the H1 criterion combines dominance with the PBI criterion, while the H2 criterion associates dominance with the Euclidean distance d2.
The Systematic Sampling Approach
--------------------------------
The systematic sampling approach proposed by Das and Dennis[@SystematicApproach] is usually used to generate weight vectors in MOEAs. In this approach, weight vectors are sampled from a unit simplex. Let $\omega=(\omega_1,...,\omega_M)^T$ is a given weight vector, $\omega_j(1\leqslant j\leqslant M)$ is the $jth$ component of $\omega$, $\delta_j$ is the uniform spacing between two consecutive $\omega_j$ values, and $1/\delta_j$ is an integer. The possible values of $\omega_j$ are sampled from $\{0,\delta_j,...,K_j\delta_j\}$, where $K_j=(1-\sum_{i=1}^{j-1}\omega_i)/\delta_j$. In a special case, all $\delta_j$ are equal to $\delta$. To generate a weight vector, the systematic sampling approach starts with sampling from $\{0,\delta,2\delta,...,1\}$ to obtain the first component $\omega_1$, and then from $\{0,\delta,2\delta,...,K_2\delta\}$ to get the second component $\omega_2$ and so forth, until the $Mth$ component $\omega_M$ is generated. Repeat such a process, until a total of $$\label{nWeightVectors}
N(D,M)=\left(
\begin{array}{c}
D+M-1 \\
M-1
\end{array}
\right)$$ different weight vectors are generated, where $D > 0$ is the number of divisions considered along each objective coordinate.
\[level distance=11mm, every node/.style=[inner sep=2.5pt]{}, level 1/.style=[sibling distance=20mm]{}, level 2/.style=[sibling distance=8.5mm]{}, level 3/.style=[sibling distance=4mm]{}\] child [node\[circle,draw\] child [node\[circle,draw\] child [node\[circle,draw\] edge from parent node\[left\][**1**]{} ]{} edge from parent node\[left,above\][**0**]{} ]{} child [node\[circle,draw\] child [node\[circle,draw\] edge from parent node\[\][**0.5**]{} ]{} edge from parent node\[\][**0.5**]{} ]{} child[node\[circle,draw\] child[node\[circle,draw\] edge from parent node\[right\][**0**]{} ]{} edge from parent node\[right,above\][**1**]{} ]{} edge from parent node\[left,above\][**0**]{} ]{} child [node\[circle,draw\] child [node\[circle,draw\] child [node\[circle,draw\] edge from parent node\[right\][**0.5**]{} ]{} edge from parent node\[left\][**0**]{} ]{} child [node\[circle,draw\] child [node\[circle,draw\] edge from parent node\[right\][**0**]{} ]{} edge from parent node\[right\][**0.5**]{} ]{} edge from parent node\[\][**0.5**]{} ]{} child[node\[circle,draw\] child[node\[circle,draw\] child [node\[circle,draw\] edge from parent node\[right\][**0**]{} ]{} edge from parent node\[right\][**0**]{} ]{} edge from parent node\[right,above\][**1**]{} ]{};
The approach can be illustrated by Fig.\[SSApproach\], in which each level represents one component of $\omega$, and each path from the root to one of the leaves represents a possible weight vector. Therefore, all weight vectors included in the tree can be listed as follows. $$\begin{array}{lcr}
(0 ,& 0 ,& 1 ) \\
(0 ,& 0.5,& 0.5) \\
(0 ,& 1 ,& 0 ) \\
(0.5,& 0 ,& 0.5) \\
(0.5,& 0.5,& 0 ) \\
(1 ,& 0 ,& 0 )
\end{array}$$
A recursive algorithm for MOEAs to generate weight vectors using the systematic sampling approach can be found in section III. Here, we consider two cases of D taking a large value and a small value respectively. As discussed in [@MOEADD] and [@SystematicApproach], a large D would add more computational burden to a MOEA, and a small D would be harmful to the population diversity. To avoid this dilemma, [@NSGAIII] and [@MOEADD] present a two-layer weight vector generation method. At first, a set of $N_1$ weight vectors in the boundary layer and a set of $N_2$ weight vectors in the inside layer are generated, according to the systematic sampling approach described above. Then, the coordinates of weight vectors in the inside layer are shrunk by a coordinate transformation as $$v^{j}_{i}=\frac{1-\tau}{M}+\tau\times \omega^{j}_{i},$$ where $\omega^{j}_{i}$ is the ith component of the jth weight vectors in the inside layer, and $\tau\in [0,1]$ is a shrinkage factor set as $\tau=0.5$ in [@NSGAIII] and [@MOEADD]. At last, the two sets of weight vectors are combined to form the final set of weight vectors. Denote the numbers of the weight vectors generated in the boundary layer and the inside layer as D1 and D2 respectively. Then, the number of the weight vectors generated by the two-layer weight vector generation method is $N(D1,M)+N(D2,M)$.
Local update and its advantages
-------------------------------
Most of the decomposition-based MOEAs update the population with an offspring generated by the reproduction operators to replace the individuals worse than the offspring in the current neighborhood. Such an update strategy can be named as a local update(LU) strategy since it involves only the individuals in the current neighborhood. The decomposition-based MOEAs with the LU strategy have at least two advantages. The first one is that the LU strategy can help the algorithms to converge to the PF faster than other algorithms with non-local update strategies, which helps them achieve great success on a lot of MOPs in the past ten years. The second one is that the time complexities of the decomposition-based MOEAs are usually lower than those of MOEAS with non-local update strategies. This allows them to have a great advantage in solving complicated problems or MOPs with many objectives, since the running time taken by a MOEA to solve a given MOP becomes much longer as the number of the objective functions increases.
In spite of the above advantages, the decomposition-based MOEAs with the LU strategy have their own disadvantages. The first disadvantage is that when the algorithms deal with some problems such as DTLZ4, the population may lose its diversity. As we can see from Fig.\[MOEADonDTLZ4\], a running instance of MOEA/D on DTLZ4 with 3 objectives generates well-distributed results, while the solution set of the other one degenerates nearly to an arc on a unit circle. What’s worse, the solution set of some running instances of MOEA/D even degenerates to a few points on a unit circle in our experiments.
Notice that a call of the LU procedure replaces all individuals worse than the newly generated offspring within the current neighborhood, which might be the reason resulting in the loss of the population diversity. Therefore, to avoid the loss of the population diversity, one can modify the LU procedure to replace at most one individual at a call. But the problem is how to decide which one individual is to be replaced when there are multiple individuals worse than the newly generated offspring. One of the simplest replacement policies is to randomly choose one individual in the current neighborhood and judge whether or not the offspring is better than it. If the selected individual is worse, it will be replaced by the offspring. Or else, the offspring will be abandoned.
Fig.\[MOEADAndItsMV\] shows the results of the original MOEA/D and its modified version with the modified LU strategy described above on DTLZ1 with 3 objectives. As it can been seen from Fig.\[MOEADAndItsMV\], the modified LU strategy lowers down the convergence speed of the algorithm, indicating that it is not a good update strategy.
The second disadvantage of the decomposition-based MOEAs with the LU strategy is that they don’t consider the individuals beyond the current neighborhood. As we can see, such a LU strategy allows the MOEAS to update the population with less time, but it might ignore some important information leading to better convergence. Fig.\[secondDisadvantage\] illustrates this viewpoint. Although the newly generated individual is better than individual A, it will be abandoned by the decomposition-based MOEAs with the LU strategy, since it is only compared to the individuals in the current neighborhood.
(0,0)–(0,1); (0,0)–(1,0); (0,0)–(0.966,0.259); (0,0)–(0.866,0.5); (0,0)–(0.707,0.707); (0,0)–(0.5,0.866); (0,0)–(0.259,0.966); (0,0.65) ellipse \[x radius=0.3, y radius=0.08\]; (0.6,0.45)–(0.85,0.6); at (0.85,0.6) ; (0.65,0.05) circle \[radius=0.015\]; (0.65,0.2) circle \[radius=0.015\]; (0.6,0.3) circle \[radius=0.015\]; (0.4,0.45) circle \[radius=0.015\]; (0.35,0.6) circle \[radius=0.015\]; (0.2,0.55) circle \[radius=0.015\]; at (0.2,0.55)[**A**]{} ; (0.05,0.65) circle \[radius=0.015\];
(0.1,0.4) circle \[radius=0.015\];
Asafuddoula et al have noticed this disadvantage of the decomposition-based MOEAs with the LU strategy[@IDBEA]. The update strategy of their algorithm involves all of the individuals in the current population, which has been demonstrated to be effective on the DTLZ and WFG test suites to some extent. We call such an update strategy a global update(GU) strategy, since each call of the update procedure considers all the individuals in the population, and replaces at most one individual. In Fig.\[secondDisadvantage\], individual A will be replaced by the newly generated individual if a decomposition-based MOEA adopts the GU strategy instead of the LU strategy.
The third disadvantage of the decomposition-based MOEAs with the LU strategy relates to the individuals and their attached weight vectors. As a simple example, consider the case where an individual x and a newly generated offspring c are attached to a weight vector $\omega_x$, and an individual y is attached to $\omega_y$, so that $g^{PBI}(c|w_x,z^{*})<g^{PBI}(x|w_x,z^{*})$ and $g^{PBI}(x|w_y,z^{*})<g^{PBI}(y|w_y,z^{*})$ are satisfied. In other words, c is better than x and x is better than y, when the weight vector $\omega_x$ and the weight vector $\omega_y$ are considered as the reference weight vector respectively. Therefore, x will be replaced by c in a typical decomposition-based MOEA. But so far, there is no decomposition-based MOEA considering x as a replacement for y.
In order to deal with the three disadvantages of the decomposition-based MOEAs with the LU strategy, we propose a MOEA with the GLU strategy(i.e. MOEA/GLU) mentioned before, which is presented in Section III.
Proposed Algorithm-MOEA/GLU
===========================
Algorithm Framework
-------------------
The general framework of MOEA/GLU is presented in Algorithm \[algFramework\]. As it is shown in the general framework, a *while* loop is executed after the initiation procedure, in which a $for$ loop is included. In the $for$ loop, the algorithm runs over N weight vectors ,generates an offspring for each weight vector in the reproduction procedure, and updates the population with the offspring in the GLU procedure.
Final Population. Initialization Procedure. Reproduction Procedure. The GLU procedure
Initialization Procedure
------------------------
The initialization procedure includes four steps. In the first step, a set of uniformly distributed weight vectors are generated using the systematic approach proposed in [@SystematicApproach]. A recursive algorithm for generating the weight vectors is presented in algorithm \[algSSA\] and \[RecursiveBodyofSSA\].
\[algSSA\] D:the number of divisions, M:the number of objectives. A set of uniform weight vectors. $\omega=(0,...,0)$; $Gen\_ith\_Level(\omega,0,0,D,M)$;
Algorithm \[algSSA\] calls the recursive function $Gen\_ith\_Level$ described in algorithm \[RecursiveBodyofSSA\] with $\omega=(0,...,0)$, $K=0$, and $i=0$, to generate weight vectors. At the ith level of the recursive function, the ith component of a weight vector is generated. As discussed before, the value of each component of a weight vector ranges from 0 to 1 with the step size $1/D$, and all components of a weight vector sum up to 1. In other words, all components of a weight vector share D divisions. Therefore, if K=D(K is the number of divisions that have been allocated), then the rest of the components are all set to zero. In addition, if $\omega[i]$ is the last component, i.e., $i=M-1$, then all the remaining divisions are assigned to it. Both the two cases indicate the end of a recursive call, and a generated weight vector is output.
\[RecursiveBodyofSSA\] $\omega[i],...,\omega[M-1]\leftarrow 0$; output($\omega$); return; $\omega[i]\leftarrow (D-K)/D$; output($\omega$); return; $\omega[i]\leftarrow j/D$; $Gen\_ith\_Level(\omega,K+j,i+1,D,M)$;
One of the main ideas of MOEA/GLU is that each individual is attached to a weight vector and a weight vector owns only one individual. Meanwhile, each weight vector determines a neighborhood. In the second step, the neighborhoods of all weight vectors are generated by calculating the Euclidean distances of the weight vectors using Eq.(\[TwoDists\]). Subsequently, a population of N individuals is initialized randomly and attached to N weight vectors in order of generation in the third step. Finally, the ideal point is initialized in the fourth step, which can be updated by every offspring in the course of evolution.
Reproduction Procedure
----------------------
The reproduction procedure can be described as follows. Firstly, a random number r between 0 and 1 is generated. If r is less than a given selection probability $P_s$, then choose two individuals from the neighborhood of the current weight, or else choose two individuals from the whole population. Secondly, the SBX operator is applied on the two individuals to generate two intermediate individuals. Notice that, if both of the two individuals are evaluated and used to update the population, then the number of individuals evaluated at each generation will be twice as many as that of the individuals in the whole population. However, the number of individuals evaluated at each generation in many popular MOEAs such as NSGA-III and MOEADD etc., is usually the same as the size of the population. Therefore, one of the two intermediate individuals is abandoned at random for the sake of fairness. Finally, the polynomial mutation operator is applied on the reserved intermediate individual to generate an offspring, which will be evaluated and used to update the current population in the following GLU procedure.
The GLU procedure
-----------------
a new offspring c, the current population P. bFlag=true; $Find\_Attathed\_Weight(c)\rightarrow i$; Swap(c,P\[i\]); bFlag=false;
The GLU procedure is illustrated in Algorithm \[GLU\], which can described as follows. Each individual is attached to a weight vector, which has the shortest perpendicular distance to the weight vector. $Find\_Attathed\_Weight(c)$ is designed to find the attached weight of c, in which the perpendicular distance is calculated by Eq.(\[TwoDists\]). Denote the perpendicular distance of the $ith$ individual $P[i]$ to the $jth$ weight vector as $d_{ij}$. A given weight vector maintains only one slot to keep the best individual attached to it generated so far from the beginning of the algorithm. The minimum value of $\{d_{i1},d_{i2},...,d_{iN}\}$ can be expected to be $d_{ii}$ after the algorithm evolves enough generations. However, in the initialization stage, all the individuals are generated randomly, and attached to the weight vectors in order of generation. In other words, the $ith$ weight vector may not be the one, to which its attached individual $P[i]$ has the shortest perpendicular distance. Supposed that the minimum value of $\{d_{i1},d_{i2},...,d_{iN}\}$ is still $d_{ij}$ at a certain generation, and the offspring c is better than $P[i]$ . Then, $P[i]$ will be replaced out by c, and considered as a candidate to take the place of the individual hold by the $jth$ weight vector, i.e., $P[j]$.
Discussion
----------
This section gives a simple discussion about the similarities and differences of MOEA/GLU, MOEA/D, and MOEA/DD.
1. Similarities of MOEA/GLU, MOEA/D, and MOEA/DD. MOEA/GLU and MOEA/DD can be seen as two variants of MOEA/D to some extent, since all of the three algorithms employ decomposition technique to deal with MOPs. In addition, a set of weight vectors is used to guide the selection procedure, and the concept of neighborhood plays an important role in all of them.
2. Differences between MOEA/GLU and MOEA/D. Firstly, MOEA/D uses a LU strategy, and MOEA/GLU employs the so-called GLU strategy, which considers all of the individuals in the current population at each call of the update procedure. Secondly, to judge whether or not an individual is better than the other, MOEA/D compares the fitness values of them, while other criteria for comparing individuals can also be used in MOEA/GLU. Thirdly, once a individual is generated in MOEA/D, all the individuals in the current neighborhood that worse than it will be replaced. However, each individual is attached to one weight vector in MOEA/GLU, and a newly generated individual is only compared to the old one attached to the same weight vector. The replacement operation occurs only when the new individual is better than the old one.
3. Differences between MOEA/GLU and MOEA/DD. In the first place, one weight vector in MOEA/DD not only defines a subproblem, but also specifies a subregion that can be used to estimate the local density of a population. In principle, a subregion owns zero, one, or more individuals at any generation. In MOEA/GLU, each individual is attached only to one weight vector, and a weight vector can hold only one individual. In the second place, the dominance criterion can be taken into account in MOEA/GLU, the way that it is used is different from that of MOEA/DD. In MOEA/GLU, the dominance between the newly generated individual and the old one attached to the same weight vector can be used to judge which of the two is better, while the dominance criterion is considered among all individuals within a subregion in MOEA/DD.
Time Complexity
---------------
The function *Find\_Attathed\_Weight* in the GLU procedure runs over all weight vectors, calculates the perpendicular distances between the newly generated offspring and all weight vectors, and finds the weight vector, to which the offspring has the shortest perpendicular distance. Therefore, it takes $O(MN)$ times of floating-point calculations for the function *Find\_Attathed\_Weight* to find the attached weight vector of the offspring, where M is the number of the objective functions and N is the size of the population.
As it is indicated before, the *while* loop is designed to help the individuals in the initial stage of the algorithm to find their attached weight vectors quickly. The fact that the individuals at a certain generation do not attach to their corresponding weight vectors causes extra entries into the function *Find\_Attathed\_Weight*. However, once all of the individuals are attached to their corresponding weight vectors, the function *Find\_Attathed\_Weight* will be entered at most two times. Let the entries into the function *Find\_Attathed\_Weight* be $(1+N_i)$ times at each call of the GLU procedure, and denote the number of the generations as G. Since $\sum_{i}{N_i}\leq N$ and the GLU procedure is called $NG$ times in the whole process of MOEA/GLU, the time complexity of the algorithm is $O(MN^2G)$, which is the same as that of MOEA/DD, but worse than that of MOEA/D.
Experimental Results
====================
Performance Metrics
-------------------
### Inverted Generational Distance(IGD)
Let S be a result solution set of a MOEA on a given MOP. Let $R$ be a uniformly distributed representative points of the PF. The IGD value of S relative to R can be calculated as[@IGD] $$IGD(S,R)=\frac{\sum_{r\in R}d(r,S)}{|R|}$$ where d(r,S) is the minimum Euclidean distance between r and the points in S, and $|R|$ is the cardinality of R. Note that, the points in R should be well distributed and $|R|$ should be large enough to ensure that the points in R could represent the PF very well. This guarantees that the IGD value of S is able to measure the convergence and diversity of the solution set. The lower the IGD value of S, the better its quality[@MOEADD].
### HyperVolume(HV)
The HV value of a given solution set S is defined as[@HV] $$HV(S)=vol\left( \bigcup_{x\in S}\left[ f_1(x),z_1 \right]\times \ldots \times\left[ f_M(x),z_M \right]\right),$$ where $vol(\cdot)$ is the Lebesgue measure,and $z^r=(z_1,\ldots,z_M)^T$ is a given reference point. As it can be seen that the HV value of S is a measure of the size of the objective space dominated by the solutions in S and bounded by $z^r$.
As with [@MOEADD], an algorithm based on Monte Carlo sampling proposed in [@HYPE] is applied to compute the approximate HV values for 15-objective test instances, and the WFG algorithm [@WFGalgorithm] is adopted to compute the exact HV values for other test instances for the convenience of comparison. In addition, all the HV values are normalized to $[0,1]$ by dividing $\prod_{i=1}^{M}z_i$.
Benchmark Problems
------------------
### DTLZ test suite
Problems DTLZ1 to DTLZ4 from the DTLZ test suite proposed by Deb et al[@DTLZ] are chosen for our experimental studies in the first place. One can refer to [@DTLZ] to find their definitions. Here, we only summarize some of their features.
- DTLZ1:The global PF of DTLZ1 is the linear hyper-plane $\sum_{i=1}^{M}f_i=0.5$. And the search space contains $(11^k-1)$ local PFs that can hinder a MOEA to converge to the hyper-plane.
- DTLZ2:The global PF of DTLZ2 satisfys $\sum_{i}^{M}f_i^2=1$. Previous studies have shown that this problem is easier to be solved by existing MOEAs, such as NSGA-III, MOEADD, etc., than DTLZ1, DTLZ3 and DTLZ4.
- DTLZ3:The definition of the glocal PF of DTLZ3 is the same as that of DTLZ2. It introduces $(3^k-1)$ local PFs. All local PFs are parallel to the global PF and a MOEA can get stuck at any of these local PFs before converging to the global PF. It can be used to investigate a MOEA’s ability to converge to the global PF.
- DTLZ4:The definition of the global PF of DTLZ4 is also the same as that of DTLZ2 and DTLZ3. This problem can be obtained by modifying DTLZ2 with a different meta-variable mapping, which is expected to introduce a biased density of solutions in the search space. Therefore, it can be used to investigate a MOEA’s ability to maintain a good distribution of solutions.
To calculate the IGD value of a result set S of a MOEA running on a MOP, a set $R$ of representative points of the PF needs to be given in advance. For DTLZ1 to DTLZ4, we take the set of the intersecting points of weight vectors and the PF surface as $R$. Let $f^*=(f_{1}^*,...,f_{M}^*) $ be the intersecting point of a weight vector $w=(w_1,...,w_M)^T$ and the PF surface. Then $f_i^*$ can be computed as[@MOEADD] $$f_i^*=0.5\times\frac{w_i}{\sum_{j=1}^{M}w_j}$$ for DTLZ1, and $$f_i^*=\frac{w_i}{\sqrt{\sum_{j=1}^{M}w_j}}$$ for DTLZ2, DTLZ3 and DTLZ4.
### WFG test suite[@WFGProblems; @WFG]
This test suite allows test problem designers to construct scalable test problems with any number of objectives, in which features such as modality and separability can be customized as required. As discussed in [@WFGProblems; @WFG], it exceeds the functionality of the DTLZ test suite. In particular, one can construct non-separable problems, deceptive problems, truly degenerative problems, mixed shape PF problems, problems scalable in the number of position-related parameters, and problems with dependencies between position- and distance-related parameters as well with the WFG test suite.
In [@WFG], several scalable problems, i.e., WFG1 to WFG9, are suggested for MOEA designers to test their algoritms, which can be described as follows. $$\begin{split}
Minimize \quad F(X)&=(f_1(X),...,f_M(X))\\
f_i(X)&=x_M+2ih_i(x_1,...,x_{M-1}) \\
X&=(x_1,...,x_M)^T
\end{split}$$ where $h_i$ is a problem-dependent shape function determining the geometry of the fitness space, and $X$ is derived from a vector of working parameters $Z=(z_1,...,z_n)^T, z_i\in [0,2i]$ , by employing four problem-dependent transformation functions $t_1$, $t_2$, $t_3$ and $t_4$. Transformation functions must be designed carefully such that the underlying PF remains intact with a relatively easy to determine Pareto optimal set. The WFG Toolkit provides a series of predefined shape and transformation functions to help ensure this is the case. One can refer to [@WFGProblems; @WFG] to see their definitions. Let $$\begin{split}
Z''&=(z''_1,...,z''_m)^T
=t_4(t_3 (t_2 (t_1(Z'))))\\
Z'&=(z_1/2,...,z_n/2n)^T.
\end{split}$$ Then $x_i=z''_i(z''_i-0.5)+0.5$ for problem WFG3, whereas $X=Z''$ for problems WFG1, WFG2 and WFG4 to WFG9.
The features of WFG1 to WFG9 can be summarized as follows.
- WFG1:A separable and uni-modal problem with a biased PF and a convex and mixed geometry.
- WFG2:A non-separable problem with a convex and disconnected geometry, i.e., the PF of WFG2 is composed of several disconnected convex segments. And all of its objectives but $f_M$ are uni-modal.
- WFG3:A non-separable and uni-modal problem with a linear and degenerate PF shape, which can be seen as a connected version of WFG2.
- WFG4:A separable and multi-modal problem with large “hill sizes”, and a concave geometry.
- WFG5:A separable and deceptive problem with a concave geometry.
- WFG6:A nonseparable and uni-modal problem with a concave geometry.
- WFG7:A separable and uni-modal problem with parameter dependency, and a concave geometry.
- WFG8:A nonseparable and uni-modal problem with parameter dependency, and a concave geometry.
- WFG9:A nonseparable, deceptive and uni-modal problem with parameter dependency, and a concave geometry.
As it can be seen from above, WFG1 and WFG7 are both separable and uni-modal, and WFG8 and WFG9 have nonseparable property, but the parameter dependency of WFG8 is much harder than that caused of WFG9. In addition, the deceptiveness of WFG5 is more difficult than that of WFG9, since WFG9 is only deceptive on its position parameters. However, when it comes to the nonseparable reduction, WFG6 and WFG9 are more difficult than WFG2 and WFG3. Meanwhile,problems WFG4 to WFG9 share the same EF shape in the objective space, which is a part of a hyper-ellipse with radii $r_i = 2i$, where $i\in\{1,...,M\}$.
Parameter Settings
------------------
The parameter settings of MOEA/GLU are listed as follows.
1. Settings for Crossover Operator:The crossover probability is set as $p_c=1.0$ and the distribution index is $\eta_c=30$.
2. Settings for Mutation Operator:The mutation probability is set as $p_m=0.6/n$, and is different from that of MOEA/DD, which is 1/n. The distribution index is set as $\eta_m=20$.
3. Population Size:The population size of MOEA/GLU is the same as the number of the weight vectors that can be calculated by Eq.(\[nWeightVectors\]). Since the divisions for 3- and 5-objective instances are set to 12 and 6, and the population sizes of them are 91 and 210, respectively. As for 8-, 10- and 15-objective instances, two-layer weight vector generation method is applied. The divisions and the population sizes of them are listed in Table \[nDivisions\].
4. Number of Runs:The algorithm is independently run 20 times on each test instance, which is the same as that of other algorithms for comparison.
5. Number of Generations: All of the algorithms stopped at a predefined number of generations. The number of generations for DTLZ1 to DTLZ4 is listed in Table \[nGens\], and the number of generations for all the instances of WFG1 to WFG9 is 3000.
6. Penalty Parameter in PBI: $ \theta= 5.0$.
7. Neighborhood Size: $T = 20$.
8. Selection Probability: The probability of selecting two mating individuals from the current neighborhood is set as $p_s = 0.9$.
9. Settings for DTLZ1 to DTLZ4:As in papers[@IDBEA; @MOEADD], the number of the objectives are set as $M \in \{3,5,8,10,15\}$ for comparative purpose. And the number of the decision variables is set as $n = M + r-1$, where $r = 5$ for DTLZ1, and $r = 10$ for DTLZ2, DTLZ3 and DTLZ4. To calculate the HV value we set the reference point to $(1,...,1)^T$ for DTLZ1, and $(2,...,2)^T$ DTLZ2 to DTLZ4.
10. Settings for WFG1 to WFG9: The number of the decision variables is set as n = k + l, where $k = 2\times(M-1)$ is the position-related variable and $l = 20$ is the distance-related variable. To calculate the HV values for problems WFG1 to WFG9, the reference point is set to $(3,...,2M+1)^T$.
\[nDivisions\]
M D1 D2 Population Size
---- ---- ---- -----------------
3 12 - 91
5 6 - 210
8 3 2 156
10 3 2 275
15 2 1 135
: Number of Population Size
problem $M=3$ $M=5$ $M=8$ $M=10$ $M=15$
--------- ------- ------- ------- -------- --------
DTLZ1 400 600 750 1000 1500
DTLZ2 250 350 500 750 1000
DTLZ3 1000 1000 1000 1500 2000
DTLZ4 600 1000 1250 2000 3000
: Number OF Generations
\[nGens\]
Performance Comparisons on DTLZ1 to DTLZ4
-----------------------------------------
\[IGDonDTLZs\]
\[HVonDTLZs\]
We calculate the IGD values and HV values of the same solution sets found by MOEA/GLU, and compare the calculation results with those of MOEA/DD, NSGA-III, MOEA/D and GrEA obtained in[@MOEADD].
1. DTLZ1:From the calculation results listed in Table \[IGDonDTLZs\] and Table \[HVonDTLZs\], it can be seen that MOEA/GLU and MOEA/DD perform better than the other three algorithms on all of the IGD values and most of the HV values. Specifically, MOEA/GLU wins in the best and median IGD values of the 3-, 8-, 10- and 15-objective instances, and MOEA/DD wins in the worst IGD values of the 3-, 8-, 10- and 15-objective instances. As for the 5-objective instance, MOEA/GLU wins in all of the IGD values. When it comes to the HV values, MOEA/GLU performs the best on the 3-, 5- and 15-objective instances, and MOEA/DD shows the best performance on the 10-objective instance as listed in Table \[HVonDTLZs\]. In addition, MOEA/GLU wins in the median and worst HV values of the 8-objective instance, and NSGA-III wins in the best HV value of it. Although all of the values obtained by MOEA/GLU and MOEA/DD are close, MOEA/GLU wins in most of the IGD and HV values. Therefore, MOEA/GLU can be considered as the best optimizer for DTLZ1.
2. DTLZ2:As it can be seen from Table \[IGDonDTLZs\], MOEA/D, MOEA/GLU and MOEA/DD are significantly better than the other two on all of the IGD values of DTLZ2. As for the IGD values, MOEA/GLU performs the best on the 3-, 5- and 8-objective instances, and MOEA/D performs the best on the 10-objective instance. In addition, MOEA/GLU wins in the best value of the 15-objective instance, and MOEA/DD wins in the median and worst values of it. When it comes to the HV values, MOEA/GLU performs the best on the 3-, 5- and 15-objective instances, MOEA/DD performs the best on the 10-objective instance and wins in the worst value of the 8-objective instance, and GrEA wins in the best and median values of the 8-objective instance. On the whole, the differences of MOEA/GLU, MOEA/DD and MOEA/D are not significant on DTLZ2, but MOEA/GLU wins in more values than both MOEA/D and MOEA/DD. Therefore, MOEA/GLU can also be considered as the best optimizer for DTLZ2.
3. DTLZ3:Again, MOEA/GLU and MOEA/DD are the best two optimizer for DTLZ3, and their performances are also close. As for the IGD values, MOEA/GLU performs the best on the 5- and 8-objective instances, MOEA/DD performs the best on the 10-objective instance, MOEA/GLU wins in the best and median values of the 3-objective instance and the best value of the 15-objective instance, MOEA/DD wins in the median and worst values of the 15-objective instance, and the worst value of the 3-objective. As far as the HV values are concerned, MOEA/GLU performs the best on the 3-, 5-, 8- and 15-objective instances, and MOEA/DD performs the best on the 10-objective instance. Since MOEA/GLU wins in more values than the other four algorithms, it can be considered as the best optimizer for DTLZ3.
4. DTLZ4:It is clear that MOEA/GLU performs the best on all of the IGD values of DTLZ4. However, it is hard to distinguish the better one from MOEA/GLU and MOEA/DD when it comes to the HV values. Interestingly, the performance of MOEA/GLU and MOEA/DD are so close that all of the HV values of the 3-objective instance, the best and median HV values of the 8-objective instance obtained by them are equal in terms of 6 significant digits. But taking the performance on the IGD values into consideration, MOEA/GLU is the best optimizer for DTLZ4.
Performance Comparisons on WFG1 to WFG9
---------------------------------------
\[HVonWFGs1To5\]
\[HVonWFG6toWFG9\]
The HV values of MOEA/GLU, MOEA/DD, MOEA/D and GrEA on WFG1 to WFG5 are listed in Table \[HVonWFGs1To5\], and the HV values on WFG6 to WFG9 are listed in Table \[HVonWFG6toWFG9\]. The comparison results can be concluded as follows.
1. WFG1:MOEA/DD wins in all the values of WFG1 except the worst value of the 3-objective instance, and hence be regarded as the best optimizer for WFG1.
2. WFG2:MOEA/GLU shows the best performance on the 3-objective instance, while MOEA/DD performs the best on the 8-objective instance. In addition, MOEA/GLU wins in the best and median values of 5- and 10-objective instances, while MOEA/DD wins in the worst values of them. Obviously, MOEA/GLU and MOEA/DD are the best two optimizer for WFG2, but it is hard to tell which one is better, since the differences between them are not significant.
3. WFG3:MOEA/GLU performs the best on the 10-objective instance, MOEA/DD shows the best performance on the 3-objective instance, and GrEA wins in the 5- and 8-objective instances. The values obtained by the three algorithms are very close. They all have their own advantages.
4. WFG4:MOEA/GLU shows the best in all values of WFG4, and is considered as the winner.
5. WFG5:Like in WFG4, MOEA/GLU is the winner of WFG5, since it wins in all values except the median and worst values of the 3-objective instance.
6. WFG6:MOEA/GLU and GrEA are the best two optimizer of WFG. The values obtained by them are not significant with ups and downs on both sides. Specifically, MOEA/GLU wins in the 3-objective instance, the best values of the 5- and 8-objective instances, the median and worst values of the 10-objective instance. GrEA wins in all the other values.
7. WFG7:MOEA/GLU wins in all the values of WFG7, and is considered as the best optimizer.
8. WFG8:MOEA/GLU wins in most of the values of WFG8 except the best value of the 5-objective instance and the median value of the 8-objective instance. Therefore, it can also be regarded as the best optimizer for WFG8.
9. WFG9:The situation of WFG9 is a little bit complicated, but it is clear that MOEA/GLU, MOEA/DD and GrEA are all better than MOEA/D. To be specific, GrEA wins in the 8-objective instance, and it might be said that MOEA/DD performs the best on the 3- and 5-objective instance although the worst value of it on the 3-objective instance is slightly worse than that of MOEA/GLU. In addition, the median and worst values of MOEA/GLU on the 10-objective instance are far better than those of other algorithms, while the best value is sightly worse than that of GrEA.
On the whole, MOEA/GLU shows a very competitive performance on the WFG test suite, especially WFG4, WFG5, WFG7 and WFG8, of which MOEA/GLU wins almost all the HV values.
Performance Comparisons of Algorithms with Different Criteria for Comparison
----------------------------------------------------------------------------
\[HVsofDiffCriteria\]
In this subsection, we compare the HV values of MOEA/GLU with different criteria for comparing solutions on WFG1 to WFG9 with different objectives. The HV values are listed in Table \[HVsofDiffCriteria\]. The comparison results can be concluded as follows. For the sake of convenience, we denote MOEA/GLU with the PBI, H1, and H2 criteria as MOEA/GLU-PBI, MOEA/GLU-H1, and MOEA/GLU-H2, respectively.
- WFG1:MOEA/GLU-H2 is the best optimizer since it performs the best on all instances of WFG1.
- WFG2:MOEA/GLU-H2 wins in all values of WFG2 except the best value of the 3-objective instance. Therefore, it is considered to be the best optimizer for WFG2.
- WFG3:The situation is a little bit complicated for WFG3. Specifically, MOEA/GLU-PBI wins on the worst value of the 5-objective instance and the best value of the 8-objective instance. And MOEA/GLU-H1 wins in the best and median values of the 3-objective instance, while MOEA/GLU-H2 is the best on other values. Therefore, MOEA/GLU-H2 can be considered the best optimizer for WFG3.
- WFG4: MOEA/GLU-H1 is the best optimizer that wins in all the values of WFG4.
- WFG5: MOEA/GLU-PBI has the best median value for the 3-objective instance of WFG5, and MOEA/GLU-H2 wins in the worst value of the 3-objective instance, while MOEA/GLU-H1 performs the best on all the other values. Therefore, MOEA/GLU-H1 is considered the best optimizer for WFG5.
- WFG6: MOEA/GLU-PBI has the best median value for the 5-objective instance of WFG6, and MOEA/GLU-H2 wins in the best value of the 8-objective instance and the worst value of the 10-objective instance, while MOEA/GLU-H1 performs the best on all other values. Therefore, MOEA/GLU-H1 is considered the best optimizer for WFG6.
- WFG7: Since MOEA/GLU-H1 wins in all the values of WFG7 except the worst value of its 3-objective instance, it is considered the best optimizer.
- WFG8: MOEA/GLU-PBI performs the worst on all the values of WFG8. MOEA/GLU-H1 wins in its 3-objective instance, while MOEA/GLU-H2 wins in the 10-objective instance. As for 5- and 8-objective instance, MOEA/GLU-H1 wins in the median and worst value, and MOEA/GLU-H2 wins in the best value. It is clear that MOEA/GLU-PBI is the worst optimizer for WFG8. However, as for MOEA/GLU-H1 and MOEA/GLU-H2, it is still hard to say which one of the two is better for WFG8.
- WFG9: MOEA/GLU-PBI wins in the best value of the 5-objective instance, the worst value of the 8-objective instance , and the median value of 10-objective instance of WFG9. MOEA/GLU-H2 only wins in the worst value of the 10-objective instance, while MOEA/GLU-H1 wins in all the other values of WFG9. Therefore, MOEA/GLU-H1 can be considered the best optimizer for WFG9.
On the whole, MOEA/GLU-H1 is the best optimizer for WFG4 to WFG7, and MOEA/GLU-H2 is the best for WFG1 to WFG3, and WFG9. As for WFG8, Both MOEA/GLU-H1 and MOEA/GLU-H2 are better than MOEA/GLU-PBI, but it is hard to say which one of the two is better. These indicate that the running results of MOEA/GLU are affected by the criterion for comparing solutions it adopts.
Conclusion
==========
In this paper, we propose a MOEA with the so-called GLU strategy, i.e., MOEA/GLU. The main ideas of MOEA/GLU can be concluded as follows. Firstly, MOEA/GLU employs a set of weight vectors to decompose a given MOP into a set of subproblems and optimizes them simultaneously, which is similar to other decomposition-based MOEAs. Secondly, each individual is attached to a weight vector and a weight vector owns only one individual in MOEA/GLU, which is the same as that in MOEA/D, but different from that in MOEA/DD. Thirdly, MOEA/GLU adopts a global update strategy, i.e. the GLU strategy. Our experiments indicate that the GLU strategy can overcome the disadvantages of MOEAs with local update strategies discussed in section II, although it makes the time complexity of the algorithm higher than that of MOEA/D. These three main ideas make MOEA/GLU a different algorithm from other MOEAs, such as MOEA/D, MOEA/DD, and NSGA-III, etc. Additionally, the GLU strategy is simpler than the update strategies of MOEA/DD and NSGA-III. And the time complexity of MOEA/GLU is the same as that of MOEA/DD, but worse than that of MOEA/D.
Our algorithm is compared to several other MOEAs, i.e., MOEA/D, MOEA/DD, NSGA-III, GrEA on 3, 5, 8, 10, 15-objective instances of DTLZ1 to DTLZ4, and 3, 5, 8, 10-objective instances of WFG1 to WFG9. The experimental results show that our algorithm wins in most of the instances. In addition, we suggest two hybrid criteria for comparing solutions, and compare them with the PBI criterion. The empirical results show that the two hybrid criteria is very competitive in 3, 5, 8, 10-objective instances of WFG1 to WFG9.
Our future work can be carried out in the following three aspects. Firstly, it is interesting to study the performances of MOEA/GLU on other MOPs, such as the ZDT test problems, the CEC2009 test problems, combinatorial optimization problems appeared in[@Zitzler1999Multiobjective; @Ishibuchi2010Many], and especially some real-world problems with a large number of objectives. Secondly, it might be valuable to apply the two hybrid criteria for comparing solutions to other MOEAs. Thirdly, improve MOEA/GLU to overcome its shortcomings. As we can see, the algorithm contains at least two shortcomings. One is that all of its experimental results on WFG1 are worse than those of MOEA/DD except for the best HV value of the 3-objective instance. The other is that its time complexity is worse than that of MOEA/D. Further research is necessary to be carried out to try to overcome these two shortcomings.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Qingfu Zhang and Ke Li for their generously giving the java codes of MOEA/D and MOEA/DD.
[Yingyu Zhang]{} received the B.Eng. degree in computer science and technology from Changsha University of Science and Technology, Changsha, China, in 2002, and the M.Eng. and Ph.D. degrees in computer science from Huazhong University of Science and Technology, Wuhan, China, in 2007, and 2011, respectively. He is now a lecturer with the School of Computer Science, Liaocheng University, Liaocheng, China. His research interests includes quantum optimization, evolutionary multi-objective optimization, machine learning, and cloud computing.
[Bing Zeng]{} received the B.Econ. degree in economics from Huazhong Agricultural University, Wuhan, China, in 2004, and received the B.Eng. degree in computer science and technology, the M.Eng. and Ph.D. degrees in information security from Huazhong University of Science and Technology, Wuhan, China, in 2004, 2007 and 2012, respectively. He is currently an Assistant Professor at South China University of Technology, Guangzhou, China. His research interests are in cryptography and network security, with a focus on secure multiparty computation.
[Yuanzhen Li]{} received his PhD degree from Beijing University of Posts and Telecommunications in 2010. He is now an associate professor in the Department of Computer Science and Technology, Liaocheng University, China. His research interests include wireless communications, evolutionary computation and Multi-objective optimization.
[Junqing Li]{} received the B.Sc. and Ph.D. degrees from Shandong Economic University, Northeastern University in 2004 and 2016, respectively. Since 2004, he has been with the School of Computer Science, Liaocheng University, Liaocheng, China, where he became an Associate Professor in 2008. He also works with the School of information and Engineering, Shandong Normal University, Jinan, Shandong, China, where he became the doctoral supervisor. His current research interests include intelligent optimization and scheduling.
[^1]: This work was supported by the National Natural Science Foundation of China under Grant 61773192.
[^2]: Y. Zhang is with the School of Computer Science, Liaocheng University, Liaocheng 252000, China (e-mail:zhangyingyu@lcu-cs.com).
[^3]: B. Zeng is with the School of Software Engineering, South China University of Technology, Guangzhou 510006, China.
[^4]: Y. Li is with the School of Computer Science, Liaocheng University, Liaocheng 252000, China.
[^5]: J. Li is with the School of information science and engineering, Shandong Normal University, Jinan 250014, China, and also with the School of Computer Science, Liaocheng University, Liaocheng 252000, China.
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abstract: 'Recent observations indicate that many if not all galaxies host massive central black holes. In this paper we explore the influence of black holes on the lensing properties. We model the lens as an isothermal ellipsoid with a finite core radius plus a central black hole. We show that the presence of the black hole substantially changes the critical curves and caustics. If the black hole mass is above a critical value, then it will completely suppress the central images for all source positions. Realistic central black holes likely have masses below this critical value. Even in such sub-critical cases, the black hole can suppress the central image when the source is inside a zone of influence, which depends on the core radius and black hole mass. In the sub-critical cases, an additional image may be created by the black hole in some regions, which for some radio lenses may be detectable with high-resolution and large dynamic-range VLBI maps. The presence of central black holes should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.'
author:
- |
Shude Mao,$^1$[^1] Hans J. Witt$^2$ and Leon V.E. Koopmans$^1$\
$^1$University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire SK11 9DL, UK\
$^2$Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
date: 'Accepted ........ Received .......; in original form .......'
title: The Influence of Central Black Holes on Gravitational Lenses
---
epsf
gravitational lensing - cosmology: theory - dark matter - galaxies: structure - galaxies: nuclei
INTRODUCTION
============
Recent observations suggest that many if not all nearby galaxies host massive central black holes (e.g. Kormendy & Richstone 1995; Magorrian et al. 1998). The existence of such central black holes can be accommodated by hierarchical structure formation theories (e.g. Silk & Rees 1998; Kauffmann & Haehnelt 2000). The effects of central black holes on gravitational lensing have not been studied in detail, although it is commonly believed that the central singularity can suppress the central image. This is important because lensing theories predict that any [*non-singular*]{} lens should have an odd number of images while the observed lenses always show an even number of images (Burke 1981; see also Schneider et al. 1992 for a general review on gravitational lensing); the only possible exception is APM 08279+5255 (Ibata et al. 2000). The lack of central images could indicate either that the potential is singular or that the central surface density in lenses is so high that the central image is highly demagnified and thus unobservable in current surveys (Narayan et al. 1984). The purpose of this paper is to clarify, through simple examples, the effects of central black holes on lensing properties. We show that the presence of black holes can not only suppress but also create additional images. These additional images may be observable in some cases. Central black holes also introduce somewhat unusual critical curves and caustic structures. The outline of this paper is as follows. In section 2, we first outline the lensing basics and derive some analytical results for the critical curves, caustics and image properties for an isothermal sphere plus a black hole. We then generalize to the case of an isothermal ellipsoid plus a black hole. In section 3, we discuss the implications of black holes on gravitational lenses, including the constraints on the core radius.
EFFECTS OF CENTRAL BLACK HOLES ON LENSING
=========================================
We model a lensing galaxy as an isothermal ellipsoid plus a central black hole. The isothermal ellipsoid model is not only analytically tractable but also consistent with models of individual lenses, lens statistics, stellar dynamics, and X-ray galaxies (e.g. Fabbiano 1989; Maoz & Rix 1993; Kochanek 1995, 1996b; Grogin & Narayan 1996; Rix et al. 1997). The surface density distribution of an isothermal ellipsoid is given by \[eq:kappa\] [\_[cr]{}]{} = [1 2 q]{} [1 ]{} , where $\rc$ is the core radius, $q$ is the axial ratio, and $\Sigma_{\rm
cr}=c^2 \Ds/(4\pi G\Dl\Dls)$ is the critical surface density, $\Dl$, $\Ds$ are angular diameter distances from the observer to the lens and source, respectively, and $\Dls$ is the angular diameter distance from the lens to the source. Notice that all the lengths ($x, y, \rc$) are expressed in units of the critical radius, $\rcr$, \[eq:units\] =b, b 4([c]{})\^2, where the critical angle $b$ is the angle extended by the critical radius on the sky ($b \sim 0.2-3$ arcsecond for typical lens galaxies; e.g. Jackson et al. 1998), and the velocity dispersion $\sigma$ in eq. (\[eq:units\]) is related to, but not identical to, the observable line of sight velocity dispersion; we shall ignore this minor complication in our analysis and simply treat it as a parameter (see Keeton, Kochanek & Seljak 1997 for further discussions).
The lensing properties of the isothermal ellipsoid have been given by Kassiola & Kovner (1993), Kormann, Schneider & Bartelmann (1994), and Keeton & Kochanek (1998). The lens equation of an isothermal ellipsoid plus a black hole is $$\begin{aligned}
u&=&x- {1 \over \sqrt{1-q^2}}\,\tan ^{-1}\left(\sqrt{1-q^2}\, x
\over {\varphi + \rc}\right)-m {x \over r^2}\,, \nonumber\\ v &=& y-
{1 \over \sqrt{1-q^2}}\,\tanh^{-1}\left(\sqrt{1-q^2}\, y \over
{\varphi +q^2 \rc}\right)-m {y \over r^2}\,, \label{eq:lens}\end{aligned}$$ where $\varphi^2=q^2(\rc^2+x^2)+y^2, r^2=x^2+y^2$, and the dimensionless black hole mass is given by \[eq:bhUnits\] m=[M\_0]{}, M\_0 , with $\Mbh$ being the mass of the black hole. Physically, $M_0$ is the mass of the galaxy contained within a cylinder with radius $\rcr$ and hence $m_0$ is essentially the ratio of the black hole mass to the mass of the galaxy in the inner parts (within $\rcr$). The magnification ($\mu$) of a given image is given by \[eq:jacobian\] \^[-1]{} = [ ]{} [ ]{} - [ ]{} [ ]{}. For definitiveness, we shall adopt a lens redshift of 0.5 and a source redshift of 2. In the Einstein-de Sitter universe ($\Omega_0=1$ with no cosmological constant), the angular and length units are given by b=1\^[”]{} ([250]{})\^2, =3.6([250]{})\^2, where $h$ is the Hubble constant in units of $100\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ and $\sigma$ is in units of ${\rm km\,s^{-1}}$. The dimensionless black hole mass for the given lens and source redshifts is \[eq:bh\] m=2.4 10\^[-3]{} h( [250]{} )\^[3.27]{}, where we have used the tight correlation of the black hole mass with velocity dispersion found by Ferrarese & Merritt (2000), $\Mbh\approx
4\times 10^8 M_\odot (\sigma/250)^{5.27}$ (see Gebhardt et al. 2000 for a different scaling.)[^2]
The lens equation (eq. [\[eq:lens\]]{}) can be readily solved numerically. In general there are several curves in the image plane along which the magnification is infinite ($\mu^{-1}=0$). These are called “critical curves”, and they map to “caustics” in the source plane. Caustics mark discontinuities in the number of images, so in order to determine the number of images produced by a lens model, it is sufficient to examine the caustics. We shall consistently employ this idea in this paper. In order to gain some analytical insights into the influence of central black holes, we start with the spherical case (i.e. $q=1$) in the next subsection.
Spherical Isothermal Models with Black Holes
--------------------------------------------
For the spherical case ($q=1$), the surface density is given by = [1 2 ]{}. This profile has been studied quite extensively by Hinshaw & Krauss (1987).
The lens equation for this profile in the presence of a black hole is quite simple, $$\label{eq:lenseq}
\rs = r-\alpha(r), \qquad \alpha(r)= {\sqrt{r^2+\rc^2}-\rc \over r} +
{m \over r},$$ where $\rs, \rc$ are the source position and core radius in units of the critical radius and $m$ the dimensionless mass of the central black hole given in eq. (\[eq:bhUnits\]).
### Critical Curves and Caustics
The critical curves are given by the equation \^[-1]{} = [r]{} [ddr]{}=0. The solution satisfies either $\rs/r=0$ or $d\rs/dr=0$. One can verify that the first condition ($\rs/r=0$) always yields one single critical curve, \[eq:poly1\] r\^2=[1 2]{} , which maps to a degenerate caustic point at the origin. The second condition ($d\rs/dr=0$) can be manipulated into a polynomial $$\begin{aligned}
\label{eq:poly2}
r^6+(2 m - 2 \rc+\rc^2)r^4 & + & \nonumber \\
(m^2-2m \rc + \rc^2+2 m \rc^2 - 2 \rc^3)r^2 & + & \nonumber \\
m^2 \rc^2 - 2 m \rc^3 & = & 0. \end{aligned}$$ This equation is cubic in $r^2$, and so can be solved analytically. One finds that beyond a critical black hole mass, there is no physical solution (i.e. positive solution in $r^2$), while below the critical mass, there are always two physical solutions. The critical mass can be found by examining eq. (\[eq:poly2\]); some algebra yields the critical mass \[eq:crit\] = (1 +) - 3 ([\^2 2]{})\^[2/3]{}. For a given core radius, the central images are always suppressed by the black hole if its mass exceeds the critical value, $\mcr \approx
\rc$ when $\rc\ll1$. This can be understood intuitively as follows: when the mass is comparable to $\rc$, the mass of the black hole is comparable to the galaxy mass enclosed by the core radius, so the black hole makes the deflection angle nearly flat at $r \sim \rc$, similar to the singular isothermal case, for which we have only two images (cf. Fig. 2). Notice, however, that even when the black hole mass is below the critical value, there is a region in the source plane where the central image is suppressed (see below).
We can obtain approximate but more illuminating solutions for the positions of critical curves and caustics when $\rc \ll 1$ and $m\ll\mcr \approx \rc$. The first assumption ($\rc \ll 1$) is supported by the lack of central images in lenses (e.g. Wallington & Narayan 1993; Kochanek 1996a; see also §2.1.2), while the second condition is valid for most galaxies as can be seen from eq. (\[eq:bh\]), and if $\rc \ga 0.002$ ($\sim 10$ pc for $\sigma=250\kms$). Under these two assumptions, the inner most critical curve satisfies $r \ll \rc$ (justified below), the lens equation (eq. \[eq:lenseq\]) simplifies to \[eq:lenseqsimple\] =(1-)r-[mr]{}, where $\kc\equiv {1/(2 \rc)}$ is the central surface density of the galaxy, in the absence of the black hole. From this simplified lens equation, one can readily show that the innermost critical curve forms where $d\rs/d r=0$, i.e. \[eq:rcci\] = . The second assumption (i.e. $m\ll\mcr$) ensures that our condition, $r
\ll \rc$, is satisfied. The corresponding caustic in the source plane is \[eq:rcai\] = -2 = [-2 m \^[-1]{}]{}. The middle critical curve in general does not satisfy the condition $r
\ll \rc$. To derive its approximate position, we simplify eq.(\[eq:poly2\]) by setting $m=0$, after which we are left with a quadratic equation in $r^2$. One finds that, to the leading order of $\rc$, the positions of the critical curves and caustics are \[eq:rccii\] , 1-2 = 1-2. From eq.(\[eq:poly1\]) one finds that the outer most critical curve and its corresponding caustic are at \[eq:rcciii\] 1-, =0. For a source located outside $\rcaii$ there are only two images. When a source is between $\rcai$ and $\rcaii$, there are four images; compared with the case without a black hole, one additional image has been created very close to the black hole. For a source inside $\rcai$, there are only two images, the central image that would be present without a black hole has been suppressed. The radius $\rcai$ can therefore be regarded as a measure of the zone of influence of the black hole: for any source $\rs<\rcai \approx (2m/\rc)^{1/2}$, the central image is destroyed. Notice that this zone of influence scales as $\propto m^{1/2}$, identical to the scaling of the Einstein radius of an isolated black hole with mass.
=
Fig. 1 illustrates the evolution of critical curves and caustics with the mass of the black hole increasing from zero to 0.03. The core radius is taken to be 0.05, a somewhat arbitrary value which satisfies the upper limit inferred from the lack of central images (e.g. Wallington & Narayan 1993; Kochanek 1996a). When there is no black hole at the center (Fig. 1a) the critical curves are two circles, the outer critical curve maps to a degenerate caustic point at the origin, while the inner critical curve maps to the solid circle. Fig. 1b shows the configuration when we include a black hole mass with $m=0.002$, a value expected for a galaxy with $\sigma \approx 250{\rm\,km\,s^{-1}}$ (cf. eq. \[eq:bh\]). In addition to the two critical curves found when $m=0$, an additional tiny critical curve appears around the origin (too small to be seen), which maps to the inner solid circle. The positions of the three critical curves and their corresponding caustics are well approximated by eqs. (\[eq:rcci\]-\[eq:rcciii\]). For a source moving from infinity to the origin, it would first have two images, then four images as the source moves across the first caustic. Compared with the case without a black hole, one additional image has been created very close to the black hole. The number of images decreases to 2 again as the source moves across the second caustic. Notice that even in this case where the black hole mass is substantially below the critical mass $m=\mcr \approx 0.0175$, the zone of influence determined by $\rcai \approx 0.28$ is substantial. Within the zone of influence, the central image is completely suppressed. When the black hole mass is further increased to $m=0.01$ (Fig. 1c), the inner-most critical curve becomes larger while the middle critical curve shrinks, their corresponding caustics also approach each other. When the black hole mass is increased to the critical mass $\mcr$ (eq. \[eq:crit\]), the two critical curves merge into a single curve, so do their corresponding caustics. When the black hole mass is increased above the critical value (an example is shown for $m=0.03$ in Fig. 1d), one is left with a single critical curve and a degenerate caustic point at the center. In this case, there are only two images no matter where the source is. Notice, however, that the black hole mass for the panels 1c and 1d are somewhat unrealistic (cf. eq. \[eq:bh\].)
=
The change of image numbers may first appear puzzling, but can be easily understood in the conventional diagram of the deflection angle, $\alpha(r)$, vs. radius, shown in Fig. 2. Three examples are shown for $\rc=0.05$ corresponding to three black hole masses, $m=0, 0.01$, and 0.03. For each source position, the image positions are given by the the intercepts of the deflection angle curve with the straight line $f(r)=r-\rs$. For example, it is easy to see why for $m=0.01$, there are three critical curves. It is clear from this figure that the central black hole does not perturb the two outer images very much, as expected, it only strongly perturbs the central images.
### Image Positions and Magnifications
In this subsection, we will obtain approximate image positions and magnifications when there are four images, i.e., when the source position is between $\rcai$ and $\rcaii$. We are particularly interested in the case when the source is close to $\rcai$. For such cases, there are usually two bright outer images and two faint central images; one of the fainter central image is created by the black hole. We would like to address the question whether such images are observable.
Using the resultant technique as discussed in Witt & Mao (1995), one can verify that the four image magnifications must satisfy an exact relation \_1+\_2+\_3+\_4=2. Note that all the magnifications include their parities (signs).
The brightest image has positive parity and is located in regions where $r \gg \rc$. For this image, the lens equation can be simplified into $r^2 -\rs r +\rc -r = 0$, where we have only retained terms to the order of $(\rc/r)^2$. From this equation, we can obtain the position and magnification of the brightest image \[mu1\] r\_1 1 + -, \_1\^[-1]{} (r\_1\^2 + m-) . Similarly, we can obtain the position for the bright outer image that has negative parity \[mu2\] r\_2 -1 + +, \_2\^[-1]{} . For singular isothermal spheres ($\rc=0$), the magnifications and positions are exact for these two images, and they satisfy $\mu_1+\mu_2=2$.
For the two central images, we can estimate their positions and magnifications if they satisfy $ r \ll \rc$. In this case, the lens equation can be approximated by eq. (\[eq:lenseqsimple\]) and the image magnification is given by \[eq:mui\] = . For very small $m$ we can obtain for the positions and magnifications for the third and fourth image \[image3\] r\_3 - \_3 , . \[image4\] r\_4 - \_4\^[-1]{} - (r\_4 + 2m) = -. Note that the magnification for the third image $\mu_3 \sim \kc^{-2} =
4 \rc^2$, is independent of the source position when the black hole mass is small. Previous studies neglected the role of central black holes and therefore the lack of central images immediately implies an upper limit on the core radius. If the brightest image is magnified by a factor of few, and no central image is seen with a dynamic range (DR) of 100, then this would imply $\rc \la 0.1$, which is roughly the published upper limits on the core radius from lenses (e.g. Wallington & Narayan 1993; Kochanek 1996a). Hence the lack of central images in lenses may mean either that the core radius is small (see §2.1.2) or that the black hole mass is quite massive; we return to this point in the discussion.
The fourth central image is usually the closest to the black hole and is usually very faint. However, when the source lies very close to $\rcai$, the second term in the denominator of eq. (\[eq:mui\]), which is the square of the shear ($\gbh$) from the black hole, can become comparable to $(1-\kc)$, and then the magnification is of order unity or higher. The question is: how large is this region? To estimate this, we combine eqs. (\[eq:rcci\]) and (\[eq:mui\]) and find that the position of the image is related to its magnification by \[eq:radmu\] ()\^4 = (1-)\^[-1]{}, where $\muc$ is the magnification of an image formed well within the core radius in the absence of the black hole and is given in eq. (\[image3\]). A small displacement $\delr$ from the critical curve is related to the magnification through = . This displacement is related to a corresponding displacement of the source in the source plane, through eq. (\[eq:lenseqsimple\]). By definition the first order term of a Taylor expansion of the lens equation vanishes near the caustic. The second term, after some algebra, gives =-\^2=-. The probability of the source in a multiply-imaged system, lying inside this region (i.e. having a magnification $>\mu$), is therefore \[eq:probmu\] P\_[BH]{}(>)==. For any realistic situation, this probability is exceedingly small ($\ll10^{-6}$); the inclusion of magnification bias and extended source structure does not substantially enhance the probability of this faint central image to have magnifications above unity.
In case of high dynamic-range (DR) maps, however, central images with $\mu\ll 1$ can still be detected. In such cases eq. (\[eq:probmu\]) is no longer valid and we have to find a new estimate for the probability. To do this, we insert eq. (\[eq:radmu\]) into the simplified lens equation, eq. (\[eq:lenseqsimple\]), and arrive at = . For $\mu\rightarrow \infty$, the source position goes to $\rcai$, as expected. In case $\mu=\mu_4=-10^{-3}$ (DR$\ga10^3$; the magnification is negative because the parity of the central image created by the BH is $-$1), the source position $\rs\approx 1.15 \,\rcai$, assuming $\rc=0.05$ and $m=0.002$. The probability of observing this image is then $\approx 2\pi \rcai(\rs-\rcai)/(\pi\rcaii^2)$, which is $\sim$8% for the above given values of $\rc$ and $m$. If $\rc\sim0.05$, the currently known sample of 20–30 radio graviational lens systems could therefore contain several systems with central images that are observable, if the dynamic range in radio observations can be improved by a factor of 5–10. To separate the two central images, which are only a few milli-arcseconds apart, one obviously requires high-resolution VLBI observations.
Elliptical Isothermal Density Distribution With Black Holes
-----------------------------------------------------------
=
The spherical case we studied in the previous subsection is idealized, since galaxies nearly always show some ellipticity. In this section, we will study the critical curves and caustics for a more realistic elliptical density distribution plus a black hole. The critical curves can be found by solving eq. (\[eq:jacobian\]) numerically and the caustics are found using the lens mapping (eq. \[eq:lens\]). Fig. 3 shows the critical curves and caustics structure as a function of black hole mass for an elliptical density mass distribution with $q=0.7$ and again $\rc=0.05$. When there is no black hole at the center (Fig. 3a), one sees the well known critical curves and caustics. When a source is inside the diamond caustic, there are five images. When a source is outside the diamond but inside the elliptical caustic, there are three images. When the source moves further outside, there is only one single image. The image configurations for three source positions are illustrated in the same panel. Usually the central images are quite faint and difficult to observe. Now as we include a black hole mass with $m=0.002$ (cf. eq. \[eq:bh\]), an additional small critical curve appears very close to center (too small to be seen), and this maps to the middle elliptical caustic. A source outside the diamond but inside this middle elliptical caustic has two images. This is analogous to Fig. 1b. When a source moves from infinity to the origin, the image number changes according to $2\rightarrow 4 \rightarrow 2 \rightarrow
4$, with each caustic crossing either increases or decreases the number of images by two. When the black hole mass is further increased to $m=0.01$ (Fig. 3c) the inner-most critical curve becomes larger while the middle critical curve shrinks, their two corresponding caustics also approach each other. When the black hole mass is further increased to or above a critical value, $\approx 0.02$ (a value slightly larger than that in the spherical case), the two inner critical curves and their corresponding caustics cancel each other, and one is left with an elliptical critical curve and a diamond caustic at the center (an example is shown for $m=0.03$); this caustic separates the outer two image regions from the inner four image regions. In either region, the central images have been suppressed by the black hole. The behavior in the elliptical density distribution case is qualitatively similar to the spherical case, the main difference is that the ellipticity breaks the spherical symmetry and changes the central degenerate caustic point in the spherical case into the central diamond (compare Figs. 1 and 3.); the caustic change due to ellipticity is similar to the case without central black holes (see e.g. Schneider et al. 1992).
Discussion
==========
As clearly demonstrated in Figs. 1–3, central black holes in gravitational lenses introduces qualitatively new features in the critical curves and caustics. For realistic black hole masses, the black hole introduces a new region in the source plane that has four images. However, this additional four-image region does not help to resolve the problem of the apparent excess of quadruple lenses relative to double lenses (King & Browne 1996; Kochanek 1996b). The reason is that these four image configurations have two very faint images at the center and two bright images outside, which are very different from the observed quadruple lenses where the images are roughly on a circle from the lens center. The image closest to the black hole is the faintest image; for the case shown in Fig. 3b, this image is about a factor of $10^{-5}$ times fainter than the brightest one. As shown in section 2.1.1, there is only a negligible probability that the central images can reach magnifications of order unity. However, we find that there is a non-negligible probability ($\sim$8% for $\rc=0.05$ and $m=0.002$) that the central image can reach magnification $\ga 10^{-3}$, with the two distinct central images separated by about $10^{-3}\,b$ ($\sim$ milli-arcseconds). Such images can in principle be detected with future high dynamic range and high resolution VLBI observations. Note that this probability obviously depends on the values of $m$ and $\rc$, and is uncertain since for the latter only upper limits are available at present.
The massive black holes also complicate the interpretation of core radius inferred due to the lack of central images. The central image will be completely suppressed for all source positions if the black hole mass exceeds the critical value as given in eq. (\[eq:crit\]). This critical mass is reached when the ratio of the mass of the black hole to the galaxy mass in the inner parts (more precisely, $M_0$ in eq. \[eq:bhUnits\]) is approximately equal to the core radius in units of the critical angle $b$. For a galaxy with velocity dispersion of $250\kms$, the mass of the central black hole is $\sim 4\times 10^{8}M_\odot$, and $M_0 \sim 1.6 \times
10^{11}h^{-1}M_\odot$. The black hole is super-critical when $\rc \la
0.002$; in physical units, this implies an upper limit on the core radius of about $\sim 10$ pc for typical lens configuration with $\rcr
\sim {\rm few} \kpc$. This upper limit is usually not reached except for lens systems with favorable image geometry such as B1030+074 (Norbury et al. 2000, in preparation.) However, the presence of central black holes can also suppress the central image even when it is much below the critical value in some regions (see Figs. 1b and 3b). The zone of influence of the black hole is characterized by the $\rcai$ (eq. \[eq:rcai\]) in the spherical case: when the source is inside the zone ($\rs<\rcai \approx (2m/\rc)^{1/2}$), the central image is destroyed. For a given source position $\rs$, the central image disappears when the black hole mass satisfies \[eq:mlimit\] m , where in the last step we have replaced the source position on the right hand side by the magnification of the brightest image $\mu_1$ using eq. (\[mu1\]). For a galaxy with $\sigma=250\kms$, $m\approx
0.002$, and $\mu_1 \sim 4$, the central image will be destroyed when $\rc <0.04$ ($\sim 100\pc$ for a typical critical radius of a few kpc). More generally, the influence of black holes on the core radius constraint has to be obtained from detailed modelling. Eq. (\[eq:mlimit\]) also highlights that the best lenses to search for central images are systems with low magnifications. In such systems, the source is closer to outer caustic ($\rcaii$) and hence more likely to be outside the zone of influence. As a result, the central images are less likely to be suppressed by the black hole. Unfortunately, these systems also have large flux ratios between the two bright out images (cf. eqs. \[mu1\]-\[mu2\]). Since lens surveys favor systems with large magnifications due to magnification bias and systems with small flux ratio, it is not surprising that very few lenses discovered so far show central images, perhaps as a result of both small core radii and central black holes in gravitational lenses.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Ian Browne for helpful discussions and Chuck Keeton for insightful criticisms on a draft of the paper.
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[^1]: e-mails: smao@jb.man.ac.uk, hwitt@aip.de,leon@jb.man.ac.uk
[^2]: We neglect any possible evolution of the black hole mass with time.
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abstract: |
We develop a comprehensive framework to model and optimize the performance of CV-QKD with a local local oscillator (LLO), when phase reference sharing and QKD are conjointly implemented with the same hardware. We first analyze the limitations of the only existing approach, called LLO-sequential, and show that it requires high modulation dynamics and can only tolerate small phase noise, leading to expensive hardware requirements. Our main contribution is to introduce two original designs to perform LLO CV-QKD with shared hardware,, respectively called LLO-delayline and LLO-displacement, and to study their performance. Both designs rely on a self-coherent approach, in which phase reference information and quantum information are coherently obtained from a single optical wavefront.
We show that these designs can lift some important limitations of the existing LLO-sequential approach. The LLO-delayline design can in particular tolerate much stronger phase noise and thus appears as an appealing alternative to LLO-sequential that can moreover be deployed with affordable hardware. We also investigate, with the LLO-displacement design, how phase reference information and quantum information can be multiplexed in a single optical pulse. By studying the trade-off between phase reference recovery and phase noise induced by displacement we however demonstrate that this design can only tolerate low phase noise. On the other hand, the LLO-displacement design has the advantage of minimal hardware requirements and can be applied to multiplex classical and quantum communications, opening practical path towards the development of coherent quantum communications systems compatible with next-generation networks requirements.
author:
- Adrien Marie
- Romain Alléaume
bibliography:
- 'bibliography.bib'
title: 'Self-coherent phase reference sharing for continuous-variable quantum key distribution'
---
Introduction
============
Quantum key distribution (QKD) [@QKDSecurity1; @QKDSecurity2; @DiamantiLoQi2016] is a promising technology that has reached the commercialization step since the last decade [@sequrenet; @idquantique]. Targeting deployment over large-scale networks, next-generation QKD should rely on affordable optical components. It will in particular consist in highly integrated systems able to operate at high rate and to be deployed over modern optical networks. Relying on standard telecommunication equipment, Continuous-Variable (CV) QKD is an attractive approach towards this new step of QKD development [@GaussianQuantumInformation; @ReviewCVQKD]. While first results towards CV-QKD practical photonics chip integration have been pursued [@PhotonicsIntegration; @OrieuxDiamanti2016], the possibility to effectively deploy CV-QKD in coexistence with intense wavelength-division multiplexing classical channels has been demonstrated [@wdm]. Furthermore, high repetition rates (up to the order of hundreds of MHz) [@YuemengChi; @ExpCVQKD-chinois] CV-QKD systemps have also been demonstrated recently. More sensitive to optical losses than discrete-variable based QKD, long distance CV-QKD has however been demonstrated by controlling excess noise [@LongDistanceCVQKD1] and developping high efficiency error correction codes [@LongDistanceCVQKD2]. These important steps in the recent development of CV-QKD are, in addition, likely to benefit from the rise of classical coherent communications [@kikuchi2016fundamentals], with the prospect of d convergence of classical and quantum communication techniques and simplified photonic integration. This positions CV-QKD and more generally quantum coherent communications as an appealing technology for the the development of modern quantum communications.
If we compare with discrete-variable quantum communications, quantum coherent communications however have to address one specific challenge, namely phase reference sharing. As a matter of fact the receiver must perform a phase-sensitive detection, using an optical beam usually called “local oscillator” whose phase drift with respect to the emitter must be controlled, or estimated and corrected. We will review in Sec. \[Sec: Previous Work\] the different methods that have been considered to perform phase reference sharing in CV-QKD, and explain why generating “locally” the local oscillator is a fundamental requirement for continuous-variable quantum key distribution (CV-QKD), both for performance and security reasons. We will first analyze the question of phase reference sharing within to the broader body of work on reference frame agreement and then focus more specifically on the issue of phase reference sharing in coherent optical communications.\
**Sharing a reference frame.** A reference frame, shared or partially shared, between an emitter and a receiver is a typical requirement in communication protocols, even though this requirement is often implicit. Information on the reference frame allows the receiver to more faithfully translate the received physical signals into logical information. It can for example consist in the knowledge of the relative angle between spatial two-dimensional cartesian reference frames [@PeresScudo2002], in the synchronization of spatially separated clocks [@einstein1905elektrodynamik], or information about the relative phase between two lasers, respectively at emitter and receiver side, when coherent optical communication is performed [@kikuchi2016fundamentals]. This latter problem, phase reference frame sharing considered in the context of CV-QKD, will be the main focus of this article .
The problem of sharing a reference frame is specific in the sense that reference frame information constitutes *unspeakable information*, that can only be shared through physical carriers exchanged between emitter and receiver [@BartlettRudolphSpekkens2007]. On the other hand, it is important to emphasize that although quantum mechanics gives a precise framework to formulate the question of reference frame sharing, in relation with quantum metrology [@BartlettRudolphSpekkens2007], this question can be solved “classically”, using macroscopic signals to exchange reference frame information. The type of questions related to phase reference sharing is not whether it is possible, but whether it can be achieved given resource constraints, dictated by the hardware resources and by the characteristics of the channel, such as losses and noise. In line with the recent work on LLO CV-QKD [@LLOexp1; @LLOexp2; @LLOexp3], we will focus on in this article on the issue of jointly performing, with the same hardware, phase reference sharing and CV-QKD.
Another question related to reference frame sharing in quantum communications consist in performing “referenceless” quantum communication in which quantum information is encoded so that it can be recovered “without reference frame” at the receiver, under some assumptions about the channel such as collective noise. Such approach can for example be used with polarization encoding, when emitter and receiver spatial reference frame are slowly rotating, by encoding quantum information over noiseless subspaces [@LaflammeSpekkens2003; @BoileauLaflamme2006; @ScaraniObrien2010]. Referenceless quantum communications can be seen as a specific approach to perform reference frame sharing. This approach however requires to encode information over entangled quantum states, and cannot be easily used to design practical optical encodings for CV-QKD. We will therefore not consider the referenceless protocols in this article.\
**Phase reference sharing in classical and quantum coherent communication.** Coherent communication systems have the advantage of offering higher sensitivity (information per photon) than systems based on direct detections (for example On-Off-Keying modulation, where the information is encoded solely in intensity), and classical coherent systems are gradually becoming more and more used in modern classical optical networks, especially in core networks, over long-distance segments. Phase reference sharing is an important requirement in coherent communication systems, in order to correct the phase drift between the phase of the emitter laser and the phase of the local oscillator laser, placed at the receiver side. The generic objective is essentially to solve this phase reference sharing problem with minimal resource overhead and minimum penalty on the associated communication protocol. An essential point is to notice that the constraints and thus the solutions that can be adopted in the classical and in the quantum cases to solve the phase reference sharing issue significantly differ.
Classical coherent detectors are designed to detect intense light pulses, typically coherent states containing a very large number of photons, while coherent detectors used in quantum communications must typically be operated in the shot-noise regime, i.e with electronic noise significantly below the signal variance associated with the detection of one photon. This limits the intensity that can be handled by shot-noise limited coherent detectors before saturation [@SaturationAttackHao]. While analogic phase lock loops where used until the 70’s in classical coherent optical systems to solve the issue of phase locking, suffering however from phase lock loop bandwidth limitations, the advent of GHz-clocked electronics and fast digital signal processing now allows to recover both signal information and phase reference information from discrete modulation, such as binary phase shift keying (BPSK) or higher order modulations [@kikuchi2016fundamentals]. Such phase recovering techniques, used for classical coherent communication systems, that require a high number of photons at reception and high-speed modulations/detections, cannot be directly applied to perform quantum coherent communications. This makes the problem of phase reference sharing in quantum communications more constrained, and requires specific approaches.
Since the problem of phase reference sharing can be solved by sending classical reference pulses, one simple approach to the problem, in the context of quantum coherent communication is indeed to use an external classical (intense) phase reference sharing scheme. Such “classical” method is always possible but will typically requires the use of two separate detectors, one shot-noise limited detector for (weak) quantum signals and a second detector, with a large linearity range, to detect classical phase reference signals. This classical method hence implies not only techniques for multiplexing and demultiplexing reference and quantum signals, but basically to deploy two separate detection hardwares.
It is natural to seek how one can lift the extra hardware requirement of the “classical” method in order to jointly perform phase reference sharing and quantum communication with the same hardware. This question has been addressed in recent works aiming at demonstrating CV-QKD operation with a local local oscillator (LLO) [@LLOexp1; @LLOexp2; @LLOexp3], however with performance limitations and constraints on the hardware.\
**Contributions of this work.** We identify and discuss the existing approaches to the phase reference sharing problem for LLO CV-QKD. Recent works [@LLOexp1; @LLOexp2; @LLOexp3] all rely on time-multiplexed quantum signals pulses with reference pulses in order to jointly perform phase recovery and quantum communication. In this work, we introduce new elements in the standard noise model of CV-QKD analysis, considering new practical constraints imposed by the simultaneous quantum signal and phase reference transmission of LLO-based CV-QKD. In particular, the amplitude modulator (AM) dynamics and the linearity range of Bob’s detector are studied and we show that the AM dynamics is a key parameter in order to compare performance of realistic implementations of LLO-based CV-QKD. As a contribution, our resulting noise model is a refined framework for realistic CV-QKD analysis, including LLO regimes. Based on this comprehensive model, we show that there exist fundamental and practical limitations in the phase noise tolerance of the designs introduced in [@LLOexp1; @LLOexp2; @LLOexp3], that we designate as LLO-sequential.
In order to go beyond that phase noise limit, we introduce the idea of self-coherence in phase reference sharing for CV-QKD implementations based on a local local oscillator. Self-coherent designs consist in ensuring the phase coherence between pairs of quantum signal and phase reference pulses by deriving both of them from the same optical wavefront at emission. This allows to perform relative phase recovery schemes with better sensitivity than in the LLO-sequential design. In particular, we propose a design, called LLO-delayline, implementing a self-coherent phase sharing design. It ensures the self-coherence using a balanced delay line interferometer split between emitter and receiver sides. We analyze how self-coherence is obtained and study the performance reachable with this design, demonstrating that they exhibit a much stronger resilience to high phase noise than the LLO-sequential design under realistic experimental parameters. While previous experimental proposals of LLO CV-QKD are limited to slowly varying reference frames regimes (ie. based on very stable lasers or high repetition rates), our newly introduced design allows phase reference sharing resilient to high phase noise regimes, using the idea of self-coherence.
A second self-coherent design, referred to as LLO-displacement, relies on an original multiplexing allowing to transmit both the quantum signal and the reference pulse within each optical pulse. The simultaneous transmission of quantum signal and phase reference can be seen as an original cryptographic primitive, considered in [@Qi2016], that can be used with different modulation schemes. In particular, this allows to optimize the resources $-$ in terms of required hardware and repetition rate $-$ in LLO-based CV-QKD experiments. We also emphasize that an important advantage of our LLO-displacement design is its experimental simplicity as we show that the multiplexing can be perform numerically on Alice’s variables. As such, no specific hardware devices are required. We study the theoretical performance of such design and exhibit its limitations.
In Sec. \[Sec: Previous Work\], we review the existing implementations of phase reference sharing CV-QKD. In Sec. \[Sec: CVQKD\], we introduce the CV-QKD model. In particular, the phase reference sharing issue in CV-QKD is formally introduced and discussed and we also introduce our comprehensive noise model. In Sec. \[Sec: Towards self-coherence\], we highlight practical limitations of existing local local oscillator based CV-QKD and introduce the idea of self-coherence for reference sharing in CV-QKD. In Sec. \[Sec: LLO-delayline\] and Sec. \[Sec: LLO-displacement\], we respectively introduce the LLO-delayline and LLO-displacement designs and study their performance. Conclusion and perspectives are presented in Sec. \[Sec: Conclusion\].
Implementing phase reference sharing in CV-QKD: previous work {#Sec: Previous Work}
=============================================================
The procedure used for phase reference sharing in quantum coherent communication is often not tackled explicitly in experiments. As mentioned in the introduction, this follows from the idea that this question can in principle be solved independently of the quantum communication protocol itself, with classical techniques. This sometimes motivates to only perform phase reference sharing by placing emitter and receiver in the same location and using locally one single laser source both for quantum signal preparation and as local oscillator. Such proof of principle implementations have been used in early CV-QKD demonstrations [@grosshans2003high] and, more recently, in experimental demonstrations of measurement device independent (MDI) CVQKD [@pirandola2015high].
In more realistic experimental demonstrations, emitter and receiver must be placed in distant locations and some specific design must be used in order to obtain a local oscillator, at the receiver side, phase locked with the emitter laser. The simplest experimental approach is actually to use the laser at the emitter side to generate the local oscillator, and to send it to the receiving side, using adapted multiplexing schemes. This procedure is called the transmitted LO design (TLO) and we will review its principle and its limitations. As we will see, TLO suffers from a fundamental weakness in the cryptographic context of CV-QKD, due to the security loophole associated with LO manipulation as it propagates on a public channel. This has lead to implement CV-QKD with a true local local oscillator (LLO), and we will review the recent work in this direction.
The transmitted local oscillator (TLO) design {#Sec: TLO}
---------------------------------------------
In most implementations of CV-QKD performed so far [@ExpCVQKD-chinois2; @ExpCVQKD-chinois; @ExpCVQKD-Sequrenet; @ExpCVQKD-Qi], the phase reference is directly transmitted from Alice to Bob through the optical channel as a bright optical pulse multiplexed in time and polarization with each quantum signal pulse and is used as the LO pulse at reception. Such implementation is detailed in Fig. \[Scheme: TLO\] and is referred to as the Transmitted LO (TLO) design. The main advantage of this scheme is the guarantee, by design, of a stable relative phase between quantum signal and LO at reception by producing both of them from a single laser L$_A$ placed at Alice’s side. An interferometric setup, based on polarization delay-line interferometers, is used to multiplex (M) and demultiplex (D) the quantum signal and the LO, hence ensuring a low relative phase noise at reception. The only limitation in terms of tolerable phase noise is that the phase of the laser can be considered as stable over the duration of a single optical pulse, resulting in $\Delta\nu /f \sim 10$, where $\Delta \nu$ is the spectral linewidth of the laser, $f$ is the repetition rate and we assume a typical pulse duration of $0.1/f$. Despite it is the most implemented GMCS protocol, security weaknesses of such implementations have however been demonstrated in practice by manipulating the LO intensity [@grosshans2007; @ma2014; @CalibAttacks] or wavelength [@weedbrook2013] on the quantum channel.
Furthermore, based on a coherent detection at reception, such protocols rely on the use of a bright LO at reception (around $10^8$ photons per pulse at reception are required to ensure that the coherent detection can be operated with low electronic to shot noise ratio in [@ExpCVQKD-Sequrenet]). For long distance or high speed (where the pulse duration is short), the requirements in terms of launch power at emission creates practical issues. Because of limited power of lasers as well as Brillouin effect and non-linear effects in optical fibers [@brillouin1; @brillouin2; @nonlinearoptics], there is a typical limit of few tens of milliwatts on the launched power of each involved laser for CV-QKD purposes. In the TLO design, this limit is a major limitation of the LO intensity at reception, especially for long distances. This will in particular limit the possibility of using the TLO design on shared optical fibers at long distance and high-rate operation, i.e. situations where the requirements on LO power at emission would be extremely large.
The local local oscillator (LLO) sequential design {#Sec: LLO-seq}
--------------------------------------------------
In order to lift the important limitations (both theoretical and practical) of CV-QKD implementations relying on the TLO design, a new CV-QKD method relying on a “local local oscillator” (LLO) has recently been independently introduced in [@LLOexp1; @LLOexp2; @LLOexp3]. This method, implementing the Gaussian modulated coherent state protocol, consists in using a second laser at Bob’s side in order to produce local LO pulses for coherent detections. One crucial advantage of implementing CV-QKD in a LLO configuration is to close, by design, any potential security loophole linked to the possibility of manipulating the LO as it propagates on the public optical channel between Alice and Bob. Implementing LLO CV-QKD allows on the other hand to ensure by design that the LO is fully trusted, and in particular that the LO amplitude (that requires careful calibration) cannot be manipulated. Another important advantage of LLO CV-QKD stems from the fact that in this configuration, repetition rate and distance do not affect the LO intensity at detection. A LO power sufficient to ensure high electronic to shot noise ratio may thus be obtained, independently of the propagation distance.
Implementing CV-QKD in the LLO configuration however comes with new experimental challenges. The main issue in LLO-based CV-QKD is to be able to perform CV-QKD despite the potentially important drift of the relative phase between Alice’s emitter laser L$_A$ and Bob’s local oscillator laser L$_B$, see Fig. \[Scheme: LLO-Sequential\]. The relative phase at reception is, in the case of LLO-based CV-QKD, the relative phase between the two free-running lasers L$_A$ and L$_B$. As such, Bob’s raw measurement outcomes are *a priori* decorrelated from Alice’s quadratures and a phase correction process has to be performed in order to allow secret key generation. The goal of the phase reference sharing in the context of LLO CV-QKD is then to ensure a low enough phase noise so that the excess noise is significantly below the threshold imposed by security proofs [@ReviewCVQKD].
Recent works [@LLOexp1; @LLOexp2; @LLOexp3] have demonstrated the possibility of implementing the GMCS protocol using a local local oscillator, by introducing an experimental design, depicted on Fig. \[Scheme: LLO-Sequential\], that we will call LLO-sequential. In the LLO-sequential design, Alice sequentially sends, at a repetition rate $f/2$, consecutive pairs $(|\alpha_\mathrm{S} \rangle, | \alpha_\mathrm{R} \rangle)$ of coherent states where $|\alpha_\mathrm{S} \rangle$ is a GMCS quantum signal pulse and $|\alpha_\mathrm{R } \rangle = | E_\mathrm{R} \rangle$ is a phase reference pulse with a fixed phase set to $0$ and an amplitude $E_\mathrm{R}$. Phase reference pulses are relatively bright pulses compared to the signal and have a fixed phase in Alice’s phase reference frame, that is publicly known so that it carries information on Alice’s reference frame. At reception, Bob performs sequential coherent detections of quantum signal and phase reference, using a single detector, operated with a “local local oscillator”, placed at Bob. Bob can thus estimate the relative phase using the phase reference pulse and a phase correction can be performed on Alice and Bob’s signal data in order to generate secret key.
In [@LLOexp2] a 250 kHz-clocked proof-of-principle experiment of the LLO-sequential designed is performed, however with only one single laser playing the role of both emitter and LO, and two consecutive uses of a homodyne detector used to emulate a heterodyne measurement. In [@LLOexp1], another proof-of-principle experiment with two lasers and Alice and Bob connected by a 25 km optical fiber is performed with a 50 MHz-clocked system. The authors demonstrate that phase correction can be implemented with a residual excess noise compatible with CV-QKD security threshold. Joint operation of CV-QKD (requiring weak quantum signals) together with the phase correction mechanism (requiring bright phase reference pulses) was studied through a simulation, which left aside the question of the hardware requirements for both CV-QKD and phase reference sharing. [@LLOexp3] provides a whole experimental demonstration of an implementation of LLO-sequential CV-QKD over 25 km, with a 100 MHz-clocked system, and a 1 GHz-bandwidth shot-noise limited homodyne detection. We should however emphasize that these strong experimental performances have been hove we obtained expensive hardware, namely two low phase noise ECL lasers, as emitter and LO, and an amplitude modulator with 60 dB of dynamics. Such hardware is typically not available in standard telecom environment and the issue of considering LLO CV-QKD implementation with realistic hardware should be addressed in order to study the ability to be ubiquitously integrate CV-QKD within modern optical networks.
CV-QKD: protocol and noise model {#Sec: CVQKD}
================================
Different CV-QKD protocols have been proposed so far including protocols based on squeezed states of the electromagnetic field [@GarciaCerf2009] or on discrete modulations of coherent states [@GrangierLeverrier2009; @BecirWahiddin2010]. However, squeezed states are experimentally challenging to produce and security analysis for discrete modulation CV-QKD are less advanced. Due to its experimental convenience [@ExpCVQKD-Sequrenet; @ExpCVQKD-Qi; @ExpCVQKD-chinois] and its good security analysis understanding [@LeverrierThesis; @GarciaPatronThesis], the Gaussian-modulated coherent states (GMCS) protocol is the most implemented CV-QKD protocol and has reached the step of commercialization [@sequrenet].
It is therefore natural to consider GMCS CV-QKD in order to study CV-QKD with a local local oscillator and to perform early experiment, as it has been the case in [@LLOexp1; @LLOexp2; @LLOexp3]. We then also focus our analysis on GMCS CV-QKD and introduce new elements in the noise model in order to account for the important constraints that drive the performance of CV-QKD in the regime of a local local oscillator. This in particular allows us to discuss the limitations of LLO-sequential design when implemented with realistic hardwares.
GMCS protocol and secret key rate
---------------------------------
In the Gaussian-modulated coherent states protocol, Alice encodes classical Gaussian variables $(x_A,p_A)$ on the mean values of the two conjugate quadratures of coherent states $|\alpha \rangle = |x_A,p_A \rangle$. Coherent states are then sent to Bob through an insecure channel controlled by an eavesdropper Eve. At reception, Bob performs a coherent detection of either one quadrature (homodyne detection) or both quadratures (heterodyne detection) of the received pulse and calculates estimators $(x_B,p_B)$ of Alice’s variables. As Eve’s optimal attacks are Gaussian [@GarciaPatronThesis; @LeverrierThesis], we can model the logical channels between Alice and Bob’s data as additive white Gaussian noise channels [@FossierThesis]: $$\begin{aligned}
\begin{array}{ccc}
x_B & = & \sqrt{\frac{G}{\delta_\mathrm{det}}} \cdot \left( x_A + x_0 + x_c \right)\\
p_B & = & \sqrt{\frac{G}{\delta_\mathrm{det}}} \cdot \left( p_A + p_0 + p_c \right)
\end{array}
\label{Eq: Gaussian channel model}\end{aligned}$$
where $(x_c,p_c)$ is the total noise of the channel and we note $(\chi_x,\chi_p)$ its variance. In Eq. \[Eq: Gaussian channel model\], $G$ is the total intensity transmission of the channel, $\delta_\mathrm{det}$ stands for the detection used at reception ($\delta_\mathrm{det}=1$ for a homodyne detection and $\delta_\mathrm{det}=2$ for a heterodyne detection), $x_0$ and $p_0$ are Gaussian variables of variance $N_0$ modelling the shot noise quadratures. In general [@LeverrierThesis], it is assumed that the channel noise is symmetric and $\chi=\chi_x=\chi_p$ where the variance $\chi$ is referred to Alice’s input. The variance $\chi$ of the total noise can be expressed as [@FossierThesis; @GarciaPatronThesis]: $$\begin{aligned}
\chi & = & \frac{\delta_\mathrm{det}-G}{G} + \xi
\label{Eq: Total noise}\end{aligned}$$ where the first term is the loss-induced vacuum noise and $\xi$ is the overall excess noise variance of the channel referred to Alice’s input. Thereby, using Eq. \[Eq: Gaussian channel model\] and Eq. \[Eq: Total noise\], we can see that the Gaussian channel between Alice and Bob is fully characterized by the two parameters $G$ and $\xi$. In a real-world experiment, Alice and Bob can estimate $G$ and $\xi$ from the correlations between their respective variables by revealing a fraction of their data and are then able to characterize the propagation channel and generate secret key. We discuss and model the different contributions to the excess noise $\xi$ in practical CV-QKD in the next paragraph. Finally, the secret key rate available to Alice and Bob in the reverse reconciliation scheme can be expressed as [@FossierThesis; @GarciaPatronThesis]: $$\begin{aligned}
k & = & \beta\cdot I_{AB} - Q_{BE}\end{aligned}$$
where $0 \leq \beta \leq 1$ is the reconciliation efficiency, $I_{AB}$ is the mutual information between Alice and Bob’s classical variables and $Q_{BE}$ stands for Eve’s maximal accessible information on Bob’s measurements, capturing assumptions on Eve’s behaviour [@GarciaPatronThesis]. In this work, we restrict the security analysis to individual attacks [@FossierThesis; @GarciaPatronThesis] and $Q_{BE}$ is then the classical information $I_{BE}$ between Bob’s measurements and Eve’s data. In [@ExpCVQKD-Sequrenet; @LLOexp1], it is assumed that Eve does not have access to Bob’s electronics. In this work however, we use the stronger security model of [@LLOexp2] assuming that Eve is able to control the noise of Bob’s detector.
Noise model {#Sec: Noise model}
-----------
Implementing CV-QKD with a local local oscillator comes with new challenges. The main challenge is related to the fact that Alice and Bob must use a procedure to compensate efficiently the phase drift between two different lasers, used respectively as emitter and local oscillator, in order to be able to perform CV-QKD with a tolerable noise level. Another more specific aspect of the challenge is related to the objective targeted in this paper: propose and study practical implementation schemes for LLO CV-QKD with shared and affordable hardware: this leads to consider practical limitations that had been previously overlooked, and allows to study resource trade-off.\
#### Relative phase noise.
In LLO-based CV-QKD, the main challenge is to create a reliable phase reference between emitter and receiver because the relative phase drift between the two involved lasers may fully decorrelate Alice’s variables and Bob’s measurements thus preventing any secret key rate generation. We define the *signal relative phase* $\theta_\mathrm{S}$ as the phase difference between the LO pulse $|\alpha_\mathrm{LO } \rangle$ and the signal pulse $|\alpha_\mathrm{S} \rangle$ at reception: $$\begin{aligned}
\theta_\mathrm{S} & = & \varphi_\mathrm{LO}-\varphi_\mathrm{S}
\label{Eq: Relative Phase}\end{aligned}$$ where $\varphi_\mathrm{S}$ is the signal phase and $\varphi_\mathrm{LO}$ is the phase of the local oscillator at reception. Using the notations of Eq. \[Eq: Gaussian channel model\] and in presence of a relative phase $\theta_\mathrm{S}$, we can write Bob’s measurement outcomes, when performing an heterodyne as: $$\begin{aligned}
\begin{pmatrix} x_B \\ p_B \\ \end{pmatrix} & = & \sqrt{\frac{G}{\delta_\mathrm{2}}} \cdot \left[
\begin{pmatrix}\cos \theta_\mathrm{S} & \sin \theta_\mathrm{S}\\
-\sin\theta_\mathrm{S} & \cos \theta_\mathrm{S}\\ \end{pmatrix} \cdot
\begin{pmatrix} x_A \\ p_A \\ \end{pmatrix} +
\begin{pmatrix} x_0 + x_c \\ p_0 + p_c \\ \end{pmatrix} \right]
\label{Eq: Bob's measurement outcomes}\end{aligned}$$
where $x_c$ and $p_c$ capture all excess noise sources but the phase noise. The relative phase $\theta_\mathrm{S}$ acts as the selector of the measured quadrature. In the TLO design, the relative phase $\theta_\mathrm{S}$ is, by design, always close to $0$. However, in the case of two free-running lasers, $\theta_\mathrm{S}$ depends on the relative phase $\theta$ between the two lasers. Assuming that the two lasers L$_A$ and L$_B$ are centered around the same optical frequency and have spectral linewidths $\Delta \nu_A$ and $\Delta \nu_B$, we can model [@Schulze2005; @YarivYeh2006; @Bittner2010] the relative phase $\theta=\varphi_B-\varphi_A$ ($\varphi_A$ and $\varphi_B$ are respectively the phase of L$_A$ and L$_B$) as a Gaussian stochastic process $\{\theta_t\}_t$ characterized by the variance of the drift between two times $t_i$ and $t_{i+1}$: $$\begin{aligned}
\mathrm{var}\left( \theta_{i+1} | \theta_i \right) & = & 2\pi \cdot (\Delta \nu_A + \Delta \nu_B) \cdot |t_{i+1} - t_{i}|
\label{Eq: Laser Phase Drift Model}\end{aligned}$$ where $\theta_{i}$ and $\theta_{i+1}$ correspond to the relative phase at consecutive times $t_i$ and $t_{i+1}$.
We can see from Eq. \[Eq: Bob’s measurement outcomes\] that this implies a decorrelation between Alice’s data and Bob’s measurements which can be seen as a contribution, noted $\xi_\mathrm{phase}$, to the excess noise $\xi$. The principle of phase reference sharing schemes considered in the article consists in using a reference pulse to build an estimate $\hat{\theta}_\mathrm{S}$ of the actual relative phase $\theta_\mathrm{S}$ of the signal (relative means relative with respect to local oscillator), and to apply a phase correction $- \hat{\theta}_\mathrm{S}$ on the signal, in order to compensate for the phase drift. In a reverse reconciliation scheme, this correction has to be performed on Alice’s data as a rotation of her data: $$\begin{aligned}
\begin{pmatrix} \tilde{x_A} \\ \tilde{p_A} \\ \end{pmatrix} & = & \begin{pmatrix} \cos \hat{\theta}_\mathrm{S} & \sin \hat{\theta}_\mathrm{S}\\
-\sin \hat{\theta}_\mathrm{S} & \cos \hat{\theta}_\mathrm{S}\\ \end{pmatrix} \cdot \begin{pmatrix} x_A \\ p_A \\ \end{pmatrix}
\label{Eq: Alice phase correction}\end{aligned}$$
We can show from Eq. \[Eq: Total noise\], \[Eq: Bob’s measurement outcomes\], \[Eq: Alice phase correction\] that the remaining excess noise $\xi_\mathrm{phase}$ due to phase noise (after correction) depends on the modulation format and is, in general, not symmetric on the two quadratures . However, in the case of the GMCS protocol (with modulation variance $V_A$) and assuming that the remaining phase noise $\theta_\mathrm{S} - \hat{\theta}_\mathrm{S}$ after correction is Gaussian, the phase noise $\xi_\mathrm{phase}$ can then be written as: $$\begin{aligned}
\xi_\mathrm{phase} & = & 2V_A \cdot \left(1 - e^{-V_\mathrm{est}/2} \right)
\label{Eq: GMCS Phase noise}\end{aligned}$$ where we define the variance $V_\mathrm{est}$ of the *remaining phase noise* (after reference quadrature measurement, relative phase estimation and correction) as: $$\begin{aligned}
V_\mathrm{est} & \hat{=} & \mathrm{var}\left(\theta_\mathrm{S} - \hat{\theta}_\mathrm{S}\right)
\label{Eq: General phase estimation variance}\end{aligned}$$ Eq. \[Eq: GMCS Phase noise\] (derived in Annex. \[Annex: Phase excess noise\]) is a generalization of the phase noise expression given in [@LLOexp2; @Qi2016] for the case of small phase noise.
An important challenge to perform LLO-based CV-QKD is therefore to calculate a precise estimator $\hat{\theta}_\mathrm{S}$ of the relative phase $\theta_\mathrm{S}$ in order to minimize $V_\mathrm{est}$ and thus $\xi_\mathrm{phase}$. The general scheme for phase reference sharing design in LLO CV-QKD can be modeled in the following way: Alice generates two coherent states, a quantum signal pulse $|\alpha_\mathrm{S} \rangle$ and a reference pulse $|\alpha_\mathrm{R} \rangle$ and sends them on the optical channel using some multiplexing scheme. At reception, Bob performs demultiplexing, and uses the received reference pulse to derive an estimate $\hat{\theta}_\mathrm{R}$ the relative phase $\theta_\mathrm{R}$ between the reference pulse and the local oscillator at reception. The phase sharing designs (cf. sections \[Sec: Towards self-coherence\], \[Sec: LLO-delayline\], \[Sec: LLO-displacement\]) give guarantees that the relative phase of the reference pulse is close to the relative phase of the quantum signal, i.e. that $\theta_\mathrm{R} \approx \theta_\mathrm{R}$. Therefore, the estimated value $\hat{\theta}_\mathrm{R}$ can be used to approximate and then correct the relative phase of the signal $\theta_\mathrm{S}$.
A general picture of the phase estimation process, and of the sources of deviations, is depicted in Fig. \[Figure: General phase estimation process\]. We can express the quantum signal relative phase $\theta_\mathrm{S}$ (with respect to local oscillator) as the sum of the relative phase $\theta_\mathrm{S}^\mathrm{A}$ at emission and the phase $\theta_\mathrm{ch}^\mathrm{S}$ accumulated by the coherent state $|\alpha_\mathrm{S} \rangle$ on the optical channel: $$\begin{aligned}
\theta_\mathrm{S} & = & \theta_\mathrm{S}^\mathrm{A} + \theta_\mathrm{S}^\mathrm{ch}
\label{Eq: Signal relative phase}\end{aligned}$$
Similarly and using the same notations for the reference pulse $|\alpha_\mathrm{R} \rangle$, we can express the relative phase $\theta_\mathrm{R}$ at reception of a reference pulse $|\alpha_\mathrm{R} \rangle$ as: $$\begin{aligned}
\theta_\mathrm{R} & = & \theta_\mathrm{R}^\mathrm{A} + \theta_\mathrm{R}^\mathrm{ch}
\label{Eq: Ref relative phase}\end{aligned}$$
At reception, Bob measures both quadratures of $|\alpha_\mathrm{R} \rangle$ using a heterodyne detection (in the remaining of the paper, we only consider heterodyne detections at reception with $\delta_\mathrm{det}=2$). The estimator $\hat{\theta}_\mathrm{R}$ of $\theta_\mathrm{R}$ can be calculated from the heterodyne measurement outcomes $x_B^{\mathrm{(R)}}$ and $p_B^{\mathrm{(R)}}$ as: $$\begin{aligned}
\hat{\theta}_\mathrm{R} = \tan^{-1}\left(\frac{p_B^{\mathrm{(R)}}}{x_B^{\mathrm{(R)}}}\right)\end{aligned}$$ Due to the fundamental shot noise and to the experimental noise on the heterodyne detection, $\hat{\theta}_\mathrm{R}$ differs from$\hat{\theta}_\mathrm{R}$ by an error $\theta_\mathrm{error}$ characterized by its variance: $$\begin{aligned}
V_\mathrm{error} & \hat{=} & \mathrm{var} \left(\hat{\theta}_\mathrm{R} - {\theta}_\mathrm{R} \right)\end{aligned}$$ We can show that, in the case of a reference pulse of the form $|\alpha_\mathrm{R} \rangle = |E_\mathrm{R} \rangle$: $$\begin{aligned}
V_\mathrm{error} & = & \frac{\chi + 1}{E_\mathrm{R}^2}
\label{Eq: Single pulse phase estimation}\end{aligned}$$
where $E_\mathrm{R} = |\alpha_\mathrm{R}|$ is the amplitude of the reference pulse and $\chi$ is defined in Eq. \[Eq: Total noise\]. Finally, Bob uses the relative phase estimate $\hat{\theta}_\mathrm{S}$ to apply a phase correction $- \hat{\theta}_\mathrm{R}$ to the quantum signal. The overall process is schematically represented in Fig. \[Figure: General phase estimation process\]. It results, after phase correction, to a remaining phase noise $V_\mathrm{est}=\mathrm{var}\left(\hat{\theta}_\mathrm{R} - \theta_\mathrm{S}\right)$ which can be expressed using Eq. \[Eq: Signal relative phase\] and Eq. \[Eq: Ref relative phase\] as:
$$\begin{aligned}
V_\mathrm{est} & = & V_\mathrm{error} + V_\mathrm{drift} + V_\mathrm{channel}
\label{Eq: Total remaining phase noise}\end{aligned}$$
where: $$\begin{aligned}
V_\mathrm{drift} & \hat{=} & \mathrm{var}\left( \theta_\mathrm{R}^\mathrm{A} - \theta_\mathrm{S}^\mathrm{A}\right) \\
V_\mathrm{channel} & \hat{=} & \mathrm{var}\left( \theta_\mathrm{R}^\mathrm{ch} - \theta_\mathrm{S}^\mathrm{ch}\right)
\label{Eq: Vchannel}\end{aligned}$$
The term $V_\mathrm{drift}$ corresponds to the variance of the relative phase drift $ \theta_\mathrm{drift}=\theta_\mathrm{R}^\mathrm{A} - \theta_\mathrm{S}^\mathrm{A}$ between the two free-running lasers L$_A$ and L$_B$ between time $t_\mathrm{S}$ at which $|\alpha_\mathrm{S} \rangle$ is emitted and time $t_\mathrm{R}$ at which $|\alpha_\mathrm{R}\rangle$ is emitted. From Eq. \[Eq: Laser Phase Drift Model\], we can express the phase noise due to laser phase drift between times $t_\mathrm{S}$ and $t_\mathrm{R}$ as: $$\begin{aligned}
V_\mathrm{drift} = & 2\pi \cdot (\Delta \nu_A + \Delta \nu_B) \cdot |t_\mathrm{R} - t_\mathrm{S}|
\label{Eq: Relative phase drift R-S}\end{aligned}$$
We can observe that the time delay between signal and phase reference emissions implies a decorrelation between the corresponding relative phases and, thus, introduce a noise on the phase estimation process. This leads to the main limitation of the LLO-sequential approach as explained in next section.\
The term $V_\mathrm{channel}$ corresponds to the relative phase drift due to the difference of the phase accumulated by $|\alpha_\mathrm{S} \rangle$ and $|\alpha_\mathrm{R} \rangle$ during propagation. In practice, we assume that this term is dominated by the difference between the optical path lengths of $|\alpha_\mathrm{S} \rangle$ and $|\alpha_\mathrm{R} \rangle$.
In the remaining of this article we will study and discuss the performance of existing as well as newly introduced LLO based CV-QKD designs, relying on different relative phase sharing designs. For each of these designs, we will explicit the expressions of the different contributions to the remaining phase noise $V_\mathrm{est}$ of Eq. \[Eq: Total remaining phase noise\].\
#### Electronic to shot noise ratio.
Intrinsic electronic noise of Bob’s detector induces a noise of variance $v_\mathrm{elec}$ in shot noise unit (SNU) on Bob’s quadrature measurements. As the shot noise value is linear with the LO intensity, it is however possible to reduce the effective electronic to shot noise ratio $\xi_\mathrm{elec}$ by increasing the LO intensity. We model $\xi_\mathrm{elec}$, referred to Alice’s input, as: $$\begin{aligned}
\xi_\mathrm{elec} & = & \frac{\delta_\mathrm{det}}{G} \cdot \frac{E^2_\mathrm{LO,cal} \cdot v_\mathrm{elec}}{E_\mathrm{LO}^2}
\label{Eq: electronic noise}\end{aligned}$$ where $E_\mathrm{LO,cal}^2$ is the photon number in the LO at which the electronic noise is $v_\mathrm{elec}$ at Bob side and $E_\mathrm{LO}^2$ is the actual photon number per LO pulse at reception. Furthermore, we consider that Eve is able to manipulate the electronic noise which corresponds to a strong security scenario [@LLOexp2].\
#### Amplitude modulator finite dynamics.
Amplitude modulators efficiency are limited by their dynamics restricting the range of the achievable transmission coefficient. Recent works [@LLOexp1; @LLOexp2; @LLOexp3] have proposed to conjointly communicate weak quantum signals and relatively bright reference pulses using a single experimental setup and, in particular, a single amplitude modulator (AM). This directly adresses the issue of the AM dynamics at emission, limitating the maximal amplitude that Alice can output and introducing a leakage on the amplitude modulated. The ratio between the maximal and minimal amplitudes $E_{max}$ and $E_{min}$ that Alice can output is characterized by the dynamics $\mathrm{dyn}_\mathrm{dB}$ of the AM defined as: $$\begin{aligned}
\mathrm{dyn}_\mathrm{dB} & = & 10 \cdot \log_{10} \left(\frac{E_{max}^2}{E_{min}^2} \right)\end{aligned}$$ From this equation, one can model Alice’s modulator imperfection as an amplitude leakage on each optical pulse resulting in an excess noise which can be approximated as: $$\begin{aligned}
\xi_\mathrm{AM} & = & E_\mathrm{max}^2\cdot 10^{-\mathrm{dyn}_\mathrm{dB}/10}
\label{Eq: AM excess noise}\end{aligned}$$ where $E_\mathrm{max}$ is the maximal amplitude to be modulated. The finite dynamics of Alice’s AM thus adds a noise proportional to the amplitude $E_\mathrm{max}$. This imperfection is then a limitation to the maximal amplitude of the phase reference pulses in LLO-based CV-QKD designs.\
#### Linearity range of the reception detector.
In practice, Bob’s detector response is linear with the input number of photons within a finite range. Beyond a threshold, the output of the detector is no longer linear and the security can be broken [@SaturationAttackHao]. Thereby, this threshold can be seen as a limitation on the amplitude of the reference pulse used to transmit the phase reference. However, as discussed in Annex. \[Annex:Linearity\_Range\_Detector\], typical values of this threshold are sufficiently large to allow precise relative phase estimation and, thus, are not a limitation to the reference amplitude.\
#### Technical noise.
In order to simulate the experimental imperfections that one can not calibrate within a typical CV-QKD implementations, we introduce a technical excess noise which is typically a fraction of the shot noise ($\xi_\mathrm{tech} = 0.01 N_0$).\
The necessity of exchanging both weak and intense optical signals in CV-QKD based on a local local oscillator using only one experimental setup is limited by the finite AM dynamics. These effects are not considered in standard TLO designs because only weak quantum signals are modulated and detected but we will show that they are key parameters in order to compare realistic implementations of LLO-based CV-QKD in terms of secret key rate. To our knowledge, the amplitude modulator and linearity range issues have not been taken into account so far in CV-QKD analysis. Based on this refined model, we first analyze practical limitations of the LLO-sequential design and, then, we compare the LLO-sequential implementations with our newly proposed designs.
Towards improved phase reference sharing designs {#Sec: Towards self-coherence}
=================================================
In this section, we highlight limitations of the LLO-sequential design in terms of tolerable phase noise due to the underlying phase reference sharing scheme. We then propose the novel idea of self-coherence to go beyond that phase noise limit.
Limitations of the LLO-sequential design {#Sec: Fundamental limit on LLO-seq}
----------------------------------------
Since signal and reference pulses follow the same optical path, the estimation process in the LLO-sequential design can be schematically shown using Fig. \[Figure: General phase estimation process\] where $\theta_\mathrm{S}^\mathrm{ch}=\theta_\mathrm{R}^\mathrm{ch}$. Thus, we have $V_\mathrm{channel} \approx 0$ and the phase noise stems from two contributions: $ V_\mathrm{est} = V_\mathrm{drift} + V_\mathrm{error}$.
One important motivation of the present work is related to the fact that there exists a minimal amount of phase noise $V_\mathrm{est}$ (Eq. \[Eq: Total remaining phase noise\]) that can be reached with the LLO-sequential design. The main limitation is due to the fact that signal and reference pulses are emitted with a time delay $1/f$ , leading to a phase noise that cannot be compensated, of variance $$\begin{aligned}
V_\mathrm{drift} & = & 2\pi \cdot \frac{\Delta \nu_A + \Delta \nu_B}{f}\end{aligned}$$
The phase variance associated to reference pulse phase estimation error, $V_\mathrm{error}$ can be minimized by choosing the amplitude $E_\mathrm{R}$ as large as possible. However the value $E_\mathrm{R}$ that can be chosen in practice is limited by the finite dynamics of her amplitude modulator, and the existence of an associated optical leakage, whose excess noise $\xi_\mathrm{AM}$ is proportional to the amplitude $E_\mathrm{R}$ as discussed in Eq. \[Eq: AM excess noise\]. This in practice leads to a compromise regarding the value of $E_\mathrm{R}$, in order to minimize the total excess noise.
The excess noise due to imperfect phase reference sharing reads as (Eq. \[Eq: GMCS Phase noise\]): $$\begin{aligned}
\xi_\mathrm{phase} & = & 2V_A\cdot(1-e^{-V_\mathrm{est}/2})\end{aligned}$$ In the regimes of low $V_\mathrm{est}$, it simplifies to $\xi_\mathrm{phase} = V_A \cdot V_\mathrm{est}$. In order to ensure a tolerable value $\xi_\mathrm{phase} \leq 0.1$ (typical value of the null key threshold according to security proofs [@SaturationAttackHao; @ExpCVQKD-chinois2]),this imposes that $V_\mathrm{drift} \lesssim 0.1/V_A$.
We can finally express a lower bound on the total excess noise, sum of the excess noise $\xi_\mathrm{phase}$ and the excess noise $\xi_\mathrm{AM}$ in the LLO-sequential design as: $$\begin{aligned}
\xi_\mathrm{phase} + \xi_\mathrm{AM} & \geq & V_A \cdot \left( V_\mathrm{drift} + \frac{\chi + 1}{E_\mathrm{R}^2} \right) + E_\mathrm{R}^2 \cdot 10^{-\mathrm{dyn}_\mathrm{dB}/10}
\label{Eq: lower bound}\end{aligned}$$
We can quantitatively understand from Eq. \[Eq: lower bound\] that increasing the amplitude $E_\mathrm{R}$ can reduce the $\xi_\mathrm{phase}$ contribution at the cost of increasing the $\xi_\mathrm{AM}$ contribution. The LLO-sequential design thus requires to the experimental regimes where $V_\mathrm{drift} \ll 1$ and where the amplitude modulator dynamics is large. In other words, the LLO-sequential has to be implemented with performant hardware. Current bandwidth limitation of shot-noise limited coherent detectors typically leads to choose $f$ below 100 MHz. This imposes in return strong requirements on the spectral linewidth of the lasers that can be used in order to perform LLO-sequential CV-QKD: the linewidth of the lasers must be at most of $200$ kHz, to ensure an excess noise $\xi_\mathrm{phase} $ lower than $0.1$. As a consequence, only very low phase noise lasers, such as external-cavity lasers (ECL), whose typical spectral linewidth is of a few kHz, are suitable to implement the LLO-sequential design. This is actually illustrated in [@LLOexp2], where the performance analysis is made in the low phase noise regime where $f\gg \Delta \nu_A + \Delta \nu_B$ (ie. in a regime where $V_\mathrm{drift} \approx 0$) and in [@LLOexp3] with the experimental choice ultra low noise of ECL lasers of 1.9 kHz linewidth.
Another issue is actually that finite modulation dynamics has not been taken into account in [@LLOexp2], allowing the authors to choose arbitrary large amplitudes $E_\mathrm{R}$. For instance, they show that, by choosing $E_\mathrm{R}^2 = 500 \ V_A$, a distance of $40$ km is achievable while a more realistic value $E_\mathrm{R}^2 = 20 \cdot V_A$ ($\xi_\mathrm{AM} \approx 10^{-2}$ for $\mathrm{dyn}_\mathrm{dB} = 40$ dB) restricts the protocol to less than $10$ km. This indicated that AM dynamics is an important parameter for analyzing CV-QKD within the LLO framework and the LLO-sequential design requires expensive optical equipments, which is not practical in terms of large-scale deployment for next-generation CV-QKD.
In Fig. \[Curves:LLO\_TolerablePhaseNoise\], we plot the secret key rates of the LLO-sequential design with finite AM dynamics. We can see that the AM dynamics is an important parameter as it allows to recover the relative phase with good efficiency while ensuring a low excess noise $\xi_\mathrm{AM}$. Below an AM dynamics of $30$ dB, no secret key rate can be produced beyond a distance of around $20$ km, even for a moderate relative phase drift $V_\mathrm{drift} = 10^{-3}$. On the other hand, because of the fundamental limit $V_\mathrm{drift}$ the LLO-sequential design has to be run at a minimal repetition rate to produce secret key, even in large AM dynamics regimes.
In Table. \[Table: Previous work\], we summarize the main characteristics of the two existing implementations of the GMCS protocol proposed so far. Although strong security loopholes have been demonstrated on the TLO implementation, the GMCS protocol has mainly been implemented by directly sending the LO from Alice to Bob. Recent works have however introduced the idea of LLO-based CV-QKD by proposing the experimental LLO-sequential design, hence fixing security weaknesses by generating the LO pulses at Bob side. We have however shown that the LLO-sequential design has strong limitations in terms of implementability in realistic regimes. In the next sections, we investigate how these limitations in term of hardware requirements can be lifted by proposing the idea of self-coherence for phase reference sharing designs.
Self-coherent phase reference sharing schemes {#Sec: Self-coherence}
---------------------------------------------
Performing CV-QKD protocols in the LLO regime can be seen as the issue of conjointly $-$ in the sense of using the same hardware $-$ sharing a phase reference between two remote lasers and performing CV-QKD between the two parties Alice and Bob holding lasers L$_A$ and L$_B$. A first method to perform such task is the LLO-sequential design [@LLOexp1; @LLOexp2; @LLOexp3]. Indeed, the specific modulation of the sequential optical pulses allows one to perform CV-QKD on signal pulses while sharing the phase reference on specific pulses. We have however shown in Sec. \[Sec: Fundamental limit on LLO-seq\] a fundamental limitation in terms of tolerable relative phase drift in the LLO-sequential design. As an unspeakable information, the phase reference has to be encoded over physical carriers, photons in this case. However, by design, the time delay between the emission of quantum signal photons and phase reference photons introduce a decoherence between signal and reference which can prevent any secret key generation in high phase noise regimes.
**Design** **Trusted LO** **Tolerable phase noise** **Hardware requirements**
---------------------------------------------------------------------------- ---------------- --------------------------------------------- -------------------------------
Transmitted LO (Fig. \[Scheme: TLO\]) No $ \Delta\nu / f \sim 10$ Stable interferometric set-up
[@ExpCVQKD-Sequrenet; @ExpCVQKD-Qi; @ExpCVQKD-chinois; @ExpCVQKD-chinois2]
LLO-sequential (Fig. \[Scheme: LLO-Sequential\]) Yes $V_\mathrm{drift} \sim 10^{-1}$ ($60$dB AM) High AM dynamics
[@LLOexp1; @LLOexp2; @LLOexp3] $V_\mathrm{drift} \sim 10^{-3}$ ($30$dB AM)
We now introduce the novel idea of self-coherence for quantum coherent communication protocols. In order to prevent the phase decorrelation between signal and reference due to sequential emissions, we propose to derive both the signal and reference from the same optical wavefront at emission thus ensuring the physical coherence between signal and phase reference pulses, ensuring that the relative phase drift from Eq. \[Eq: Total remaining phase noise\] is $V_\mathrm{drift}=0$. We call *self-coherent* such a design. The relative phase between signal and phase reference is then not affected by the relative phase drift of the lasers and a stable relation between the quantum signal and the LO phases at reception can be provided. As the relative phase estimation does no longer depend on the relative phase drift, self-coherent designs allow Alice and Bob to perform more efficient phase reference sharings. This new method however comes with the challenge of coherently sending $-$ ie. by conserving the stable phase relation $-$ the quantum signal and the phase reference from Alice to Bob. This challenge can be seen as a multiplexing issue. In the remaining of this work, we propose two designs to realize GMCS CV-QKD relying on such self-coherent phase reference sharing designs.
Our first proposal to implement self-coherent CV-QKD is to split a single optical pulse into two pulses used to respectively carry signal and reference information. As output of the same optical pulse, the relative phase between the two pulses at reception only depends on the phases accumulated on their optical paths between emission and reception. This phase reference sharing design relies on the balancing of remote delay line interferometers and we refer to it as the LLO-delayline design. We describe and study its performance in Sec. \[Sec: LLO-delayline\].
A second idea, that we first introduced in [@MarieAlleaume2016], to directly ensure self-coherence between signal and phase reference is to encode both of them within the same optical pulse at emission while recovering both information at reception. In Sec. \[Sec: LLO-displacement\], we propose such a design, the LLO-displacement, in which the phase reference information is encoded over a displacement of the quantum signal modulation. Altough we show that LLO-displacement is restricted to low phase noise, an advantage of this design is that the experimental setup is drastically simplified compare to LLO-delayline, which is a major advantage in the optics of the integration of LLO-based CV-QKD.
Self coherent design based on delay-line interferometer {#Sec: LLO-delayline}
=======================================================
The idea of the LLO-delayline design is to derive consecutive pulse pairs with fixed relative phase, using a balanced delay line interferometer, hence ensuring a self-coherence property. This design does not suffer from the drift limitation of LLO-sequential and can allow Bob to recover the relative phase with better precision.\
**The protocol.** The protocol can be decomposed in successive cycles at the repetition rate $f/2$. We note $2\tau = 2/f$ the time interval between two consecutive cycles. Each cycle consists in producing and measuring a self-coherent pair of pulses: one quantum signal pulse and one phase reference pulse. We here describe the protocol for one cycle while Fig. \[fig:scheme LLO-delayline\] details the overall design.
At the beginning of a cycle, Alice produces a coherent state $|\alpha _\mathrm{source} \rangle$ which has an optical phase $\varphi_\mathrm{source}^A$. From that single optical pulse, she derives two coherent optical pulses in the following way: she splits the state $|\alpha _\mathrm{source} \rangle$ into two optical pulses, using an unbalanced delayline interferometer:
- The weak pulse $|\alpha _\mathrm{S} \rangle$ is modulated as the GMCS quantum signal and propagates through an optical path of length $l_A$.
- The strong pulse $|\alpha _\mathrm{R} \rangle$ is delayed by a time $\tau = 1/f$ on a optical path of length $l_A + \delta l_A$ and is referred to as the reference pulse.
Alice then recombines the two pulses $|\alpha _\mathrm{S} \rangle$ and $|\alpha _\mathrm{R} \rangle$ resulting in consecutive optical pulses. A major point is that the relative phase between phase reference and quantum signal does no longer depend on the phase drift between Alice and Bob’s lasers. An other advantage of this scheme is that the amplitude modulator only modulates the quantum signal which removes the constraints on the AM dynamics. The two optical pulses are then successively sent to Bob through the optical channel, resulting in a repetition rate $f$.
At reception, Bob produces coherent LO pulse pairs using a similar delay line technique used at Alice side. He produces an optical pulse $|\beta _\mathrm{source} \rangle$ with phase $\varphi_\mathrm{source}^B$ and derives two pulses on a 50/50 beamsplitter:
- The pulse $|\beta _\mathrm{S} \rangle$ that goes through an optical path of length $l_B$.
- The pulse $|\beta _\mathrm{R} \rangle$ that is delayed and follows an optical path of length $l_B + \delta l_B$.
Bob uses the $|\beta _\mathrm{S} \rangle$ and $|\beta _\mathrm{R} \rangle$ pulses as LO pulses to successively measure the received $|\alpha _\mathrm{S} \rangle$ and $|\alpha _\mathrm{R} \rangle$ pulses. This experimental setup can thus be seen as a remote delay-line interferometer split between Alice and Bob sides.
The reference pulse measurement outcomes allows Bob to calculate an estimation $\hat{\theta}_\mathrm{R}$ of the relative phase $\theta_\mathrm{R}$ at reference measurement and, thus, infer an estimation of the relative phase $\theta_\mathrm{S}$ at signal measurement. Alice can then correct her data to decrease the induced excess noise according to Eq. \[Eq: Alice phase correction\].\
**Excess noise evaluation.** In order to study the performance of the LLO-delayline design and calculate the achievable secret key rate, one can note that Alice modulates the quantum signal according to the standard GMCS modulation. Thus, the usual secret key rates formulas of [@FossierThesis] can be used. We then have to express the excess noise of the propagating channel in this design, in particular the amplitude modulator noise and the remaining relative phase noise $V_\mathrm{est}$.
In the LLO-delayline design, the finite dynamics of Alice’s amplitude modulator only induces a small contribution to the excess noise $\xi$ in affordable hardware regimes. As it only modulates the quantum signal, the maximal amplitude $E_\mathrm{max}$ of Eq. \[Eq: AM excess noise\] does not depend on the reference pulse amplitude. The excess noise $\xi_\mathrm{AM}$ is then independant of the reference amplitude $E_\mathrm{R}$ and the intensity $E_\mathrm{max}^2$ of Eq. \[Eq: AM excess noise\] only has to be a few times larger than $V_A$ [@FossierThesis], resulting in a moderate contribution of $\xi_\mathrm{AM}$ to the excess noise $\xi$ ($\xi_\mathrm{AM} \sim 10^{-2}$ for $V_A = 4$, $E_\mathrm{max}^2=10 \ V_A$ and dyn$_\mathrm{dB} = 30$).
We now quantify the excess noise $\xi_\mathrm{phase}$ by expressing the remaining phase noise $V_\mathrm{est}$ on Bob’s estimation of the relative phase. By design, the simultaneous emission on the source pulse of $|\alpha _\mathrm{S} \rangle$ and $|\alpha _\mathrm{R} \rangle$, ie. $t_\mathrm{S}=t_\mathrm{R}$, implies $\theta_\mathrm{drift} = 0$ and, thus, $V_\mathrm{drift}=0$, which corresponds to the self-coherence property. The variance $V_\mathrm{est}$ can then be written as the sum of the phase estimation efficiency $V_\mathrm{error}$ (given in Eq. \[Eq: Single pulse phase estimation\]) and the variance $V_\mathrm{channel}$ (Eq. \[Eq: Vchannel\]) of the difference between the accumulated phases on the channel: $$\begin{aligned}
V_\mathrm{est} & = & V_\mathrm{error} + V_\mathrm{channel}
\label{Eq: Remaining phase noise DLI}\end{aligned}$$ In this design, signal and phase reference pulses propagate through different optical path. Then, the former term depends on the stability of the delayline interferometer. As introduced in Sec. \[Sec: Noise model\], this corresponds to the variance: $$\begin{aligned}
V_\mathrm{channel} & = & \mathrm{var}\left( \theta_\mathrm{R}^\mathrm{ch} - \theta_\mathrm{S}^\mathrm{ch} \right)\end{aligned}$$
where $\theta_\mathrm{S}^\mathrm{ch}$ and $\theta_\mathrm{R}^\mathrm{ch}$ respectively correspond to the phases accumulated by $|\alpha_\mathrm{S}\rangle$ and $|\alpha_\mathrm{R}\rangle$ through their propagation. Therefore, one wants to express the quantity $\theta_\mathrm{channel} =\theta_\mathrm{R}^\mathrm{ch} - \theta_\mathrm{S}^\mathrm{ch}$. Using the definition of the relative phase of Eq. \[Eq: Relative Phase\], we can first write the relative phase as the difference between the phases respectively accumulated by the two interfering LO and signal pulses: $$\begin{aligned}
\begin{array}{rcl}
\theta_\mathrm{S}^\mathrm{ch} & = & \varphi_{\beta ,\mathrm{S}}^\mathrm{acc} - \varphi_{\alpha, \mathrm{S}}^\mathrm{acc} \\
\theta_\mathrm{R}^\mathrm{ch} & = & \varphi_{\beta , \mathrm{R}}^\mathrm{acc} - \varphi_{\alpha , \mathrm{R}}^\mathrm{acc}
\label{Eq: Accumulated phase}
\end{array}\end{aligned}$$
where, for instance, $\varphi_{\beta ,\mathrm{S}}^\mathrm{acc}$ stands for the phase accumulated by the LO pulse $|\beta_\mathrm{S} \rangle$ during its propagation. We model the accumulated phase as a linear function $\varphi^\mathrm{acc}(l)$ of the optical path length $l$, then we can derive the following expressions: $$\begin{aligned}
\begin{array}{rcl}
\varphi_{\alpha , \mathrm{S}}^\mathrm{acc} & = & \varphi^\mathrm{acc}(l_A) \\
\varphi_{\alpha , \mathrm{R}}^\mathrm{acc} & = & \varphi^\mathrm{acc}(l_A) + \varphi^\mathrm{acc}(\delta l_A) \\
\varphi_{\beta , \mathrm{S}}^\mathrm{acc} & = & \varphi^\mathrm{acc}(l_B) \\
\varphi_{\beta , \mathrm{R}}^\mathrm{acc} & = & \varphi^\mathrm{acc}(l_B) + \varphi^\mathrm{acc}(\delta l_B)
\end{array}\end{aligned}$$
Using the previous equations, we can finally express $\theta_\mathrm{channel} = \theta_\mathrm{R}^\mathrm{ch} - \theta_\mathrm{S}^\mathrm{ch}$ as: $$\begin{aligned}
\theta_\mathrm{channel} & = & \varphi^\mathrm{acc}(\delta l_B) - \varphi^\mathrm{acc}(\delta l_A)
\label{Eq: Relative phase drift dli}\end{aligned}$$
As we can see, the relative phase drift $\theta_\mathrm{channel}$ only depends on the difference of the accumulated phases between the delayline optical paths $\delta l_A$ and $\delta l_B$. Due to experimental imperfections as thermal fluctuations, we model $\delta l_A$ and $\delta l_B$ as a stochastic processes over time around the same mean value $\langle \delta l_A \rangle = \langle \delta l _B \rangle = c\tau$ ($c$ being the speed of the light). The phase $\theta_\mathrm{channel}$ then only depends on the fluctuations of the processes $\delta l_A$ and $\delta l_B$ corresponding to the interferometer balancing efficiency. An important point is that previous experimental demonstrations of CV-QKD [@ExpCVQKD-Qi; @ExpCVQKD-Sequrenet; @ExpCVQKD-chinois2] with transmitted LO, that rely on such delay line interferometers, have proven that phase fluctuations (with frequency typically of order of Hz) associated to interferometer path length fluctuations can be kept low in frequency and amplitude when sampled at CV-QKD repetition rate and do not prevent to perform CV-QKD with repetition rates in the MHz (or above) and we consider that $V_\mathrm{channel}= \mathrm{var}(\theta_\mathrm{channel}) \approx 0$. We can then consider that the variance $V_\mathrm{est}$ from Eq. \[Eq: Remaining phase noise DLI\] is dominated by the phase measurement efficiency: $$\begin{aligned}
V_\mathrm{est} & = & V_\mathrm{error} \end{aligned}$$
The LLO-delayline design ensures self-coherence at interference using delayline interferometers and the dependence on the relative phase drift of the lasers is removed. Bob gets self-coherent outcome measurements and is able to estimate the phase drift in the same way as in the LLO-sequential design but with higher efficiency. Finally, this results in the following excess noise: $$\begin{aligned}
\xi_\mathrm{phase} & = & 2V_A\cdot (1 - e^{-V_\mathrm{error}/2}) \end{aligned}$$
**Performance analysis.** Based on the previous excess noise analysis, we can now study the performance of the LLO-delayline design in terms of secret key rate and compare its performance with the LLO-sequential design. As the quantum signal modulation is the same as in the LLO-sequential, we can equivalently compare the achievable secret key rate or the excess noise contributions.
The LLO-delayline design allows to remove the relative phase drift $V_\mathrm{drift}$ from the excess noise expression. However, the relative phase drift between the two lasers should be stable within the duration of a single optical pulse and imposes that $V_\mathrm{drift} \lesssim 10$ (Sec. \[Sec: TLO\]). The remaining phase noise is only limited by the efficiency $V_\mathrm{error}$ of the phase reference estimation. In particular, the LLO-delayline allows to perform CV-QKD stronger phase noise regime than the LLO-sequential design. Furthermore, as the amplitude modulator excess noise $\xi_\mathrm{AM}$ does not depend on the phase reference amplitude, the AM dynamics do no longer restrict the relative phase measurement efficiency $V_\mathrm{error}$. The reference amplitude $E_\mathrm{R}$ can then be chosen as large as possible in the limit of the saturation limit of Bob’s detector and of the launched power limit without increasing the excess noise $\xi_\mathrm{AM}$. In practice, these limits allow an very efficient phase measurement.
In Fig. \[Curves: LLO-delayline\], we plot the expected key rates for both the LLO-sequential and LLO-delayline designs for different relative phase drift and AM dynamics. We can see that the LLO-delayline design is more resilient to both a decrease of the AM dynamics and to an increase of the relative phase drift. Tthe self-coherence between quantum signal and phase reference allows to reach stronger phase noise regime than LLO-sequential with similar optical hardwares. We can see on the left figure of Fig. \[Curves: LLO-delayline\] that with a $50$ dB AM, the LLO-delayline design allows secret key generation at $50$ km with $V_\mathrm{drift} = 0.1$ when LLO-sequential can only reach $30$ km. Furthermore, in the LLO-delayline design, the amplitude of the reference pulses can be chosen much larger than in the LLO-sequential design because it does not increse the excess noise $\xi_\mathrm{AM}$. Thus, the relative phase is estimated with better precision reducing the induced excess noise. For instance, the LLO-delayline allows to perform CV-QKD at a distance of $50$ km with affordable $30$ dB amplitude modulators even in the regime of standard DFB lasers (linewidths of order of MHz) which is not possible with the LLO-sequential design. We have thus shown that the LLO-delayline allows to perform LLO-based CV-QKD in the regime of affordable optical hardware regimes, which is an improvement towards LLO-based CV-QKD based on standard optical equipments.
Self coherent designs based on a modulation displacement {#Sec: LLO-displacement}
========================================================
In this section, we propose a second design (firstly proposed in [@MarieAlleaume2016]), the LLO-displacement design, implementing CV-QKD with a self-coherent phase reference sharing. This design is based on a method for jointly encoding both quantum signal and phase reference information over each optical pulse produced by Alice’s laser L$_\mathrm{A}$ at emission.\
**The protocol.** The main idea is to displace, in the phase space, the modulation sent by Alice with a fixed displacement of amplitude $\Delta$ and phase $\phi_\Delta$. Given her standard GMCS variables $(x_A,p_A)$, Alice produces and sends to Bob the displaced coherent state: $$\begin{aligned}
|\alpha \rangle _{\overrightarrow{\Delta}} & = & |x_A + \Delta \cos\phi_\Delta, p_A + \Delta \sin\phi_\Delta \rangle\end{aligned}$$ The amplitude $\Delta$ and the phase $\phi_\Delta$ of the displacement are publicly known so that it carries information on Alice’s phase reference. At reception, Bob measures both $\hat{x}$ and $\hat{p}$ quadratures of each received optical pulse using a heterodyne detection and gets measurement outcomes $(x_{B},p_{B})$: $$\begin{aligned}
\begin{array}{ccc}
x_{B} & = & \sqrt{\frac{G}{2}} \cdot \left[ (x_A + \Delta\cos\theta_\Delta) \cdot \cos\theta + (p_A + \Delta\sin\theta_\Delta) \cdot \sin\theta + x_0 + x_c \right]\\
p_{B} & = & \sqrt{\frac{G}{2}} \cdot \left[ - (x_A + \Delta\cos\theta_\Delta) \cdot \sin\theta + (p_A + \Delta\sin\theta_\Delta) \cdot \cos\theta + p_0 + p_c \right]
\end{array}\end{aligned}$$
Using the displacement of Alice’s modulation, Bob is able to measure an estimator $\hat{\theta}_\mathrm{S}$ of the relative phase $\theta_\mathrm{S}$ by using his measurement outcomes $(x_B,p_B)$ as detailed in Sec. \[Sec: Noise model\]. Furthermore, using the $i$ indexes for successive pulses, Bob can calculate a more precise estimator $\hat{\theta}_\mathrm{filter}^{(i)}$ by averaging each estimator $\hat{\theta}_\mathrm{S}^{(i)}$ with the previous filtered estimator $\hat{\theta}_\mathrm{filter}^{(i-1)}$ using optimized weighted coefficients. Finally, the estimator $\hat{\theta}_\mathrm{filter}^{(i)}$ allows Alice and Bob to correct their data using Eq. \[Eq: Alice phase correction\].\
**Security of the protocol.** In order to study the security of the design LLO-displacement and calculate the secret key rate, one can observe that security proofs for the GMCS protocols [@LeverrierThesis; @GarciaPatronThesis] do not rely on the mean value of Alice’s quadrature because it is fully described using the covariance matrices formalism. However, as we will show, the excess noise induced by the phase noise on a displaced modulation is asymmetric. The secret key rates then has to be calculated using a specific method which is detailed in Annex. \[Annex:secret key rate formulas\].
We now quantify the remaining phase noise $V_\mathrm{est}$ of Eq. \[Eq: Total remaining phase noise\]. As both quantum signal and phase reference are encoded and transmitted within the same optical pulse, we can directly write $V_\mathrm{drift}=0$ and $V_\mathrm{channel} = 0$. Finally, the only term contributing to the remaining phase noise $V_\mathrm{est}$ is the variance $V_\mathrm{error}$. Due to the the particular modulation scheme however, the variance of the estimates $\hat{\theta}_\mathrm{S}^{(i)}$ is not expressed as Eq. \[Eq: Vchannel\]. In this case, Alice’s modulation can be seen as a noise in the phase estimation process, resulting in: $$\begin{aligned}
V_\mathrm{error} & = & \frac{V_A + \chi + 1}{\Delta^2}\end{aligned}$$ Furthermore, the filtering technique $-$ hence based on all previous relative phase estimates $-$ allows to use correlations of the phase drift over time to recover the relative phase with better precision than $V_\mathrm{error}$. In the asymptotic regime (when $i$ is large), we can show that the successive variances $V_\mathrm{filter}^{(i)}$ tend to an asymptotic limit and, finally, one can write: $$\begin{aligned}
V_\mathrm{est} & = & \sqrt{V_\mathrm{error}} \cdot \sqrt{V_\mathrm{drift}}
\label{Eq: optimal filter size 1 approx}\end{aligned}$$ where $V_\mathrm{drift}$ is the relative phase drift between lasers L$_A$ and L$_B$ between two consecutive pulses, ie. it is expressed as Eq. \[Eq: Relative phase drift LLO-sequential\].
We can now express the excess noise $\xi_\mathrm{phase}$ due to phase noise in the LLO-displacement regime (using Annex. \[Annex: Phase excess noise\]). It can be simplified when $V_\mathrm{est} \ll 1$ and reads as: $$\begin{aligned}
\begin{array}{ccl}
\xi_\mathrm{phase}^{(x)} & = & (V_A + \Delta^2 \cdot \sin^2 \phi_\Delta ) \cdot V_\mathrm{est} \\
\xi_\mathrm{phase}^{(p)} & = & (V_A + \Delta^2 \cdot \cos^2 \phi_\Delta ) \cdot V_\mathrm{est}
\label{Eq: Phase Excess Noise Displacement}
\end{array}\end{aligned}$$
where the expression of $V_\mathrm{est}$ is defined using Eq. \[Eq: optimal filter size 1 approx\]. A crucial point in the noise analysis of the LLO-displacement design is that the displacement of Alice’s modulation creates an asymmetry on the excess noise on each quadrature. For instance, if $\phi_\Delta = 0$ (displacement according to the $\hat{x}$ quadrature), the displacement induces an increasing of the excess noise on the $\hat{p}$ quadrature.
Furthermore, in this design, the maximal amplitude $E_\mathrm{max}$ of the AM excess noise (Eq. \[Eq: AM excess noise\]) can be approximate as $\sqrt{V_A + \Delta^2}$, resulting in: $$\begin{aligned}
\xi_{AM} & = & (V_A + \Delta^2) \cdot 10^{-\mathrm{dyn}_\mathrm{dB}}\end{aligned}$$
However, one can note that the excess noise $\xi_\mathrm{phase}$ is a more restricting limitation to the displacement amplitude than the excess noise $\xi_{AM}$ for realistic parameters and, in practice, the AM dynamics do not restrict the amplitude displacement.\
**Performance analysis.** In [@MarieAlleaume2016], we only considered the $\xi_\mathrm{phase}^{(x)}$ contribution in the case of $\phi_\Delta =0$, resulting in a too optimistic key rate. Although the displacement decreases the variance $V_\mathrm{error}$ and, thus, the remaining phase noise $V_\mathrm{est}$, it also increases its impact on the excess noise according to Eq. \[Eq: Phase Excess Noise Displacement\]. Unfortunately, this result is a strong limitation to the achievable $\Delta$ and, thus, to the tolerable phase noise in the LLO-displacement design.
We here consider the case where $\phi_\Delta = 0$ (other cases can be treated in a similar way by observing that the sum $\xi_\mathrm{phase}^{(x)}+\xi_\mathrm{phase}^{(p)}$ does not depend on $\phi_\Delta$). From Eq. \[Eq: Phase Excess Noise Displacement\], one can note that the phase excess noise induced on the $\hat{p}$ quadrature is proportional to the displacement mean photon number $\Delta^2$. This sets a strong constraints on the achievable value of $\Delta$ and, thereby, on the achievable value of $V_\mathrm{error}$. There is a trade-off in terms of the $\Delta$ value between the remaining phase noise $V_\mathrm{est}$ $-$ the displacement $\Delta$ decreases $V_\mathrm{error}$ $-$ and the excess noise $\xi_\mathrm{phase}^{(p)}$. An optimal value for the displacement can be found and is calculated in our simulations. However, the optimal value of the $\Delta$ only allows secret key generation for low values of $V_\mathrm{drift}$. This means that, solely based on the single estimates $\hat{\theta}_\mathrm{S}$, the LLO-displacement design does not allow a low enough excess noise $\xi_\mathrm{phase}$ and, as such, requires strong correlations, ie. low values of $V_\mathrm{drift}$, between consecutive relative phases to recover phase information based on filtering techniques.
In Fig. \[Curves: LLO-displacement\], we plot the expected key rates for both the LLO-sequential and the LLO-displacement designs. The displacement value is optimized to maximize the overall secret key rate. In low phase noise regimes, we can see that the LLO-displacement is better in terms of secret key rate generation. This is due to the fact that the LLO-displacement relies on the whole repetition rate to generate secret key and the filtering technique combined to the displacement allows a more efficient relative phase recovery. For higher phase noise regimes however, the displacement value required to estimate the relative phase is limited by the excess noise $\xi_\mathrm{phase}^{(p)}$ and, finally, the relative phase estimation can not be performed with a good efficiency.\
We have shown that the coherence coming from the simultaneous encoding of both the quantum signal and phase reference in this design comes with new challenges. In particular, the displacement of the modulation increases the excess noise induced by the relative phase noise and creates an asymmetry on the excess noise on each quadratures. An interesting issue is then to optimize the LLO-displacement design in order to increase its performance, especially in terms of phase noise resilience. This study is however kept for future works. We however emphasize that LLO-displacement design relies on an extremely convenient experimental scheme as the phase reference encoding is performed simultaneously with the quantum signal modulation. To our knowledge, the simultaneous quantum signal and phase reference transmission introduced in the LLO-displacement design is a new primitive which has not been studied so far in quantum communication regimes and can be applied to different signal modulations as BPSK or higher order modulations. Since both quantum signal and phase reference information are sent over the same optical pulse, the key rate obtained with the LLO-displacement design is moreover not lowered by time multiplexing, as it is the case with LLO-sequential. Thereby, unlike all other proposals for locally generated local oscillator based CV-QKD designs, it allows to use the whole repetition rate for secret key generation.
Conclusion and perspectives {#Sec: Conclusion}
===========================
In order to lift security loophole issues, the local oscillator should not be directly sent through the optical channel in CV-QKD experiments and LLO-based CV-QKD protocols have been introduced. We have however shown that the only design proposed to date, the LLO-sequential design [@LLOexp1; @LLOexp2; @LLOexp3], requires to use ultra low noise lasers and high dynamics modulators. This strong requirements in terms of hardware performance are a limitation to the ability to deploy LLO-based CV-QKD over large-scale optical networks. In this work, we have addressed the issue of performing CV-QKD with a local local oscillator, using affordable hardware.
**Design** **Trusted LO** **Tolerable phase noise** **Hardware requirements**
---------------------------------------------------------------------------- ---------------- --------------------------------- -------------------------------
Transmitted LO
[@ExpCVQKD-Sequrenet; @ExpCVQKD-Qi; @ExpCVQKD-chinois; @ExpCVQKD-chinois2] No $\Delta \nu/f \sim 10$ Stable interferometric set-up
(Section \[Sec: TLO\])
LLO-sequential
[@LLOexp1; @LLOexp2; @LLOexp3] Yes $V_\mathrm{drift} \sim 10^{-2}$ High AM dynamics
(Section \[Sec: LLO-seq\])
LLO-delayline
(Section \[Sec: LLO-delayline\]) Yes $V_\mathrm{drift} \sim 10$ Stable interferometric set-up
LLO-displacement
(Section \[Sec: LLO-displacement\]) Yes $V_\mathrm{drift} \sim 10^{-4}$ $\emptyset$
The main challenge of LLO CV-QKD is that the phase drift between emitter laser and local oscillator laser, placed at Bob, induces a phase noise on the quantum communication, that has to be efficiently corrected. In Table \[table\], we summarize the performance and requirements of existing as well as newly proposed designs for CV-QKD. The LLO-sequential design is intrinsically limited to low phase noise regimes. This puts important constraints on the type of lasers that can be used both as emitter and LO. An other limitation of the LLO-sequential is the efficiency of the relative phase estimation process, which is limited in practice by Alice’s amplitude modulator dynamics.
Our results imply that next generation CV-QKD, implemented with a local LO, is possible even with low cost DFB lasers and standard amplitude modulators. Such features are made possible by the newly introduced self-coherent phase reference sharing design, the LLO-delayline design, and are essential to progress towards photonic integration and wide deployment of CV-QKD. The LLO-delayline design relies on the self-coherence between the quantum signal and phase reference and on the interferometric stability of two delay lines at short time scale. The finite dynamics of the amplitude modulator does no longer restrict the reference pulse amplitude and, by consequence, the relative phase estimation process. These characteristics allow the LLO-delayline design to be more resilient to phase noise than the previously proposed LLO-sequential design. In Fig. \[Curves: LLO-delayline\], we can see that the LLO-delayline design is able to reach a distance of $50$ km in a regime of high phase noise,$V_\mathrm{drift}=0.1$, while the reachable distance with the LLO-sequential design is below $25$ km even with large AM dynamics. Furthermore, we emphasize that remote delay-line interferometric stability has already been demonstrated in practice on several CV-QKD implementations [@ExpCVQKD-Sequrenet; @ExpCVQKD-chinois2] paving the way to the demonstration of LLO CV-QKD with cheap hardware, using the LLO-delayline design.
As another contribution, we have investigated a scheme, LLO-displacement, allowing to simultaneously transmit the quantum signal and the phase reference information on the same optical pulse. We have however observed that the implementation of the LLO-displacement design with Gaussian modulated coherente states (GMCS CV-QKD protocol) leads to an overall excess noise that increases with displacement, which restricts its use to low phase noise regimes. Simultaneous transmission of quantum and phase reference information had however not been studied so far and our results can be of interest in view of performing a joint optimization of classical and quantum coherent communication systems, operating with the same hardware. The optimization of such protocols is then an interesting open question and is kept for future works.
**Aknowledgements.** The authors thank Bing Qi for noticing an important limitation of the LLO-displacement design, in terms of additional excess noise, as well as for the fruitful exchanges and discussions on the LLO-displacement design.
Annex
=====
Excess noise due to phase noise {#Annex: Phase excess noise}
-------------------------------
Alice sends the coherent state $|\alpha \rangle = |x_A + x_0, p_A + p_0 \rangle$, where, in the general case, we suppose that $x_A \sim \mathcal{N}(0,V_x)$ and $p_A \sim \mathcal{N}(0,V_p)$ while Bob uses a heterodyne detection at reception. In order to estimate the phase noise induced excess noise, we consider in this analysis that the relative phase noise is the only noise source. Bob then gets the following measurement outcomes $({x}_m,{p}_m)$: $$\begin{aligned}
\left( \begin{array}{c} {x}_m \\ {p}_m\end{array} \right) & = & \sqrt{\frac{G}{2}} \cdot \left[ \begin{pmatrix} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{pmatrix} \cdot \left( \begin{array}{c} x_A + x_0 \\ p_A + p_0 \end{array} \right) \right]\end{aligned}$$
We suppose that Bob gets an estimator $\hat{\theta} \sim \mathcal{N}(\theta, V_{\varphi})$. He sends his estimator to Alice which corrects her data and, in the reverse reconciliation scheme, Alice then estimates Bob’s measurements as: $$\begin{aligned}
\begin{pmatrix} \tilde{x_A} \\ \tilde{p_A} \\ \end{pmatrix} & = & \sqrt{\frac{G}{2}} \cdot \begin{pmatrix} \cos \hat{\theta} & \sin \hat{\theta}\\
-\sin \hat{\theta} & \cos \hat{\theta}\\ \end{pmatrix} \cdot \begin{pmatrix} x_A + x_0 \\ p_A + p_0 \\ \end{pmatrix}
\label{quad estimator v1}\end{aligned}$$
We can then express the excess noise on each quadrature as: $$\begin{aligned}
\begin{array}{rcl}
\xi_x & = & \mathrm{var}(x_m-\tilde{x}_A) \\
\xi_p & = & \mathrm{var}(p_m-\tilde{p}_A)
\end{array}\end{aligned}$$
These two quantities depends on the remaining relative phase $\varphi = \theta - \hat{\theta}$. Assuming that the variable $\varphi$ is a Gaussian variable such that $\varphi \sim \mathcal{N}(0,V_{\varphi})$, the above expressions can be calculated from the characteristic function of the Gaussian function and, after calculations, we obtain the following expressions: $$\begin{aligned}
\begin{array}{ccc}
\xi_\mathrm{phase}^{(x)} & = & V_x \cdot (1+e^{-V_\varphi}-2e^{-V_\varphi /2}) + (V_x + x_0^2)\cdot(\frac{1}{2}+\frac{1}{2}e^{-2V_\varphi} - e^{-V_\varphi}) + (V_p + p_0^2)\cdot(\frac{1}{2} - \frac{1}{2} e^{-2V_\varphi})\\
\xi_\mathrm{phase}^{(p)} & = & V_p \cdot (1+e^{-V_\varphi}-2e^{-V_\varphi /2}) + (V_p + p_0^2)\cdot(\frac{1}{2}+\frac{1}{2}e^{-2V_\varphi} - e^{-V_\varphi}) + (V_x + x_0^2)\cdot(\frac{1}{2} - \frac{1}{2} e^{-2V_\varphi})
\end{array}\end{aligned}$$
Linearity range of the coherent detector {#Annex:Linearity_Range_Detector}
----------------------------------------
We consider that Bob relies on a single coherent detector which addresses the issue of the linearity range of the detector when considering both quantum signal and phase reference transmission. A typical homodyne detector is presented in Fig. \[fig:scheme HD\]. The response of the integrator circuit is proportional to the incoming number of electrons over a finite range. Beyond a certain threshold, the response of the integrator circuit is no longer linear and the security can be broken by specific attacks [@SaturationAttackHao]. We define this threshold as the maximal number of electrons $N_\mathrm{sat}$ per electrical pulses that can be detected in a linear regime. The number of electrons in each electrical pulse is $N_\mathrm{e} = 0.5 \cdot G \cdot E_\mathrm{S} \cdot E_\mathrm{LO}$ (see Fig. \[fig:scheme HD\]) where $E_\mathrm{S}$ and $E_\mathrm{LO}$ are the amplitudes of the signal and the LO so that the saturation hypothesis imposes: $$\begin{aligned}
\frac{G}{2} \cdot E_\mathrm{S} \cdot E_\mathrm{LO} \leq N_\mathrm{sat}
\label{saturation}\end{aligned}$$ A example of saturation threshold $N_\mathrm{sat} = 10^6$ has been experimentally evaluated in [@SaturationAttackHao]. For quantum signal of intensity of order of the shot noise, this threshold is not important and has not been considered so far in CV-QKD analysis. In LLO-based CV-QKD however, the relatively large amplitude of phase reference pulses imposes to consider the saturation threshold as a limit on the reference pulses amplitude.
Eq. \[saturation\] implies a trade-off, in the LLO-sequential design, between the signal amplitude $-$ in particular the reference amplitude $E_\mathrm{R}$ $-$ and the local oscillator amplitude $-$ used to decrease the electronic to shot noise ratio. If we want to maximize these two quantities, one has to saturate Eq. \[saturation\] by choosing: $$\begin{aligned}
E_\mathrm{LO} & = & \frac{2}{G} \cdot \frac{N_\mathrm{sat}}{E_\mathrm{R}}\end{aligned}$$ The electronic to shot noise ratio of Eq. \[Eq: electronic noise\] is then written as: $$\begin{aligned}
\xi_\mathrm{elec} & = & \frac{G\cdot 10^6}{2} \cdot \frac{E_\mathrm{R}^2}{N_\mathrm{sat}^2}\end{aligned}$$ where $v_\mathrm{elec}=0.01$ and $E_\mathrm{LO,cal}^2 = 10^8$. In Fig. \[fig:linearity\], we plot the expected key rate of the LLO-sequential design for different values of the threshold $N_\mathrm{sat}$. As we can see, only low values of $N_\mathrm{sat}$ (two order of magnitude lower than the experimental value of [@SaturationAttackHao]) are limitations to this design. As a typical value of $N_\mathrm{sat}=10^6$ photons is sufficiently large to allow a precise relative phase sharing, the saturation threshold will not be a limitation to the reference amplitude.
Secret key rate formulas for CV-QKD {#Annex:secret key rate formulas}
-----------------------------------
In this work, we focus on the Gaussian-modulated coherent state (GMCS) protocol. In this protocol, Alice encodes zero-mean gaussian classical variables $x_A$ and $p_A$ on both the $\hat{x}$ and $\hat{p}$ quadratures of coherent states [@GaussianQuantumInformation] before sending them to Bob through an insecure optical channel controlled by an eavesdropper Eve. In the Gaussian model, the channel between Alice and Bob is fully characterized by the intensity transmission $G$ and the excess noise $\xi_x$ and $\xi_p$ (*a priori* different on each quadratures. We also assume that Bob uses a heterodyne detection at reception.
**Symmetric channel.** We first detail the secret key rate formulas used for symmetric excess noise $\xi = \xi_x = \xi_p$. This formulas are used for the LLO-sequential and the LLO-delayline designs. We consider that Eve controls the whole excess noise and we also consider individual attacks. The secret key rates is written as [@FossierThesis]: $$\begin{aligned}
k & = & \beta \cdot I_{AB} - I_{BE}
\label{Eq: Secret key rate ANNEX}\end{aligned}$$
where: $$\begin{aligned}
I_{AB} & = & \frac{1}{2} \cdot \log_2 \left( \frac{V+\chi}{1+\chi} \right)\end{aligned}$$ $$\begin{aligned}
I_{BE} & = & \frac{1}{2} \cdot \log_2 \left( \frac{G\cdot(V+\chi)\cdot(V+\chi_E)}{(\chi_E+1)\cdot(V+1)} \right)\end{aligned}$$
with: $$\begin{aligned}
V & = & V_A +1 \\
\chi & = & \frac{2-G}{G} + \xi \\
\chi_E & = & \frac{G\cdot(2-\xi)^2}{(\sqrt{2-2G+G\xi}+\sqrt{\xi})^2}+1\end{aligned}$$
**Asymmetric channel.** We have shown in Sec. \[Sec: LLO-displacement\] that the excess noise induced by the phase noise in the case of the displaced modulation is asymmetric in the two quadratures. We then need to derive specific secret key rate expressions. From [@GarciaPatronThesis; @FossierThesis], we can write: $$\begin{aligned}
k & = & k_x + k_p\end{aligned}$$ where $k_x$ and $k_p$ represent the respective key rates on the logical channel corresponding to each quadrature. Each of these two key rates can then be obtained using the secret key formulas from Eq. \[Eq: Secret key rate ANNEX\], using the corresponding expression $\xi_x$ and $\xi_p$ from Eq. \[Eq: Phase Excess Noise Displacement\].
|
---
abstract: 'It has been shown that very light or even massless neutralinos are consistent with all current experiments, given non-universal gaugino masses. Furthermore, a very light neutralino is consistent with astrophysical bounds from supernov[æ]{} and cosmological bounds on dark matter. Here we study the cosmological constraints on this scenario from Big Bang nucleosynthesis taking gravitinos into account and find that a very light neutralino is even favoured by current observations.'
author:
- 'Herbi K. Dreiner'
- Marja Hanussek
- Jong Soo Kim
- Subir Sarkar
title: Gravitino cosmology with a very light neutralino
---
Introduction
============
Within the minimal supersymmetric Standard Model (MSSM), the photon and the $Z^0$ boson, as well as the two neutral CP-even Higgs bosons, have SUSY spin-1/2 partners which mix. The resulting mass eigenstates are denoted neutralinos, $\chi^0_i$, with $i=1, \ldots,4$, and are ordered by mass $m_{\chi^0_1}<\ldots<m_ {\chi^0_4}$ [@Martin:1997ns]. The Particle Data Group quotes a lower mass bound on the lightest neutralino [@Nakamura:2010zzi] $$m_{\chi^0_1} > 46 \textrm{ GeV}\,,
\label{pdg}$$ which is derived from the LEP chargino search under the assumption of gaugino mass universality: $$M_1=\frac{5}{3}\tan^2\theta_\text{W} M_2\,.$$ Here $M_{1,2}$ are the supersymmetry (SUSY) breaking bino mass and wino mass, respectively and $\theta_\text{W}$ is the electroweak mixing angle. If we relax this latter assumption, the bound (\[pdg\]) no longer applies. In fact for any value of $M_2,\,\mu,$ and $\tan\beta$ there is always a $M_1$ $$\begin{aligned}
M_1 &=& \frac{M_2 M_Z^2\sin(2\beta)\sin^2\theta_\text{W}}{\mu
M_2 - M_Z^2 \sin(2\beta)\cos^2\theta_\text{W}} \label{massless}\\
&\simeq& 2.5 \textrm{ GeV} \left( \frac{10}{\tan\beta} \right)
\left( \frac{150 \textrm{ GeV}}{\mu} \right)\,,\end{aligned}$$ such that the lightest neutralino is massless [@Gogoladze:2002xp; @Dreiner:2009ic]. Here $M_Z$ is the mass of the $Z^0$ boson, $\tan\beta$ is the ratio of the vacuum expectation values of the two $CP$–even neutral Higgs bosons in the MSSM and $\mu$ is the Higgs mixing parameter of the superpotential. A very light or massless neutralino is necessarily predominantly bino–like since the experimental lower bound on the chargino mass, sets lower limits on $M_2$ and $\mu$ [@Choudhury:1999tn; @Dreiner:2006sb]. Although Eq. (\[massless\]) holds at tree–level, there is always a massless solution even after including quantum corrections to the neutralino mass [@Dreiner:2009ic].
Such a light or even massless neutralino is consistent with all laboratory data. The processes considered include the invisible width of the $Z^0$, electroweak precision observables, direct pair production, associated production, and rare meson decays. Note that a bino-like neutralino does not couple directly to the $Z^0$. The other production processes, including the meson decays, thus necessarily involve virtual sleptons or squarks. If these have masses of ${\cal
O}(200)$ GeV or heavier, then all bounds are evaded — for details on the individual analyses see Refs. [@Choudhury:1999tn; @Dedes:2001zia; @Gogoladze:2002xp; @Barger:2005hb; @Dreiner:2003wh; @Dreiner:2006sb; @Dreiner:2009er; @Dreiner:2009ic]. The best possible laboratory mass measurement can be performed at a linear collider via selectron pair production with an accuracy of order 1 GeV, depending on the selectron mass [@Conley:2010jk].
Light neutralinos can lead to rapid cooling of supernov[æ]{}, so are constrained by the broad agreement between the expected neutrino pulse from core collapse and observations of SN 1987A [@Ellis:1988aa]. The neutralinos would be produced and interact via the exchange of virtual selectrons and squarks. For a massless neutralino which ‘free-streams’ out of the supernova, the selectron must be heavier than about 1.2 TeV and the squarks must be heavier than about 360 GeV. For light selectrons or squarks of mass $\sim 100-300$ GeV, the neutralinos instead diffuse out of the supernova just as the neutrinos do and thus play an important role in the supernova dynamics. Hence lacking a detailed simulation which includes the effects of neutralino diffusion, no definitive statement can presently be made [@Ellis:1988aa; @Grifols:1988fw; @Kachelriess:2000dz; @Dreiner:2003wh]. Recently the luminosity function of white dwarfs has been determined to high precision [@A8Silvestri:2006vz; @A8Isern:2008fs] and this may imply interesting new bounds on light neutralinos, just as on axions.
If a neutralino is stable on cosmological time scales it can contribute to the dark matter (DM) of the universe. If ‘cold’, then its mass is constrained from below by the usual Lee-Weinberg bound [@Lee:1977ua] which depends only on the self-annihilation cross-section. This limit has been widely discussed in the literature in the framework of the $\Lambda$CDM cosmology [@Hooper:2002nq; @Belanger:2002nr; @Belanger:2003wb; @Bottino:2003iu] and various values are quoted for a MSSM neutralino: $M_{\chi^0_1}>
12.6\,$ GeV [@Vasquez:2010ru; @arXiv:1108.1338] and $M_{
\chi^0_1}>9\,$ GeV [@Fornengo:2010mk; @arXiv:1108.2190]. The low mass range is particularly interesting because the DAMA [@Bernabei:2010mq] and CoGeNT [@Aalseth:2010vx] direct detection experiments have presented evidence for annual modulation signals suggestive of a DM particle with mass of ${\cal O}(10)$ GeV.
A light neutralino with a much smaller mass is also viable as ‘warm’ or ‘hot’ DM but this possibility has been less discussed. The observed DM density $\Omega_\text{DM} h^2 \approx 0.11$ can in principle be entirely accounted for with warm dark matter (WDM) in the form of neutralinos having a mass of a few keV [@Profumo:2008yg]. However the usual assumption of radiation domination and entropy conservation prior to big bang nucleosynthesis (BBN) then needs to be relaxed otherwise the relic neutralino density is nominally much larger than required. This scenario requires a (unspecified) late episode of entropy production or, equivalently, reheating after inflation to a rather low temperature of a few MeV. Although models of baryogenesis with such reheating temperatures exist [@Allahverdi:2010im; @Kohri:2009ka], the necessary baryon number violating interactions would result in rapid decay of the proton to (the lighter) neutralinos. This makes such models very difficult to realise in this context, although the situation may be somewhat eased since the maximum temperature during reheating can be higher than the final thermalisation temperature [@Davidson:2000dw].
In this paper we focus on a light neutralino which acts as hot dark matter (HDM)[^1], *i.e.* can suppress cosmic density fluctuations on small scales through free-streaming. In order for its relic abundance to be small enough to be consistent with the observed small-scale structure we require [@Dreiner:2009ic] following Ref.[@Cowsik:1972gh]: $$m_{\chi^0_1}\lesssim 0.7 \textrm{ eV}\,.$$ Such ultralight neutralinos affect BBN by contributing to the relativistic degrees of freedom and thus speeding up the expansion rate of the universe; consequently neutron-proton decoupling occurs earlier and the mass fraction of primordial ${}^4$He is increased [@Sarkar:1995dd]. The resulting constraint on new relativistic degrees of freedom is usually presented as a limit on the number of additional effective $SU(2)$ doublet neutrinos: $$\Delta N^{\textrm{eff}}_\nu{({\chi^0_1})}
\equiv N^{{\rm eff}}_\nu-3\,.$$ In §\[DeltaNuNeut\], we calculate this number in detail and compare it with observational bounds on $\Delta N^{\textrm{eff}}_\nu$ from BBN [@Cyburt:2004yc].
Until recently, the BBN prediction and the inferred primordial $^4$He abundance implied according to some authors [@Barger2003; @Simha:2008zj] $$\Delta N^{\rm eff}_\nu\lesssim0\,.$$ This is however in tension with recent measurements of the cosmic microwave background (CMB) anisotropy by WMAP, which suggest a larger value of [@Komatsu:2010fb; @Hamann:2010bk] $$\mathrm{WMAP:}\quad \Delta N^{\rm eff}_\nu = 1.34\,^{+0.86}_{-0.88}\,.
\label{CMBNeff}$$ Recent measurements of the primordial $^4$He abundance are also higher than reported earlier, implying [@Izotov:2010ca; @Aver:2010wq]: $$\mathrm{BBN:}\quad \Delta N^{\rm eff}_{\nu}=0.68\,^{+0.8}_{-0.7} \,.
\label{boundNeff}$$ Given these large uncertainties, a very light neutralino is easily accommodated, and even favoured, by the BBN and CMB data. In the near future, the Planck mission [@Tauber:2011ah] is foreseen to determine $N^{\rm eff}_\nu$ to a higher precision of about $\delta
N^{\rm eff}_\nu = \pm 0.26$ [@Hamann:2010bk], thus possibly constraining the light neutralino hypothesis.
Local SUSY models necessarily include a massive gravitino [@Freedman:1976xh]. Depending on its mass, the gravitino can also contribute to $\Delta N^{\textrm{eff}}_\nu$ as we discuss in §\[DeltaNuNeutGrav\]. This effect is only relevant for sub-eV mass gravitinos (for models see *e.g.* Ref. [@Brignole:1997pe]). More commonly the gravitino has electroweak-scale mass and its decays into the light neutralino will result in photo-dissociation of light elements, in particular ${}^4$He [@Sarkar:1995dd]. The resulting (over) production of ${}^2$H and ${}^3$He is strongly constrained observationally and we present the resulting bounds in §\[gravitinodecays\]. In §\[quasistable\] we examine under which conditions the gravitino itself can be a viable DM candidate in the presence of a very light neutralino. Conclusions are presented in §\[conclusion\].
Light neutralinos and nucleosynthesis {#DeltaNuNeut}
=====================================
In global SUSY models, or local SUSY models with a non–relativistic gravitino, the sub–eV neutralino is the only relativistic particle present at the onset of nucleosynthesis apart from the usual photons, electrons and 3 types of neutrinos.
The contribution of the neutralino to the number of effective neutrino species is [@Sarkar:1995dd]: $$\Delta N_{\nu}^{\rm eff}(\chi^0_1) = \frac{g_{{\chi^0_1}}}{2} \left(
\frac{T_{{\chi^0_1}}}{T_{\nu}}\right)^4,
\label{eq:delta_nu}$$ where $g_{{\chi^0_1}}$ is the number of internal degrees of freedom, equal to 2 due to the Majorana character of the neutralino. The ratio of temperatures is given by $$\frac{T_{{\chi^0_1}}}{T_{\nu}} = \left[ \frac{g^* (T_{\rm
fr}^{\nu})}{g^*(T_{\rm fr}^{{\chi^0_1}})} \right]^{1/3}\,,
\label{eq:TchiOverTnu}$$ where $T_{\rm{fr}}^i$ is the freeze–out temperature of particle $i$ and $$g^*(T)=\sum_{\rm{bosons}} g_i\cdot \left(\frac{T_i}{T}\right)^4 +
\frac{7}{8} \sum_{\rm{fermions}} g_i \cdot\left(\frac{T_i}{T}\right)^4.$$ with $g_i$ being the internal relativistic degrees of freedom at temperature $T$. Usually $T_i$ for a decoupled particle species $i$ is lower than the photon temperature $T$. because of subsequent entropy generation.
The freeze-out temperature of $SU(2)$ doublet neutrinos is $T_{\rm
fr}^{\nu} \sim 2$ MeV [@Dicus:1982bz]. The interaction rate $\Gamma_{\chi^0_1}$ of the lightest neutralino is suppressed relative to that of neutrinos [@Dreiner:2009ic] because the SUSY mass scale $m_{\rm SUSY} >M_\text{W}$, where $m_{\rm SUSY}$ denotes the relevant SUSY particle mass involved in the neutralino reactions. Hence the freeze-out temperature of the very light neutralino will generally be higher than $T_{\rm fr}^{\nu}$.
Estimating the thermally-averaged neutralino annihilation cross-section via an effective vertex, we obtain the approximate interaction rate $$\Gamma_{\chi^0_1}(T) = 2\frac{3}{4}\frac{\zeta(3)}{\pi^2} G_{\rm
SUSY}^2\,T_{\chi^0_1}^5,$$ where $G_{\rm SUSY}/\sqrt{2}=g^2/(8 m_{\rm SUSY}^2)$. Equating this to the Hubble expansion rate [@Sarkar:1995dd] $$H (T)=\sqrt{\frac{4\pi^3g^*(T)}{45}}\frac{T^2}{M_{\rm{Pl}}} ,
\label{hubble}$$ where $g^*$ counts the relativistic degrees of freedom, yields the approximate freeze-out temperature: $$T_{\rm fr}^{{\chi^0_1}} \approx 3 \; \left(\frac{m_{\rm SUSY}}{200\textrm{
GeV}}\right)^{4/3} T_{\rm fr}^{\nu} \,.
\label{eq:TNeutEstimate}$$ Thus, for sparticle masses below $\sim 3$ TeV, the neutralinos freeze–out below the temperature at which muons annihilate [@Dreiner:2009ic].
We now calculate the freeze–out temperature of a pure bino–like neutralino more carefully, considering all annihilation processes into leptons which are present at the time of neutralino freeze–out: $${\chi^0_1}{\chi^0_1}\rightarrow \ell\bar
\ell,\quad \ell=e,\nu_e,\nu_\mu,\nu_\tau.
\label{eq:annihilation}$$ Assuming that sleptons and sneutrinos have a common mass scale $m_{\mathrm{slepton}}$, the following relations hold $$\begin{aligned}
\sigma ({\chi^0_1}{\chi^0_1} \rightarrow \ell_R \bar \ell_L) & = &
16\sigma ({\chi^0_1}{\chi^0_1} \rightarrow \ell_L\bar\ell_R)
\nonumber\\ & = & 16\sigma({\chi^0_1}{\chi^0_1} \rightarrow
\nu\bar\nu),\end{aligned}$$ so the total annihilation cross section into leptons is given by $$\sigma({\chi^0_1}{\chi^0_1}\rightarrow \ell\bar\ell) = 20 \sigma
({\chi^0_1}{\chi^0_1}\rightarrow \ell_L\bar \ell_R)\,,$$ where we have taken the electron to be massless. The thermally-averaged cross-section is then given by $$\langle \sigma({\chi^0_1}{\chi^0_1}\rightarrow \ell\bar
\ell) v\rangle= \frac{20}{9
\zeta(3)^2}\frac{2^5}{3}I(1)^2\hat\sigma T^2,
\label{eq:thermally_averaged}$$ where $$I(n)=\int_0^{\infty}\frac{y^{n+2}}{\exp(y)+1}$$ and $$\hat\sigma=\frac{e^4}{8\pi\cos^4\theta_ W}\frac{1}{
m_{\mathrm{slepton}}^4}$$ for $m_{\rm slepton}\gg T$. In calculating the cross-section (\[eq:thermally\_averaged\]), we have neglected the Pauli blocking factors in the final state statistics [@Gherghetta:1996fm].
Relating the reaction rate (\[eq:thermally\_averaged\]) to the Hubble expansion rate (\[hubble\]), we can now obtain the freeze–out temperature for a bino–like neutralino, shown in Fig. \[plot:decoupling\] as a function of the common mass scale $m_{\mathrm{slepton}}$. Note that for $m_{\mathrm{slepton}}$ below a few TeV, the neutralino decouples below the muon mass as noted earlier. Thus neutrinos and neutralinos will have the same temperature, $$T_{{\chi^0_1}}=T_{\nu}\,,$$ hence during BBN, $$\Delta N_\nu^{\rm eff}({\chi^0_1}) = 1\;.
\label{eq:rough_result_neutralino1}$$
![Freeze-out temperature of the pure bino–like neutralino as a function of the common mass scale $m_{\mathrm{slepton}}$.[]{data-label="plot:decoupling"}](decoupling.eps){width="\columnwidth"}
However, for slepton masses above a few TeV, the neutralino freeze–out temperature is close to the muon mass, and muon annihilation will influence the neutralino and neutrino temperature differently. For $T^{{\chi^0_1}}_{\rm fr}\gtrsim m_\mu$, the neutrinos are heated by the muon annihilations, whereas this affects the neutralinos only marginally. Therefore $T_{{\chi^0_1}}/T_{\nu}$ is reduced due to the conservation of comoving entropy. The muons contribute to $g^*(T_{\chi^0_1})$, such that $$\frac{T_{{\chi^0_1}}}{T_{\nu}} = \left[ \frac{g_\gamma +
\frac{7}{8}(\, g_e + 3 g_\nu)}{g_\gamma + \frac{7}{8} (g_e + 3
g_\nu + g_\mu)} \right]^{1/3} = \left (\frac{43}{57} \right)^{1/3}
.$$ Thus employing Eq. (\[eq:delta\_nu\]) we obtain $$\Delta N_\nu^{\rm eff}({\chi^0_1}) = 0.69 \;,
\label{eq:rough_result_neutralino2}$$ which is interestingly close to the observationally inferred central value of 0.68 in Eq. (\[boundNeff\]). The LHC already restricts the masses of strongly coupled SUSY particles (squarks and gluinos) to be above several hundred GeV [@daCosta:2011qk; @Chatrchyan:2011ek; @Bechtle:2011dm] and the supernova cooling argument requires the selectron mass to also be above a TeV for a massless neutralino [@Dreiner:2003wh], so the picture is consistent.
Even for a neutralino freeze–out temperature somewhat below the muon mass, the effects from muon annihilation are notable. We now determine the equivalent number of neutrino species more carefully using the Boltzmann equation as in Refs. [@Dicus:1982bz; @Ellis:1985fp], in order to determine the effect for arbitrary slepton masses. Consider a fiducial relativistic fermion $x$ which is decoupled during $\mu
\bar\mu$ annihilation, so that its number density, $n_x$, satisfies $$\dot n_x+ \frac{3 \dot R}{R} n_x=0\,.$$ The Boltzmann equation controlling the number density of the lightest neutralino can then be written as $$\frac{d}{dt} \left(\frac{n_{{\chi^0_1}}}{n_x}\right)=n_x\langle\sigma v
\rangle \: \left[\left(\frac{n_\mu}{n_x}\right)^2-f(T_{\chi^0_1})
\left(\frac{n_{{\chi^0_1}}}{n_x}\right)^2\right],
\label{eq:boltzmann}$$ where $$f (T_{{\chi^0_1}}) =
\left[\frac{n_\mu(T_{{\chi^0_1}})}{n_{{\chi^0_1}}(T_{{\chi^0_1}})}\right
]^2_{\rm{equilibrium}}.$$ The cross-section $\mu\bar\mu\rightarrow{\chi^0_1}{\chi^0_1}$ is given by $$\begin{aligned}
16\pi s^2&&\frac{\cos\theta_{\rm
W}^4}{e^4}\;\sigma(\mu_R\bar\mu_L\rightarrow
{\chi^0_1}{\chi^0_1})=\\ 2(m_{\tilde \mu}^2 -&&m_\mu^2)\,
\ln\left(\frac{2(m_{\tilde \mu}^2
-m_\mu^2)+s-\sqrt{s}\sqrt{s-4m_\mu^2}}{2(m_{\tilde \mu}^2
-m_\mu^2)+s+\sqrt{s}\sqrt{s-4m_\mu^2}}\right)\nonumber\\ &&+\sqrt{s}\sqrt{s-4m_\mu^2}\;
\frac{2(m_{\tilde \mu}^2 -m_\mu^2)^2+m_{\tilde \mu}^2s}{(m_{\tilde \mu}^2
-m_\mu^2)^2+m_{\tilde \mu}^2s} .\end{aligned}$$ Since this involves a cancellation between the two terms, we Taylor expand to ensure numerical stability: $$\begin{aligned}
16\pi\frac{\cos\theta_{\rm W}^4}{e^4}\sigma(\mu_R\bar\mu_L\rightarrow
{\chi^0_1}{\chi^0_1})&\approx&
\frac{\sqrt{1-\frac{4m_\mu^2}{s}}(s-m_\mu^2)}{3(m_{\tilde
\mu}^2-m_\mu^2)^2}, \nonumber\end{aligned}$$ then take the thermal average $\langle\sigma v\rangle$ following Ref. [@Gondolo:1990dk].
In order to reformulate Eq. (\[eq:boltzmann\]) in terms of dimensionless quantities, we define $$\delta \equiv \frac{T_{{\chi^0_1}}-T_x}{T_x}, \quad \epsilon \equiv
\frac{T_{\gamma}-T_x}{T_x}\,, \quad y \equiv \frac{m_\mu}{T_\gamma}\,.$$ Here $\delta$ measures the temperature difference between the decoupled particle $x$ and the lightest neutralino and thus quantifies the heating of the lightest neutralino due to $\mu\bar\mu$ annihilation. We now evaluate ${n_\mu}/{n_x}$ numerically and expand ${n_ {{\chi^0_1}}}/{n_x}\approx1+3\delta$ so Eq. (\[eq:boltzmann\]) can be written as [@Dicus:1982bz; @Ellis:1985fp] $$\frac{d\delta}{dy}\approx ay^{-2}(\epsilon-\delta)\,,
\label{eq:delta-eq}$$ for $\delta\ll1$, *i.e.* for small temperature differences. The prefactor $a$ depends on the size of the annihilation cross-section, and thus on $y$ and the slepton mass: $$a(y, m_{\tilde l}) = \frac{ 5.67 \times 10^{17} }{\sqrt{g^*}} \frac{
\langle \sigma v\rangle }{\textrm{ GeV}^{-2}}\,.$$ We approximate the drop in $g^*$ when the muons become non–relativistic by a step-function with $g^*(y<1)=16$ and $g^*(y>1) = 12.34$.
Now $T_x$ and the photon temperature $T_\gamma$ are related through entropy conservation [@Sarkar:1995dd]: $$\frac{T_x}{T_\gamma}=\left(\frac{43}{57}\right)^{1/3}\left[\zeta(y)
\right]^{1/3},
\label{eq:TxTgamma}$$ where $$\begin{aligned}
\zeta(y)&=&1+ \frac{180}{43\pi^4} \nonumber\\ & & \times
\int_0^{\infty} x^2 \frac{\sqrt{x^2+y^2}+\frac{x^2}{3
\sqrt{x^2+y^2}}}{e^{\sqrt{x^2+y^2}}+1}\, \text{d}x. \label{eq:zeta}\end{aligned}$$ We use Eqs. (\[eq:TxTgamma\]) and (\[eq:zeta\]) to numerically evaluate $\epsilon(y)$ and then solve the differential equation (\[eq:delta-eq\]) for $\delta(y, m_{\tilde l})$. The solution asymptotically approaches a limit \[denoted by $\delta_{\rm
max}(m_{\tilde l})$\] for $y \gtrsim 10$ because for temperatures far below the muon mass there is no further heating of the neutralinos from muon annihilation. This improves our estimate (\[eq:rough\_result\_neutralino1\]) to: $$\Delta N_\nu^{\rm eff}({\chi^0_1})=\left(\frac{T_{{\chi^0_1}}}
{T_\nu}\right)^4
=0.69\,[1+\delta_{\rm max}(m_{\tilde l})]^4.$$ In Fig. \[plot:deltanu\], we show $\Delta N_\nu^{\rm
eff}({\chi^0_1})$ as a function of the common slepton mass $m_{\mathrm{slepton}}$. We see that for slepton masses above 3 TeV, our previous result of 0.69 in Eq.(\[eq:rough\_result\_neutralino2\]) is not modified. This is because if the interaction between the neutralinos and muons is too weak, then the neutralinos cannot stay in thermal contact with the muons. For slepton masses around 1 TeV, we get again 1 additional effective neutrino species. (Our numerical approximation is valid only for $\delta\ll1$, so holds down to $m_{\mathrm{slepton}}=0.5$ TeV when $\delta\simeq 0.1$.)
![Contribution of the pure bino–like neutralino to the effective number of neutrinos versus the slepton mass.[]{data-label="plot:deltanu"}](deltanu_mu.eps){width="\columnwidth"}
Summarizing, the neutralino contribution to the effective number of neutrinos lies between 0.69 and 1, depending on the slepton mass as seen in Fig. \[plot:deltanu\]. Thus, a very light neutralino is easily accommodated by BBN and CMB data and is in fact favoured by the recent observational indication (\[boundNeff\]) that $N_\nu \gtrsim
3$ .
A very light neutralino and a very light gravitino {#DeltaNuNeutGrav}
==================================================
A very light gravitino (as realized *e.g.* in some models of gauge-mediated SUSY breaking) can constitute HDM. For its relic density to be small enough to be consistent with the observed small–scale structure requires [@Feng:2010ij]: $$m_{\tilde G} \lesssim 15 - 30 \textrm{ eV}\,.
\label{gravitino-mass}$$ If the gravitino is heavier than the (very light) neutralino it will decay into it plus a photon with a lifetime $\gtrsim 10^{38}$ s \[see Eq. (\[gravdecay\]) below\] which is well above the age of the universe $\sim 4\times10^{17}$ s. Conversely if the gravitino is lighter than the neutralino, the latter will decay to a gravitino and a photon with lifetime [@Covi:2009bk] $$\tau_{\chi^0_1} \simeq 7.3 \times 10^{41} \textrm{ s}
\left(\frac{m_{\chi^0_1}}{1 \textrm{ eV}} \right)^{-5}
\left(\frac{m_{\tilde G}}{0.1 \textrm{ eV}} \right)^{2} \,,$$ assuming that there is no near–mass degeneracy between the neutralino and the gravitino. Again the lifetime is well above the age of the universe, therefore we can consider both the gravitino and the very light neutralino as effectively stable HDM.
The presence of a very light gravitino thus affects the primordial $^4$He abundance analogously to a very light neutralino. However, the contribution of the gravitino to the expansion rate depends on its mass, since it couples to other particles predominantly via its helicity–1/2 components with the coupling strength $\Delta
m^2/(m_{\tilde G} m_{\rm Pl})$, where $\Delta m^2$ is the squared mass splitting of the superpartners [@Fayet:1977vd]. For a very light gravitino, the interaction cross-section can be of order the weak interaction, leading to later decoupling. Hence it can have a sizeable effect on BBN.
The freeze-out temperature of a very light gravitino can be estimated from the conversion process with cross-section [@Fayet:1979yb] $$\sigma(\tilde G e^\pm\rightarrow e^\pm{\chi^0_1})=\frac{\alpha}{9}\frac{s}{m_{\rm
Pl}^2m_{{\tilde G}}^2}.
\label{eq:conversion}$$ We neglect self–annihilations, ${\tilde G}{\tilde G}\rightarrow\ell
\bar\ell,\gamma\gamma$ since the annihilation rate into photons is $\propto m_{\chi^0_1}^4$ [@Gherghetta:1996fm; @Bhattacharya:1988ey] hence suppressed for a light neutralino, while the annihilation rate into leptons is $\propto T^6$ [@Gherghetta:1996fm] so falls out of equilibrium much earlier than the conversions.
![Contour lines for the ratio of cross-sections for neutralino self–annihilation (\[eq:annihilation\]) and conversion (\[eq:conversion\]), in the gravitino–slepton mass plane. The shaded area indicates where $\Delta N_\nu^ {\rm total}=2$. []{data-label="plot:ratio"}](contour.eps){width="\columnwidth"}
After thermal averaging of the conversion rate (\[eq:conversion\]) as before, we find $$\begin{aligned}
T_{\rm{fr}}^{\rm conversion}&\simeq& 7.51\, m_{\tilde
G}^{2/3}m_{\rm{Pl}}^{1/3} g^{*^{1/6}} \nonumber \\ &\approx& 100
\,g^{*^{1/6}} \left(\frac{m_{\tilde G}}{10^{-3} \textrm{ eV}}\right)^{2/3}
\textrm{ MeV}.
\label{eq:gravfreezeout1}\end{aligned}$$ Since the goldstino coupling is enhanced for decreasing gravitino mass, the freeze-out temperature of the gravitino increases with its mass. For a gravitino mass of $5.6\times10^ {-4} $ eV ($7.8\times
10^{-4}$ eV) its freeze-out temperature equals the muon (pion) mass, so for heavier gravitinos the contribution to $\Delta N_\nu^{\rm eff}$ will decrease. We also consider the case $m_{\tilde G}=10$ eV which gives a freeze-out temperature of ${\cal O} (100)$ GeV, thus a negligible effect on $\Delta N_\nu^{\rm eff}$. (Note however that $T_{\rm{fr}}^{\rm {\tilde G}}$ will now depend on the SUSY mass spectrum because above temperatures of a GeV or so other SUSY processses can also be in thermal equilibrium [@Fayet:1982gg; @Chung:1997rq] and Eq. (\[eq:gravfreezeout1\]) may not apply.)
We can now evaluate the contribution of the gravitino, in conjunction with the very light neutralino, to the effective number of neutrino species. We need to keep in mind that the gravitino can affect neutralino decoupling since for very large slepton masses and/or very light gravitinos, the neutralino annihilation process ${\chi^0_1}{\chi^0_1}\rightarrow\ell\bar\ell$ becomes sub–dominant to the conversion process $\tilde G e^\pm\rightarrow e^\pm{\chi^0_1}$ and therefore neutralino freeze–out is also governed by Eq. (\[eq:gravfreezeout1\]).
In Fig. \[plot:ratio\], we show contour lines for the ratio of the cross-sections for neutralino annihiliation (\[eq:annihilation\]), and the conversion process (\[eq:conversion\]), in the slepton–gravitino mass plane. For a ratio less than 0.1, the freeze–out temperature of both particles is determined via the conversion process (\[eq:conversion\]) and $T_{{\tilde G}}=T_{\chi^
0_1}$. Hence $\Delta N_\nu^{\rm eff}({\tilde G},{\chi^0_1})=1/0.69/
0.57$, the latter two cases corresponding to gravitino masses above $5.6\times10^ {-4} $ eV and $7.8\times 10^{-4}$ eV, respectively \[corresponding to a freeze–out temperature below the muon and the pion mass, as determined from Eq. (\[eq:gravfreezeout1\])\]. The corresponding equivalent number of neutrino species is: $$\Delta N_\nu^{\rm total}\equiv \Delta N_\nu^{\rm eff}({\tilde G}) + \Delta
N_\nu^{\rm eff}({\chi^0_1})=2/1.38/1.14 \,.$$ Thus a very light gravitino is strongly constrained by the BBN bound (\[boundNeff\]), a mass below $5.6\times10^ {-4} $ eV being excluded at 3$\sigma$. As the gravitino mass increases, $\Delta
N_\nu^{\rm total}$ decreases because the gravitino and neutralino freeze-out earlier, hence are colder than the neutrinos at the onset of BBN.
One can see from Fig. \[plot:ratio\] that a further increase of the gravitino mass (or smaller slepton mass) accesses parameter regions where the neutralino annihilation process dominates over the conversion process. When the ratio of their rates exceeds $\sim 10$, the freeze–out of the neutralino and the gravitino is governed by the processes (\[eq:annihilation\]) and (\[eq:conversion\]) respectively. For a slepton mass above $\sim 3$ TeV, the lightest neutralino decouples above the muon mass hence yields $\Delta
N_\nu^{\rm eff}({\chi^0_1})=0.69$. Fig. \[plot:deltanu\] shows that with decreasing slepton mass, this increases to $\Delta N_\nu^{\rm
eff}({\chi^0_1})=1$ as before. Hence we obtain the same bounds on the gravitino mass for $\Delta N_\nu^{\rm eff}({\tilde G})=1/0.69/0.57$.
In summary for a slepton mass below $\sim1$ TeV $$\Delta N_\nu^{\rm total}=2/1.69/ 1.57\, ,$$ while for a slepton mass above $\sim 3$ TeV $$\Delta N_\nu^
{\rm total}=1.69/1.38/1.26\,;$$ for intermediate slepton masses, there is a continuous transition between the two cases.
If the gravitino mass increases further its effect on the expansion rate continues to decrease, *e.g.* for $m_{\tilde G} = 10$ eV (corresponding to $T_{\rm fr}^{\tilde G}\approx100$ GeV), we find $g^*
= 395/4$ or $\Delta N_\nu^{\rm eff}({\tilde G})\simeq 0.05$. Thus, gravitinos with mass $\gtrsim$ eV do not significantly affect the expansion rate.
Summarising, $\Delta N_\nu^{\rm total}$ is between 1.14 and 2 for scenarios with both a relativistic neutralino and a relativistic gravitino (when their freeze-out temperature lies between the freeze-out temperature of the neutrino and the pion mass). As before we can use the Boltzmann equation if necessary to obtain exact values for $\Delta N_\nu^{\rm eff}$ around the mass thresholds. From Eq. (\[boundNeff\]), $N_\nu^{\rm total}> 4.9$ is excluded at $3\sigma$ implying a lower bound on the gravitino mass of $5.6\times10^{-4}$ eV, *cf.* Fig. \[plot:ratio\]. This bound is two orders of magnitude weaker than the one stated in Ref. [@Gherghetta:1996fm] where a model with a very light gravitino but a heavy neutralino was considered. This is because the gravitino annihilation into di-photons or leptons is the relevant process when there is no light neutralino, also Ref. [@Gherghetta:1996fm] assumed a more stringent BBN limit: $N_\nu^{\rm total}<3.6$.
Decaying Gravitinos {#gravitinodecays}
===================
So far we have considered the increase in the expansion rate caused by sub–eV neutralinos and gravitinos which are quasi–stable (*cf.* §\[quasistable\]). We now consider a gravitino with a mass above ${\cal O}(100\,\mathrm{GeV})$ as would be the case in gravity mediated SUSY breaking where the gravitino sets the mass scale of SUSY partners.
As the gravitino mass increases, the relative coupling strength of the helicity–1/2 components, $\Delta m^2/(m_{\tilde G} m_{\rm Pl})$ decreases and the helicity–3/2 components come to dominate. These are however also suppressed by 1/m$_{\rm Pl}$ hence gravitinos decouple from thermal equilibrium very early. During reheating, gravitinos are produced thermally via two–body scattering processes (dominantly QCD interactions) and the gravitino abundance is proportional to the reheating temperature $T_\text{R}$ [@Ellis:1984eq]. The gravitino is unstable and will decay subsequently into the very light neutralino and a photon with lifetime [@Ellis:1984eq; @Ellis:1984er; @Ellis:1990nb; @Kawasaki2005], $$\tau_{\tilde G} \simeq 4.9\times 10^8
\;\left(\frac{m_{3/2}}{100\,\rm{GeV}}\right)^{-3}\rm{s}\,,
\label{gravdecay}$$ where we have assumed for simplicity that the gravitino is the next-to-lightest-SUSY particle (NLSP) while the neutralino is the lightest-SUSY particle (LSP). If the gravitino decays around or after BBN, the light element abundances are affected by the decay products whether photons or hadrons. In particular there is potential overproduction of D and $^3$He from photodissociation of (the much more abundant) $^4$He [@Ellis:1984er; @Ellis:1990nb], while for short lifetimes, decays into hadrons have more effect [@Kawasaki2005].
Therefore, the observationally inferred light element abundances constrain the number density of gravitinos. For a gravitino lifetime of $\mathcal{O}(10^8\,$sec) one obtains [@Kohri2006a; @Cyburt2009a] a severe bound on the abundance $Y_{3/2}\equiv n_{3/2}/s$: $$Y_{3/2} \lesssim 10^{-14} \;
\left(\frac{100\,\rm{GeV}}{m_{\tilde G}}\right).$$ This is proportional to the reheating temperature through [@Ellis:1984eq; @Ellis:1984er; @Ellis:1990nb; @Kawasaki2005] $$\left(\frac{T_\text{R}}{10^{10}\,\rm{GeV}} \right) \approx 3.0\times10^{11}
\; Y_{3/2}\,,$$ hence the latter is constrained to be $$T_\text{R} \lesssim 3.0\times10^{7} \textrm{ GeV} \times
\left(\frac{100\,\rm{GeV}}{m_{3/2}}\right)\;.$$ Note that a reheating temperature below $\mathcal{O}(10^8$ GeV) is not consistent with thermal leptogenesis, which typically requires $T_\text{R}\sim10^{10}$ GeV [@Fukugita:1986hr]. There are however other possible means to produce the baryon asymmetry of the universe at lower temperature [@Allahverdi:2010im; @Davidson:2000dw; @Kohri:2009ka].
The contribution to the present neutralino relic density from gravitino decays is $$\Omega^{\textrm{decay}}_{\chi_1^0}h^2\approx0.28 \; Y_{3/2}\left(
\frac{m_{\chi_1^0}}{1\,\rm{eV}}\right)\,.
\label{eq:upperbounds}$$ *i.e.* negligible, such that the Cowsik–McClelland bound on the neutralino mass is unaffected.
Quasi–stable Gravitinos {#quasistable}
=======================
As mentioned in §\[gravitinodecays\], when the gravitino mass is below $\sim 100$ MeV its lifetime is longer than the age of the universe so it is quasi–stable and can constitute warm dark matter. Decaying gravitino DM is constrained by limits on the diffuse $\gamma$–ray background. For a mass between $\sim100$ keV and $\sim
100$ MeV the gravitino decays to a photon and a neutralino, and the photon spectrum is simply $$\frac{dN_{\gamma}}{dE} = \delta(E - \frac{m_{\tilde G}}{2}) \,.$$ The $\gamma$–flux from gravitions decaying in our Milky Way halo dominates [@Buchmuller:2007ui; @Bertone:2007aw] over the redshifted flux from gravitino decays at cosmological distances. Using a Navarro-Frenk-White profile for the distribution of DM in our galaxy, we obtain $$\begin{aligned}
&E^2& \frac{{\rm d}J}{{\rm d}E}|_{\rm halo} \equiv \frac{2 E^2}{8 \pi
\tau_{{\tilde G}} m_{{\tilde G}}} \frac{{\rm d}N_\gamma}{{\rm d}E}
\int_{\rm l.o.s} \langle \rho_{\rm halo}(\vec\ell) d\vec\ell\,
\rangle / \Delta\Omega \nonumber\\ &\;&=31.1 \left( \frac{m_{\tilde
G}}{1 \textrm{ MeV}} \right)^4 \delta(E - \frac{m_{\tilde
G}}{2}) \frac{\textrm {MeV}}{\textrm{cm}^2 \textrm{ str s}}\,.\end{aligned}$$ We compare this to the measurements of the $\gamma$-ray background by COMPTEL, EGRET and Fermi [@Strong:2004ry; @Strong:2005zx; @Abdo:2010nz] and extract a conservative upper bound of $3 \times 10^{-2}$ cm$^{-2}$str$^{-1}$s$^{-1}$MeV on the $\gamma$-ray flux from the inner Galaxy in the relevant mass region below $\sim 100$ MeV. This implies that gravitinos with mass above $\sim 250$ keV would generate a flux exceeding the observed galactic $\gamma$-ray emission. On the other hand, constraints from small–scale structure formation set a lower mass bound on WDM of $\mathcal{O}$(keV) [@Viel:2007mv; @deVega:2009ku; @Boyarsky:2008ju].
Now we consider the relic density of those gravitinos. Due to the presence of the very light neutralino, all sparticles will decay into the latter before the onset of BBN. Therefore the gravitino will only be produced thermally with relic density [@Pradler:2007ne] $$\Omega_{3/2} h^2 \approx \left(\frac {1 \textrm{ keV}}{m_{\tilde G}}\right)
\left(\frac{T_\text{R}}{10 \textrm{ TeV}}\right) \left(\frac {M_{\rm SUSY}}{200
\textrm{ GeV}}\right)^2 .
\label{eq:omega}$$ This further restricts the gravitino mass and/or the reheating temperature in order not to exceed the observed value $\Omega_{\rm DM}
h^2\approx 0.11$. The least restrictive upper bound on the reheating temperature from Eq. (\[eq:omega\]) is $\mathcal{O}(10^5\,
\mathrm{GeV})$ for gravitino and gaugino masses of order 100 keV and 100 GeV, respectively. This could be alleviated if the gravitino density is diluted by the decay of particles (such as moduli fields [@Kawasaki2005] or the saxion from the axion multiplet [@Hasenkamp:2010if; @Pradler:2006hh]). In this context, there have been several detailed studies on gravitinos as light DM [@Kawasaki:1996wc; @Baltz:2001rq; @Jedamzik:2005ir; @Gorbunov:2008ui; @Heckman:2008jy].
Summary {#conclusion}
=======
We have studied the cosmology of the gravitino in the presence of a very light neutralino. Even a massless neutralino is compatible with all laboratory data, while the strictest astrophysical constraint is imposed by supernova cooling and requires selectrons to be heavy ($m_{\tilde e}\gtrsim1\,$TeV). Here we have considered the effect of a stable very light neutralino arising on the effective number of neutrino species during big bang nucleosynthesis. For slepton masses above $\sim\!3$ TeV, $\Delta
N_\nu^{\mathrm{eff}}(\chi^0_1)$ is 0.69 and this increases as the slepton mass decreases, reaching 1 for slepton masses below $\sim\!0.5$ TeV.
Next, we have considered constraints on the gravitino mass in the context of local SUSY with a very light neutralino. A very light gravitino will affect the expansion rate of the universe similarly to a light neutralino. We have identified the mass range where a gravitino has a sizeable effect on the effective number of neutrino species as $\sim 10^{-4}-10$ eV. Within this range, we obtain values for $\Delta N_\nu^{\mathrm{eff}}(\chi^0_1 \,\& \,{\tilde G})$ between 0.74 and 1.69, depending on the gravitino and slepton masses. Values around 0.7 are favored by recent BBN measurements. However, the uncertainties in the determination of ${}^4$He are still sufficiently large that we need to await data from Planck to pin down the allowed gravitino and slepton mass.
If the gravitino is heavier than $\sim 100$ MeV, it decays to the neutralino and a photon with a lifetime smaller than the age of the universe. This results in photo-dissociation of the light elements, which is strongly constrained observationally and translates into an upper bound on the reheating temperature of the universe of $\sim
10^7$ GeV for typical gravity mediated SUSY breaking models. Note that neither the neutralino nor the gravitino can constitute [[ the complete]{}]{} dark matter in the scenarios considered so far.
The mass range where the gravitino can constitute warm dark matter is constrained by bounds from the diffuse $\gamma$-ray background, from the formation of structure on small-scales, and from the observed DM abundance, leaving a small window of allowed gravitino mass between 1 and 100 keV for a reheating temperature below $10^5$ GeV.
We thank Manuel Drees and Nicolas Bernal for many useful discussions. MH and JSK would like to thank the Rudolf Peierls Center of Theoretical Physics at the University of Oxford for their hospitality. JSK also thanks the Bethe Center of Theoretical Physics and the Physikalisches Institut at the University of Bonn for their hospitality. This work has been supported in part by the ARC Centre of Excellence for Particle Physics at the Terascale and by the Initiative and Networking Fund of the Helmholtz Association, contract HA-101 (ÒPhysics at the TerascaleÓ), the Deutsche Telekom Stiftung and the Bonn–Cologne Graduate School.
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|
---
abstract: 'Wide-field ($\sim$8$''$ $\times$ 8$''$) and deep near-infrared ($JHKs$ bands) polarization images of the Orion Nebula are presented. These data revealed various circumstellar structures as infrared reflection nebulae (IRN) around young stellar objects (YSOs), both massive and low-mass. We found the IRN around both IRc2 and BN to be very extensive, suggesting that there might be two extended ($>$0.7 pc) bipolar/monopolar IRN in these sources. We discovered at least 13 smaller-scale ($\sim$0.01-0.1 pc) IRN around less-massive YSOs including the famous source $\theta$[$^{2}$]{} Ori C. We also suggest the presence of many unresolved ($<$690 AU) systems around low-mass YSOs and young brown dwarfs showing possible intrinsic polarizations. Wide-field infrared polarimetry is thus demonstrated to be a powerful technique in revealing IRN and hence potential disk/outflow systems among high-mass to substellar YSOs.'
author:
- 'M. Tamura, R. Kandori, N. Kusakabe, Y. Nakajima, J. Hashimoto'
- 'C. Nagashima, T. Nagata, T. Nagayama, H. Kimura, T. Yamamoto,'
- 'J. H. Hough, P. Lucas, A. Chrysostomou, J. Bailey'
- '[*Astrophysical Journal Letters, in press*]{}'
title: NEAR INFRARED POLARIZATION IMAGES OF THE ORION NEBULA
---
INTRODUCTION
============
The Orion Nebula and its associated population of stars are amongst the best studied objects in the sky. As a component of the ridge of molecular cloud (OMC-1) extending north-south, M42 is the HII region excited by the Trapezium, a group of massive (OB-type) young stars, near the surface of the molecular cloud. Of the Trapezium stars, $\theta$[$^1$]{} Ori C is the most dominant photo-ionizing source. One arcmin to the north-west on the sky are the two massive YSOs, IRc2 and BN, whose masses are suggested to be 25 and [$\gtrsim$]{}7 [$M$]{}$_{\sun}$, respectively (see Genzel & Stutzki 1989; Jiang et al. 2005). In fact IRc2 is a cluster of sources seen at infrared wavelengths, with additional nearby sources detected at different wavelengths (see e.g., Beuther et al. 2004) such as the radio source (source I) and the submillimetre source (SMA1). For convenience we refer to the sources responsible for the major outflows as the “IRc2 prtostars”. At least two outflows are known to originate from the region near the IRc2 protostars; one high-velocity outflow in the SE-NW direction, observed in radio molecular lines, and in optical and near-infrared shocked lines (Allen & Burton 1993; Chernin & Wright 1996), and one lower-velocity outflow in the NE-SW direction best seen in thermal SiO and H$_{2}$O maser emission, as well as some H$_{2}$ bow shocks (Genzel & Stutzki 1989; Chrysostomou et al. 1997). The driving sources of these outflows are uncertain. Situated to the south is the Bright Bar, which corresponds to the tangential region of the wall of the ionization front sculpting OMC-1. Distributed over the entire nebula is the Orion Nebular Cluster (ONC; aka. Trapezium cluster), which is composed of some 3500 low-mass YSOs and about 150 stars with protoplanetary disks revealed by $HST$ imaging (O’Dell 2001).
Many of these features are clearly seen with near-infrared intensity imaging (see e.g., Hillenbrand & Carpenter 2000; see also Figure 1). Near-infrared wavelengths are useful because of a combination of the low temperature of the YSOs and the extensive dust-extinction over the star forming region. However, near-infrared radiation from massive star forming regions is varied in its origin: not only direct radiation from YSOs but also gas free-free continuum emission, dust thermal emission, and dust scattering from various illuminating sources. The last scattered component from both diffuse nebulae over the whole region and the local environs around YSOs are best studied with polarized radiation.
However, polarimetric studies covering the central few arcmin of M42, where a variety of activities are seen have been limited so far. Imaging polarimetry has been conducted either only at optical wavelengths (Pallister et al. 1977) or toward a small region near IRc2 or BN at near-infrared wavelengths (Minchin et al. 1991; Jiang et al. 2005; Simpson et al. 2006). In this [Letter]{}, we present for the first time the near-infrared polarization images of a $\sim$1 pc region of M42 and describe newly detected features seen as IR nebulae on various scales.
OBSERVATIONS AND DATA REDUCTION
===============================
The 1.25 ($J$ band), 1.63 ($H$ band), and 2.14 ($Ks$ band) $\mu$m imaging polarimetry of M42 and a sky-field were obtained simultaneously with the SIRIUS camera (Nagayama et al. 2003) and its polarimeter on the 1.4-m IRSF telescope in South Africa, on the night of 2005 December 26. The instrument is among the first ones that provide deep and wide-field infrared polarimetric images, which can in principle measure polarizations of all the 2MASS-detected sources within a field-of-view of 7$\farcm$7 $\times$ 7$\farcm$7 in the $JHKs$ bands simultaneously with 1% polarization accuracy. See Kandori et al. (2006) for the details of the polarimeter.
The total integration time per wave plate position was 900 s and the resultant stellar seeing size was 1$\farcs$5. The pixel scale was 0$\farcs$45. The exposures were performed at four position angles (PAs) of the halfwave plate, in the sequence of PA = 0$\arcdeg$, 45$\arcdeg$, 22$\arcdeg$.5, and 67$\arcdeg$.5 to measure the Stokes parameters. After image calibrations in the standard manner using IRAF (dark subtraction, flat-fielding with twilight-flats, bad-pixel substitution, and sky subtraction), the Stokes parameters $(I,\, Q,\, U)$, the degree of polarization $p$, and the polarization angle $\theta$ were calculated (Kandori et al. 2006). Software aperture polarimetry was carried out for a number of point sources in the field of view. The aperture radius was 3 pixels. First the point-like sources were selected after the subtraction of the smooth nebulous components. Then the local background was subtracted using the mean of a circular annulus around the source on the original images. We rejected all the sources whose photometric errors were greater than 0.1 mag. The polarization percentages were debiased (Wardle & Kronberg 1974). In this [Letter]{}, we discuss only $H$ band data for the aperture polarimetry, as the extended nebula contamination is less than in the $J$ band and the scattering efficiency is higher than in the $Ks$ band. In the $H$ band image, 498 sources were measured in total. The full list of the aperture polarimetry at $JHKs$ will be given in a separate paper.
RESULTS AND DISCUSSION
======================
Polarized Intensity Images of M42 and Large Scale Infrared Nebulae
------------------------------------------------------------------
The outflows produced by both massive and low-mass YSOs are or were powerful enough to open a cavity within their parent cloud or core, in a direction that tends to be perpendicular to dense disk-like structures around the star. The radiation from the star preferentially escapes in the polar direction of the cavity, and is then scattered by the dust at the wall of the cavity, forming so-called polarized infrared reflection nebulae (IRN; see e.g., Hodapp 1984; Sato et al. 1985; Weintraub, Goodman, & Akeson 2000) Figure 1 shows the three-color composite, intensity and polarized intensity images in the $J$, $H$, and $Ks$ bands. The IRN revealed by our polarization images is a signature of circumstellar structures of various scales (from $\sim$1 pc down to less than the seeing size, $<$690 AU). The infrared polarization has been used to derive the morphology of the IRN associated with past or present outflow phenomena (Tamura et al. 1991; Sato et al. 1985; Hodapp 1984), either via spatially resolving the IRN or via detecting significant levels of integrated (i.e., unresolved) polarization.
Our polarization images reveal a new picture of the geometry in this region. The most dominant feature is the bipolar IRN centered on the IRc2 protostars, seen as red nebulae extending predominantly to the east and west in the color composite image (note that an arc to the north belongs to another IRN illuminated by the BN object as described below) and the $Ks$ band polarization vector map in Figure 2 (see also a sketch in Figure 3). This most likely traces the cavity carved out by the molecular outflow from the IRc2 protostars. First, the highly polarized (up to $p$ $\sim$ 40 % at $Ks$) IRN is more extended than previously thought; the extent of the outflow traced by our polarization images is $\sim$0.7 pc east-west, while the extent of the H$_{2}$ outflow has been suggested to be $\sim$0.3 pc to the NNW and only $\sim$0.1 pc to the west (Allen & Burton 1993; Schultz et al. 1999). Therefore, the molecular cloud near the IRc2 protostars might already have had a cavity formed with a size-comparable to the diameter of the ionized region around the Trapezium. Second, the morphology is different between the H$_{2}$ emission and the IRN; in H$_{2}$ emission the well known “fingers” (Allen & Burton 1993) most prominently extend to the NW, while in polarized emission the extension is to the west. The western nebula clearly shows a V-shaped cavity structure with a wide opening angle ($\sim$50$\arcdeg$), which is typical for IRN associated with molecular outflows (Tamura et al. 1991; Staude & Els[ä]{}sser 1993). This, together with the extension to the east, makes the total outflow quite symmetric around the IRc2 protostars in both direction and size. Note that the cavity direction is perpendicular to the dense ridge of OMC-1. Once UV radiation from the IRc2 protostars becomes powerful enough, the cavity evacuated by the outflow might become a circular-symmetric HII region, as is seen around the Trapezium.
In addition to the IRN illuminated by the IRc2 protostars, there is another large IRN illuminated by BN (Simpson et al. 2006) as also seen in the polarization vector pattern near BN (Figure 2). This IRN extends to the north (seen as a reddish arc with its apex coincident with BN), but no clear counterpart lobe is seen in our images. Although the IRN could simply be due to dust in the cloud that is illuminated by BN, we suggest that the IRN associated with BN might represent an independent outflow, whose direction is distinct from that of the IRc2 protostars. Such a large (0.3 pc) and curved morphology starting from BN whose direction is consistent with the polar region of the compact polarization disk (Jiang et al. 2005) could be carved out by a massive outflow associated with BN. The misalignment of the BN and IRc2 “outflows” is not surprising if BN is a “run-away” star from the Trapezium region (Tan 2004).
Medium Scale Infrared Nebulae around YSOs
-----------------------------------------
A number of smaller-scale ($\sim$0.01-0.1 pc) IRN are also seen in our polarization images; there are at least 13 “resolved” IRN (see Figure 3). They are around low- to intermediate-mass YSOs. Most prominent are those near OMC-1 S and $\theta$[$^2$]{} Ori C. The former is seen as a reddish and irregular pattern in the polarized images (see Hashimoto et al. 2006 for $JKs$ data and more discussion). The latter shows a bipolar-pattern in the polarized intensity and a centro-symmetric polarization vector pattern. Although $\theta$[$^2$]{} Ori C is a well-known object and has been observed at many times, no circumstellar structure has been suggested previously. Note, however, that care must be taken in determining the exact geometry of IRN from the polarized intensity images in Figure 1 alone, because a combination of circumstellar scattering of light from the central star and illumination by the Trapezium stars, could systematically change the apparent polarization pattern. These (as well as unresolved, highly polarized sources described below) might serve as the best targets for future high-resolution imaging with adaptive optics (e.g., Perrin et al. 2004).
Unresolved Infrared Nebulae around Low-Mass YSOs and Young Brown Dwarfs
-----------------------------------------------------------------------
There are numerous “unresolved” (point-like, $<$690 AU) sources detected in our near-infrared images, most of which are low-mass YSOs and comprise part of the ONC cluster. The circumstellar structure is not resolved in our 1$\farcs$5 resolution images. However, software aperture polarimetry of these sources can provide geometrical evidence for circumstellar structures (Sato et al. 1985; Tamura & Sato 1989). Interestingly, 86 (17%) sources show an aperture polarization larger than 6%, while 58 have a larger $p(H)/(H-K)$ value than that for BN. Such a large near-infrared polarization is likely to arise from scattering in the small-scale IRN system around each source, rather than originating from interstellar or intra-cloud extinction, except for extremely red sources. The details of these polarized sources are described elsewhere (N. Kusakabe et al. in preparation), but in this [Letter]{}, we just point out that some young brown dwarf candidates, whose masses have been estimated to be less than 0.08 [$M$]{}${_{\odot}}$ from [*both*]{} photometry and spectroscopy (Slesnick, Hillenbrand, & Carpenter 2004), appear to have intrinsic polarizations because their $p(H)/$[$\tau$]{}$(H)$ values are greater than that for the dichroic polarization due to extinction in the molecular clouds (Jones, Klebe, & Dickey 1992). In particular, the $H$ band polarizations of HC022 (0.05 $M$$_{\sun}$, $A_{V}$ = 2 mag) and HC064 (0.03 $M$$_{\sun}$, $A_{V}$ = 4 mag) are 1.49 $\pm$ 0.06 % and 5.82 $\pm$ 0.09 %, respectively, and are most likely due to scattering in unresolved IRN. It has been suggested that young brown dwarfs are associated with disks based on indirect evidence such as excess IR emission (e.g., Muench et al. 2001). Besides a few examples of disk imaging by $HST$ (Bally, O’Dell, & McCaughrean 2000) and a detection of an outflow by spectro-astrometry (Whelan et al. 2005), our results serve as direct “geometrical” evidence for IRN associated with disk-outflow systems around young brown dwarfs.
Polarized Orion Bar
-------------------
The polarized radiation at shorter wavelengths (our $J$ band data and previous optical data by Pallister et al. 1977) is dominated by scattered light from the Trapezium stars rather than that from the IRc2 protostars, as evidenced from the polarization vector patterns (Figure 2). Our polarization images show exactly where the scattering of this diffuse nebula (seen as white) occurs; it originates from the boundary between the ionization front and the background molecular cloud, OMC-1. This is evidenced by the systematic displacement of the position of the Orion Bar seen in the intensity image (corresponding to the ionization front) and that seen in the polarized intensity image (corresponding to the scattering region), and the very low polarization near the Trapezium. The distance between the ionization front and the dust scattering region is $\sim$0.02 pc. The typical polarization level in the Bar region is 5-10% in the $J$ band. The scattering dust is mostly concentrated at the boundary because the dust is swept-up by stellar radiation pressure from the Trapezium stars and slowed down quickly by gas drag in the molecular cloud (Ferland 2001).
We appreciate discussions with M. Kurita, S. Sato, and Z. Jiang on various aspects of this study. We also thank an anonymous referee for helpful comments. This work is supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (16077101, 16077204), by JSPS (16340061), and by PPARC.
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|
---
author:
- 'Hideo [Kodama]{}$^{1,}$[^1] and Akihiro [Ishibashi]{}$^{2,}$[^2]'
title: |
Master equations for perturbations of generalised\
static black holes with charge in higher dimensions
---
Introduction
============
In recent years, motivated by developments in higher-dimensional unification theories, the behaviour of gravity in higher dimensions has become one of the major subjects in fundamental physics. In particular, the proposals of TeV gravity theories in the context of large extra dimensions and warped compactification have led to the speculation that higher-dimensional black holes might be produced in colliders and in cosmic ray events. Although fully nonlinear analysis of the classical and quantum dynamics of black holes will eventually be required to test this possibility by experiments, linear perturbation analysis is expected to give valuable information concerning some aspects of the problem, such as the stability of black holes, an estimation of gravitational emission during black hole formation, and the determination of the greybody factor for quantum evaporation of black holes. The linear perturbation theory of black holes can be used also in the quasi-normal mode analysis of the AdS/CFT issues and to obtain some insight into whether the uniqueness theorems of asymptotically flat regular black holes in four dimension (see Ref. for a review) and in higher dimensions can be extended to the asymptotically de Sitter and anti-de Sitter cases.
With the motivation provided by the developments described above, in a previous paper, Ref. (Paper I), we developed a formulation that reduces the linear perturbation analysis of generalised static black holes in higher-dimensional spacetimes with or without a cosmological constant to the study of a single second-order ODE of the Schrödinger-type for each type of perturbation. Here, a generalised black hole is considered to be a black hole whose horizon geometry is described by an Einstein metric. This includes a maximally symmetric black hole, i.e. a black hole whose horizon has a spatial section with constant curvature, such as a spherically symmetric black hole, as a special case. Then, in Ref. (Paper II), we studied the stability of such black holes using this formulation and proved the perturbative stability of asymptotically flat static black holes in higher dimensions as well as asymptotically de Sitter and anti-de Sitter static black holes in four dimensions. We also showed that the other types of maximally symmetric and static black holes might be unstable only for scalar-type perturbations.
One of the main purposes of the present paper is to extend the formulation given in Paper I to a generalised black hole with charge and analyse its stability. This extension is non-trivial, because perturbations of the metric and the electromagnetic field couple in the Einstein-Maxwell system. Hence, the main task is to show that the perturbation equations for the Einstein-Maxwell system can be transformed into two decoupled equations by an appropriate choice of the perturbation variables, as in the four-dimensional case[@Moncrief.V1974; @Zerilli.F1974; @Chandrasekhar.S1983B]. Since higher-dimensional unified theories based on string/M theories contain various $U(1)$ gauge fields, this extension is expected to be useful in studying generic black holes in unified theories.
The other purpose of the present paper is to give explicit expressions for the source terms of the master equations. This information is necessary to apply the formulation to the estimation of graviton and photon emissions from mini-black holes formed by colliders.
This paper is organized as follows. In the next section, we first make clear the basic assumptions regarding the unperturbed background, and then we give a general argument concerning the tensorial decomposition of perturbations. We also give basic formulas for the perturbation of electromagnetic fields that are used in later sections. Then, in the subsequent three sections, we derive decoupled master equations with a source for the Einstein-Maxwell system in a generalised static background with a static electric field for tensor-type, vector-type and scalar-type perturbations. In §6, using the formulations given in the previous sections, we analyse the stability of generalised static black holes with charge. Section 7 is devoted to summary and discussion.
Background Spacetime and Perturbation
=====================================
In this section, we first explain the assumptions concerning the unperturbed background spacetimes considered in the present paper and present the basic formulas concerning them. Then, we give general arguments on the types of perturbations and the expansion of perturbations in harmonic tensors on the Einstein space by supplementing the argument of Gibbons and Hartnoll given in Ref. with some fine points associated with scalar and vector perturbations. Finally, we give the basic perturbation equations of electromagnetic fields and formulas used in the subsequent sections.
Unperturbed background {#subsec:BG}
----------------------
In the present paper, we assume that the background manifold has the structure ${\cal M}\approx {\cal N}^2 \times \K^n$ locally and its metric is given by Here, $d\sigma_n^2=\gamma_{ij}(z)dz^idz^j$ is the metric of the $n$-dimensional Einstein space $\K^n$ with where $\hat R_{ij}$ is the Ricci tensor of the metric $\gamma_{ij}$. When the metric represents a black hole spacetime, the space $\K^n$ describes the structure of a spatial section of its horizon. In the case in which $\K^n$ is a constant curvature space, $K$ denotes its sectional curvature, while in a generic case, $K$ is just a constant representing a local average of the sectional curvature. Because an Einstein space of dimension smaller than four always has a constant curvature and is maximally symmetric, this difference arises only for $n\ge4$, or, equivalently, when the spacetime dimension $d=n+2$ is greater than or equal to 6. In the present paper, we assume that $\K^n$ is complete with respect to the Einstein metric, and we normalise $K$ so that $K=0,\pm1$.
The non-vanishing connection coefficients of the metric are and the curvature tensors are where $\hat R^i{}_{jkl}$ is the curvature tensor of $\gamma_{ij}$. From this and , we obtain where $D_a$, $\Box$ and ${}^2\! R$ are the covariant derivative, the D’Alermbertian and the scalar curvature for the metric $g_{ab}$, respectively. Thus, the Ricci tensor takes the same form as in the case in which $\K^n$ is maximally symmetric, as first pointed out by Birmingham[@Birmingham.D1999].
As the background source for the gravitational field, we consider an electromagnetic field whose field strength tensor $\F_{\mu\nu}$ has the structure Then, from the Maxwell equation $d\F=0$, we obtain and from $\nabla_\nu \F^{\mu\nu}=0$, These equations imply that the electric field $E_0$ takes the Coulomb form, and $\hat\F=\frac{1}{2}\F_{ij}(z)dz^i\wedge dz^j$ is a harmonic form on $\K^n$. In general, there may exist such a harmonic form that produces an energy-momentum tensor consistent with the structure of the Ricci tensors in , if the second Betti number of $\K^n$ is not zero. The monopole-type magnetic field in four-dimensional spacetime provides such an example. However, since scalar and vector perturbations become coupled if such a background field exists, in the present paper we consider only the case $\F_{ij}=0$.
With this assumption, the energy-momentum tensor for the electromagnetic field, is written and the background Einstein equations, are reduced to where From , and the identity it follows that Hence, we obtain When $\nabla r\not=0$, these equations give the black hole type solution with To be precise, the spacetime described by this metric contains a regular black hole for $\lambda_c\le \lambda<0$ if $K=0$ or $K=-1$, and for $\lambda_{c1}\le \lambda < \lambda_{c2}$ and $Q^2/M^2<(n+1)^2/(4n)$ if $K=1$, where $\lambda_c,\lambda_{c1}$ and $\lambda_{c2}$ are functions of $Q,M$ and $n$. (For details, see Appendix \[Appendix:D\].)
Next, let us consider the case in which $r=a$ is constant. Here, we obtain the Nariai-type solution[@Nariai.H1950; @Nariai.H1961] where and the constant $a$ is determined as a solution to
Tensorial decomposition of perturbations and the Einstein equations {#subsec:TensorialDecomposition}
-------------------------------------------------------------------
In general, as tensors on $\K^n$, the metric perturbation variables $h_{\mu\nu}=\delta g_{\mu\nu}$ are classified into three groups of components, the scalar $h_{ab}$, the vector $h_{ai}$ and the tensor $h_{ij}$. Unfortunately, this grouping is not so useful, since components belonging to different groups are coupled through contraction with the metric tensor and the covariant derivatives in the Einstein equations. However, in the case that $\K^n$ is maximally symmetric, if we further decompose the vector and tensor as the Einstein equations are decomposed into three groups, each of which contains only variables belonging to one of the three sets of variables $\{h_{ab}, h_a, h_L, h^{(0)}_T\}$, $\{h^{(1)}_{ai},
h^{(1)}_{Ti}\}$ and $\{h^{(2)}_{Tij}\}$. Variables belonging to each set are called the scalar-type, the vector-type and the tensor-type variables, respectively. This phenomenon arises because the metric tensor $\gamma_{ij}$ is the only non-trivial tensor in the maximally symmetric space, and as a consequence, the tensorial operations on $h_{\mu\nu}$ to construct the Einstein tensors preserve this decomposition. Moreover, for the same reason, the covariant derivatives are always combined into the Laplacian in the Einstein equations after this decomposition. Thus, the harmonic expansion of the perturbation variables with respect to the Laplacian is useful.
In the case in which $\K^n$ is of the Einstein type, the Laplacian preserves the transverse condition, and leads to the relations Hence, the tensorial decomposition is still well-defined if these equations can be solved with respect to $h_a$, $h^{(0)}_T$ and $h^{(1)}_{Ti}$. We assume that this condition is satisfied in the present paper.
The relations also show that tensor operations on $h_{\mu\nu}$ that lower the rank as tensors on $\K^n$ preserve the tensorial decomposition into the scalar type and vector type, because the Weyl tensor $\hat C^i{}_{jkl}$ of $\K^n$ is of second order with respect to differentiation and does not take part in such operations. It is also clear that tensor operations on $h_{ab}$ and $h_{ai}$ that preserve or increase the rank have the same property. Furthermore, the covariant derivatives are always combined into the Laplacian in the tensor equations for the scalar-type and vector-type variables obtained through these operations. Therefore the difference between the perturbation equations in the Einstein case and the maximally symmetric case can arise only through the operations that produce the second-rank terms in $(\delta G_{ij})_T$ from $h_{ij}$. As shown by Gibbons and Hartnoll, these terms are given by The Lichnerowicz operator $\hat\triangle_L$ defined by this equation preserves the transverse and trace-free property of $h_{ij}$ as pointed out in Ref. . Furthermore, we can easily check that the following relations hold: Hence, the Lichnerowicz operator also preserves the tensorial types. One can also see that only the Laplacian appears as a differential operator in the perturbed Einstein equations for the scalar-type and vector-type components after the tensorial decomposition. Thus, we can utilize the expansion in terms of scalar and vector harmonics for these types of perturbations, and the structure of the Einstein space affects the perturbation equations only through the spectrum of the Laplacian $\hat D\cdot\hat D$. Similarly, for tensor-type perturbations, if we expand the perturbation variable $h^{(2)}_{Tij}$ in the eigenfunctions of the Lichnerowicz operator, we obtain the perturbation equation for the Einstein case from that for the maximally symmetric case by replacing the eigenvalue $k_T^2$ for $-\hat D\cdot\hat D$ in the latter case by $\lambda_L-2nK$, where $\lambda_L$ is an eigenvalue of $\hat \triangle_L$.
Perturbation of electromagnetic fields {#subsec:EMFperturbation}
--------------------------------------
The Maxwell equations consist of two sets of equations for the electromagnetic field strength $\F=(1/2)\F_{\mu\nu}dx^\mu\wedge
dx^\nu$, $d\F=0$ and $\nabla_\nu \F^{\mu\nu}=J^\mu$. If we regard $(\delta \F)_{\mu\nu}=\delta \F_{\mu\nu}$ as the basic perturbation variable, the perturbation of the first equation does not couple to the metric perturbation, and it is simply given by This is simply the condition that $\delta\F$ is expressed in terms of the perturbation of the vector potential, $\delta \A$, as $\delta\F=d\delta\A$. Next, perturbation of the second equation gives two set of equations, where $h_{\mu\nu}=\delta g_{\mu\nu}$, and the external current $J^\mu$ is treated as a first-order quantity.
The perturbation variable $\delta\F$ transforms under an infinitesimal coordinate transformation $x^\mu \tend x^\mu
+\bar\delta x^\mu$ as To be explicit, writing $\bar\delta x^\mu$ as we have
As in the case of the metric perturbation, the perturbation of the electromagnetic field, $\delta\F_{\mu\nu}$, can be decomposed into different tensorial types. The only difference is that no tensor perturbation exists for the electromagnetic field, because it can be described by a vector potential. After harmonic expansion, the Maxwell equations are decomposed into decoupled gauge-invariant equations for each type in the background consisting of and , because the Weyl tensor of $\K^n$ does not appear in the Maxwell equations. This gauge-invariant formulation is given in the subsequent sections for each type. Here, we only note that $\delta\F$ is gauge-invariant for a vector perturbation, since from the gauge transformation of $\delta\F$ does not depend on $L^i$. In contrast, for a scalar perturbation, these perturbation variables must be combined with perturbation variables for the metric to construct a basis for gauge-invariant variables for the electromagnetic fields.
Finally, we give expressions for the contribution of electromagnetic field perturbations to the energy-momentum tensor:
Tensor-type Perturbation
========================
Because an electromagnetic field perturbation does not have a tensor-type component, the electromagnetic fields enter the equations for a tensor perturbation only through their effect on the background geometry.
As explained in §\[subsec:TensorialDecomposition\], tensor perturbations of the metric and the energy-momentum tensor can be expanded in terms of the eigentensors $\THB_{ij}$ of the Lichnerowicz operator $\hat\triangle_L$ defined by as The expansion coefficients $H_T$ and $\tau_T$ themselves are gauge-invariant, and the Einstein equations for them are obtained from those in the maximally symmetric case considered in Ref. through the replacement $k_T^2
\tend \lambda_L-2nK$. The result is expressed by the single equation where $\lambda_L$ is the eigenvalue of the Lichnerowicz operator,
For the Nariai-type background , the above perturbation equation simplifies to For the black hole background , if we introduce the master variable $\Phi$ by can be put into the canonical form where In particular, for $f(r)$ given by , $V_T$ is expressed as Note that with describes the behaviour of a tensor perturbation if the background metric has the form . Hence, it may apply to a system more general than the Einstein-Maxwell system considered in the present paper.
Vector-type Perturbation
========================
We expand vector perturbations in vector harmonics $\VHB_i$ satisfying and the symmetric trace-free tensors $\VHB_{ij}$ defined by Note that this tensor is an eigentensor of the Lichnerowicz operator, but it is not an eigentensor of the Laplacian in general.
In this paper, we assume that the Laplacian $-\hat D\cdot\hat D$ is extended to a non-negative self-adjoint operator in the $L_2$-space of divergence-free vector fields on $\K^n$, in order to guarantee the completeness of the vector harmonics. Because $-\hat D\cdot\hat
D$ is symmetric and non-negative in the space consisting of smooth divergence-free vector fields with compact support, it always possesses a Friedrichs extension that has the desired property. With this assumption, $k_V^2$ is non-negative.
One subtlety that arises in this harmonic expansion concerns the zero modes of the Laplacian. If $\K^n$ is closed, from the integration of the identity $\hat D^i(V^j\hat D_i V_j)-\hat D^i V^j
\hat D_i V_j=V^j\hat D\cdot\hat D V_j$, it follows that $\hat D_i
\VHB_j=0$ for $k_V^2=0$. Hence, we cannot construct a harmonic tensor from such a vector harmonic. We obtain the same result even in the case in which $\K^n$ is open if we require that $\VHB^j\hat
D_i \VHB_j$ fall off sufficiently rapidly at infinity. In the present paper, we assume that this fall-off condition is satisfied. From the identity $\hat D^j\hat D_i V_j=\hat D_i\hat
D^jV_j+(n-1)K\hat V_i$, such a zero mode exists only in the case $K=0$. We can further show that vector fields satisfying $\hat D_i
V_j=0$ exist if and only if $\K^n$ is a product of a locally flat space and an Einstein manifold with vanishing Ricci tensor.
More generally, $\VHB_{ij}$ vanishes if $\VHB_i$ is a Killing vector. In this case, from the relation $k_V^2$ takes the special value $k_V^2=(n-1)K$. Because $k_V^2\ge0$, this occurs only for $K=0$ or $K=1$. In the case $K=0$, this mode corresponds to the zero mode discussed above. In the case $K=1$, since we are assuming that $\K^n$ is complete, $\K^n$ is compact and closed, as known from Myers’ theorem[@Myers.S1941], and we can show the converse, i.e. that if $k_V^2=(n-1)K$, then $\VHB_{ij}$ vanishes, by integrating the identity $0= {{\VHB}^i}^* \hat{D}^j
\VHB_{ij}=\hat{D}^j({{\VHB}^i}^*\VHB_{ij})+k_V
{{\VHB}^{ij}}^*\VHB_{ij}$ over $\K^n$. Furthermore, using the same identity for ${{\VHB}^i}^* \hat{D}^j
\VHB_{ij}$, we can show that there is no eigenvalue in the range $0<k_V^2<n-1$.
The Einstein equations with a source
------------------------------------
In terms of vector harmonics, perturbations of the metric and the energy-momentum tensor can be expanded as For $m_V\equiv k^2-(n-1)K\not=0$, the matter variables $\tau_a$ and $\tau_T$ are themselves gauge-invariant, and the combination can be adopted as a basis for gauge-invariant variables for the metric perturbation. The perturbation of the Einstein equations reduces to where $\epsilon_{ab}$ is the two-dimensional Levi-Civita tensor for $g_{ab}$, and This should be supplemented by the perturbation of the energy-momentum conservation law
Now, note that, for $m_V=0$, the perturbation variables $H_T$ and $\tau_T$ do not exist. The matter variable $\tau_a$ is still gauge-invariant, but concerning the metric variables, only the combination $F^{(1)}$ defined in terms of $f_a$ in is gauge invariant. In this case, the Einstein equations are reduced to the single equation , and the energy-momentum conservation law is given by without the $\tau_T$ term.
Now, we show that these gauge-invariant perturbation equations can be reduced to a single wave equation with a source in the two-dimensional spacetime $\N^2$. We first treat the case $m_V\not=0$. In this case, by eliminating $\tau_T$ in with the help of , we obtain From this it follows that $F^a$ can be written in terms of a variable $\tilde \Omega$ as Inserting this expression into , we obtain the master equation Next, we consider the special modes with $m_V=0$. For these modes, from with $\tau_T=0$, it follows that $\tau_a$ can be expressed in terms of a function $\tau^{(1)}$ as Inserting this expression into with $\epsilon^{cd}D_c(F_d/r)$ replaced by $F^{(1)}/r$, we obtain Taking account of the freedom of adding a constant in the definition of $\tau^{(1)}$, the general solution can be written Hence, there exists no dynamical freedom in these special modes. In particular, in the source-free case in which $\tau^{(1)}$ is a constant and $K=1$, this solution corresponds to adding a small rotation to the background solution.
Einstein-Maxwell system
-----------------------
As shown in §\[subsec:EMFperturbation\], all components of $\delta\F_{\mu\nu}$ are invariant under a coordinate gauge transformation for a vector perturbation. In order to find an independent gauge-invariant variable, we expand $\delta\F_{ai}$ in vector harmonics as $\delta\F_{ai}=\A_a\VHB_i$. Then, since $\delta\F_{ab}=0$ for a vector perturbation, the $(a,b,i)$-component of the Maxwell equation is written Hence, $\A_a$ can be expressed in terms of a function $\A$ as $\A_a=D_a\A$. Then, the $(a,i,j)$-component of is written This implies that $\delta\F_{ij}$ can be expressed as where $C_{ij}$ is an antisymmetric tensor on $\K^n$ that does not depend on the $y$-coordinates. Finally, from the $(i,j,k)$-component of , it follows that $C_{ij}dz^i\wedge dz^j$ is a closed form on $\K^n$. Hence, $C_{ij}$ can be expressed in terms of a divergence-free vector field $W_i$ as $C_{ij}=\partial_i W_j-\partial_j W_i$. With the expansion in vector harmonics, we can assume that $W_i$ is a constant multiple of $\VHB_i$, without loss of generality. Hence, this term can be absorbed into $\A$ by redefining it through the addition of a constant. Thus, we find a vector perturbation of the electromagnetic field can be expressed in terms of the single gauge-invariant variable $\A$ as Next, we express the remaining Maxwell equations in terms of this gauge-invariant variable. For a vector perturbation, only Eq. is non-trivial. If we expand the current $J_i$ as it can be written Hence, from the identity the gauge-invariant form for the Maxwell equation is given by In order to complete the formulation of the basic perturbation equations, we must separate the contribution of the electromagnetic field to the source term in the Einstein equation . Because for a vector perturbation, is expressed as the contributions of the electromagnetic field to $\tau_a$ and $\tau_T$ are given by Hence, the Einstein equations for the Einstein-Maxwell system can be obtained by replacing $\tau_a$ in by where the second term represents the contribution from matter other than the electromagnetic field.
In order to rewrite the basic equations obtained to this point in simpler forms, we treat the generic modes and the exceptional modes separately.
### Generic modes
For modes with $m_V\equiv k_V^2-(n-1)K\not=0$, the above simple replacement yields a wave equation for $\tilde\Omega$ with $\Box\A$ in the source term. This second derivative term can be eliminated if we extract the contribution of the electromagnetic field to $\tilde
\Omega$ as The result is Equations and are replaced by Finally, inserting this expression for $F^a$ into and using , we obtain Thus, the coupled system of equations consisting of and provides the basic gauge-invariant equations with source for a vector perturbation with $k_V^2\not=(n-1)K$.
### Exceptional modes
For exceptional modes with $k_V^2=(n-1)K$, from the definition of $\tau^{(1)}$ and , $\tau^{(1)}$ can be expressed as Hence, can be rewritten as Inserting this into , we obtain Therefore, only the electromagnetic perturbation is dynamical.
Decoupled master equations
--------------------------
The basic equations for the generic modes are coupled differential equations and are not useful. Fortunately, they can be transformed into a set of two decoupled equations by simply introducing master variables written as linear combinations of the original gauge-invariant variables. Such combinations can be found through simple algebraic manipulations.
### Black hole background
For the black hole background , appropriate combinations are given by with where $\Delta$ is a positive constant satisfying When expressed in terms of these master variables, and are transformed into the two decoupled wave equations Here, where and For $n=2, K=1$ and $\lambda=0$, the variables $\Phi_+$ and $\Phi_-$ are proportional to the variables for the axial modes, $Z^{(-)}_1$ and $Z^{(-)}_2$ given in Ref. , and $V_{V+}$ and $V_{V-}$ coincide with the corresponding potentials, $V^{(-)}_1$ and $V^{(-)}_2$, respectively.
Here, note that in the limit $Q\tend0$, $\Phi_+$ becomes proportional to $\A$ and $\Phi_-$ to $\Omega$. Hence, $\Phi_+$ and $\Phi_-$ represent the electromagnetic mode and the gravitational mode, respectively. In particular, in the limit $Q\tend0$, the equation for $\Phi_-$ coincides with the master equation for a vector perturbation on a neutral black hole background derived in Paper I.
### Nariai-type background
For the Nariai-type background , the combinations with give the decoupled equations where
Scalar-type Perturbation {#sec:ScalarPerturbation}
========================
We expand scalar perturbations in scalar harmonics satisfying As in the case of vector harmonics, we assume that $-\hat\triangle$ is extended to a non-negative self-adjoint operator in the $L^2$-space of functions on $\K^n$. Hence, $k^2\ge0$. In the present case, such an extension is unique, since we are assuming that $\K^n$ is complete, and it is given by the Friedrichs self-adjoint extension of the symmetric and non-negative operator $-\hat\triangle$ on $C_0^\infty(\K^n)$. If $\K^n$ is closed, the spectrum of $\hat\triangle$ is completely discrete, each eigenvalue has a finite multiplicity, and the lowest eigenvalue is $k^2$=0, whose eigenfunction is a constant. A perturbation corresponding to such a constant mode represents a variation of the parameters $\lambda, M$, and $Q$ of the background, as seen from the argument in §\[subsec:BG\]. For this reason, we do not consider the modes with $k^2=0$, although there may be a non-trivial eigenfunction with $k^2=0$ in the cases in which $\K^n$ is open. Note that when $k^2=0$ is contained in the full spectrum but does not belong to the point spectrum, as in the case $\K^n=\RF^n$, it can be ignored without loss of generality.
For modes with $k^2>0$, we can use the vector fields and the symmetric trace-free tensor fields defined by to expand vector and symmetric trace-free tensor fields, respectively. Note that $\SHB_i$ is also an eigenmode of the operator $\hat D\cdot\hat D$, i.e., while $\SHB_{ij}$ is an eigenmode of the Lichnerowicz operator: In the case of scalar harmonics, the modes with $k^2=nK$ are exceptional. Given our assumption, these modes exist only for $K=1$. Because $\K^n$ is compact and closed in this case, from the identity we have $\hat D_j\SHB^j_i=0$. From this, it follows that $\int d^nz
\sqrt{\gamma}{\SHB_{ij}}^*\SHB^{ij}=0$. Hence, $\SHB_{ij}$ vanishes identically.
Metric perturbations
--------------------
In terms of scalar harmonics, perturbations of the metric and the energy-momentum tensor are expanded as Following Ref. , we adopt the following combinations of these expansion coefficients as the basic gauge-invariant variables for perturbations of the metric and the energy-momentum tensor: Here, and we have used the background value for $T^a_b$ given in .
Now, note that for the exceptional modes with $k^2=n$ for $K=1$, $H_T$ and $\tau_T$ are not defined, because a second-rank symmetric tensor cannot be constructed from $\SHB$, as mentioned above. For such a mode, we define $F, F_{ab}, \Sigma_{ab},\Sigma_a$ and $\Sigma_L$ by setting $H_T=0$ in the above definitions. These quantities defined in this way are, however, gauge dependent. These exceptional modes are treated in Appendix \[Appendix:C\].
Maxwell equations
-----------------
Because $X_a$ defined in transforms under as from it follows that the following combinations $\E$ and $\E_a$ can be used as basic gauge-invariants for perturbations of the electromagnetic field: Here, note that $\delta\F_{ij}=0$ for a scalar perturbation, since $\delta\F_{ij}$ can be written as $\delta\F_{ij}=\hat
D_i\delta\A_j-\hat D_j\delta\A_i$, from the Maxwell equations, and $\delta\A_i$ is just the gradient of a scalar perturbation.
By expanding the current $J_i$ as the Maxwell equations can be written while is expressed as Note that and give the current conservation law
We can reduce these equations to a single wave equation for a single master variable. First, from and , we have Therefore, if we define $\tilde J_a$ by $\E_a$ can be expressed in terms of a function $\A$ as Then, the insertion of this into yields Therefore (adding some constant to $\A$ in its definition, if necessary), we obtain Thus, the gauge-invariant variables $\E$ and $\E_a$ can be expressed in terms of the single master variable $\A$. Finally, by inserting these expressions into , we obtain the following wave equation for $\A$: Next, we derive an expression in terms of $\A$ for the contribution of the electromagnetic field to the perturbation of the energy-momentum tensor. First, for a scalar perturbation, can be written in terms of $\E$, $\E_a$ and the metric perturbation variables as Inserting these into – and using and , we obtain the following expressions for the corresponding gauge-invariant variables $\Sigma_{ab}$, $\Sigma_a$ and $\Sigma_L$ in terms of $\A$, $F$ and $F_{ab}$:
Einstein-Maxwell system: Black hole background
----------------------------------------------
For generic modes of scalar perturbations, the Einstein equations consist of four sets of equations of the forms (For the definitions of $E_{ab}, E_a, E_L$ and $E_T$, see Eqs. (63)–(66) in Ref..) If we introduce the variables $\tilde F^a_b$ and $\tilde F$ as the equation for $E_T$ is algebraic and can be written where Hence, if we introduce $X,Y$ and $Z$ defined by as in Paper I, the original variables are expressed as In contrast, for the exceptional modes, is not obtained from the Einstein equations. However, this equation with $S_T=0$ can be imposed as a gauge condition, as shown in Paper I. Under this gauge condition, all equations derived in this section hold without change. However, the variables still contain some residual gauge freedom, and we must eliminate it in order to extract physical degrees of freedom. This is done in Appendix \[Appendix:C\].
To obtain the expressions of the remaining Einstein equations in terms of $X,Y$ and $Z$, we introduce $\hat E^a_b$, $\hat E_a$ and $\hat E_L$ defined by and separate the contributions of the electromagnetic field to the perturbation of the energy-momentum tensor as Then, the remaining Einstein equations are written As shown in Paper I, if the equations corresponding to $\hat E_a$, $\hat E^r_t$ and $\hat E^r_r$ are satisfied, the other equations are automatically satisfied, as seen from the Bianchi identities, provided that the energy momentum tensor satisfies the conservation law, which in the present case is given by the two equations The explicit expressions of the relevant equations in terms of $X,Y$ and $Z$ are as follows. First, the $\hat E_a$ parts and the $\hat
E^r_t$ part are After the Fourier transformation with respect to the time coordinate $t$, setting $\partial_t( X,Y,Z)=-i\omega (X,Y,Z)$ and solving with respect to $(X',Y',Z')=\partial_r(X,Y,Z)$, we obtain Next, $\hat E^r_r$ is expressed as Applying the Fourier transformation with respect to time to the corresponding equation and eliminating $X',Y'$ and $Z'$, with the help of , we obtain the following linear constraint on $X,Y$ and $Z$: Using almost the same method as in Paper I, we can reduce this constrained system for $X,Y$ and $Z$ to a single second-order ODE with source. The master variable in the present case is given by where This master variable coincides with that for the neutral and source-free case in Paper I for $Q=0$ and $S_T=0$. The master equation for $\Phi$ is Here, the effective potential $V_S$ is expressed as and the source term $S_\Phi$ has the following structure: where $P_{S1}, P_{S2}$ and $P_{S3}$ are functions of $r$ expressed as polynomials of $x,y$ and $z$. Their explicit expressions are given in Appendix \[Appendix:B\].
The basic variables $X,Y$ and $Z$ are expressed in terms of the master variable $\Phi$ as where $X_s,Y_s$ and $Z_s$ are contributions from the source terms given by Here, $P_{X0},P_{X1},P_{Y0},P_{Y1},P_{Z},P_{XA},P_{X2},P_{YA}$ and $P_{Y2}$ are functions of $r$ and are given in Appendix \[Appendix:B\].
In particular, $\tilde F$ is expressed as Hence, using the relation we can rewrite the Maxwell equation as where Thus, the two coupled second-order ODEs and represent the master equations with a source for a scalar perturbation of the Einstein-Maxwell system in the background .
Finally, we note that in terms of the variable can be rewritten as where and These equations are formal, since they contain the factor $i/\omega$, which is equivalent to integration with respect to the time coordinate $t$. However, when $S_a=\kappa^2 E_0\tilde J_a$ and $S^r_t=0$, this factor can be eliminated through the replacements $\tilde S_t \tend i\omega \kappa^2E_0\A$ and $\epsilon^{ab}D_a(r^2\tilde S_b) \tend -i\omega (n-2)\kappa^2r^2E_0
\A$. In this case, provides expressions that are manifestly covariant as equations in the 2-dimensional spacetime $\N^2$. The source term $S_\Phi$ for $\Phi$ can also be rewritten in such a covariant form with the help of the energy-momentum conservation laws : The comment made above regarding covariance applies to this expression as well as to .
Reduction to decoupled equations
--------------------------------
As in the vector perturbation case, we can transform the master equations and into two decoupled second-order ODEs by introducing new master variables written as linear combinations of $\Phi$ and $\A$. The only difference in this case is that the coefficients are not constant. These new variables are given by with where $\mu$ is a positive constant satisfying The reduced master equations have the canonical forms Here, if we define the parameter $\delta$ by the effective potentials $V_{S\pm}$ are expressed as with and The source terms $S_{S\pm}$ are linear combinations of $\bar
S_\Phi=S_\Phi|_{\A=0}$ given in and $S_\A$ given in : Here, note that the following relations hold: From these relations, we find that $Q=0$ corresponds to $\delta=0$, and in this limit, $\Phi_-$ coincides with $\Phi$, and its equation coincides with the mater equation for the master variable $\Phi$ derived in Paper I. Hence, $\Phi_-$ and $\Phi_+$ represent the gravitational mode and the electromagnetic mode, respectively. For $n=2, K=1$ and $\lambda=0$, these variables $\Phi_+$ and $\Phi_-$ are proportional to the variables for the polar modes, $Z^{(+)}_1$ and $Z^{(+)}_2$, appearing in Ref. , and $V_{S+}$ and $V_{S-}$ coincide with the corresponding potentials, $V^{(+)}_1$ and $V^{(+)}_2$, respectively.
Einstein-Maxwell system: Nariai background
------------------------------------------
We can derive the master equations for the Einstein-Maxwell system in the Nariai-type background in almost the same way as in the black hole case. Actually, the calculations are much simpler in this case. For this reason, we give only key equations. As in the previous case, the elimination of the residual gauge freedom for the exceptional modes is discussed in Appendix \[Appendix:C\].
First, by defining the variables $X,Y$ and $Z$ by , after the Fourier transformation, the equations corresponding to $\hat E_\rho,\hat E_t$ and $\hat E^\rho_t$ can be written as The constraint equation obtained from the $E^\rho_\rho$ equation is These equations can be reduced to a single second-order ODE for $F$, where Hence, in the present case, we can use $F$, which is related to $X$ and $Y$ by as the master variable. $X,Y$ and $Z$ are expressed in terms of $F$ as Finally, the Maxwell equation also holds in the present case, in which it becomes where By taking linear combinations of this equation and and introducing the variables we obtain the decoupled equations with where $\bar S_\Phi=S_\Phi|_{\A=0}$, $\mu$ is a positive constant satisfying and
Stability of Black Holes
========================
![The parameter ranges in which the spacetime contains a regular black hole (the shaded regions). []{data-label="fig:BHcondition"}](BHcond_K1.eps){width="6cm"}
![The parameter ranges in which the spacetime contains a regular black hole (the shaded regions). []{data-label="fig:BHcondition"}](BHcond_K-1.eps){width="6cm"}
In this section, we consider what we are able to conclude at this time about the stability of generalised static black holes with charge from the results of our formulation. In the present paper, we consider only the stability in the static region outside the black hole horizon with respect to a perturbation whose support is compact on the initial surface. This region is represented as $r>r_H$ for $\lambda\le0$ and $r_H<r<r_c$ for $\lambda>0$. As mentioned in §\[subsec:BG\], such a region exists only for restricted ranges of the parameters $M,Q$ and $\lambda$. These parameter ranges are shown in Fig. \[fig:BHcondition\] (see Appendix \[Appendix:D\] for details).
To study the stability in this region, as in Paper II, we utilize the fact that for any perturbation type, the perturbation equations in the static region of the spacetime are reduced to an eigenvalue problem of the type where $A$ is a self-adjoint operator, with $V(r)$ being equal to $V_T(r)$, $V_{V\pm}(r)$ or $V_{S\pm}(r)$. We regard the black hole to be stable if the spectrum of $A$, i.e., $\omega^2$, is non-negative.
To be precise, we must specify a boundary condition for $\Phi$ at $r=\infty$ in the case $\lambda<0$, for which the range of $r_*$ has an upper bound. In the present paper, we adopt the simplest condition, $\Phi\tend 0$ as $r \tend \infty$, in this case, which corresponds to the Friedrichs extension of $A$. For $\lambda\ge0$, the range of $r_*$ is $(-\infty,+\infty)$ and the operator $A$ is essentially self-adjoint; i.e., it has a unique self-adjoint extension.
In general, if $\Phi(r)$ is a function with compact support contained in $r>r_H$ (or $r_H<r<r_c$ for $\lambda>0$), we can rewrite the expectation value of $A$, $(\Phi,A\Phi)$, as For the Friedrichs extension of $A$, the lower bound of this quantity coincides with the lower bound of the spectrum of $A$ with domain $C^\infty_0(r_*)$. Hence, if we can show that the right-hand side of is non-negative, then we can conclude that the system is stable. In particular, if $V$ is non-negative, this condition is trivially satisfied. However, such a lucky situation is not realized in most cases. One powerful method that can be used to show the positivity of $A$ beyond such a simple situation is to deform the right-hand side of by partial integration in terms of a function $S$ as where We call this procedure [*the $S$-deformation of $V$*]{} in this paper. As in Paper II, this is the main tool for the analysis in the present paper.
Tensor perturbation
-------------------
If we apply the $S$-deformation with to , we obtain irrespective of the $r$-dependence of $f(r)$. Hence, the effective potential $\tilde V_T$ is positive, and the system is perturbatively stable with respect to a tensor perturbation if In particular, this guarantees the stability of maximally symmetric black holes for $K=1$ and $K=0$, since $\lambda_L$ is related to the eigenvalue $k_T^2$ of the positive operator $-D\cdot D$ as $\lambda_L=k_T^2+2nK$ when $\K^n$ is maximally symmetric. In contrast, even in the maximally symmetric case, $\tilde V_T$ becomes negative in the range $0<k_T^2<2$ for $K=-1$, and we cannot conclude anything about the stability for $K=-1$ from this argument alone.
Note that the condition is just a sufficient condition for stability, and it is not a necessary condition in general. In fact, for a tensor perturbation, we can obtain stronger stability conditions directly from the positivity of $V_T$ if we restrict the range of parameters. For example, for $K=1$ and $\lambda=0$, it is easy to see that $V_T$ is positive if for $M^2\ge Q^2>8n(n-1)M^2/(3n-2)^2$, and for $Q^2\le 8n(n-1)M^2/(3n-2)^2$. Thus, if we do not restrict the range of $Q^2$, we obtain the same sufficient condition for stability as , but for the restricted range $Q^2\le 8n(n-1)M^2/(3n-2)^2$, we obtain the stronger sufficient condition , which coincides with the condition obtained in Paper II for the case $Q=0$.
![Ranges of the value $x_H$.[]{data-label="fig:xh"}](xh_K1.eps){width="6cm"}
![Ranges of the value $x_H$.[]{data-label="fig:xh"}](xh_NL.eps){width="5cm"}
Similarly, for $K=-1$ and $\lambda<0$, rewriting $V_T$ as we obtain a sufficient condition for stability stronger than , from $V_T>0$ if we restrict the range of $\lambda$ to This condition is sufficient to guarantee the stability of a maximally symmetric black hole with $K=-1$ for $n\ge2$. However, if we extend the range of $\lambda$ to the whole allowed range, i.e., that satisfying $\lambda\ge \lambda_{c-}$, is the strongest condition that can be obtained only from $V_T>0$.
Vector perturbation
-------------------
For $V=V_{V\pm}$ in and we obtain Hence, from $k_V^2\ge(n-1)K$, we find that $\tilde V_{V+}$ is always positive, and static charged black holes are stable with respect to the electromagnetic mode of the vector perturbation. In contrast, from it is seen that $\tilde V_{V-}$ may become negative: Because $h$ is a monotonically increasing function of $r$, $\tilde V_{V-}$ is positive if and only if $h(r_H)\ge0$.
### The case $\lambda\ge0$
In this case, the background spacetime contains a regular black hole only for $K=1$, and the static region outside the black hole is given by $r_H<r<r_c$ $(\le+\infty)$. In this region, from , we have $2M/r^{n-1}<x_H\le x_{H,{\rm max}}\le (n+1)M^2/(nQ^2)$. Hence, $h>0$, and the black hole is stable in this case.
### The case $\lambda<0$
In this case, as shown in Appendix \[Appendix:D\], the spacetime contains a regular black hole if $\lambda\ge\lambda_c$ and $2M/r^{n-1}<x_H\le x_{H,{\rm
max}}$. Hence, from , we obtain the relation For $K=0,1$, $h>0$ follows from this. Hence, the black hole is stable. In contrast, for $K=-1$, the right-hand side of this inequality becomes negative for $k_V^2<n-1$. Hence, $\tilde V_{V-}$ can become negative near the horizon if $\lambda$ is sufficiently close to $\lambda_{c-}$, provided that the spectrum of $k_V^2$ extends to $k_V^2<n-1$.
![Examples of $V_{S+}$ for $K=1$ and $\lambda<0$.[]{data-label="fig:VSplus"}](PotSpK1LN.eps){width="5cm"}
Scalar perturbation
-------------------
By applying the $S$-deformation to $V_{S+}$ with we obtain Since this is positive definite, the electromagnetic mode $\Phi_+$ is always stable for any values of $K$, $M$, $Q$ and $\lambda$, provided that the spacetime contains a regular black hole, although $V_{S+}$ has a negative region near the horizon when $\lambda<0$ and $Q^2/M^2$ is small (see Fig. \[fig:VSplus\]).
Using a similar transformation, we can also prove the stability of the gravitational mode $\Phi_-$ for some special cases. For example, the $S$-deformation of $V_{S-}$ with leads to For $n=2$, this is positive definite for $m>0$. When $K=1$, $\lambda\ge0$ and $n=3$ or when $\lambda\ge0,Q=0$ and the horizon is $S^4$, from $m\ge n+2$ ($l\ge2$) and the behaviour of $x_h$ (see Fig. \[fig:xh\]), we can show that $\tilde V_{S-}>0$. Hence, in these special cases, the black hole is stable with respect to any type of perturbation.
However, for the other cases, $\tilde V_{S-}$ is not positive definite for generic values of the parameters. The $S$-deformation used to prove the stability of neutral black holes in Paper II is not effective either. This is because $V_{S-}$ has a negative region around the horizon for the extremal and near extremal cases, as shown in Fig. \[fig:VSminus\], and the $S$-deformation cannot remove this negative region if $S$ is a regular function at the horizon. Hence, determination of the stability for these generic cases with $n\ge3$ is left as an open problem.
![Examples of $V_{S-}$.[]{data-label="fig:VSminus"}](PotSmK1L0.eps){width="6cm"}
![Examples of $V_{S-}$.[]{data-label="fig:VSminus"}](PotSmK1LN.eps){width="6cm"}
Summary and Discussion
======================
In the present paper, we have extended the formulation for perturbations of a generalised static black hole given in Paper I to an Einstein-Maxwell system in a generalised static spacetime with a static electric field, and we have shown that the perturbation equations for vector and scalar perturbations can be reduced to two decoupled second-order ODEs for the gravitational mode and the electromagnetic mode, irrespective of the value of the cosmological constant and the curvature of the horizon. In particular, we have found that the coupling between the perturbations of the metric and the electromagnetic field produces significant modifications of the effective potentials for the gravitational mode and the electromagnetic mode when the black hole is charged. Our formulation also provides an extension of the corresponding formulation for the Reissner-Nordstrom black hole in four dimensions[@Moncrief.V1974; @Zerilli.F1974; @Chandrasekhar.S1983B] to asymptotically de Sitter and anti-de Sitter cases.
-------- ------------- ----------- ------- ----------- ------- ----------------------------------------------- -----------------------------------------------
$Q=0$ $Q\not=0$ $Q=0$ $Q\not=0$ $Q=0$ $Q\not=0$
$K=1$ $\lambda=0$ OK OK OK OK OK $\begin{array}{l}
d=4,5\ \text{OK} \\ d\ge6\ \text{?}
\end{array}$
$\lambda>0$ OK OK OK OK $\begin{array}{l} $\begin{array}{l}
d\le6\ \text{OK} \\ d\ge7\ \text{?} d=4,5\ \text{OK} \\ d\ge6\ \text{?}
\end{array}$ \end{array}$
$\lambda<0$ OK OK OK OK $\begin{array}{l} $\begin{array}{l}
d=4\ \text{OK} \\ d\ge5\ \text{?} d=4\ \text{OK} \\ d\ge5\ \text{?}
\end{array}$ \end{array}$
$K=0$ $\lambda<0$ OK OK OK OK $\begin{array}{l} $\begin{array}{l}
d=4\ \text{OK} \\ d\ge5\ \text{?} d=4\ \text{OK} \\ d\ge5\ \text{?}
\end{array}$ \end{array}$
$K=-1$ $\lambda<0$ OK ? OK ? $\begin{array}{l} $\begin{array}{l}
d=4\ \text{OK} \\ d\ge5\ \text{?} d=4\ \text{OK} \\ d\ge5\ \text{?}
\end{array}$ \end{array}$
-------- ------------- ----------- ------- ----------- ------- ----------------------------------------------- -----------------------------------------------
: Stabilities of generalised static black holes. In this table, “$d$” represents the spacetime dimension, $n+2$. The results for tensor perturbations apply only for maximally symmetric black holes, while those for vector and scalar perturbations are valid for black holes with generic Einstein horizons, except in the case with $K=1,Q=0,\lambda>0$ and $d=6$.[]{data-label="tbl:stability"}
With the help of this formulation and the method used in Paper II, we have analysed the stability of generalised static black holes with charge. The results are summarised in Table \[tbl:stability\]. As shown there, maximally symmetric black holes are stable with respect to tensor and vector perturbations over almost the entire parameter range; the exceptional case corresponds to a rather exotic black hole, whose horizon is a hyperbolic space. In contrast, for a scalar perturbation, we were not able to prove even the stability of asymptotically flat black holes with charge in generic dimensions, due to the existence of a negative region in the effective potential around the horizon in the extremal and near extremal cases, in contrast to the neutral case. Whether this negative ditch produces an unstable mode or not is uncertain. Hence, the stability of asymptotically flat and asymptotically de Sitter black holes for $d\ge 6$ and of asymptotically anti-de Sitter black holes for $d\ge5$ are left as open problems. In connection to this, it should be noted that the existence of a negative region in the effective potential near the horizon may have a significant influence on the frequencies of the quasi-normal modes and the greybody factor for the Hawking process, even if these black holes are found to be stable.
In the present paper, we have also given explicit expressions for the source terms in the master equations. As mentioned in the introduction, this information will be necessary in the estimation of gravitational and electromagnetic emission from black holes in higher dimensions. In addition to this practical application, we also expect that these master equations with source terms can be used in the analysis of static singular perturbations of black holes associated with some singular source such as a string or a membrane. For example, if one treats the C-metric as a perturbation of a spherically symmetric solution, it is found that this perturbation obeys an equation with a string source. Hence, it is expected that one can obtain some information concerning higher-dimensional analogue of the C-metric by studying singular solutions to the master equation with a singular source.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Gary Gibbons, Sean Hartnoll and Toby Wiseman for conversations. AI is a JSPS fellow, and HK is supported by the JSPS grant No. 15540267.
Parameter range for the existence of a regular black hole {#Appendix:D}
=========================================================
In this section, we determine the parameter range in which the background metric contains a regular black hole. Here, by a regular black hole, we mean a degenerate Killing horizon or bifurcating Killing horizons at $r=r_H$ that separate(s) a regular region with $f(r)>0$ and a singular region containing the singularity at $r=0$.
First, we consider parameter values satisfying $\lambda\ge0$ and $K\le0$. In this range, if $Q=0$, there exists no horizon, because $f$ is negative everywhere. If $Q\neq0$, from $r^{n-1}f$ is a monotonically decreasing function of $r$. Further, Hence, the spacetime has no region with $f>0$ that is separated from the singular region by Killing horizons. Therefore, the spacetime contains regular black holes only for $K=1$ or $\lambda<0$.
The case $Q=0$
--------------
In this case, since $f\tend -\infty$ as $r\tend+0$, the spacetime contains a regular black hole if there is a region in which $f>0$. From it is seen that the result depends on $\lambda$.
### $\lambda<0$ or $K=1$ and $\lambda=0$
Because $f$ is monotonically increasing and becomes positive as $r\tend+\infty$, $f$ has a single zero at $r=r_H$, and $f>0$ for $r>r_H$. Thus, there is a regular black hole.
### $K=1$ and $\lambda>0$
Because $f$ has a single maximum at $r=a$, the spacetime contains a regular black hole if and only if $f(a)>0$. Then, since $f'(a)=0$ is equivalent to this condition can be written as In terms of $\lambda$, it is expressed as $f$ has two zeros, at $r=r_H$ and $r=r_c$ ($r_H<r_c$), and $f>0$ for $r_H<r<r_c$,
The case $Q\not=0$
------------------
When $Q\not=0$, since $f\tend+\infty$ as $r\tend+0$, there must exist a point $r=a$ such that $f'(a)=0$ and $f(a)\le0$ in order for the spacetime to contain a regular black hole. From $\lambda$ is expressed in terms of $a$ as From this, $f(a)$ can be written as Hence, the condition $f(a)\le0$ requires and under this condition, $f(a)\le0$ if $a$ satisfies Note that $g(a)$ has a maximum at $a=a_m$, where and it is monotonic everywhere except at this point. Further, $g(+0)=-\infty$ and $g(+\infty)=0$. Hence, $\lambda<0$ for $a<a_0$, where and $\lambda>0$ for $a>a_0$.
### $K=1$ and $\lambda=0$
In this case, the spacetime contains a regular black hole if and only if $M^2\ge Q^2$, and the horizon is at $r^{n-1}=r_H^{n-1}=M+\sqrt{M^2-Q^2}$.
### $K=1$ and $\lambda>0$
Because the sign of $f'(r)$ is the same as the sign of $g(r)-\lambda$, the condition that there is a point $r=a$ such that $f'(a)=0$ is equivalent to the relation Further, from the condition $D\ge0$, we have Under these conditions, $f'(a)=0$ has in general two solutions, $a=a_1,a_2$ $(a_0<a_1<a_m<a_2)$, and the spacetime contains a regular black hole if and only if $f(a_1)\le 0<f(a_2)$.
First, from $f(a_2)>0$, we obtain In terms of $\lambda$, this can be expressed as Next, because $f(a)$ becomes minimal at $a=a_m$, from $f(a_0)=1-M^2/Q^2$, we have $f(a_1)\le f(a_0)\le0$ if $Q^2\le M^2$. Also, for $M^2<Q^2\le (n+1)^2/(4n)M^2$, $f(a_1)\le0$ is equivalent to or, in terms of $\lambda$, to Here, if we vary $Q^2$ with $M$ fixed, we have Hence, $\lambda_{c2}-\lambda_{c1}$ is a monotonically increasing function of $Q$ for fixed $M$, and $\lambda_{c1}=\lambda_{c2}$ at $Q^2=(n+1)^2M^2/(4n)$. From these results, it follows that $\lambda_{c1}<\lambda_{c2}$ for $Q^2<(n+1)^2/(4n)M^2$. Further, $\lambda_{c1}$ vanishes at $Q^2=M^2$. Therefore, for $\lambda>0$ and $K=1$, the spacetime contains a regular black hole if and only if
$K=1$ $K=0$ $K=-1$
------------- ----------------------------------------- --------------------------- ----------------------------
$\lambda=0$ $Q^2\le M^2$ $\not\exists$ $\not\exists$
$\lambda>0$ For $Q^2\le M^2$
$\lambda<\lambda_{c2}$ $\not\exists$ $\not\exists$
For $M^2<Q^2<(n+1)^2M^2/(4n)$
$\lambda_{c1}\le\lambda<\lambda_{c2}$
$\lambda<0$ $Q^2<M^2$ and $\lambda_{c1}\le \lambda$ $\lambda_{c0}\le \lambda$ $\lambda_{c-} \le \lambda$
: The parameter range in which the spacetime contains a regular black hole.[]{data-label="tbl:BHcondition"}
### $\lambda<0$
In this case, $f'(a)=0$ always has a single solution, and $f\tend+\infty$ as $r\tend +0, +\infty$. Hence, the spacetime contains a regular black hole if and only if $f(a)\le0$. Because $a<a_0<a_m$, this condition is equivalent to or in terms of $\lambda$, This, together with $D\ge0$, leads to the following conditions:
Finally, we determine the range of $x_H=2M/r_H^{n-1}$. In general, $\lambda$ is expressed in terms of $r_H$ as From this, we have First, for $K=1$, this can be written in terms of $a_{c1}$ and $a_{c2}$ as From this and the relation $a_{c2}<a_2<r_H <a_1<a_{c1}$, it follows that $\lambda$ is a monotonically increasing function of $r_H$ for fixed $M$ and $Q$. Hence, from the constraint $\lambda_{c1}\le \lambda <\lambda_{c2}$, we obtain where $x_{H,\min}$ and $x_{H,\max}$ are the values of $x_H$ for $\lambda=\lambda_{c2}$ and $\lambda=\lambda_{c1}$, respectively: Next, for $K=0$ or $K=-1$, $d\lambda/dr_H$ can be written in terms of $a_c$ as From $r_H\le a\le a_c$, this is non-negative. Hence, from $\lambda_{c}\le\lambda<0$, we obtain The allowed ranges of $x_H$ given in and in the $(x_H,Q^2/M^2)$ plane are displayed in Fig. \[fig:xh\].
Expressions for $\hat E^t_t$ and $\hat E_L$ {#Appendex:A}
===========================================
We have the following expressions for $\hat E^t_t$ and $\hat E_L$:
Expressions for coefficient functions {#Appendix:B}
=====================================
We have the following:
Note that these functions satisfy the following relations: The following relations hold for these functions:
Gauge-invariant treatment of the exceptional modes {#Appendix:C}
==================================================
In the present paper, we imposed the Einstein equation $F^a_a+2(n-2)F=0$ as the gauge condition for the exceptional modes of the scalar perturbation with $k^2=n$ for $K=1$. As discussed in Paper I, this gauge condition does not fix the gauge freedom completely, and it leaves the residual gauge freedom represented by $\bar\delta z^i=L\SHB^i$ satisfying Hence, the master variables introduced in § \[sec:ScalarPerturbation\] contain unphysical degrees of freedom for the exceptional modes, although the master equations themselves are gauge-invariant. In this appendix, we express the master equations in terms of genuinely gauge-invariant variables.
We define all quantities for an exceptional mode corresponding to the gauge-invariant variables for a generic mode introduced in the text by the same expressions with $H_T=0$. Then, for the general gauge transformation , $X_a$ transforms as $X_a \tend X_a
+\bar\delta X_a$ with From this, the following transformation laws of $F$ and $F^a_b$ are obtained: In particular, we have Hence, from and , $\E$ and $\E_a$ transform as From this and the definition of $\A$, , we find that $\A$ transforms as Like these variables, the matter variables $\Sigma_{ab}, \Sigma_a$ and $\Sigma_L$ are gauge dependent. However, $S_a, S^a_b$ and $S_L$ are gauge-invariant, because they represent perturbations of quantities whose background values vanish, like $J_a$.
In order to proceed further, we have to treat the black-hole-type background case and the Nariai-type background case separately.
Black hole background
---------------------
In the black hole background , the gauge transformation of $F^a_b$ is written In particular, we have Hence, the master variable $\Phi$ transforms as
For $k^2=n$ and $K=1$, $\mu=M$ and $\Phi_\pm$ are written From this, we find that $\Phi_+$ is gauge-invariant. In contrast, as is seen from the relation $\Phi_-$ is not gauge-invariant. Nevertheless, the master equation for $\Phi_-$ is gauge-invariant. This becomes evident if we rewrite this equation in terms of the gauge-invariant combinations Then, taking account of the fact that the master equation was derived under the gauge condition $F^a_a+2(n-2)F=0$, we obtain Here, for $k^2=n$ and $K=1$, we have Since this equation is written only in terms of gauge-invariant variables, it is valid in any gauge. Thus, the master equation for $\Phi_-$ gives an algebraic relation among the gauge-invariant variables $\Phi_+$, $\hat F$ and $\hat F^a_b$.
The definition of $\Phi$ yields another relation, Further, the expressions for $X$ and $Z$ provide two more relations:
where The four equations , and can be solved to yield expressions for $\hat F$ and $\hat F^a_b$ in terms of $\Phi_+$ and the gauge-invariant matter source terms $S_a$, $S^a_b$ and $\tilde J_a$. Therefore, the only dynamical gauge-invariant variable for the exceptional modes is $\Phi_+$. Note that we can impose the gauge condition $\Phi=0$, and for this gauge, $\hat F^a_b=F^a_b$ and $\hat
F=F$ hold and $\Phi_+$ is proportional to $\A$.
Nariai-type background
----------------------
The case of the Nariai-type background can be treated in almost the same way. First, the gauge transformations of $F^a_b$ and $F$ are written For $m=0$ and $K=1$, $\sigma$ and $\mu$ have the simple expressions Hence, $\Phi_\pm$ are written From this, we find that $\Phi_+$ is gauge-invariant, while $\Phi_-$ is not gauge-invariant, as seen from the relation as in the black hole background case.
We can construct the following gauge-invariant quantities from $F^a_b$ and $F$: Under the gauge condition $F^a_a+2(n-2)F=0$, we have From this, it follows that the master equation for $\Phi_-$ can be expressed in terms of the gauge invariant variables as Further, the equations for $X$ and $Z$ can be rewritten as The corresponding expression for $\hat F^\rho_\rho$ can be obtained from those for $\hat F^t_t$ and $\hat F^a_a$. Under the gauge condition $F=0$, $\hat F^a_b$ coincides with $F^a_b$, and $\Phi_+$ becomes a constant multiple of $\A$.
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|
---
author:
- 'Qing-Lin Xu'
- Yang Qiu
- Jin Tian
- Qi Liu
date: 'Received 2009 month day; accepted 2009 month day'
title: 'Interference coupling analysis based on a hybrid method: application for radio telescope system$^*$ '
---
Introduction {#sect:intro}
============
While radio astronomy service has been assigned the use of many frequency bands, the radio telescope is still faced with the radio frequency interference (RFI), such as unwanted out-of-band transmissions from adjacent and nearby bands (Waterman [@wate84]). On the other hand, the electromagnetic interference (EMI) generated by devices at radio telescope site is also a major threat (Ambrosini et al. [@ambr10]). When these interferences couple into the radio telescope system through multiple paths, the ability of radio astronomical observations would be reduced and even the radio telescope cannot work normally.
In order to protect the radio telescope from RFI, the threshold levels of detrimental interference in radio astronomy bands have been given in Recommendation ITU-R RA.769. Therefore, radio telescope sites are usually chosen in remote area so as to satisfy the threshold levels (Driel [@driel09]; Umar et al. [@umar14]). Further, shielded cabinets and Faraday cages have been used to attenuate EMI (Abeywickrema et al. [@abey15]). In order to determine the level of attenuation of mitigation techniques, it is necessary to analyze these interferences quantificationally. Some measurements have been performed to estimate the absolute interference level (Bolli et al. [@boll13]; Hidayat et al. [@hida14]). However, it usually needs a long time to carry out the measurements for a radio telescope system because of its large volume and complex structure. Even some measurements cannot be performed because the test conditions cannot be met. Therefore, the interference prediction based on numerical and analytical methods has been investigated by different authors.
In order to predict the interference coupled into a complex system, numerical methods, such as finite-difference time-domain method (Georgakopoulos et al. [@geor01]) and method of moments (Audone & Balma [@audo89]; Araneo & Lovat [@aran09]), have been used to calculate the interference response. However, they often require much computing time and memory because of the detailed mesh generation in electrically large system (Tzeremes et al. [@tzer04]), moreover it is difficult to investigate the variation of interference response with design parameters. Analytical formulations, although approximate, provide a better efficiency, and enable the variation with design parameters to be investigated, but it is difficult to handle complex geometry and material (Nie et al. [@nie11]). Among these methods, electromagnetic topology (EMT) owns the specific advantage to analyze electromagnetic interaction with complex electronic systems. The EMT method was first proposed by Baum ([@baum74]) and later improved by Tesche ([@tesc78]). In this method, interaction sequence diagram (Tesche & Liu [@tesc86]) and Baum-Liu-Tesche (BLT) (Baum et al. [@baum78]) equation are used to conduct qualitative and quantitative analysis of electromagnetic interactions respectively. Stated in matrix form, BLT equation can be easily programmed and promoted to network BLT equation (Parmantier et al. [@parm90]). After that many authors worked on various problems raised by application of the method. The fundamental BLT equation has been used by Parmantier ([@parm04]) to carry out the topological analysis of a system, and a strategy enlarging the scope of the entire simulation by combining several specific numerical tools has been proposed. In Kirawanich et al. ([@kira05]), a methodology has been proposed to simulate external interaction problems on very large complex electronic systems, and the response of the cables to lightning and electromagnetic pulses has been studied using an EMT-based code, which relies on transmission line theory to determine the transfer function. After that experiment and EMT-based simulation have been performed to study the effect of wideband electromagnetic pulse on a cable behind a slot aperture, and validated the numerical results by comparing with the experimental results (Kirawanich et al. [@kira08]). However such a sequential method cannot solve multiple responses simultaneously. Therefore, it is of great interest to think of developing a hybrid method to quickly predict the electromagnetic interference.
In this paper, we present a hybrid method that combines BLT equation and transfer function to predict the interference response of a radio telescope system. In Section \[sect:Ana\], a coupling intensity criterion is proposed by analyzing the conditions which BLT equation simplifies to transfer function. In Section \[sect:App\], based on the EMT theory, the coupling paths and volumes of radio telescope system are determined. By the criterion proposed in Section \[sect:Ana\], the coupling path of radio telescope is divided into strong coupling and weak coupling sub-paths, and the solving process of system response is given. In Section \[sect:The\], the responses of a typical model obtained by the proposed method are compared with those obtained by numerical method. Finally, some conclusions are made in Section \[sect:Conclusion\].
Analytical method {#sect:Ana}
=================
Generalized BLT equation {#sect:Gen}
------------------------
The conventional BLT equation is a frequency domain method for solving the interference response of transmission-line load (Tesche & Liu [@tesc86]). Similar to the propagation of electromagnetic field along the transmission line, the electromagnetic field in free space can also be described with incident wave and reflected wave, so it can be regarded as a virtual transmission line. Therefore, the generalized BLT equation can be established to solve the voltage response of transmission line and the electric field response of the space simultaneously.
The flux of interference into the entire system can be described with a topological network. The network is constituted by tubes related to each other through junctions. A propagation equation can be defined to describe the relationship between reflected wave $ W_{i,m} ^{ref} $ and incident wave $ W_{i,n} ^{inc} $ at each extremity of tube $ i $: $$W_{i,m} ^{ref}=P_{m:n}^{i} \cdot W_{i,n} ^{inc}-W_{s_{i}}$$
The scattering equation is thereby defined to describe the relationship between each reflected wave $ W_{i,m} ^{ref} $ and incident wave $ W_{j,m} ^{inc} $ at junction $ J_{m} $: $$W_{i,m} ^{ref}=S_{i:j}^{m} \cdot W_{j,m} ^{inc}$$ Where $ P_{m:n}^{i} $ and $ S_{i:j}^{m} $ are propagation and scattering parameters, respectively. $ W_{s_{i}} $ is excitation source applied on tube $ i $. By establishing propagation and scattering supermatrices involving all the waves on the network, the propagation and scattering equations of the entire network can be defined as: $$\emph{\textbf{W}}^{ref}=\emph{\textbf{P}} \cdot \emph{\textbf{W}}^{inc}-\emph{\textbf{W}}_{s}$$ $$\emph{\textbf{W}}^{ref}=\emph{\textbf{S}} \cdot \emph{\textbf{W}}^{inc}$$ Where $ \emph{\textbf{W}}^{inc} $, $ \emph{\textbf{W}}^{ref} $ and $ \emph{\textbf{W}}_{s} $ are incident wave, reflected wave and excitation source supervectors, respectively, $ \emph{\textbf{P}} $ and $ \emph{\textbf{S}} $ are the propagation and scattering supermatrices of the network, respectively. The generalized BLT equation of the entire network is: $$\emph{\textbf{W}}=\emph{\textbf{W}}^{inc}+\emph{\textbf{W}}^{ref}=(\emph{\textbf{E}}+\emph{\textbf{S}})(\emph{\textbf{P}}-\emph{\textbf{S}})^{-1}\emph{\textbf{W}}_{s}$$ Where $ \emph{\textbf{E}} $ is the unit supermatrix. By substituting the corresponding parameters into generalized BLT equation, the response of each extremity of the tube can be expressed as $ W_{i,m}=W_{i,m}^{inc}+W_{i,m}^{ref} $, i.e. the voltage or electric field that interference induces on junction $ J_{m} $ through tube $ i $, and the response of each junction can be expressed as $ W_{Jm}=\sum\limits_{i}W_{i,m} $, i.e. the sum of the voltages or electric fields that interference induces on junction $ J_{m} $ through all the tubes connected to this junction.
Strong coupling and weak coupling paths {#sect:Str}
---------------------------------------
When a topological network is solved with generalized BLT equation, the order of propagation and scattering matrices is twice the number of all tubes. Hence, it is very hard to solve the system response with the increase of the coupling paths. Besides, a coupling path of EMI consists of multiple tubes with different coupling modes, so that it is also difficult to solve the response accurately by using a single method. If a coupling path could be divided into sub-paths which can be analyzed respectively, the overall response of the entire system could be obtained by integrating the responses of each sub-path, which would greatly reduce the order of propagation and scattering matrices as well as improve the efficiency of solving the system response.
According to the transmission line theory, the electromagnetic wave in a tube is the superposition of the incident wave and reflected wave, which embodies as bi-directional coupling. Under this condition, by solving the generalized BLT equation, the response of junction $ J_{m} $ can be expressed as: $$W_{Jm}=\sum_{i}W_{i,m}=\frac{\sum\limits_{j}\left[\left(1+\sum\limits_{i}S_{i:j}^{m}\right)\sum\limits_{k}W_{s_{k}}A_{k:j}\right]}{|\emph{\textbf{P}}-\emph{\textbf{S}}|}$$ Where $ S_{i:j}^{m} $ is the scattering parameter of $ J_{m} $, $ k $ is the tube on which excitation source $ W_{s_{k}} $ applied, and $ A_{k:j} $ is the algebraic complement of the element in row and column corresponding to tubes $ k $ and $ j $ of $ |\emph{\textbf{P}}-\emph{\textbf{S}}| $. Because the value of denominator $ |\emph{\textbf{P}}-\emph{\textbf{S}}| $ is associated with the propagation and scattering parameters of all junctions, the response of junction $ J_{m} $ is related to all junctions of the network. In this case, the path between the interference source and the junction to be solved is defined as the strong coupling path.
The response of $ J_{m} $ can also be expressed by: $$W_{Jm}=\frac{W_{Jm}}{W_{Jz}} \cdot \cdots \cdot \frac{W_{Jy}}{W_{Jx}} \cdot W_{Jx}$$
Where the ratio of $ W_{Jy} $ to $ W_{Jx} $ can be expressed as: $$\frac{W_{Jy}}{W_{Jx}}=\frac{\sum\limits_{i}W_{i,y}}{\sum\limits_{i}W_{i,x}}=\frac{\sum\limits_{j}\left[\left(1+\sum\limits_{i}S_{i:j}^{y}\right)\sum\limits_{k}W_{s_{k}}A_{k:j}\right]}{\sum\limits_{j}\left[\left(1+\sum\limits_{i}S_{i:j}^{x}\right)\sum\limits_{k}W_{s_{k}}A_{k:j}\right]}$$
Through the derivation, if the interference is directed from $ J_{x} $ to $ J_{y} $ and the scattering parameter $ S_{i:j}^{y} (i>j) $ or propagation parameter $ P_{y:y+1} $ or $ P_{y+1:y} $ is much less than 1, the electromagnetic wave in $ J_{y} $ has only one direction, which embodies as unidirectional coupling. If the interference is directed from $ J_{y} $ to $ J_{x} $ and the scattering parameter $ S_{i:j}^{x} (i<j) $ or propagation parameter $ P_{x-1:x} $ or $ P_{x:x-1} $ is much less than 1, the same characteristic shows in $ J_{x} $, which also embodies as unidirectional coupling. In these cases, algebraic complement $ A_{k:j} $ can be expressed as the product of three determinants which correspond to the sub-path from interference source to $ J_{x} $, the sub-path from $ J_{x} $ to $ J_{y} $, and the sub-path from $ J_{y} $ to the end of the sub-path. By dividing out the same part, the ratio of $ W_{Jy} $ to $ W_{Jx} $ is only related to the two junctions and the junctions between them. In this case, the path between the interference source and the junction to be solved is defined as the weak coupling path. Described above set a criterion defining a weak coupling path, which is named as coupling intensity criterion. It is to be noted that, if there is a function to describe the bi-directional coupling characteristic of a strong coupling path, the strong coupling path can be handled as a weak coupling path.
From the above analysis, the coupling path can be expressed as a cascade of multiple weak coupling sub-paths and strong coupling sub-paths. By solving the different sub-paths with corresponding methods, the response of the system could be solved more effectively.
Transfer function of weak coupling path {#sect:Weak}
---------------------------------------
Transfer function is the ratio function of response phasor to excitation phasor at different extremities. Transfer function only considers the incidence and propagation of the electromagnetic wave, i.e. unidirectional coupling. Because each weak coupling sub-path is independent, that is, it would not be affected by other sub-paths, a transfer function can be established to describe the relationship between two ends of a weak coupling sub-path. By choosing different outputs and inputs, four transfer functions can be defined to represent the coupling efficiency of the electromagnetic fields between two volumes, the coupling efficiency of the voltages/currents between two points on wires or circuits, and the coupling efficiencies between the electromagnetic field and the voltage/current, respectively. (Qiu et al. [@qiu17]).
The transfer function of the electromagnetic fields between two volumes is established as: $$(E_{Jy},H_{Jy})=T_{x:y}^{r} \cdot (E_{Jx},H_{Jx})$$ Where $ (E_{Jx},H_{Jx}) $ is the incident field in volume $ V_{Jx} $, $ (E_{Jy},H_{Jy}) $ is the coupled field in volume $ V_{Jy} $, $ T_{x:y}^{r} $ is the transfer function of field-to-field coupling from $ V_{Jx} $ to $ V_{Jy} $ (superscript denotes the coupling mode). In the same manner, the transfer function of field-to-wire coupling from volume $ V_{Jx} $ to wire or circuit $ L_{Jy} $, the transfer function of radiation coupling from wire or circuit $ L_{Jx} $ to volume $ V_{Jy} $, and the transfer function of conduction coupling from point $ L_{Jx} $ on wire or circuit to another point $ L_{Jy} $ are established as: $$(U_{Jy},I_{Jy})=T_{x:y}^{r:c} \cdot (E_{Jx},H_{Jx})$$ $$(E_{Jy},H_{Jy})=T_{x:y}^{c:r} \cdot (U_{Jx},I_{Jx})$$ $$(U_{Jy},I_{Jy})=T_{x:y}^{c} \cdot (U_{Jx},I_{Jx})$$
When the coupling path between $ J_{x} $ and $ J_{m} $ is the cascade of multiple weak coupling sub-paths, the response of junction $ J_{m} $ can be represented as: $$W_{Jm}=T_{z:m} \cdot \cdots \cdot T_{x:y} \cdot W_{Jx}$$
For a complex system whose coupling path contains multiple strong and weak sub-paths, the transfer function of each weak coupling sub-path can be established to solve the output response of the sub-path, then the response treated as excitation source of strong coupling sub-paths is substituted into generalized BLT equation. The final response of the system is given by: $$\emph{\textbf{W}}=(\emph{\textbf{E}}+\emph{\textbf{S}}')[(\emph{\textbf{P}}'-\emph{\textbf{T}})-\emph{\textbf{S}}']^{-1}\emph{\textbf{W}}_{s}$$ Where $ \emph{\textbf{P}}' $ and $ \emph{\textbf{S}}' $ are propagation and scattering supermatrices of strong coupling sub-paths, respectively. $ \emph{\textbf{T}} $ is the transfer supermatrix of interferences which apply on strong coupling sub-paths through the weak coupling sub-paths.
In order to solve the response of the system accurately, it is necessary to describe different sub-paths with the corresponding models and determine the types of transfer functions and the corresponding calculation methods. For example, for the interference which penetrates into the shield, equivalent transmission line theory, equivalent circuit method (Tesche et al. [@tesc97]; Yin & Du [@yin16]) and scattering theory (Tsang et al. [@tsang00]) can be used; for the interference through aperture, dyadic green’s function method (Yang & Volakis [@yang05]), method of moments (Audone & Balma [@audo89]; Araneo & Lovat [@aran09]) and finite-difference time-domain method (Georgakopoulos et al. [@geor01]) can be used; for the interference coupled to the antenna and cable, field-to-wire coupling theory (Agrawal et al. [@agra80]) can be used; for the interference between two cables, crosstalk theory (Mohr [@mohr67]) and spectral analysis theory can be used.
Application of the hybrid method to radio telescope system {#sect:App}
==========================================================
Coupling characteristics of interference in radio telescope system {#sect:Cou}
------------------------------------------------------------------
Radio telescope system is a complex system which contains high sensitive receivers, high power drive subsystem, control and monitoring subsystem and other subsystems (Wang [@wang14]). The system and its electromagnetic environment are shown in Figure \[fig:1\]. Both electromagnetic emission signals of the system in operation and various electromagnetic signals in the environment will bring interference to the radio telescope.
![Radio telescope system and its electromagnetic environment.[]{data-label="fig:1"}](ms0118fig1.eps){width="95.00000%"}
The electromagnetic interference to the radio telescope system, essentially, is the interference to sensitive components or circuits of receiver subsystem and control subsystem. Based on the electromagnetic topology theory, the volumes of a radio telescope system are divided. According to the coupling paths of EMI in the system, the interference sequence diagram is obtained, as shown in Figure \[fig:2\]. The external interference sources are in volume $ V_{1.1} $, and the internal interference sources are in volume $ V_{2.1} \sim V_{2.5} $. The sensitive circuit of control subsystem is analog circuit, in volume $ V_{2.5} $, and the sensitive component of receiver subsystem is cryogenic amplifier, in volume $ V_{2.6} $. In order to analyze easily, the radio telescope system is simplified, and the interference which penetrates into the shield is ignored.
![Interference sequence diagram of a radio telescope system.[]{data-label="fig:2"}](ms0118fig2.eps){width="\textwidth"}
In Figure \[fig:2\], the coupling path of EMI consists of multiple tubes with different coupling modes, including field-to-antenna coupling, field-to-wire coupling, field-to-field coupling, and conduction coupling. The external interferences and internal interferences can couple into the feed by field-to-antenna coupling, then couple to sensitive components or circuits by conduction coupling. Besides, the internal interferences can also couple into the interior of the receiver subsystem and control subsystem by field-to-field coupling, then couple to the conduction sub-path by field-to-wire coupling.
Because the receiver passively receives electromagnetic radiation of celestial body, considering the principle of dual-reflector antenna, on most occasions, the reflection of the antenna can be ignored. Therefore, the field-to-antenna coupling sub-path can be handled as weak coupling sub-path.
Because of the various structure factors (enclosure, shape of the shield, material, aperture, etc.) and circuit factors (component, wire, grounding method, power, intensity of the circuit, etc.) of radio telescope system, both the field-to-field coupling and the conduction coupling have different coupling strengths. When the interference encounters a metal shield or transfers to a circuit of impedance mismatch, the reflection will occur, and the sub-path can be handled as strong coupling sub-path. On the other hand, when the interference encounters a non-metallic shield or transfers to a circuit of impedance match, the reflection can be ignored, and the sub-path can be handled as weak coupling sub-path. For the field-to-wire coupling, if the length of transmission line is much greater than the distance between the conductors of the transmission line, the secondary radiation of the transmission line can be ignored, and the sub-path can also be handled as weak coupling sub-path.
The analysis of subsystem {#sect:Sub}
-------------------------
Through the above analysis, the coupling path of radio telescope system can be divided into strong coupling and weak coupling sub-paths which can be modeled respectively. By establishing transfer functions of interferences, the responses of strong coupling and weak coupling sub-paths can be integrated as the response of the entire system, and the quantitative analysis of interference in radio telescope system is realized.
According to the dividing criterion of coupling intensity, the coupling path of radio telescope system is divided. Combining with the interference sequence diagram, the topological network of radio telescope system is established, as shown in Figure \[fig:3\]. In Figure \[fig:3\], two junctions are connected by a tube which are associated to unidirectional or bi-directional branch.
![Topological network associated to the interaction sequence diagram.[]{data-label="fig:3"}](ms0118fig3.eps){width="\textwidth"}
In Figure \[fig:3\], based on the composition of the system, the radio telescope system is preliminarily divided into four subsystems (the subsystem $ D_{11} \sim D_{13} $ which have the same coupling mode have been combined into one subsystem). The coupling path of each subsystem is composed of strong coupling sub-paths and weak coupling sub-paths, and the coupling paths of two subsystem are connected by weak coupling sub-paths. In addition, junctions $ J_{1.2} $, $ J_{2.1} $, $ J_{3.1} $ and $ J_{4.1} $ represent volume $ V_{1.2} $ in the four subsystems respectively. Similarly, volume $ V_{2.1} $ and wire or circuit $ L_{4.5} $ are represented by two junctions respectively.
In the four subsystems above, the subsystem 1 which completely contains all the typical coupling modes is taken as an example here. In subsystem 1, the sub-paths from external interference region $ V_{1.1} $ ($ J_{1.1} $) and internal interference region $ V_{1.2} $ ($ J_{1.2} $) to feed $ L_{4.2} $ ($ J_{1.7} $), the sub-path from internal interference region $ V_{1.2} $ ($ J_{1.2} $) to wire or circuit $ L_{4.4} $ ($ J_{1.11} $) and the sub-path from the interior of receiver $ V_{2.6} $ ($ J_{1.4} $) to sensitive component $ L_{4.3} $ ($ J_{1.9} $) are classified as weak coupling sub-paths. The sub-path from internal interference region $ V_{1.2} $ ($ J_{1.2} $) through the enclosure of the receiver $ S_{1.2:2.6} $ ($ J_{1.3} $) to the interior of the receiver $ V_{2.6} $ ($ J_{1.4} $) and the sub-path from feed $ L_{4.2} $ ($ J_{1.7} $) through interface $ S_{4.2:4.3} $ ($ J_{1.8} $) to sensitive component $ L_{4.3} $ ($ J_{1.9} $) are classified as strong coupling sub-paths. Among them, $ J_{1.9} $ and $ J_{1.11} $ are the junctions where field-to-wire coupling occurs, and junction $ J_{1.5} $ is the end of the shield.
According to topological network, propagation and scattering supermatrices of strong coupling sub-paths are obtained: $$\emph{\textbf{P}}=diag(\emph{\textbf{P}}^{1.4},\emph{\textbf{P}}^{1.5},\emph{\textbf{P}}^{1.6},\emph{\textbf{P}}^{1.8},\emph{\textbf{P}}^{1.9},\emph{\textbf{P}}^{1.10},\emph{\textbf{P}}^{1.11},\emph{\textbf{P}}^{1.12},\emph{\textbf{P}}^{1.13})$$ $$\emph{\textbf{S}}=diag(S_{1.4:1.4}^{1.2},\emph{\textbf{S}}^{1.3},\emph{\textbf{S}}^{1.4},S_{1.6:1.6}^{1.5},S_{1.8:1.8}^{1.6},\emph{\textbf{S}}^{1.7},\emph{\textbf{S}}^{1.8},\emph{\textbf{S}}^{1.9},\emph{\textbf{S}}^{1.10},\emph{\textbf{S}}^{1.11},S_{1.13:1.13}^{1.12})$$ Where $ \emph{\textbf{S}}^{m}=\left[\begin{array}{cc}S_{i:i}^{m}& S_{i:j}^{m}\\S_{j:i}^{m}& S_{j:j}^{m}\end{array}\right] $ is the scattering matrix of junction $ J_{m} $, $ \emph{\textbf{P}}^{i}=\left[\begin{array}{cc}0& P_{m:n}^{i}\\P_{n:m}^{i}& 0\end{array}\right] $ is the propagation matrix between $ J_{m} $ and $ J_{n} $ connected by tube $ i $. Furthermore, transfer supermatrix of interference which applies on conduction sub-path through field-to-wire coupling is obtained: $$\emph{\textbf{T}}=diag(0,0,0,\emph{\textbf{T}}^{1.4\sim1.10},0,0,0)$$ $$\emph{\textbf{T}}^{1.4\sim1.10}=
\left[\begin{array}{lllll;{2pt/2pt}cc}
%& & & & & &\\
\multicolumn{5}{c;{2pt/2pt}}{\raisebox{0ex}[0pt]{\Large0}}& & \multicolumn{1}{c}{\raisebox{0ex}[0pt]{\Large0}}\\
\hdashline[2pt/2pt]
%\vspace{-6pt}
%& & & & & &\\
(1+S_{1.5:1.5}^{1.4}) \cdot T_{1.4:1.8}^{1.10} & & & (1+S_{1.6:1.6}^{1.4}) \cdot T_{1.4:1.8}^{1.10} & & & \multirow{4}{*}{\raisebox{0ex}[0pt]{\Large0}}\\
(1+S_{1.5:1.5}^{1.4}) \cdot T_{1.4:1.9}^{1.10} & & & (1+S_{1.6:1.6}^{1.4}) \cdot T_{1.4:1.9}^{1.10} & & &\\
(1+S_{1.5:1.5}^{1.4}) \cdot T_{1.4:1.9}^{1.11} & & & (1+S_{1.6:1.6}^{1.4}) \cdot T_{1.4:1.9}^{1.11} & & &\\
(1+S_{1.5:1.5}^{1.4}) \cdot T_{1.4:1.10}^{1.11}& & & (1+S_{1.6:1.6}^{1.4}) \cdot T_{1.4:1.10}^{1.11}& & &
\end{array}\right]$$ Where $ T_{m:n}^{i} $ is the transfer function of field-to-wire coupling from junction $ J_{m} $ to the extremity where tube $ i $ connects to junction $ J_{n} $. Substituting the supermatrices into generalized BLT equation, the response of subsystem 1 is obtained: $$\begin{aligned}
\emph{\textbf{W}}=(&W_{1.4,1.2},W_{1.4,1.3},W_{1.5,1.3},W_{1.5,1.4},W_{1.6,1.4},W_{1.6,1.5},\\
&W_{1.8,1.6},W_{1.8,1.7},W_{1.9,1.7},W_{1.9,1.8},W_{1.10,1.8},W_{1.10,1.9},\\
&W_{1.11,1.9},W_{1.11,1.10},W_{1.12,1.10},W_{1.12,1.11},W_{1.13,1.11},W_{1.13,1.12})^{T}
\end{aligned}$$
The response of junction $ J_{1.9} $ can be represented as: $$\begin{aligned}
W_{J1.9}&=W_{1.10,1.9}\!+\!W_{1.11,1.9}\\
&=\frac{(1\!+\!S_{1.10:1.10}^{1.9}\!+\!S_{1.11:1.10}^{1.9})\sum\limits_{k}W_{s_{k}}A_{k:1.10}\!+\!(1\!+\!S_{1.10:1.11}^{1.9}\!+\!S_{1.11:1.11}^{1.9})\sum\limits_{k}W_{s_{k}}A_{k:1.11}}{|\emph{\textbf{P}}-\emph{\textbf{S}}|}
\end{aligned}$$
Considering only the coupling path from junction $ J_{1.2} $ to sensitive component through aperture, the response of junction $ J_{1.9} $ can be represented as: $$W_{J1.9}=\frac{(1\!+\!S_{1.10:1.10}^{1.9}\!+\!S_{1.11:1.10}^{1.9})W_{1.4}A_{1.4:1.10}\!+\!(1\!+\!S_{1.10:1.11}^{1.9}\!+\!S_{1.11:1.11}^{1.9})W_{1.4}A_{1.4:1.11}}{|\emph{\textbf{P}}-\emph{\textbf{S}}|}$$
By means of simulation and measurement, the propagation and scattering parameters of each junction in the subsystem can be obtained, then the response of the subsystem can be solved. When the junction to be solved and the interference source are in the same subsystem, only this subsystem needs to be solved. But when they are in different subsystems, the transfer function between subsystems needs to be built to solve the response of the merged path synthetically.
The analysis of system {#sect:Sys}
----------------------
Because multiple interference sources couple to sensitive component of receiver subsystem by multiple paths and each path can be expressed as the cascade of strong coupling and weak coupling sub-paths, according to the superposition principle of the circuit, the final response of junction $ J_{1.9} $ can be represented as: $$%\begin{small}
\begin{aligned}
%&(U_{J1.9},I_{J1.9})=T_{1.1:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} \cdot (E_{J1.1},H_{J1.1})\\
%&+ M_{3.6:3.1}^{r} \cdot T_{3.1:1.2}^{r} \cdot (T_{1.2:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} + M_{1.2:1.4}^{r} \cdot T_{1.4:1.9}^{r:c} + T_{1.2:1.11}^{r:c} \cdot M_{1.11:1.9}^{c}) \cdot (E_{J3.6},H_{J3.6})\\
%&+ M_{3.8:3.1}^{r} \cdot T_{3.1:1.2}^{r} \cdot (T_{1.2:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} + M_{1.2:1.4}^{r} \cdot T_{1.4:1.9}^{r:c} + T_{1.2:1.11}^{r:c} \cdot M_{1.11:1.9}^{c}) \cdot (E_{J3.8},H_{J3.8})\\
%&+ M_{3.10:3.1}^{r} \cdot T_{3.1:1.2}^{r} \cdot (T_{1.2:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} + M_{1.2:1.4}^{r} \cdot T_{1.4:1.9}^{r:c} + T_{1.2:1.11}^{r:c} \cdot M_{1.11:1.9}^{c}) \cdot (E_{J3.10},H_{J3.10})\\
%&+ M_{3.12:3.1}^{r} \cdot T_{3.1:1.2}^{r} \cdot (T_{1.2:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} + M_{1.2:1.4}^{r} \cdot T_{1.4:1.9}^{r:c} + T_{1.2:1.11}^{r:c} \cdot M_{1.11:1.9}^{c}) \cdot (E_{J3.12},H_{J3.12})\\
%&+ M_{4.3:4.1}^{r} \cdot T_{4.1:1.2}^{r} \cdot (T_{1.2:1.7}^{r:c} \cdot M_{1.7:1.9}^{c} + M_{1.2:1.4}^{r} \cdot T_{1.4:1.9}^{r:c} + T_{1.2:1.11}^{r:c} \cdot M_{1.11:1.9}^{c}) \cdot (E_{J4.3},H_{J4.3})
&(U_{J1.9},I_{J1.9})=T_{1.1:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!\cdot\!(E_{J1.1},H_{J1.1})\\
&+\!M_{3.6:3.1}^{r}\!\cdot\!T_{3.1:1.2}^{r}\!\cdot\!(T_{1.2:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!+\!M_{1.2:1.4}^{r}\!\cdot\!T_{1.4:1.9}^{r:c}\!+\! T_{1.2:1.11}^{r:c}\!\cdot\!M_{1.11:1.9}^{c})\!\cdot\!(E_{J3.6},H_{J3.6})\\
&+\!M_{3.8:3.1}^{r}\!\cdot\!T_{3.1:1.2}^{r}\!\cdot\!(T_{1.2:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!+\!M_{1.2:1.4}^{r}\!\cdot\!T_{1.4:1.9}^{r:c}\!+\! T_{1.2:1.11}^{r:c}\!\cdot\!M_{1.11:1.9}^{c})\!\cdot\!(E_{J3.8},H_{J3.8})\\
&+\!M_{3.10:3.1}^{r}\!\cdot\!T_{3.1:1.2}^{r}\!\cdot\!(T_{1.2:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!+\!M_{1.2:1.4}^{r}\!\cdot\!T_{1.4:1.9}^{r:c}\!+\! T_{1.2:1.11}^{r:c}\!\cdot\!M_{1.11:1.9}^{c})\!\cdot\!(E_{J3.10},H_{J3.10})\\
&+\!M_{3.12:3.1}^{r}\!\cdot\!T_{3.1:1.2}^{r}\!\cdot\!(T_{1.2:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!+\!M_{1.2:1.4}^{r}\!\cdot\!T_{1.4:1.9}^{r:c}\!+\! T_{1.2:1.11}^{r:c}\!\cdot\!M_{1.11:1.9}^{c})\!\cdot\!(E_{J3.12},H_{J3.12})\\
&+\!M_{4.3:4.1}^{r}\!\cdot\!T_{4.1:1.2}^{r}\!\cdot\!(T_{1.2:1.7}^{r:c}\!\cdot\!M_{1.7:1.9}^{c}\!+\!M_{1.2:1.4}^{r}\!\cdot\!T_{1.4:1.9}^{r:c}\!+\! T_{1.2:1.11}^{r:c}\!\cdot\!M_{1.11:1.9}^{c})\!\cdot\!(E_{J4.3},H_{J4.3})
\end{aligned}
%\end{small}$$ Where $ M_{m:n} $ represents the strong coupling sub-path, and $ T_{m:n} $ represents the weak coupling sub-path. According to the above process, by establishing the generalized BLT equation of strong coupling sub-paths and the transfer functions of weak coupling sub-paths and substituting the excitation sources, the final response of junction $ J_{1.9} $ can be obtained. In this case, the order of established propagation and scattering supermatrices is reduced from 90 to 46, where the 90-order supermatrices correspond to all the paths of the topological network in Figure \[fig:3\], and the 46-order matrices correspond to the strong coupling sub-paths between the interference source and the junction to be solved. It is observed that this method can reduce the order of matrices to be solved, thereby reducing the computation time and the difficulty of solving. In the same manner, the interference coupled to control subsystem can also be analyzed.
Numerical results and discussion {#sect:The}
================================
In this section, similar to the coupling path through aperture of subsystem 1 described in section \[sect:Sub\], a typical model of a two-conductor transmission line inside a rectangular shielding cavity with aperture is considered. The proposed hybrid method is used to solve the electric field response of an observation point in the cavity and the induced current of transmission-line load, and the accuracy is verified by comparing the responses with the results obtained by [cst]{}.
The geometry of the model is shown in Figure \[fig:4\]. Inside dimensions of the rectangular cavity are 300mm$\times$120mm$\times$260mm, and the thickness is $t=1$mm. A rectangular aperture with dimensions of 40mm$\times$20mm is on the front wall. The length of transmission line is $L_{t}=100$mm, the radius of each conductor is $a_{1}=a_{2}=0.5$mm, and the distance between two conductors is $D=10$mm. The distance between transmission line and aperture is $q=215$mm. Point $ P $ of coordinates $ (245,85,-215) $ denotes the observation point located in the center of the transmission line. The incident plane wave propagates along the negative $z$-axis and has an E-filed oriented along the $y$-axis with the amplitude $ V_{0}=1$V m$^{-1} $ and with the frequency from 0 to 3GHz.
![Model of transmission line inside rectangular shielding cavity with aperture.[]{data-label="fig:4"}](ms0118fig4.eps){width="60.00000%"}
In this model, the coupling path can be expressed as the cascade of field-to-field coupling sub-path, field-to-wire coupling sub-path and conduction coupling sub-path. As described in section \[sect:Sub\], the field-to-field coupling sub-path and conduction coupling sub-path are classified as strong coupling sub-paths, and the field-to-wire coupling sub-path is classified as weak coupling sub-path. On the other hand, there is a function that describes the bi-directional coupling characteristic of the field-to-field coupling sub-path, so this sub-path can also be handled as weak coupling sub-path. Therefore, we considered two different scenarios.
In scenario 1, by establishing the equivalent circuit model, the propagation and scattering parameters of each junction can be obtained to build the BLT equation of the field-to-field coupling and conduction coupling sub-paths. A transfer function of field-to-wire coupling is established and substituted into the BLT equation. In this case, according to Robinson algorithm (Robinson et al. [@robi98]), free space can be represented by a transmission line whose characteristic impedance and propagation constant are $ Z_{0} $ and $ k_{0} $. Aperture can be represented by a coplanar transmission line shorted at the end whose characteristic impedance is $ Z_{oS} $. Cavity can be represented by a rectangular waveguide shorted at the end whose characteristic impedance and propagation constant are $ Z_{g} $ and $ k_{g} $. The two-conductor transmission line in the cavity is uniform and lossless and its characteristic impedance is $ Z_{t} $. By superimposing multiple resonant modes, the y-component of the electric field at arbitrary point $ (x_{p},y_{p},z_{p}) $ in the cavity can be obtained as $ W_{Total}=\sum\limits_{m,n}W_{J}\sin(m \pi x_{p}/a)\cos(n \pi y_{b}/b) $. Figure \[fig:5\] shows the responses of the observation point and the load solved by [cst]{} simulation and hybrid method combining BLT equation and the transfer function of field-to-wire coupling. By comparing with [cst]{}, it can be seen that the hybrid method has a high precision.
In scenario 2, the BLT equation of field-to-field coupling is replaced by a transfer function. In this case, according to Lee ([@lee86]), the aperture can be represented by equivalent electric dipole and magnetic dipole, then the dyadic Green’s function of a rectangular cavity can be used to solve the coupled field inside the cavity. Figure \[fig:6\] shows the responses of the observation point and the load solved by [cst]{} simulation and hybrid method combining BLT equation and transfer function of field-to-field coupling and field-to-wire coupling. It can be seen that, because the equivalent model calculates the radiation field of the magnetic dipole, the results have a higher precision in high frequency.
Table \[Tab:1\] shows the time and resource usage taken by [cst]{} simulation software and the hybrid method programmed with [matlab]{} to calculate the typical model. All simulations and calculations are performed on the same computer, which has a 3-GHz AMD Athlon(tm) ¢ò X4 640 Processor and 4-GB RAM. It can be seen that the hybrid method coded by [matlab]{} takes less computing time and resource than [cst]{}.
[lcc]{} Method & Run time/s & CPU utility/%\
& 9712 & 99\
hybrid method of scenario 1 & 60.34 & 25\
hybrid method of scenario 2 & 6.59 & 25\
Conclusion {#sect:Conclusion}
==========
In this paper, a hybrid method for predicting the EMI response of a radio telescope system is proposed. Based on the conditions that BLT equation simplifies to transfer function, the coupling intensity criterion is proposed. Further, the coupling path of radio telescope system is divided into strong coupling and weak coupling sub-paths. Transfer function is used to solve the responses of weak coupling sub-paths, which are treated as the excitation sources of strong coupling sub-paths. Through adopting BLT equation to solve the strong coupling sub-paths, the integrated response of a system is obtained. Finally, the proposed method is used to analyze a simple typical problem. The responses of the observation point and the transmission-line load have been obtained and compared with the results obtained by [cst]{}. The results show that this hybrid methodology is accurate and efficient to solve the response of the system.
This work was funded by the National Basic Research Program of China under No.2015CB857100 and by the National Natural Science Foundation of China under No.11473061 and No.11103056.
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\[lastpage\]
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---
abstract: 'We develop a thermodynamic description of particles held at a fixed surface potential. This system is of particular interest in view of the continuing controversy over the possibility of a fluid-fluid phase separation in aqueous colloidal suspensions with monovalent counterions. The condition of fixed surface potential allows in a natural way to account for the colloidal charge renormalization. In a first approach, we assess the importance of the so called “volume terms”, and find that in the absence of salt, charge renormalization is sufficient to stabilize suspension against a fluid-fluid phase separation. Presence of salt, on the other hand, is found to lead to an instability. A very strong dependence on the approximations used, however, puts the reality of this phase transition in a serious doubt. To further understand the nature of the instability we next study a Jellium-like approximation, which does not lead to a phase separation and produces a relatively accurate analytical equation of state for a deionized suspensions of highly charged colloidal spheres. A critical analysis of various theories of strongly asymmetric electrolytes is presented to asses their reliability as compared to the Monte Carlo simulations.'
author:
- 'Yan Levin$^{1,2}$[^1], Emmanuel Trizac$^2$[^2], Lydéric Bocquet$^3$[^3]'
title: 'On the fluid-fluid phase separation in charged-stabilized colloidal suspensions '
---
Introduction
============
There is a long standing debate in the field of colloidal science concerning a possibility of phase separation in aqueous colloidal suspensions containing monovalent counterions . Charged colloids provide a particular challenge to the theorists. The long ranged Coulomb interaction and the extremely large asymmetry between the polyions and the counterions makes it very difficult to apply to this system the traditional methods of the liquid state theory. In particular, such a well established tool of condensed matter theorists as integral equations is found to be useless when applied to strongly asymmetric electrolytes at large couplings (low temperatures). For a wide range of parameters of physical interest the integral equations fail to even converge. The original hope of associating the lack of convergence with an underlying physical instability has proven to be unfounded[@Fis81; @Bel93].
Recently, a linearized density functional theory has been used to study charged colloidal suspensions [@vRoij1; @vRoij2]. The theory predicted a thermodynamic instability which manifested itself as a fluid-fluid phase separation. However, the underlying approximation of the theory put its conclusions in doubt [@Diehl; @Grunberg; @Deserno]. Specifically the linearization of the density functional lowers the statistical weight of the configurations in which the counterions are in a close vicinity of the polyions. This effect can be partially accounted for through the renormalization of the bare colloidal charge [@Alexander; @Belloni]. Unlike the bare charge, which can be very large, the effective (renormalized) charge is found to be bounded by the saturation value controlled —in a given solvent— by the colloidal size, temperature, and salt concentration. The renormalization of colloidal charge was argued to wash out the phase transition predicted by the linear theories [@Diehl]. Furthermore, numerical solutions of the full non-linear Poisson-Boltzmann (PB) equation inside a Wigner Seitz (WS) cell shows absence of any instability[@Grunberg; @Deserno; @Tellez]. On the other hand, linearization of the PB equation leads to a non-convex pressure, as a function of colloidal concentration, similar to the one observed in other linear theories. All these suggest that the phase instability predicted by the linear theories might be an artifact of the underlying approximations.
To further explore these interesting points, we have investigated the thermodynamics of particles fixed at constant surface potential[@Trizac1; @Trizac2; @Levin]. Relevance of the constant potential ensemble follows from the observation that unlike the bare charge, the effective charge of colloidal particles does not grow indefinitely but instead saturates. The saturation value is such that the effective electrostatic potential $\phi_s$ in the vicinity of colloidal surface is $\beta q \phi_s \approx 4$, where $\beta=1/k_B T$ and $q$ is the elementary charge. Thus, although the colloidal charge can be very large, the potential near the colloidal surface, i.e. within the Debye length, does not increase beyond $\beta q \phi_s \approx 4$. This simple observation is sufficient to construct a consistent thermodynamic description of colloids with a state dependent effective charge. Since the WS cell description of colloidal suspension[@Marcus] does not lead to a fluid-fluid phase separation (see [@Tellez] for a general argument), to further understand the mechanism of the instability observed within the linearized theories, we focus on this alternative treatment.
We shall first (section \[sec:theory\]), study the effect of charge renormalization on the polyion-microion interaction free energy, i.e. the volume term that appears when the original mixture of colloids, coions and counterions is mapped onto an effective one component system of dressed colloids [@vRoij1; @vRoij2; @Levin1]. In section \[sec:jellium\] an alternative derivation of the thermodynamic equation of state based only on the far field considerations is presented. The corresponding pressure-density isotherms, in this case, do not exhibit criticality at any salt concentration. In order to assess the reliability of various approaches, in section \[sec:discussion\] we compare the corresponding pressures to the results of the Monte Carlo simulations of Linse [@Linse]. We also consider the, recently proposed, symmetric Poisson-Boltzmann [@Bhuiyan] and the “boot-strap” Poisson-Boltzmann [@BSPB] theories. Conclusions are drawn in section \[sec:conclusion\].
Role of volume terms {#sec:theory}
=====================
State dependent effective charges
---------------------------------
Consider a colloidal suspension at concentration $\rho_p$, containing spherical polyions of charge $-Zq$ and radius $a$ in contact with a monovalent salt reservoir at concentration $c_s$. Now suppose one colloid is fixed at $r=0$. In a continuum approximation consisting of smearing out the charge of other colloids and linearizing the Poisson-Boltzmann equation[@Levin], the electrostatic potential at distance $r$ from the center of colloid is $$\begin{aligned}
\label{1}
\phi(r)=-\frac{Zq \theta(\kappa a) e^{-\kappa r}}{\epsilon r} \;,
\;\;\;\;
\theta(x)=\frac{e^{x}}{(1+x)}\;,\end{aligned}$$ where the inverse Debye length is $$\label{2}
\kappa = \sqrt{4 \pi \lambda_B (\rho_++\rho_-}) \;,$$ and the Bjerrum length is $$\label{5}
\lambda_B=\frac{q^2}{\epsilon k_B T}\;.$$ The mean densities of coions and counterions inside the suspension are respectively $\rho_-$ and $\rho_+$. When $\rho_p \to 0$, $\kappa^2 \to \kappa_s^2 = 8 \pi \lambda_B \,c_s$.
For highly charged polyions, Eq. (\[1\]) strongly overestimates the real electrostatic potential. However, it can be made consistent with the full non-linear PB, if instead of the bare charge $Z$ an effective, renormalized, charge $Z_{eff}$ is used. The observation that for large surface potentials the electrostatics away from the colloidal surface is completely insensitive to the surface charge density allows for the “far-field” definition of the effective charge. Specifically, viewed from a distance larger than the Debye length, and provided that $\kappa a>1$, the surface potential of a strongly charged colloidal particle appears to be $\beta q \phi(a) \approx -4$ [@Trizac1; @Trizac2; @Levin]. Combining this with Eq. (\[1\]) leads directly to [@Trizac1; @Levin] $$\label{3}
Z_{eff}=\frac{4 a }{\lambda_B}\,(1+\kappa a)\;.$$ We should note that this is the saturated value of the effective charge relevant for the highly charged colloidal particles. For weekly charged particles there is little or no charge renormalization. In the infinite dilution limit[@JPA] of one colloid immersed in a 1:1 electrolyte of concentration $c_s$, the exact result for the saturation limit of $Z_{eff}$ is $$Z_{eff} = \frac{4 a }{\lambda_B} \left(\frac{3}{2} + \kappa_{s} a \right).
\label{eq:Zeff}$$ This expression –valid up to corrections of order $(\kappa a)^{-1}$, which turn out to be quite small as soon as $\kappa a>1$, is very close to the approximation (\[3\]), derived from the matching procedure detailed in [@Trizac2; @Levin].
The effective charge depends strongly on the electrolyte concentration inside the suspension. Salt screens the electrostatic interactions between the counterions and the polyions and leads to an increase in the colloidal effective charge. While the effective charge of colloidal particles is a strongly state dependent function, the effective surface potential is not. Therefore, inside suspension, colloids behave as if their surface potential was effectively fixed. It is interesting, therefore, to study the thermodynamics of colloidal particles at fixed surface potential.
Thermodynamics of particles at fixed surface potential
------------------------------------------------------
The change of the thermodynamic ensemble from the constant charge to the constant surface potential allows us in a natural way to explore the role of charge renormalization in the framework of a linear theory. The surface potential is related to the effective colloidal charge through the Eq. (\[1\]) $$\label{4}
\varphi=-\beta q \phi(a)=\frac{Z \lambda_B}{a (1+\kappa a) }\;.$$
As mentioned above, the reduced surface potential within the non-linear Poisson-Boltzmann theory is found to saturate at $\varphi=4$. For the sake of generality we shall, however, keep its value arbitrary. In the subsequent analysis, $Z$ will refer to the saturation value of the colloidal effective charge. An implicit assumption is therefore that the bare charge largely exceeds the effective one.
In the simplest approximation, the Helmholtz free energy of the suspension is a sum of entropic and electrostatic contributions $$\label{6}
\beta F=N_+[\ln{\rho_+ \Lambda^3}-1]+
N_-[\ln{\rho_- \Lambda^3}-1]+\beta F^{el}\;,$$ where $\Lambda$ is the de Broglie thermal wavelength, and $N_+$ and $N_-$ refer to the number of counterions and coions inside the suspension.
The electrostatic free energy results from the polyion-microion, microion-microion and the polyion-polyion interactions. For suspensions containing monovalent counterions the polyion-counterion interaction is the dominant contribution and will be the only one kept in the present exposition (we come back to this point in section \[sec:discussion\]). We find[@Levin] $$\label{7}
\beta F^{el}=\frac{Z^2 \lambda_B N_p}{2 a (1+\kappa a)}\;.$$ This expression can be obtained through the usual Debye charging process in which all the particles are simultaneously charged from $0$ to the their final charge[@Levin]. Alternatively a surface charging process, at constant Debye length can be employed[@PRE97]. It is noteworthy that in the salt free case, we recover precisely the volume term obtained in Refs [@vRoij1; @vRoij2].
If the suspension is in contact with a salt reservoir of chemical potential $\mu_s$, the effective charge of colloidal particles, as well as the number of counterions and coions, is determined by the minimum of the grand potential function $$\label{8}
\beta \Omega=\beta F-\beta\mu_s (N_++N_-)-(\varphi+\gamma) Z N_p
-\omega N_p\left[\varphi(1+\kappa a)-\frac{Z \lambda _B}{a}\right]-
\gamma(N_+-N_--ZN_p)\;.$$ In this equation, $\omega$ and $\gamma$ are the Lagrange multipliers: $\gamma$ ensures the charge neutrality of the system, while $\omega$ enforces the relationship between the surface potential and the effective charge, Eq. (\[4\]). In the biophysics literature $\gamma$ is known as the Donnan potential. It results from the inability of macroions to diffuse through a semi-permeable membrane. This is precisely the situation that we have in mind, while the microions are assumed to be in a free exchange with the reservoir, the polyions are confined to the interior of suspension. This restriction on the polyion mobility results in a potential difference $\gamma$, between the bulk of suspension and the reservoir. The colloids are then held at potential $\varphi+\gamma$ with respect to the reservoir, or equivalently at potential $\varphi$ with respect to the bulk of the suspension. Within the WS cell model, a similar prescription of constraining the potential difference between the colloidal surface and the outer (reservoir) boundary has been shown to yield a surprisingly good agreement with the full non-linear Poisson-Boltzmann equation[@Trizac2].
Minimizing the grand potential with respect to $N_+$, $N_-$ and $Z$, we find $$\label{9}
\frac{\partial \beta\Omega}{\partial N_\pm}=\frac{\partial \beta F}{\partial
N_\pm}-\mu_s-
\omega \left[\frac{2 \pi \varphi \lambda_B a \rho_p}{\kappa} \right] \mp\gamma =0\;,$$
$$\label{11}
\frac{\partial \beta\Omega}{\partial Z}=\frac{\partial \beta F}{\partial
Z}-\omega N_p \left[\frac{\lambda_B}{a} \right]-\varphi N_p =0\;,$$
and the charge neutrality condition reads $$\label{12}
N_+-N_-=\frac{\varphi a (1+\kappa a)}{\lambda_B} N_p\;.$$ Noting that $$\label{13}
\frac{\partial \beta F}{\partial Z}=\varphi N_p\;,$$ Eq. (\[11\]) simplifies to $$\label{14}
\omega=0\;.$$ Eliminating the Lagrange multiplier between Eqs. (\[9\]), we are left with two equations which govern the concentrations of counterions and coions inside suspension, $$\label{15}
\rho_+ \rho_-=c_s^2 \exp\left(\frac{2 \pi \varphi^2 a^2 \rho_p}{\kappa}\right)\;$$ and $$\label{16}
\rho_+- \rho_-=\frac{\varphi a (1+\kappa a)}{\lambda_B} \rho_p \;.$$
The equation of state
---------------------
The osmotic pressure inside suspension of colloids at fixed surface potential is $$\label{17}
P=-\frac{d \Omega}{d V}
\Big\arrowvert_{N_p,\mu_s,\omega,\gamma,\varphi}.$$ It is important to keep in mind that as $\Omega$ changes with volume, the number of coions, counterions, as well as the charge of colloidal particles are all varying. This is the reason for writing the total derivative in the expression (\[17\]) $$\label{18}
\frac{d \Omega}{d V}
\Big\arrowvert_{N_p,\mu_s,\omega,\gamma,\varphi}=
\frac{\partial \Omega}{\partial N_+}\frac{d N_+}{d V}+
\frac{\partial \Omega}{\partial N_-}\frac{d N_-}{d V}+
\frac{\partial \Omega}{\partial Z}\frac{d Z}{d V}+
\frac{\partial \Omega}{\partial V}.$$ Recalling that at the thermodynamic equilibrium $$\label{19}
\frac{\partial \Omega}{\partial N_+}=
\frac{\partial \Omega}{\partial N_-}=
\frac{\partial \Omega}{\partial Z}=0$$ expression for pressure simplifies to $$\label{20}
P=-\frac{\partial \Omega}{\partial V}
\Big\arrowvert_{N_p,N_+,N_-,Z,\mu_s,\omega,\gamma,\varphi}=
-\frac{\partial F}{\partial V}\Big\arrowvert_{N_p,N_+,N_-,Z}.$$ Equation (\[20\]) beautifully illustrates the thermodynamic principle of ensemble equivalence. The functional form of the pressure is the same weather the calculation is done in the fixed potential ensemble using the grand potential function $\Omega$, or in the fixed colloidal charge ensemble using the Helmholtz free energy $F$. We stress that simply inserting $Z_{eff}(V)$ with its state dependence into $F$ and then differentiating it with respect to volume will lead to an incorrect result. If the Helmholtz free energy is used, the variation must be performed at [*fixed*]{} colloidal charge.
In general it can be very difficult to find a suitable thermodynamic potential for a constrained system. The calculation of pressure, on the other hand, can be done very straightforwardly using the constant $Z$ ensemble, and enforcing the constraint [*a posteriori*]{}. Evaluating the partial derivative in Eq. (\[20\]), the osmotic pressure inside the suspension takes a particularly simple form $$\label{21}
\beta P= \rho_++\rho_--\frac{1}{4} \frac{a}{\lambda_B} \varphi^2 \kappa
a\rho_p \;,$$ where the concentrations of coions and counterions are determined from Eqs. (\[15\]) and (\[16\]).
In the special case of vanishing salt concentration, Eqs. (\[15\]) and (\[16\]) simplify to $$\label{22}
\rho_-=0 \;,$$ $$\label{23}
\rho_+=\varphi \rho_p\frac{a}{\lambda_B}\left[1+\frac{3}{2} \eta \varphi
+
\frac{1}{2}\sqrt{3 \eta \varphi(4+3 \eta \varphi)}\right]\;,$$ and the ratio of colloidal size to Debye length is $$\label{24}
\kappa a = \frac{3 \eta \varphi}{2} +
\frac{1}{2}\sqrt{3 \eta \varphi(4+3 \eta \varphi)}\;,$$ where $\eta = 4 \pi \rho_p a^3/ 3$ is the macroion volume fraction. For salt-free suspensions pressure becomes $$4 \pi \lambda_B a^2 \beta P = 3 \eta \varphi
\left(1+\kappa a -\frac{1}{4} \varphi \kappa a \;.
\right)
\label{eq:pressZ}$$ If $\varphi < 4$ the osmotic pressure is a convex up function of colloidal density. For real colloids with $\varphi=4$, the pressure is a linear function of colloidal density, $\beta P = 4 \rho_p a/\lambda_B$. For surface potentials strictly above $4$, a thermodynamic instability appears. It is very curious to note that the instability sets in precisely at $\varphi=4$, which is the saturation value for the surface potential obtained within the non-linear Poisson-Boltzmann theory. Renormalization of the electrostatic free energy is, therefore, sufficient to stabilize a salt free [*real*]{} colloidal suspension ($\varphi=4$) against a fluid-fluid phase separation[@Diehl]. However, the fact that the critical surface potential is precisely equal to the saturation value of the non-linear theory, suggests that the approach, most likely, is very sensitive to the approximations made. Furthermore, a relatively small amount of salt destabilizes suspension even when $\varphi=4$, see Fig. \[fig:1\]. The critical salt concentration for highly charged colloids of radius $a=1000\,$Å, is $c_s^*\approx
10^{-4}$ M, which corresponds to $\kappa_s^* a \simeq 3.3$. Since there is no explicit polyion-polyion nor microion-microion interaction, the instability is completely driven by the polyion-counterion correlations. In Fig. \[fig:10\] we show the effective charge $Z$ resulting from our approach, as a function of volume fraction. A good agreement with the Poisson-Boltzmann cell model is found.
![Pressure isotherms for different reservoir salt concentrations ($\kappa_s^2 = 8 \pi \lambda_B c_s$, $\varphi=4$)[]{data-label="fig:1"}](press4.eps){width="8cm"}
![Comparison between the effective charge (solid line) found using our variational approach, to that obtained within the Poisson-Boltzmann cell theory (circles), following the prescription proposed by Alexander [*et al.*]{} [@Alexander; @Langmuir][]{data-label="fig:10"}](zeff.eps){width="8cm"}
It is important to stress that even a minor modification of the approximations employed may have a dramatic effect on the predicted phase instability. One may wish for instance to use Eq. (\[eq:Zeff\]) for the effective charge instead of Eq. (\[3\]). It is important to note, however, that such a modification means that colloid is no longer held at fixed potential. Therefore, the grand potential $\Omega$, as written in (\[8\]), can no longer be used. However, we can compute the functional dependence of $P$ on $Z$, $\rho_p$ and $c_s$ by differentiating the Helmholtz free energy $F$ with respect to volume at [*constant*]{} $Z$, and enforcing the constraint $Z = a (4 \kappa a + 6)/\lambda_B$ [*a posteriori*]{}. Following this route, we recover the same equation of state as before \[i.e. Eq. (\[eq:pressZ\]) in the salt free case but with now a different salt dependence of $Z$\] and the critical salt concentration above which the instability sets in (see Figure \[fig:2\]), decreases by a factor of four to $\kappa_s^* a \simeq 1.77$.
![Pressure isotherms for different reservoir salt concentrations, making use of Eq. (\[eq:Zeff\]) instead of Eq. (\[3\]). The situation is now different from that of the previous constant $\varphi$ ensemble. Working is the constant $Z$ ensemble with the Helmholtz free energy (\[6\]) nevertheless allows to compute the pressure. []{data-label="fig:2"}](press6.eps){width="8cm"}
An alternative approach : a jellium approximation {#sec:jellium}
=================================================
In the previous section, we have found a fairly accurate expression for the effective colloidal charge at saturation and used it to renormalize the electrostatic free energy. It is important to remember, however, that the effective charge, is by definition related to the “far field” asymptotic properties of the electrostatic potential. Its use for the renormalization of electrostatic free energy is, therefore, questionable, since not only far field but also the near field properties of the electrostatic potential may be relevant. We now turn our attention to a simple approach which relies only on the far field features of the electrostatic potential to obtain the equation of state. As within the WS cell picture, use will be made of contact theorem, which relates the osmotic pressure to the concentration of counterions in the region where the electric field is zero[@Marcus].
We now reconsider the approach put forward at the beginning of section \[sec:theory\]. Consider one colloidal particle fixed at the origin of coordinate system. As before the charge of other microions is uniformly smeared throughout the solution. On the other hand, positions of counterions are strongly correlated with those of colloids. The system then forms a jellium, where the electrostatic potential far from the colloid $\phi_\infty$ (bulk) differs from that in the reservoir (chosen to vanish). The solution of the linearized Poisson-Boltzmann equation for $\delta \phi = \phi-\phi_\infty$ is again given by Eq. (\[1\]), where the screening length is related to the bulk salt concentration, $\kappa^2 = 4 \pi \lambda_B (\rho_+(\infty)+\rho_-(\infty))$. In the spirit of the previous discussion, the colloidal particles are held at constant surface potential $\varphi=4$ with respect to the bulk $\phi_\infty$, which again imposes $Z = 4 a (1+\kappa a)/\lambda_B$. The counterions and coions are distributed inside the jellium in accordance with the Boltzmann distribution $$\label{24a}
\rho_+(r)=c_s e^{-\beta q \phi(r)}$$ $$\label{24b}
\rho_-(r)=c_s e^{+\beta q \phi(r)}$$ where $\phi(r)$ is the local electrostatic potential with respect to the reservoir. Taking the product of Eqs.(\[24a\]) and (\[24b\]) we find the familiar condition for Donnan equilibrium $\rho_+(r) \rho_-(r) = c_s^2$. The electro neutrality constraint $\rho_+(\infty) = \rho_-(\infty) + Z \rho_p$, closes the problem[@JPCM]. The total concentration of microions inside the suspension is then related to their concentration inside the salt reservoir through $$(\kappa a)^4 = (\kappa_s a)^4 + [12 \eta (1+ \kappa a)]^2.
\label{eq:kappadonnan}$$ The osmotic pressure, within the non-linear PB theory, is determined from the concentration of microions in the region where the electric field is zero. Since the electrostatic potential decays exponentially with $r$, the electric field vanishes when $r\to \infty$. Within the jellium approximation the osmotic pressure then takes a particularly simple form, $$\label{24c}
\beta P = \rho_+(\infty) + \rho_-(\infty)=\frac{\kappa^2}{4 \pi \lambda_B}.$$ One can show that solution of Eq. (\[eq:kappadonnan\]) obeys the inequality $$\frac{\partial \kappa}{\partial \eta}\biggl|_{\kappa_s} >0$$ which ensures that the compressibility is always positive and that suspension is stable against the phase separation. In the absence of salt, we obtain a simple analytic expression $$4 \pi \lambda_B a^2 \beta P \,=\, 12 \eta \left[
1+6 \eta + \sqrt{12 \eta (1+3 \eta)}
\right],
\label{eq:pressjel}$$ that will be tested against experimental data in section \[sec:discussion\].
Discussion {#sec:discussion}
==========
Comparison with the Monte Carlo simulations
-------------------------------------------
$~~~~\lambda_B/a$ $\beta P/n$, MC $\beta P/n$, $Z=Z_{\hbox{\scriptsize eff}}$ $\beta P/n$, $Z=Z_{\hbox{\scriptsize bare}}$ $Z_{\hbox{\scriptsize eff}}/Z_{\hbox{\scriptsize bare}} $
----------------------- ------------------------- ------------------------------------------------- -------------------------------------------------- ---------------------------------------------------------------
0.022 0.98 0.99 0.99 1.0
0.044 0.95 0.96 0.97 0.99
0.089 0.89 0.88 0.92 0.95
0.178 0.71 0.68 0.78 0.82
0.356 0.45 0.41 0.46 0.53
0.712 0.26 0.21 -0.32 0.29
: Equation of state (\[eq:pressZ\]) with (third column) and without (fourth column) renormalization of volume terms, as a function of electrostatic coupling $\lambda_B/a$, for a packing fraction $\eta=0.00125$. The quantity $n$ denotes the mean total density of counterions $n=\rho_p Z_{\hbox{\scriptsize bare}}$. The MC data (second column) are taken from reference [@Linse]. Since the previous parameters do not correspond to the saturation regime of effective charges but only approach it, we have used the effective charge given by Alexander’s prescription [@Alexander] to compute the pressure from Eq. (\[eq:pressZ\]) in the third column. The corresponding ratio $Z/Z_{\hbox{\scriptsize bare}} $ is indicated in the last column[]{data-label="table:1"}
The above analysis shows that different routes to thermodynamic pressure lead to very different results. In order to decide which route is the most reliable, a comparison with “exact results” is welcome. As a benchmark, we can use the Monte Carlo (MC) pressure data of Linse [@Linse] for salt-free asymmetric electrolytes consisting of highly charged spherical macroions and point counterions. At high electrostatic couplings, this system exhibits an instability and separates into two coexisting phases of different electrolyte concentration. We shall argue that a minimum requirement for a reliable theory of phase behavior is its ability to reproduce reasonably accurately the MC equation, at least up to the transition point. This appears to be a stringent test and a necessary condition to trust any instability that a theory might predict.
We first test in table \[table:1\] the equation of state (\[eq:pressZ\]) for a charge asymmetry $Z_{\hbox{\scriptsize bare}}=40$ between colloids and counterions. It is evident that renormalization of colloidal charge significantly improves upon complete neglect of non linearities. The latter approach consists in considering $Z= Z_{\hbox{\scriptsize bare}}$ and severely fails at high $\lambda_B/a$, giving negative pressures. Our renormalized volume term captures the main effect of nonlinearities, but the agreement with MC, even if decent in view of the simplicity of the approach is nevertheless only qualitative, and does not reach the level of accuracy required to discuss phase stability.
The jellium equation of state derived in section \[sec:jellium\] only holds for saturated effective charges, i.e. in a regime of coupling that the Monte Carlo simulations, so far, have not reached (which corresponds to a very high bare charge with a large separation of scales between Bjerrum length and colloid radius, see below). We therefore directly turn to the comparison of the relative performances of the Poisson-Boltzmann cell model, symmetric PB [@Bhuiyan] and boot-strap PB (see [@BSPB] for details), with respect to Monte Carlo data (see Figures \[fig:pb1\] and \[fig:pb2\]).
To produce these figures (providing a similar comparison as Table \[table:1\]), we have chosen the lowest and the highest packing fractions investigated by Linse in [@Linse]. The striking feature revealed by Figs \[fig:pb1\] and \[fig:pb2\] is the remarkably good agreement between the PB cell pressures [@Wenner; @Holm] and the MC simulations, even at $\eta=0.00125$ where the cell model could have been anticipated to fail (see also [@Lobaskin]). The only competitive approach at $\eta = 0.08$ seems to be the boot strap PB theory [@BSPB], but this theory severely fails for low volume fractions, see Fig \[fig:pb1\], necessary to study colloidal phase-stability. At this volume fraction, the simple treatment of section \[sec:theory\] \[Eq. (\[eq:pressZ\])\], provides a better equation of state than the boot strap PB or symmetric PB (see Table \[table:1\] and Fig. \[fig:pb1\]). Within the PB cell, polyion-polyion as well as counterion-counterion correlations are discarded ; Figs \[fig:pb1\] and \[fig:pb2\] show that as far as the pressure is concerned, these contributions are small or negligible, for the parameters investigated, even at the highest couplings. This justifies their neglect in our analytical treatment.
![Monte Carlo pressures (dots) compared to those obtained within PB cell model (continuous curve), boot strap PB [@BSPB] (crosses) and symmetric PB theories [@Bhuiyan] (triangles, corresponding to virial, charging and compressibility routes). The packing fraction is $\eta=0.00125$ and $n = \rho_p Z_{\hbox{\scriptsize bare}}$. As in table \[table:1\], the charge asymmetry polyion/counterion is $Z_{\hbox{\scriptsize bare}}=40$. []{data-label="fig:pb1"}](P_et0.00125_Z40.eps){width="8cm"}
![Same as Figure \[fig:pb1\], for a packing fraction $\eta=0.08$. []{data-label="fig:pb2"}](P_et0.08_Z40.eps){width="8cm"}
Given the accuracy of the PB cell model, we are now in the position to assess the quality of the jellium approximation of section \[sec:jellium\]. The corresponding pressures are compared in Fig. \[fig:jellium\] with their PB cell counterparts for highly charged colloids (where the effective charge saturates to its upper threshold), both with and without added salt. The simple analytical expression (\[eq:pressjel\]) for salt free suspensions is found to be in good agreement with the PB data. Unfortunately the agreement deteriorates when $\kappa_s a>1$ (see the inset).
![Comparison between PB cell pressures and those for the jellium model, without added salt \[in the latter situation, the equation of state is given by expression (\[eq:pressjel\])\]. Inset: same when the suspension is dialyzed against a salt reservoir such that $\kappa_s a = 2.6$ (the quantity $P$ considered is the osmotic pressure, i.e. the reservoir contribution has been subtracted). []{data-label="fig:jellium"}](jellium.eps){width="8cm"}
Relevance for colloidal suspensions
-----------------------------------
At this point, we must conclude that the PB theory, even restricted to the cell, is superior to the competing approaches [*for aqueous suspensions with monovalent counterions*]{}. At high electrostatic couplings, corresponding to multivalent counterions in water, the MC simulations of Linse [@Linse] find an instability. This transition, however, has nothing to do with the volume terms, but is the result of strong correlations between the double layers of colloidal particles[@Levin] which produce attraction between like-charged colloids at sufficiently short separations[@Rouzina; @Levin2]. So far, this attraction has not been properly included in any of the thermodynamic theories of colloidal stability.
[*Validity of PB theory*]{}. To quantify the range of validity of the PB theory one may construct a dimensionless parameter $\Gamma_{cc}$ characterizing the importance of microions correlations, discarded within the PB theory. For monovalent microions, $\Gamma_{cc} \propto \beta q^2/(\ell \epsilon)$ where $\ell$ is the characteristic mean distance between the microions in the double layer. If the number of condensed counterions is such as to almost completely neutralize the colloidal charge, which is the case for strongly charged colloids, $\ell \simeq a / \sqrt{Z_{\hbox{\scriptsize bare}}}$, and $\Gamma_{cc}$ becomes[@Rouzina; @Levin] $$\Gamma_{cc} = \frac{\lambda_B}{a} \sqrt{\frac{Z_{\hbox{\scriptsize bare}}}{4\pi}}.$$ When $\Gamma_{cc}$ exceeds unity, PB theory is expected to break down. The value $\Gamma_{cc} \simeq 2$ has been reported to correspond to the instability threshold [@Rouzina; @Linse_corro], which and has been observed in the simulations of Linse [@Linse]. The field theoretic treatment of Netz also corroborates this conclusion [@Netz]. For particles with $Z_{\hbox{\scriptsize bare}}=40$, $\Gamma_{cc} \simeq 2$ corresponds to $\lambda_B/a \simeq 1.1$. Thus the PB theory can be expected to work quite well up to very high surface charge concentrations. Indeed, comparing the predictions of the PB cell model to the MC simulations, an excellent agreement is observed up to $\lambda_B/a \simeq 0.7$ where the MC data stop, rather close to the expected point of instability $\lambda_B/a \simeq 1.1$, see Figs. \[fig:pb1\] and \[fig:pb2\].
[*Validity of the saturation picture within the PB*]{}. The constant potential approach used in sections \[sec:theory\] and \[sec:jellium\] relies on the phenomenon of effective charge saturation exhibited by the PB theory, when $Z_{\hbox{\scriptsize bare}}$ is large enough. The saturation occurs when the electrostatic energy of the condensed counterion is significantly larger than $k_B T$. This can be characterized by a dimensionless parameter $$\Gamma_{\hbox{\scriptsize sat}} =\frac{Z_{\hbox{\scriptsize bare}}}{Z_{eff}}=\frac{Z_{\hbox{\scriptsize bare}} \lambda_B}
{4 a (1+\kappa_s a)}.$$ When $\Gamma_{\hbox{\scriptsize sat}}$ becomes larger than 1 [@JPA; @Trizac2], linearized theory fails and charge renormalization becomes important.
We must stress that large values of $\Gamma_{\hbox{\scriptsize sat}}$ are fully compatible with small values of of $\Gamma_{cc}$. Specifically, for any $a \gg \lambda_B$, there exists a range of bare colloidal charges $Z_{\hbox{\scriptsize bare}}$, such that $\Gamma_{cc}<2<\Gamma_{\hbox{\scriptsize sat}}$. For these values of $Z_{\hbox{\scriptsize bare}}$, the polyion-microion interaction is sufficiently strong that linearized theories, without charge renormalization, will certainly fail. On the other hand the counterion-counterion correlations are sufficiently weak, so that the PB theory is still applicable. To illustrate this point, we compare in Fig. \[fig:reus\] the pressure obtained at saturation within the PB cell model calculation (formally $Z_{\hbox{\scriptsize bare}} \to \infty$), to that measured experimentally by Reus [*et al*]{}, under the conditions very close to complete deionization (no salt). The agreement with PB theory had already been mentioned in [@Reus; @Trizac2], but provides an illustration of the saturation phenomenon in real suspensions. It also shows that despite its simplicity, Eq. (\[eq:pressjel\]) is fairly accurate.
It is important to stress, however, that for any colloidal size $a$, there is a maximum value of $Z_{\hbox{\scriptsize bare}}$ above which the PB theory fails. In practice, however, this break down of the PB equation never occurs for aqueous suspensions containing only monovalent counterions. The reason for this is that the hydrated ionic size provides a lower cutoff for the length $\ell$ which appears in $\Gamma_{cc}$, i.e. $\ell>d$, where $d\approx 4$ Å, is the hydrated ionic diameter. For aqueous suspension with monovalent counterions $\Gamma_{cc}$ is, therefore, always less than $\lambda_B/d$, so that $\Gamma_{cc}<2$. For monovalent counterions in water, the PB theory, therefore, should apply without any restriction. Of course, this pleasant situation changes as soon as multivalent salt is added to suspension. In this case, hydrated ionic size is no longer sufficient to restrict the strength of microion-microion correlations and $\Gamma_{cc}>2$. Under these conditions the PB theory will no longer apply and a more sophisticated approach must be used.
![Same as figure \[fig:jellium\], including a comparison with osmotic pressures of deionized bromopolystyrene particles (shown by diamonds, from ref. [@Reus]). The curve labeled “jellium” corresponds to the analytical prediction (\[eq:pressjel\]). No adjustable parameters have been used.[]{data-label="fig:reus"}](reus.eps){width="8cm"}
Summary and conclusions {#sec:conclusion}
=======================
We have proposed (section \[sec:theory\]) a linear theory to investigate the phase behavior of colloidal suspensions. The non-linear effects are partially taken into account through the postulate that highly charged polyions behave as if they were constant potential objects. The effective charge of colloidal particles is, therefore, a state dependent function. Our first goal was to develop a consistent thermodynamic approach for such state dependent charges. The results found in section \[sec:theory\] rely on a simple form for the electrostatic free energy (volume term), resulting from the polyion-microion interactions calculated using the linearized PB equation, for spherical colloids. The approach could be easily generalized to the case of cylindrical macroions. The critical behavior predicted is, however, spurious, which may be attributed to the simplicity of the volume term used, and/or the difficulty of renormalizing such terms. The jellium-like model of section \[sec:jellium\] provides a more reliable route, and allows to obtain analytically a simple equation of state for highly charged colloids in the salt free limit, see Fig. \[fig:reus\]. Unfortunately it is difficulty to see how this kind of approach can be extended to account for the polyion-polyion interactions.
>From our analysis, we conclude that for $\Gamma_{cc} = [Z_{\hbox{\scriptsize bare}}/(4 \pi)]^{1/2} \lambda_B/a < 2 $, Poisson-Boltzmann approach (PB), even restricted to the cell model, leads to more accurate predictions for the thermodynamic functions than the competing theories. This is quite remarkable, since it is by far the simplest (see the appendix of [@Langmuir] for a “ready-to-use” implementation of PB cell model). However, there is clearly a need to go beyond the PB theory when dealing with the multivalent counterions, since it is the counterion-counterion correlations that drive a phase instability for $\Gamma_{cc}>2$. Inclusion of these effects in a theoretical approach is a difficult task, since they have little signature on the pressure data up to the electrostatic coupling where suddenly they destabilize the system.
This work was supported in part by the Brazilian agencies CNPq and FAPERGS and by the french CNRS. Y. Levin acknowledges the hospitality of the Theoretical Physics Laboratory in Orsay where part of this work was performed.
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[^1]: E-mail:
[^2]: E-mail:
[^3]: E-mail:
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abstract: 'We constructed Hirota-Kimura type discretization of the classical nonholonomic Suslov problem of motion of rigid body fixed at a point. We found a first integral proving integrability. Also, we have shown that discrete trajectories asymptotically tend to a line of discrete analogies of so-called steady-state rotations. The last property completely corresponds to well-known property of the continuous Suslov case. The explicite formulae for solutions are given. In $n$-dimensional case we give discrete equations.'
author:
- Vladimir Dragović and Borislav Gajić
title: 'Hirota-Kimura type discretization of the classical nonholonomic Suslov problem'
---
Introduction
============
There are several methods for constructing discrete counterparts of integrable dynamical systems in classical mechanics. One is well-known Veselov-Moser discretization (see [@MV]). By this method, based on discrete variational principle, many of discrete integrable systems are found (see [@Sur]). In cases with discrete Lagrangian, corresponding discrete map is Poisson with respect to certain Poisson structure. Usually the discrete map is multi-valued. Recently, Hirota and Kimura constructed explicit integrable discretizations of the Euler and the Lagrange cases of motion of a heavy rigid body fixed at a point using Hirota’s bilinear method [@HK1; @HK2]. They found first integrals of motion and they solved equations in terms of elliptic functions. Suris and Petrera in [@PS] found bi-Hamiltonian structure for discrete Euler top.
Our goal is to apply Hirota-Kimura method to the classical nonholonomic Suslov problem. Suslov in [@Sus] considered motion of a rigid body fixed at a point with projection of angular velocity to an axis fixed in the body equal to zero. Solutions in terms of trigonometric and exponential functions are given as well in [@Sus]. There are various generalizations of classical Suslov problem: integrable potential perturbations ([@K; @DGJ]) and higher-dimensional generalizations([@FK; @J; @J1; @ZB]). In [@FZ] Fedorov and Zenkov presented certain discretization of Suslov problem based on the Veselov-Moser discretization and its extension to nonholonomic cases suggested in [@CM; @LMS]. Obtained map is multi-valued. Discrete trajectories asymptotically tend to discrete analogies of so-called steady-state rotations, the property characteristic for continuous case considered by Suslov. Moreover, Fedorov and Zenkov [@FZ] gave the equations in $n$-dimensional case and proved similar asymptotic behavior as in three-dimensional case.
In the present paper we use the Hirota-Kimura method and get explicit discretization of the reduced Suslov problem on the linear subspace of algebra $so(3)$. The corresponding map is not multi-valued. We gave one first integral which appears to be enough for integration. Using a linear change of variables, we have found explicite solutions in terms of exponentional functions. (Notice, that in the case of discretization given in [@FZ], the explicite solutions are not known.) Presented discrete version of Suslov case has similar asymptotic behavior as the continuous one.
The paper is organized as follows. In Section 2 basic facts about the classical Suslov problem are given. In Section 3 we present discrete equations of Hirota-Kimura type and we construct first integral of motion. Integration procedure for discrete equations is performed in Section 4. In Section 5 we give the discrete equations for $n$-dimensional Suslov case.
A brief account of the classical Suslov problem
===============================================
The classical nonholonomic Suslov problem is defined in [@Sus]. Configuration space is Lie group $SO(3)$. In a bases fixed in the body, the equations of the motion are: $$\begin{aligned}
\dot{\bold M}={\bold M}\times{\bold \Omega}+\lambda {\bold a}\\
\langle {\bold a}, {\bold \Omega} \rangle=0.
\end{aligned}
\label{s}$$ Here ${\bold \Omega}$ is the angular velocity, and ${\bold M}=I{\bold \Omega}$ is the angular momentum, $I$ is the inertia operator, ${\bold a}$ is a unit vector fixed in the body and $\lambda$ is the Lagrange multiplier. In a bases chosen such that ${\bold a}=(0,0,1)$ and $$I=\left[ \begin{matrix} I_1&0&I_{13}\\
0& I_2&I_{23}\\
I_{13}&I_{23}&I_3
\end{matrix}
\right],$$ the equations become: $$\begin{aligned}
I_1\dot \Omega_1&=-I_{13}\Omega_1\Omega_2-I_{23}\Omega_2^2\\
I_2\dot \Omega_2&=I_{13}\Omega_1^2+I_{23}\Omega_1\Omega_2\\
0&=-I_{13}\dot\Omega_1-I_{23}\dot\Omega_2+(I_1-I_2)\Omega_1\Omega_2+\lambda\\
\Omega_3&=0.
\end{aligned}
\label {s1}$$
The first two equations are closed in $\Omega_1$ and $\Omega_2$. After solving them one finds the Lagrange multiplier $\lambda$ as a function of time from the third equation. Hence, for complete integrability by quadratures, only one first integral of motion is necessary. This is the energy integral as follows from easily. Suslov in [@Sus] gave solutions of the system in terms of trigonometric and exponential functions. He observed a remarkable fact (as the referee observed to be known before to Walker and Routh): the motion of the body asymptotically tends to a line of rotations with constant angular velocities which satisfy $I_{13}\Omega_1+I_{23}\Omega_2=0$.
Hirota-Kimura type discretization of the Suslov problem
=======================================================
In the spirit of Hirota-Kimura, a discrete counterpart of the first two equations and nonholonomic constraint of is: $$\begin{aligned}
I_1&(\widetilde{\Omega}_1-\Omega_1)+I_{13}(\widetilde{\Omega}_3-\Omega_3)=\epsilon\Big[\frac{I_2}{2}(\widetilde{\Omega}_2\Omega_3+\Omega_2\widetilde{\Omega}_3)\\
&+I_{23}\Omega_3\widetilde{\Omega}_3-
\frac{I_3}{2}(\widetilde{\Omega}_2\Omega_3+\Omega_2\widetilde{\Omega}_3)-\frac{I_{13}}{2}(\widetilde{\Omega}_1\Omega_2+\Omega_1\widetilde{\Omega}_2)-
I_{23}\Omega_2\widetilde{\Omega}_2\Big]\\
I_2&(\widetilde{\Omega}_2-\Omega_2)+I_{23}(\widetilde{\Omega}_3-\Omega_3)=\epsilon\Big[\frac{I_3}{2}(\widetilde{\Omega}_1\Omega_3+\Omega_1\widetilde{\Omega}_3)\\
&-I_{13}\Omega_3\widetilde{\Omega}_3-
\frac{I_1}{2}(\widetilde{\Omega}_1\Omega_3+\Omega_1\widetilde{\Omega}_3)+\frac{I_{23}}{2}(\widetilde{\Omega}_1\Omega_2+\Omega_1\widetilde{\Omega}_2)+
I_{13}\Omega_1\widetilde{\Omega}_1\Big]\\
\widetilde{\Omega}_3&=-\Omega_3.
\end{aligned}
\label{ds}$$ Here $\Omega_i=\Omega_i(t)$, $\widetilde\Omega_i=\Omega_i(t+\epsilon)$ and $\epsilon$ is the time step. The limit when $\epsilon$ goes to $0$ should reconstruct equations of the continuous Suslov problem . Thus one concludes that $\Omega_3$ should be equal to zero, and equations become: $$\begin{aligned}
I_1(\widetilde{\Omega}_1-\Omega_1)&=\epsilon\Big[-\frac{I_{13}}{2}(\widetilde{\Omega}_1\Omega_2+\Omega_1\widetilde{\Omega}_2)-
I_{23}\Omega_2\widetilde{\Omega}_2\Big]\\
I_2(\widetilde{\Omega}_2-\Omega_2)&=\epsilon\Big[\frac{I_{23}}{2}(\widetilde{\Omega}_1\Omega_2+\Omega_1\widetilde{\Omega}_2)+
I_{13}\Omega_1\widetilde{\Omega}_1\Big]\\
\Omega_3&=0.
\end{aligned}
\label{ds1}$$ Since these equations are linear in $\widetilde{\Omega}_i$, the map defined by is explicit and unique-valued: $$\left[\begin{matrix} \widetilde{\Omega}_1\\
\widetilde{\Omega}_2
\end{matrix}\right]=
\left[\begin{matrix}1+\frac{\epsilon I_{13}}{2I_1}\Omega_2& \frac{\epsilon I_{13}}{2I_1}\Omega_1+\frac{\epsilon I_{23}}{I_1}\Omega_2\\
-\frac{\epsilon I_{13}}{I_2}\Omega_1-\frac{\epsilon I_{23}}{2I_2}\Omega_2&1-\frac{\epsilon I_{23}}{2I_2}\Omega_1
\end{matrix}\right]^{-1}
\left[\begin{matrix}\Omega_1\\ \Omega_2
\end{matrix}\right],$$ giving $$\begin{aligned}
\widetilde{\Omega}_1&=\frac{1}{\Delta}(\Omega_1-\frac{\epsilon I_{23}}{2I_2}\Omega_1^2-
\frac{\epsilon I_{13}}{2I_1}\Omega_1\Omega_2-\frac{\epsilon I_{23}}{I_1}\Omega_2^2)\\
\widetilde{\Omega}_2&=\frac{1}{\Delta}(\Omega_2+\frac{\epsilon I_{13}}{2I_1}\Omega_2^2+
\frac{\epsilon I_{23}}{2I_2}\Omega_1\Omega_2+\frac{\epsilon I_{13}}{I_2}\Omega_1^2),
\end{aligned}
\label{ds2}$$ where $$\Delta=\left(1+\frac{\epsilon I_{13}}{2I_1}\Omega_2\right)\left(1-\frac{\epsilon I_{23}}{2I_2}\Omega_1\right)+
\frac{\epsilon^2}{I_1I_2}\left(\frac{I_{13}\Omega_1}{2}+I_{23}\Omega_2\right)\left(I_{13}\Omega_1+\frac{I_{23}\Omega_2}{2}\right).$$ As we have already mentioned, in continuous case equations have the energy integral. But for considered discretization the energy is not an integral anymore, but there exists a first integral as it can be seen from the following statement.
[\[l1\]]{} The function $$F=\frac{I_1\Omega_1^2+I_2\Omega_2^2}{4I_1I_2+\epsilon^2(I_{13}\Omega_1+I_{23}\Omega_2)^2}
\label{i1}$$ is a first integral of equations
Proof follows by direct calculations.
In the limit when $\epsilon$ goes to zero, integral tends to the energy integral divided by constant.
Integration
===========
In order to integrate discrete Suslov equations, we introduce new coordinates: $$x=I_{13}\Omega_1+I_{23}\Omega_2,\ \ \ y=I_{23}I_{1}\Omega_1-I_{13}I_2\Omega_2.
\label{xy}$$ The Jacobian of the change of coordinates is: $$-(I_{13}^2I_2+I_{23}^2I_1).$$ Thus, it is equal to zero only in the case $I_{13}=I_{23}=0$ when $\widetilde{\Omega}_1=\Omega_1$ and $\widetilde{\Omega}_2=\Omega_2$, giving equilibrium position.
In the new coordinates equations are: $$\begin{aligned}
\widetilde x-x&= \frac{\epsilon}{2I_1I_2}(\widetilde xy+x\widetilde y)\\
\widetilde y-y&= -\epsilon \widetilde xx.
\end{aligned}
\label{xy1}$$ The first integral becomes $$F=\frac{I_1I_2x^2+y^2}{4I_1I_2+\epsilon^2x^2}.
\label{i2}$$ The curve $F(x,y)=h$ can be parameterized by introducing: $$\begin{aligned}
x&=2\sqrt{\frac{I_1I_2h}{I_1I_2-h\epsilon\cos^2\phi}}\cos\phi,\\
y&=2I_1I_2\sqrt{\frac{h}{I_1I_2-h\epsilon\cos^2\phi}}\sin\phi.
\end{aligned}
\label{phi}$$ Putting into the second equation of and denoting $$u=\sqrt{\frac{I_1I_2}{I_1I_2-h\epsilon^2}}\tan\phi$$ we get: $$u\sqrt{\widetilde{u}^2+1}-\widetilde{u}\sqrt{u^2+1}=\frac{2\epsilon\sqrt{I_1I_2h}}{I_1I_2-h\epsilon^2}.
\label{u}$$ Let us suppose the form of a solution of : $u(n)=\operatorname{sh}(k_1(t_0+n\epsilon)+k_2)$, where $k_1$ and $k_2$ are constants. By plugging the form into $\eqref{u}$, one gets: $$\operatorname{sh}(-k_1\epsilon)=\frac{2\epsilon\sqrt{I_1I_2h}}{I_1I_2-h\epsilon^2}.
\label{k1}$$ So, we have
\[t1\] Let constant $k_1$ satisfy and $k_2$ be arbitrary constant. Then the function $u(n)=\operatorname{sh}(k_1(t_0+n\epsilon)+k_2)$ gives solutions of equation .
As a consequence we have the following statement:
Let constant $k_1$ satisfy and $k_2$ be arbitrary constant. The functions: $$\begin{aligned}
\Omega_{1}(n)&= \frac{2\sqrt{h}}{(I_{13}^2I_1+I_{13}^2I_{2})\operatorname{ch}(k_1(t_0+n\epsilon)+k_2)}\left(\frac{I_{13}I_2}{\sqrt{I_1I_2-h\epsilon^2}}+I_{23}\operatorname{sh}(k_1(t_0+n\epsilon)+k_2)\right)\\
\Omega_{2}(n)&= \frac{2\sqrt{h}}{(I_{13}^2I_1+I_{13}^2I_{2})\operatorname{ch}(k_1(t_0+n\epsilon)+k_2)}\left(\frac{I_{23}I_1}{\sqrt{I_1I_2-h\epsilon^2}}-I_{13}\operatorname{sh}(k_1(t_0+n\epsilon)+k_2)\right)\\
\end{aligned}$$ give the solutions of equations .
From Proposition [\[t1\]]{} and one gets: $$x(n)=2\sqrt{\frac{hI_1I_2}{I_1I_2-h\epsilon^2}}\frac{1}{\operatorname{ch}(k_1(t_0+n\epsilon)+k_2)},\,\, y(n)=2\sqrt{hI_1I_2}\frac{\operatorname{sh}(k_1(t_0+n\epsilon)+k_2)}{\operatorname{ch}(k_1(t_0+n\epsilon)+k_2)}.$$ Proof follows from .
The discrete trajectories of motion of the body asymptotically tend to a line of discrete analogies of so-called steady-state rotations that satisfy: $$I_{13}\Omega_1+I_{23}\Omega_2=0.$$
In the limit $n\longrightarrow \pm\infty$, we have that $x$ goes to zero giving that $I_{13}\Omega_1+I_{23}\Omega_2$ goes to zero.
The last statement is illustrated by the following pictures. In all four cases, the line $I_{13}\Omega_1+I_{23}\Omega_2=0$ is clearly indicated. Following parameters are chosen for picture 1: $\epsilon=0.2, I_1=4, I_2=1, I_{13}=-0.5, I_{23}=-0.3$, for picture 2: $\epsilon=0.2, I_1=4, I_2=3, I_{13}=-0.4, I_{23}=-0.2$, for picture 3: $\epsilon=0.02, I_1=4, I_2=2, I_{13}=0, I_{23}=-0.2$ and for picture 4: $\epsilon=1, I_1=3, I_2=3, I_{13}=-0.2, I_{23}=-0.2$.
Equation splits on two equations: $$\widetilde{u}=-c\sqrt{u^2+1}+u\sqrt{c^2+1},\ \ \ \widetilde{u}=-c\sqrt{u^2+1}-u\sqrt{c^2+1}$$ where $c=\frac{2\epsilon\sqrt{I_1I_2h}}{I_1I_2-h\epsilon^2}$. The solutions given in Proposition \[t1\] correspond to the first equation. One easily concludes that for the second equation the limit when $\epsilon$ goes to zero is not defined well.
One can use the change of coordinates $x=I_{13}\Omega_1+I_{23}\Omega_2,
y=I_{23}I_{1}\Omega_1-I_{13}I_2\Omega_2$ also in the continuous case. Then equations become: $$\dot x=\frac{xy}{I_1 I_2},\ \ \dot y=-x^2.$$ which are simpler then original ones. One can easily see that Hirota-Kimura type discretizations of last equations are equations , as linear change of variables commutes with Hirota-Kimura type discretization.
Higher-dimensional case
=======================
As it has already been mentioned, the higher-dimensional generalization of Suslov case was suggested by Kozlov and Fedorov (see[@FK; @FZ]). The configuration space is Lie group $SO(n)$ and the nonholonomic constraints are: $$\Omega_{ij}=0,\ \ \ 1\leq i,j\leq n-1.$$ The equations of motion are: $$\dot M=[M, \Omega] +\Lambda
\label{nsus}$$ where $M=I\Omega+\Omega I$, and $$I=\left [\begin{matrix} I_{11}& 0& ...& I_{1n}\\
0&I_{22}&...&I_{2n}\\
\vdots&\vdots&...&\vdots\\
I_{1n}&I_{2n}&...&I_{nn}
\end{matrix}
\right],
\quad
\Lambda=\left [\begin{matrix} 0& \lambda_{12}& ...&\lambda_{1,n-1}& 0\\
-\lambda_{12}&0&...&\lambda_{2,n-1}&0\\
\vdots&\vdots&...&\vdots\\
-\lambda_{1, n-1}&-\lambda_{2,n-1}&...&0&0\\
0&0&...&0&0
\end{matrix}
\right],$$ From closed systems of equations in $\Omega_{in},\ 1\leq i\leq n-1$ follows: $$\begin{aligned}
(I_{ii}+I_{nn})&\dot\Omega_{in}=-I_{in}(\Omega_{1n}^2+...+\Omega_{n-1, n}^2)
&+(I_{1n}\Omega_{1n}+...+I_{n-1, n}\Omega_{n-1, n})\Omega_{in}.
\end{aligned}
\label{nsus1}$$
Similarly as in three-dimensional case, we give the Hirota-Kimura discretization in $n$ dimensions by the following system of equations: $$\begin{aligned}
(I_{ii}+I_{nn})(\widetilde{\Omega}_{in}-\Omega_{in})&=-\epsilon I_{in}(\widetilde{\Omega}_{1n}\Omega_{1n}+...+\widetilde{\Omega}_{n-1, n}\Omega_{n-1,n})\\
&+\epsilon(I_{1n}\widetilde{\Omega}_{1n}+...+I_{n-1, n}\widetilde{\Omega}_{n-1, n})\frac{\Omega_{in}}{2}\\
&+\epsilon(I_{1n}\Omega_{1n}+...+I_{n-1, n}\Omega_{n-1, n})\frac{\widetilde{\Omega}_{in}}{2}.
\end{aligned}
\label{dsn}$$
As in three-dimensional case, map defined by is explicit and unique-valued: $$\left[\begin{matrix} \widetilde{\Omega}_{1n}\\ .\\.\\.\\\widetilde{\Omega}_{n-1,n}\end{matrix}\right]=A^{-1}
\left[\begin{matrix} \Omega_{1n}\\ .\\.\\.\\\Omega_{n-1,n}\end{matrix}\right]$$ where $$A=\left[\begin{matrix} 1-\frac{\epsilon(I_{2n}\Omega_{2n}+...+I_{n-1,n}\Omega_{n-1,n})}{2(I_{11}+I_{nn})}&
...&\frac{\epsilon(2 I_{1n}\Omega_{n-1,n}-I_{n-1,n}\Omega_{1n})}{2(I_{11}+I_{nn})}\\
\frac{\epsilon(2 I_{2n}\Omega_{1n}-I_{1,n}\Omega_{2n})}{2(I_{22}+I_{nn})}&...&\frac{\epsilon (2 I_{2n}\Omega_{n-1,n}-I_{n-1,n}\Omega_{2n})}{2(I_{22}+I_{nn})}\\
\vdots\\
\frac{\epsilon(2I_{n-1,n}\Omega_{1n}-I_{1n}\Omega_{n-1,n})}{2(I_{n-1,n-1}+I_{nn})}&
...&1-\frac{\epsilon(I_{1n}\Omega_{1n}+...+I_{n-2,n}\Omega_{n-2,n})}{2(I_{n-1,n-1}+I_{nn})}
\end{matrix}
\right]$$ Let us mention that equations have a particular solution $\frac{\Omega_{1n}}{I_{1n}}=...=\frac{\Omega_{n-1,n}}{I_{n-1,n}}=c=const$, which corresponds to motion with constant angular velocity.
Acknowledgments {#acknowledgments .unnumbered}
===============
Our research is supported by the Project 144014 of the Ministry of Science of Republic of Serbia. We use the opportunity to thank Prof. Yu. N. Fedorov for useful discussions on background of the Suslov problem and its discretization from [@FZ].
[99]{}
J. Cort' es, S. Mart' inez: Nonholonomic integrators, [*Nonlinearity*]{}, [**14**]{}, (2001), 1365-1392.
V. Dragovi' c, B. Gaji' c, B. Jovanovi' c: Generalizations of classical integrable nonholonomic rigid body motion, [*J. Phys. A: Math. Gen.*]{}, [**31**]{}, (1998), 9861-9869
Yu. N. Fedorov, V. V. Kozlov: Various Aspects of $n$-Dimensional Rigid Body Dynamics, [*Amer. Math. Soc. Transl*]{}, Ser. 2, [**168**]{}, (1995), 141-171
Yu. N. Fedorov, D. V. Zenkov: Discrete nonholonomic LL systems on Lie groups, [*Nonlineairy*]{}, [**18**]{}, (2005), 2211-2241.
R. Hirota, K. Kimura: Discretization of Euler top, [*Jour. Phys. Soc. Japan*]{}, [**69**]{}, (2000), 627-630.
R. Hirota, K. Kimura, H. Yahagi: How to find conserved quantities of nonlinear discrete equations, [*J. Phys. A: Math. Gen.*]{}, [**34**]{}, (2001), 10377-10386.
B. Jovanovi' c: Some multidimensional integrable cases of nonholonomic rigid body dynamics, [*Reg. Chaotic Dynamics*]{}, [**8**]{}, (2003), 125-132.
B. Jovanovi' c: Geometry and integrability of Euler-Poincar' e-Suslov equations, [*Nonlinaerity*]{}, [**14**]{}, (2001), 1555-1567
K. Kimura, R. Hirota: Discretization of the Lagrange top, [*Jour. Phys. Soc. Japan*]{}, [**69**]{}, (2000), 3193-3199.
V. V. Kozlov: On the integration theory of equations of nonholonomic mechanics, [*Adv. Mech*]{}, [**8**]{}, (1985), 85-107, \[in Russian\].
M. de Le' on, D. Mart' in de Diego, A. Santamar' ia Merino: Geometric integrators and Nonholonomic Mechanics, [*J. Math. Phys.*]{}, 45(3), (2004), 1042-1064. q J. Moser, A.P. Veselov: Discrete version of some classical integrable systems and factorization of matrix polynomials, [*Commun. Math. Phys*]{}, [**139**]{}, (1991), 217-243.
M. Petrera, Yu. Suris: On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top, arXiv:math-ph:0707.4382v1, (2007).
Yu. Suris: [*The problem of integrable discretization: Hamiltonian approach*]{}, Progres in Mathematics, [**219**]{}, Birkhäuser Verlag, Basel, (2003).
G. Suslov: [*Theoretical mechanics*]{}, Gostekizdat, Moskow-Leningrad, (1946), \[in Russian\].
D. V. Zenkov, A. M. Bloch: Dynamics of the $n$-Dimensional Suslov problem, [*J. Geom. Phys.*]{}, [**34**]{}, (2000), 121-136.
Vladimir Dragović,\
Mathematical Institute SANU\
Kneza Mihaila 36, Belgrade\
Serbia,\
Grupo de Fisica Matematica,\
Complexo Interdisciplinar da Universidade de Lisboa\
Av. Prof. Gama Pinto, 2\
PT-1649-003 Lisboa\
Portugal\
Borislav Gaji' c\
Mathematical Institute SANU\
Kneza Mihaila 36, Belgrade\
Serbia\
|
---
author:
- Martin Aumüller
- Tobias Christiani
- Rasmus Pagh
- Francesco Silvestri
title: 'Distance-Sensitive Hashing'
---
<ccs2012> <concept> <concept\_id>10003752.10010061</concept\_id> <concept\_desc>Theory of computation Randomness, geometry and discrete structures</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003809.10010031</concept\_id> <concept\_desc>Theory of computation Data structures design and analysis</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317</concept\_id> <concept\_desc>Information systems Information retrieval</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction {#sec:introduction}
============
Optimal angular DSH {#angular}
===================
Lower bound for monotone DSH {#sec:lower:bounds}
============================
Hamming and Euclidean Space DSH {#sec:algos}
===============================
General constructions {#sec:generalconstr}
=====================
Applications {#app:applications}
============
Conclusion {#sec:conclusion}
==========
[54]{}
\#1 \#1[\#1]{}\#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{}
[AbbarAIMV13]{} . . In . .
[Abiteboul17]{} . . , (), .
[Agrawal:2001:DQP:375551.375602]{} . . In . .
[AhleAP17]{} . . In . .
[ahle2016inner]{} . . In . .
[AndoniI16]{} . . In , .
[andoni2006]{} . . In . .
[andoni2015practical]{} . . In . .
[andoni2014beyond]{} . . In . .
[andoni2017optimal]{} . . In . .
[andoni2015optimal]{} . . In . .
[andoni2016tight]{} . . In . .
[augsten2013similarity]{} . . , (), .
[BeckerDGL16]{} . . In . .
[Broder97]{} . . In . .
[broder1997syntactic]{} . . , (), .
[Charikar02]{} . . In . .
[Chierichetti15]{} . . , (), .
[Chierichetti14]{} . . (), .
[chierichetti2017distortion]{} . . In .
[christiani2017framework]{} . . In . .
[christiani2017beyond]{} . . In .
[Datar04]{} . . In . .
[de2010linear]{} . . In , Vol. . .
[Deerwester90]{} . . , (), .
[DBLP:conf/eurocrypt/FreedmanNP04]{} . . In . .
[gionis1999similarity]{} . . In . .
[goldreich2009foundations]{} . . .
[Har-PeledIM12]{} . . , (), .
[DBLP:conf/pods/HuTY17]{} . . In . .
[Indyk2003]{} . . In . .
[IndykM98]{} . . In . .
[Jain\_NIPS10]{} . . In .
[Kapralov15]{} . . In . .
[Lambert1999]{} . . , (), .
[Liu\_ICML12]{} . . In .
[motwani2007]{} . . , (), .
[neyshabur15symmetric]{} . . In . .
[odonnell2014analysis]{} . . .
[o2014optimal]{} . . , (), .
[pagh2017approximate]{} . . (), .
[pham2013fast]{} . . In . ACM, .
[pinkas2017]{} . . ().
[rahimi2007]{} . . In . .
[Riazi16]{} . . ().
[rudin1990]{} . . , .
[savage1962]{} . . , (), .
[ShrivastavaL14]{} . . In . .
[silva2010similarity]{} . . In . IEEE, .
[szarek1999]{} . . , (), .
[valiant2015finding]{} . . , (), .
[vijayanarasimhan2014hyperplane]{} . . , (), .
[wang2014lsh-survey]{} . . (). <http://arxiv.org/abs/1408.2927>
[DBLP:conf/kdd/Zhang017]{} . . In . .
|
---
abstract: 'We report on hole compact double quantum dots fabricated using conventional CMOS technology. We provide evidence of Pauli spin blockade in the few hole regime which is relevant to spin qubit implementations. A current dip is observed around zero magnetic field, in agreement with the expected behavior for the case of strong spin-orbit. We deduce an intradot spin relaxation rate $\approx$120kHz for the first holes, an important step towards a robust hole spin-orbit qubit.'
author:
- 'H. Bohuslavskyi'
- 'D. Kotekar-Patil'
- 'R. Maurand'
- 'A. Corna'
- 'S. Barraud'
- 'L. Bourdet'
- 'L. Hutin'
- 'Y.-M. Niquet'
- 'X. Jehl'
- 'S. De Franceschi'
- 'M. Vinet'
- 'M. Sanquer'
bibliography:
- 'biblio\_holeDQDv4.bib'
title: 'Pauli Blockade in a Few-Hole PMOS Double Quantum Dot limited by Spin-Orbit Interaction'
---
Since the proposal of Loss and DiVincenzo in 1998 [@Loss1998] to make quantum bits based on spins confined in semiconductor quantum dots, substantial progress has been made. First in III-V materials, where the maturity of growth techniques has allowed the emergence of top-down qubits based on the confinement of a two-dimensional electron gas in GaAs/AlGaAs hetero-structures [@Hanson2007], but also of bottom-up qubits made from nanowires (InAs or InSb) [@Nadj-Perge2010a; @Berg2013]. In all III-V qubits, the dephasing time is limited by the interaction of the electron spin with the nuclear spins present in the host material [@Bluhm2011; @Lange2010]. In contrast silicon, which presents a low natural abundance of nuclear spins and can even be isotopically purified, can be used to make electron spin qubits with extremely long dephasing time [@Maune2012; @Wu2014; @Kawakami2014; @Veldhorst2014]. An all-electrical control of single dot spin qubit by a single gate voltage microwave signal without the need of local magnetic field gradient [@Pioro-Ladriere2008] would be a clear asset for future developments. Fast and local electrical manipulation using spin-orbit interactions has already been demonstrated in III-V materials. [@Nadj-Perge2010a; @Berg2013]. Thus focusing on holes in silicon appears as an appealing strategy since valence-band holes present a limited hyperfine interaction[@Testelin2009] together with a strong spin-orbit interaction (SOI) due to their p-orbital nature. Recent experiments [@Li2015; @Voisin2016] have indeed revealed SOI-related spin properties and a hole spin qubit has even been demonstrated [@Maurand2016]. Here, we report on the implementation of a silicon hole double quantum dot (DQD) based on the technology decribed in refs. . The device is tunable in the few hole regime in which we investigate Pauli spin blockade (PSB), the key ingredient for spin qubits initialization and readout in several qubit implementations [@Hanson2007]. More specifically, we focus on the magnetic field evolution of the leakage current through the device in the PSB regime. It reveals a dip around zero magnetic field linked to spin-orbit mixing [@Danon2009]. The spin relaxation rates determined from the PSB are comparable with the values extracted for electrons in InAs nanowire double quantum dots [@Nadj-Perge2010] and are compatible with the operation of a hole spin-orbit qubit in silicon [@Maurand2016].
Our devices are nanowire field-effect transistors fabricated in a 300mm CMOS facility on silicon-on-insulator wafers with 145 nm-thick buried oxide. The 11nm-thick Si channel is doped with $ \simeq 4\times 10^{24}$ Boron.m$^{-3}$. Nanowire width down to 18 nm are achieved after patterning. Two gates (G1 and G2) in series are patterned by electron-beam lithography and are isolated from the channel by 2.5nm of SiO$_2$ and 1.9nm of HfO$_2$. Silicon nitride spacers are then deposited and etched on the sidewalls of the gates (see figure \[fig:fig1\]a,b). The spacers effectively protect the inter-gate spacing from the silicidation and dopant implantation used to reduce the access resistances [@Barraud2016]. The resulting structure, sketched in figure \[fig:fig1\]c, yields a compact DQD with optimal gate control. At low temperature, two quantum dots, QD1 and QD2, are formed by accumulation below G1 and G2 respectively (see figure \[fig:fig1\]c-d). The same process has been used to produce n-type DQD[@Kotekar2016].
![(a) Top view scanning electron micrograph of a typical DQD device after spacer etching, featuring 30nm-long gates separated by 35nm. (b) Transmission electron micrograph along the source-drain axis. (c) Schematics of the DQD made by hole accumulation below G1 and G2. (d) Schematic Pauli spin blockade for the (1h,1h)$\rightarrow$(0,2h) transition at reverse bias ([$V_{\rm{DS}}$]{}$\le$0, analogue to the (3h,3h)$\rightarrow$(2h,4h) transition in the inset of Fig. \[fig:fig4\]c). The black line is the top of the valence band. The green regions indicate the hole reservoirs in the source and drain. []{data-label="fig:fig1"}](figure1.pdf){width="\columnwidth"}
![(a) [$I_{\rm{DS}}$]{} versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} measured with [$V_{\rm{DS}}$]{}=20mV at T=60mK. White lines indicate the position of four lines of current detected at larger bias (see b). The absolute hole occupation numbers are indicated for the few hole region. The red square indicates the region studied in fig. \[fig:fig3\]a). (b) [$I_{\rm{DS}}$]{} versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} measured with [$V_{\rm{DS}}$]{}=-100mV at T=60mK in a region where no current is detected at [$V_{\rm{DS}}$]{}=20mV ((n,1)$\rightarrow$(n,2) transition line). (c)[$I_{\rm{DS}}$]{} versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} measured with [$V_{\rm{DS}}$]{}=-3mV at T=60mK in the many hole regime. (d) [$I_{\rm{DS}}$]{} versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} measured with [$V_{\rm{DS}}$]{}=-70mV at T=60mK in the region where PSB has been studied (see Fig. \[fig:fig4\]). The absolute hole occupation numbers are indicated.[]{data-label="fig:fig2"}](figure2fin.pdf){width="10cm"}
Electrical characterization was performed from T=300K down to very low temperature by recording the drain-source current [$I_{\rm{DS}}$]{} as a function of the two gate voltages [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} (stability diagram) at various drain source voltages [$V_{\rm{DS}}$]{}. In the experimental setup the source was grounded. The stability diagram shown in fig. \[fig:fig2\]a reveals overlapped bias triangles [@Wiel2002] with vertical and horizontal edges. This is a characteristic of an excellent electrostatic control of each dot by one gate. The gate capacitances associated to G1 and G2, [$C_{\rm{G1}}$]{} and [$C_{\rm{G2}}$]{} are therefore the dominant capacitances and the lever arm parameters $\frac{C_{G1}}{C_{\Sigma1}}$ and $\frac{C_{G2}}{C_{\Sigma2}}$ are close to 1 ($C_{\Sigma1}$=[$C_{\rm{G1}}$]{}+[$C_{\rm{S}}$]{}+[$C_{\rm{12}}$]{} and $C_{\Sigma2}$=[$C_{\rm{G2}}$]{}+[$C_{\rm{D}}$]{}+[$C_{\rm{12}}$]{} are the total capacitances for QD1 and QD2, [$C_{\rm{12}}$]{} being the capacitance between QD1 and QD2 and [$C_{\rm{S}}$]{} ([$C_{\rm{D}}$]{}) being the source to QD1 (drain to QD2) capacitance).
In order to precisely know $(n,m)$ -the charge state with $n(m)$ excess holes in QD1(QD2)- a large [$V_{\rm{DS}}$]{} has been applied. Even if transitions $(1,m)\rightarrow(0,m+1)$ and $(n,1)\rightarrow(n+1,0)$ have not been detected at [$V_{\rm{DS}}$]{}=20mV, they appear above $\vert$ [$V_{\rm{DS}}$]{} $\vert$ $\approx$ 100mV thanks to the enhanced tunneling through the barriers under the spacers: the latter are markedly tilted at high [$V_{\rm{DS}}$]{} so that the tunnel transparencies [$\Gamma_S$]{}, [$\Gamma_D$]{} increase significantly. In figure \[fig:fig2\]b, the drain current recorded at [$V_{\rm{DS}}$]{}=-100mV is shown in a region where no current is detected at [$V_{\rm{DS}}$]{}=20mV. This row of triangles correspond to the second (1$\rightarrow$2) transition in dot 2. The conducting parts of the triangles are replicated as a result of the ionization of dopants near the channel at large bias[@Golovach2011].
Interestingly, the charging energies are significantly larger in the few holes (up to 70 meV) than in the many holes regime ($\simeq$ 20 meV). We have, therefore, performed tight-binding calculations [@Niquet2009] in a realistic geometry in order to understand the nature of the very first low-lying hole states. The first few holes do not localize in edge states as in Ref. [@edges_NL2014] because the channel is doped with Boron atoms and the back gate is grounded, therefore the hole are not pulled in the upper corners. They might rather be bound to clusters of two or more nearby Boron impurities which exist in the doped channel. Assuming a random distribution of Boron atoms, there is indeed $>50$% (resp. $>95$%) chance of having at least two impurities closer than $d=1.5$ nm (resp. $d=2.5$ nm) under the gate. Configuration interaction calculations show that such clusters show larger binding and charging energies $E_c$ than single impurities ($E_c\sim 75$ meV at $d=1.5$ nm and $E_c<60$ meV when $d>2.5$ nm). The charging energy decreases once the deepest clusters are filled and the confinement gets dominated by the structure and gate fields. Despite doping, the SOI is mostly mediated by the silicon matrix as the probability that the holes sit on the Boron atoms is always small.
Once the first holes are added in the channel the DQD is defined by the geometry of the sample. We have simulated the stability diagram in the $(n,m)\ge(5,3)$ regime with the orthodox Coulomb blockade theory. We solved the master equation for transport [@Pierre2009] with the parameters given in table \[table\]. In addition to the capacitances defined above, we set the electronic temperature $T_e$, as well as the tunneling rates [$\Gamma_S$]{}, [$\Gamma_D$]{} and [$\Gamma_{12}$]{} associated to [$C_{\rm{S}}$]{}, [$C_{\rm{D}}$]{} and [$C_{\rm{12}}$]{}respectively. The simulation, shown in fig. \[fig:fig3\]b, reproduces the shape of the measured bias triangles.
$T_e$ 150mK
------------------------------------------------- --------------------------
[$C_{\rm{G1}}$]{}=[$C_{\rm{G2}}$]{} 7.6aF
[$C_{\rm{S}}$]{}=[$C_{\rm{D}}$]{} 0.15aF
[$C_{\rm{12}}$]{} 0.65aF
[$\Gamma_S$]{}=[$\Gamma_D$]{}=[$\Gamma_{12}$]{} $10^{-4}$$\frac{e^2}{h}$
: Numerical values used in the simulation of fig. \[fig:fig3\]b.[]{data-label="table"}
The value for [$C_{\rm{12}}$]{} is deduced from the gate voltage separation $\Delta V_{\rm G}$ between the triple points [@Wiel2002] observed at small [$V_{\rm{DS}}$]{}(see fig. \[fig:fig2\]c): $\Delta V_{\rm G}= e\frac{C_{\rm 12}}{C_{\rm G1}C_{\rm G2}} \simeq 1.8$mV. The values of [$C_{\rm{G1}}$]{} and [$C_{\rm{G2}}$]{} used in this simulation are in good agreement with a planar model for the gate capacitance of our DQD: $C_{\rm G1(2)}={\frac{\epsilon_0\epsilon_{SiO_{2}}A_{1(2)}}{EOT}}\approx$ 11aF, where $A_{1(2)}$ is the channel area covered by G1 (G2), and $EOT \simeq 2.9$ is the equivalent oxide thickness nm [@Hofheinz2006].
We now turn to the investigation of PSB. Spin blockade in a DQD arises when the current involves a transport cycle equivalent to $(0,1)\to(1,1)\to(0,2)\to(0,1)$ [@Ono2002]. Since the $(0,2)$ ground state is a spin singlet, the cycle stops as soon as the DQD enters in a (1,1) triplet state. The remaining leakage current results from spin relaxation or spin-orbit mixing mechanisms. Depending on the relevant mechanism, the leakage current will behave differently as a function of the magnetic field and detuning[@Danon2009; @Nadj-Perge2010].
Figs. \[fig:fig4\]a and \[fig:fig4\]b present current triangles in which PSB is evidenced at $T=60$mK thanks to the magnetic field dependence of the drain-source current. As expected, a reduced current is detected at the base of the bias triangles[@Hanson2007] corresponding to the $(1,3)\to(2,2)$ transition in Fig. \[fig:fig4\]a and to the $(3,3)\to(2,4)$ transition in Fig. \[fig:fig4\]b, respectively. Figs. \[fig:fig4\]c and \[fig:fig4\]d display the leakage current as a function of the out-of-plane magnetic field, $B$, and of the detuning axis in the PSB regime of Fig. \[fig:fig4\]a and Fig. \[fig:fig4\]b, respectively (detuning axis are indicated by white arrows in Figs. \[fig:fig4\]a and \[fig:fig4\]d). The leakage current decreases around $B=0$ in both cases. The current does not depend on magnetic field for the reverse polarity ($V_{\rm DS}<0$) as well as for the two other triangles shown in Fig. \[fig:fig2\]d, i.e. for $(3,2) \leftrightarrow (2,3)$ and $(2,3) \leftrightarrow (1,4)$ transitions. Note here that the $(1,1)\to(0,2)$ transition - at which PSB is also expected- was not caught even at large bias.
A cut at zero detuning taken in Fig. \[fig:fig4\]c (in Fig. \[fig:fig4\]d) is shown in figure Fig. \[fig:fig4\]e (in Fig. \[fig:fig4\]f). It reveals a current dip that can be fitted to a Lorentzian function, in line with a model assuming strong SOI[@Danon2009]: $$I = I_{\rm max}(1-{\frac{8}{9}}{\frac{B_C^2}{B_C^2+B^2}})+I_0
\label{eq:crossover}$$ with $I_{\rm max} =4e\Gamma_{\rm rel}$ the dip height, where $\Gamma_{\rm rel}$ is the spin relaxation rate among the (1,1) states, $B_{\rm C}$ is the dip width and $ I_{0}$ is a B-independent background current [@Li2015]($0.15$ pA for the (1,3)$\rightarrow$(2,2) transition and $1.3$ pA for the (3,3)$\rightarrow$(2,4) transition). $B_{\rm C}$ accounts for the cross-over between leakage currents resulting from spin relaxation at small field and spin-orbit mixing at higher field. The rate $\Gamma_{\rm SO}$ of spin-orbit mixing between (1,1) states and the (0,2) singlet can be estimated with: $$g\mu_{\rm B} B_{\rm C} \simeq h \sqrt{\Gamma_{\rm rel}\times\Gamma_{\rm SO} }
\label{eq:ct}$$
![(a) [$I_{\rm{DS}}$]{} versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} measured with [$V_{\rm{DS}}$]{}=20mV at T=60mK (region highlighted in red in fig. \[fig:fig2\]a) (b)Electrostatic simulations with the parameters given in table \[table\].[]{data-label="fig:fig3"}](figure3.pdf){width="\columnwidth"}
![Current in the PSB regime as a function of detuning and out-of-plane magnetic field $B$ at $T$=60mK. (a) Current versus [$V_{\rm{G1}}$]{} and [$V_{\rm{G2}}$]{} at [$V_{\rm{DS}}$]{}=70mV and $B$=0. (b) Same as in (a) except for [$V_{\rm{DS}}$]{}=-70mV. (c) Current versus detuning energy $\epsilon$ and magnetic field for the (1,3)$\rightarrow$(2,2) transition (white arrow in a)).(d) Same as in (c) but for the (3,3)$\rightarrow$(2,4) transition (white arrow in b)). (e) and (f) are cuts of the (1,3)$\rightarrow$(2,2) and (3,3)$\rightarrow$(2,4) transitions at $\epsilon$=0. The curves are fitted (black lines) assuming that PSB is spin-orbit mediated [@Danon2009].[]{data-label="fig:fig4"}](figure4fin.pdf){width="\columnwidth"}
Contrarily to refs. , we always see a dip of current at low magnetic field that we attribute to the dominance of spin-orbit mixing over hyperfine [@Nadj-Perge2010] or spin-flip cotunneling mechanisms [@Yamahata2012; @Morello_2011SB]. We also observe two current peaks at $B=\pm20$mT and $T= 60$mK (see Figs. \[fig:fig4\]c and \[fig:fig4\]e). Peaks of current at finite magnetic field, whose origin remains unclear, are also reported in refs. . The dip observed at zero magnetic field extends in detuning up to several meV, which indicates that the $(0,2)$ singlet-triplet splitting in our QDs -as other orbital splittings - is large. As a result PSB can be seen even at $T$=4.2K (not shown).
$\Gamma_{rel}$ and $\Gamma_{SO}$ can be estimated from the above experiments. For two QDs in series, $\Gamma_{SO}$ mainly depends on the interdot coupling. The hole $g$-factor was found to be anisotropic in similar nanowire transistors [@Voisin2016], with $g=$1.5-2.6. Eq. (\[eq:ct\]) then yields $\Gamma_{SO}$=1.4-4.3meV for the (1,3)$\rightarrow$(2,2) transition and $\Gamma_{SO}$=0.6-1.8meV for the (3,3)$\rightarrow$(2,4) transition. The spin relaxation is dominated by the spin-orbit coupling in our DQD rather than by hyperfine effects. Indeed we estimate a fluctuating Overhauser field $B_{nuc}\approx$20$\rm{\mu}$T, which is much smaller than the current dip width of $10$-$20$mT [@Yamahata2012; @Voisin2016]. $\Gamma_{SO}$ is larger than in previous reports [@Nadj-Perge2010; @Li2015] while the critical field $B_C$ is comparable to that of refs. (for electrons) and smaller than in refs. (for holes). This can be attributed to the small value of $\Gamma_{rel}$. A large $\Gamma_{SO}$ can limit qubit readout fidelity through unwanted transitions from (1,1) triplet to (0,2) singlet [@Nadj-Perge2012] and it would be favorable to reduce the interdot coupling.
$\Gamma_{rel} = 120$ kHz (resp. $ 2.0$ MHz) for the (1,3)$\rightarrow$(2,2) (resp. (3,3)$\rightarrow$(2,4)) transition is smaller than in previous reports [@Nadj-Perge2010; @Yamahata2012; @S.2013; @Li2015] where it ranges between 0.8 [@Yamahata2012] and 6MHz [@Nadj-Perge2012] (3MHz in ref. , [$I_{\rm{DS}}$]{}=6pA for $B\ge B_C$ in ref. ). $\Gamma_{rel}$, which limits the inelastic relaxation time $T_1$, should be primarily minimized for hole spin-orbit qubits. Contrarily to $\Gamma_{SO}$, $\Gamma_{rel}$ cannot be adjusted by changing the interdot coupling and its optimization is material and process dependent.
To conclude, the few-hole regime has been reached in a silicon CMOS DQD and Pauli spin blockade has been observed at different charge transitions.We found that this blockade is dominated by the SOI. By analyzing the magnetic field evolution of the leakage current in the blockade regime we deduced a small intradot spin relaxation rate ($\approx$120kHz for the first holes), an important step towards a robust hole spin-orbit qubit.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank C. Guedj and G. Audoit for extensive TEM analysis. This work is supported by the EU through the FP7 ICT SiSPIN (323841), SiAM (610637) and H2020 ICT25 MOSQUITO (688539) collaborative projects. Part of the calculations were run on TGCC/Curie thanks to a GENCI allocation.
|
---
abstract: 'If the present dark matter in the Universe annihilates into Standard Model particles, it must contribute to the gamma ray fluxes detected on the Earth. The magnitude of such contribution depends on the particular dark matter candidate, but certain features of the produced spectra may be analyzed in a rather model-independent fashion. In this communication we briefly revise the complete photon spectra coming from WIMP annihilation into Standard Model particle-antiparticle pairs obtained by extensive Monte Carlo simulations and consequent fitting functions presented by Dombriz et [*al.*]{} in a wide range of WIMP masses. In order to illustrate the usefulness of these fitting functions, we mention how these results may be applied to the so-called brane-world theories whose fluctuations, the branons, behave as WIMPs and therefore may spontaneously annihilate in SM particles. The subsequent $\gamma$-rays signal in the framework of dark matter indirect searches from Milky Way dSphs and Galactic Center may provide first evidences for this scenario.'
address:
- '$^{1}$ Astrophysics, Cosmology and Gravity Centre (ACGC), University of Cape Town, Rondebosch, 7701, South Africa.'
- '$^{2}$ Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa.'
- '$^{3}$ Departamento de Física Teórica I, Universidad Complutense de Madrid, av. Complutense s/n, E-28040 Madrid, Spain.'
author:
- 'Álvaro de la Cruz-Dombriz$^{\,1, 2}$ and Viviana Gammaldi$^{\,3}$'
title: Dark matter with photons
---
Introduction
============
According to present observations of large scale structures, CMB anisotropies and light nuclei abundances, dark matter (DM) cannot be accommodated within the Standard model (SM) of elementary particles. Indeed, DM presence is a required component on cosmological scales, but also to provide a satisfactory description of rotational speeds of galaxies, orbital velocities of galaxies in clusters, gravitational lensing of background objects by galaxy clusters and the temperature distribution of hot gas in galaxies and clusters of galaxies. The experimental determination of the DM nature will require the interplay of collider experiments and astrophysical observations. These searches use to be classified in direct or indirect searches (see [@1] and references in Introduction of [@Dombriz_wimps_PRD]). Concerning direct ones, the elastic scattering of DM particles from nuclei should lead directly to observable nuclear recoil signatures although the weak interactions between DM and the standard matter makes DM direct detection difficult.
On the other hand, DM might be detected indirectly, by observing their annihilation products into SM particles. Thus, even if WIMPs (Weakly Interacting Massive Particles) are stable, two of them may annihilate into ordinary matter such as quarks, leptons and gauge bosons. Their annihilation in different places (galactic halo, Sun, etc.) produce cosmic rays to be discriminated through distinctive signatures from the background. After WIMPs annihilation a cascade process occurs. In the end the stable particles: neutrinos, gamma rays, antimatter... may be observed through different devices. Neutrinos and gamma rays have the advantage of maintaining their original direction due to their null electric charges. This communication precisely focuses on photon production coming from WIMPs when they annihilate into SM particles. Photon fluxes in specific DM models are usually obtained by software packages such as DarkSUSY and micrOMEGAs based on PYTHIA Monte Carlo event generator [@PYTHIA] after having fixed a WIMP mass for the particular SUSY model under consideration. In this sense, the aim of this investigation is to provide fitting functions for the photon spectra corresponding to each individual SM annihilation channel and, in addition, determine the dependence of such spectra on the WIMP mass in a model independent way. This would allow to apply the results to alternative DM candidates for which software packages have not been developed. On the other hand, the information about channel contribution and mass dependence can be very useful in order to identify gamma-ray signals for specific WIMP candidates and may also provide relevant information about the photon energy distribution when SM pairs annihilate.
The paper is organized as follows: in section 2, we review the standard procedure for the calculation of gamma-ray fluxes coming from WIMP pair annihilations. Section 3 is then devoted to the details of spectra simulations performed with PYTHIA. We mention here some important issues about the final state radiation and the particular case of the top quark annihilation channel. In section 4, we introduce the fitting formulas used to describe the spectra depending upon the annihilation channel. Then, in section 5 we explicitly provide the results for two relevant annihilation channels. Section 6 is finally devoted to describe how these results may be useful in brane-world theories providing WIMP candidates.
Gamma-ray flux from DM annihilation
===================================
Let us remind that the $\gamma$-ray flux from the annihilation of two WIMPs of mass $M$ into two SM particles coming from all possible annihilation channels (labelled by the subindex $i$) is given by: $$\begin{aligned}
\label{eq:integrand}
\frac{{\rm d}\,\Phi_{\gamma}^{{\rm DM}}}{{\rm d}\,E_{\gamma}}
&=& \frac{1}{4\pi M^2}
\sum_i\langle\sigma_i v\rangle
\frac{{\rm d}\,N_\gamma^i}{{\rm d}\,E_{\gamma}}
\;\;\;\, \times \;\;\;\,
\frac{1}{\Delta\Omega} \int_{\Delta\Omega} {\rm d}\Omega
\int_{\rm l.o.s.} \rho^2 [r(s)]\ {\rm d}s\;,
\\
&&
\underbrace{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}
_{{\rm Particle~model~dependent}}\;\;\;\;\;\;\;\;\;\;
\underbrace{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}_{{\rm Dark~matter~density~dependent}}\nonumber\end{aligned}$$ where $M$ denotes the mass of the WIMP, $\langle \sigma_{i} v \rangle$ holds for the thermal averaged annihilation cross-section of two WIMPs into two ($i^{th}$ channel) SM particles and $\rho$ is the DM density. The first piece of the r.h.s. in (1) depends upon the particular particle physics model for DM annihilations. In particular, the self-annihilation cross sections are mainly described by the theory explaining the WIMP physics, whereas the number of photons produced in each decaying channel per energy interval involves decays and/or hadronization of unstable products, for instance quarks and gauge bosons. Consequently, the detailed study of these decay chains and non-perturbative effects related to QCD is an almost impossible task to be tackled by any analytical approach. The second piece in (\[eq:integrand\]) is a line-of-sight (l.o.s.) integration through the DM density distribution of the target and averaged over the detector solid angle $\Delta\Omega$. Let us discuss each of these pieces separately:
Particle Physics model
----------------------
Although annihilation cross sections are not known, they are restricted by collider constraints and direct detection. In addition, the thermal relic density in the range $\Omega_{\rm CDM}
h^2 = 0.1123\pm 0.0035$ which is determined by fitting the standard $\Lambda$CDM model to the WMAP7 data [@Komatsu:2010fb], the latest measurements from the BAO in the distribution of galaxies [@Percival:2009] and the Hubble constant measurement [@Riess:2009pu], do not allow arbitrary contributions from the DM gamma ray fluxes.
As already mentioned, the annihilation of WIMPs is closely related to SM particle production. The time scale of annihilation processes is shorter than typical astrophysical scales. This fact implies that only stable or very long-lived particles survive to the WIMP annihilations and may therefore be observed by detectors.
For most of the DM candidates, the production of mono-energetic photons is very suppressed. The main reason for such a suppression comes from the fact that DM is neutral. Therefore, the gamma-ray signal comes fundamentally from secondary photons originated in the cascade of decays of gauge bosons and jets produced from WIMP annihilations. These annihilations would produce in the end a broad energy distribution of photons, which would be difficult to be distinguished from the background. However, the directional dependence of the gamma ray intensity coming from these annihilations is mainly localized in point-like sources which may provide a distinctive signature. All those channels contributions produce a broad energy gamma ray flux, whose maximum constitutes a potential signature for its detection. On the other hand, a different strategy can be followed by taking into account the fact that the cosmic ray background is suppressed at high energies. Primary photons coming from the Weicksäcker-Williams radiation dominate the spectrum at energies close to the mass of the DM candidate and their signature is potentially observable as a cut-off [@cutoff]. This approach is less sensitive to electroweak corrections which may be important if the mass of the DM candidate is larger than the electroweak scale [@Ciafaloni:2010ti].
DM density directionality
-------------------------
The line of sight integration can be obtained from: $$\langle J \rangle_{\Delta \Omega} \doteq \frac{1}{\Delta \Omega}
\int_{\Delta \Omega} J(\psi) {\rm d}\Omega = \frac{2\pi}{\Delta\Omega}
\int_0^{\theta_{\rm max}} {\rm d}\theta\,\sin\theta
\int_{s_{\rm min}}^{s_{\rm max}}
{\rm d}s\,\rho^2 \left(\sqrt{s^2+s_0^2-2 s s_0 \cos\theta}\right)
\label{eq:jav}$$ where $J(\psi) = \int_{\rm l.o.s.} {\rm d}s\,\rho^2(r)$. The angled brackets denote the averaging over the solid angle $\Delta \Omega$, and $s_{\rm min}$ and $s_{\rm max}$ are the lower and upper limits of the line-of-sight integration: $s_0
\cos\theta \pm \sqrt{r_t^2 - s_0^2 \sin^2\theta}$. In this formula $s_0$ is the heliocentric distance and $r_t$ is the tidal radius.
Traditionally, the galactic center has attracted the attention of this type of directional analysis since standard cusped Navarro-Frenk-White halos predict the existence of a very important amount of DM in that direction [@stoehr]. However, this assumption is in contradiction with a substantial body of astrophysical evidences [@evidences], and a core profile is not sensitive to standard DM candidates. On the contrary, cusped profiles are not excluded for the Local Group dwarf spheroidals (dSphs) that constitute interesting targets since they are much more dominated by DM. In this way, directional analysis towards Canis Major, Draco and Sagittarius or Segue 1 [@dSphs] are more promising.
In any case, galaxy clusters are also promising targets [@Ref2_referee]. Other alternative strategy takes advantage of the large field of view of FERMI, that may be sensitive to the continuum photon flux coming from DM annihilation at moderate latitudes ($|b| > 10^\circ$) [@stoehr]. Other proposed targets, as the Large Magellanic Cloud [@olinto], are less interesting since their central parts are dominated by baryonic matter.
Procedure
=========
In this section, we explicitly specify how gamma rays spectra have been generated and we will discuss some issues concerning the final state radiation and the top quark channel particularities.
Spectra generation
------------------
Throughout this investigation, we have used the particle physics PYTHIA software \[3\] to obtain our results. The WIMP annihilation is usually split into two separated processes: The first describes the annihilation of WIMPs and its SM output. The second one considers the evolution of the obtained SM unstable products. Due to the expected velocity dispersion of DM, most of the annihilations happen quasi-statically. This fact allows to state that by considering different center of mass (CM) energies for the obtained SM particles pairs from WIMP annihilation process, we are indeed studying different WIMP masses, i.e. $E_{{\rm CM}} \simeq 2\,M$. The procedure to obtain the photon spectra is thus straightforward: For a given pair of SM particles which are produced in the WIMP annihilation, we count the produced number of photons. Statistics have to be large enough, in particular for highly energetic photons usually suppressed when not high enough number of annihilations is simulated.
Final State Radiation
---------------------
If the final state in the annihilation process contains charged particles, there is a finite probability of emission of an additional photon [@Bringmann]. In principle there are two types of contributions: that coming from photons directly radiated from the external legs, which is the final state radiation we have considered in the work, and that coming from virtual particles exchanged in the WIMP annihilation process. The first kind of contribution can be described for relativistic final states by means of an universal Weizsäcker-Williams term fundamentally independent from the particle physics model [@Bringmann]. On the other hand, radiation from virtual particles only takes place in certain DM models and is only relevant in particular cases, for instance, when the virtual particle mass is almost degenerate with the WIMP mass. Even in these cases, it has been shown [@Cannoni] that although this effect has to be included for the complete evaluation of fluxes of high energy photons from WIMP annihilation, its contribution is relevant only in models and at energies where the lines contribution is dominant over the secondary photons. For those reasons and since the aim of the present work is to provide model independent results for photon spectra, only final state radiation was included in our simulations.
The case for $t$ quark decay
----------------------------
The decay of top ($t$) quark is not explicitly included in PYTHIA package. We have approximated this process by its dominant SM decay, i.e. each (anti) top decays into $W^{+(-)}$ and (anti) bottom. In order to maintain any non-perturbative effect, we work on an initial four-particle state composed by $W^{+} b$ coming from the top and $W^{-} \bar{b}$ from anti-top, which keeps all kinematics and color properties from the original pair. Starting from this configuration, we have forced decays and hadronization processes to evolve as PYTHIA does and therefore, the gamma rays spectra corresponding to this channel have also been included in our analysis.
In order to verify the validity of these results, further calculations were made [@Roberto_2011] by including hadronic string between $b\bar{b}$ and improving statistics in the high energy photons range. Conclusions about $t$ quark channel remained unchanged with respect to the ones in [@Dombriz_wimps_PRD].
Analytical fits to PYTHIA simulation spectra
============================================
In this section we present the fitting functions used for the different channels. According to the PYTHIA simulations described in the previous section, three different parametrizations were required in order to fit all available data from the studied channels: one for quarks (except top quark) and leptons, a second one for $W$ and $Z$ gauge bosons and a third one for top quark.
The parameters in the following expressions were considered in principle to be WIMP mass dependent and their mass dependences were fitted by using power laws.
Quarks and leptons
------------------
For quarks (except the top), $\tau$ and $\mu$ leptons, the most general formula needed to reproduce the behavior of the differential number of photons per photon energy may be written as: $$\begin{aligned}
x^{1.5}\frac{{\rm d}N_{\gamma}}{{\rm d}x}\,=\, a_{1}{\rm exp}\left(-b_{1} x^{n_1}-b_2 x^{n_2} -\frac{c_{1}}{x^{d_1}}+\frac{c_2}{x^{d_2}}\right) + q\,x^{1.5}\,{\rm ln}\left[p(1-x)\right]\frac{x^2-2x+2}{x}
\label{general_formula_1}\end{aligned}$$ In this formula, the logarithmic term takes into account the final state radiation through the Weizsäcker-Williams expression [@Hooper2004; @Bringmann]. Nevertheless, initial radiation is removed from our Monte Carlo simulations in order to avoid wrongly counting their possible contributions.
Strictly speaking, the $p$ parameter in the Weizsäcker-Williams term in the previous formula is $(M/m_{particle})^2$ where $m_{particle}$ is the mass of the charged particle that emits radiation. However in our case, it will be a free parameter to be fitted since the radiation comes from many possible charged particles, which are produced along the decay and hadronization processes. Therefore we are encapsulating all the bremsstrahlung effects in a single Weizsäcker-Williams-like term.
Concerning the $\mu$ lepton, the expression above (\[general\_formula\_1\]) becomes simpler since the exponential contribution is absent. Thus its flux becomes $$\begin{aligned}
x^{1.5}\frac{{\rm d}N_{\gamma}}{{\rm d}x}\,=\, q\,x^{1.5}\,{\rm ln}\left[p(1-x^{l})\right]\frac{x^2-2x+2}{x}
\label{general_formula_mu}\end{aligned}$$ where the $l$ parameter in the logarithm is needed in order to fit the simulations as will be seen in the corresponding sections.
Parameters in expression (\[general\_formula\_1\]) are channel dependent as can be found in [@Dombriz_wimps_PRD]. Depending on the studied channel, these parameters may be either dependent or independent from the studied WIMP mass. For instance, the case for $c$ quark was studied in [@Dombriz_QCHSIX].
$W$ and $Z$ bosons
------------------
For the $W$ and $Z$ gauge bosons, the parametrization used to fit the Monte Carlo simulation is: $$\begin{aligned}
x^{1.5}\frac{{\rm d}N_{\gamma}}{{\rm d}x}\,=\, a_{1}\,{\rm exp}\left(-b_{1}\, x^{n_1}-\frac{c_{1}}{x^{d_1}}\right)\left\{\frac{{\rm ln}[p(j-x)]}{{\rm ln}\,p}\right\}^{q}
\label{general_formula_W_Z}
% FORMULA FOR W AND Z.\end{aligned}$$ This expression differs from the expression (\[general\_formula\_1\]) in the absence of the additive logarithmic contribution. Nonetheless, this contribution acquires a multiplicative character. The exponential contribution is also quite simplified with only one positive and one negative power laws. Moreover, $a_1$, $n_1$ and $q$ parameters appear to be independent of the WIMP mass. The rest of parameters, i.e., $b_1$, $c_1$, $d_1$, $p$ and $j$, are WIMP mass dependent and were determined in [@Dombriz_wimps_PRD] for each WIMP mass and for the $W$ and $Z$ separately. In both cases the covered WIMP mass range was from $100$ to $10^4$ ${\rm GeV}$. Nonetheless, for masses higher than 1000 GeV, no significant change in the photon spectra for both particles [@Dombriz_wimps_PRD] was observed.
$t$ quark
---------
Finally, for the top channel, the required parametrization turned out to be: $$\begin{aligned}
x^{1.5}\frac{{\rm d}N_{\gamma}}{{\rm d}x}\,=\, a_{1}\,{\rm exp}\left(-b_{1}\, x^{n_1}-\frac{c_{1}}{x^{d_1}}-\frac{c_{2}}{x^{d_2}}\right)\left\{\frac{{\rm ln}[p(1-x^{l})]}{{\rm ln}\,p}\right\}^{q}
\label{general_formula_t}\end{aligned}$$ Likewise the previous case for $W$ and $Z$ bosons, gamma-ray spectra parametrization for the top is quite different from that given by expression (\[general\_formula\_1\]). This time, the exponential contribution is more complicated than the one in expression (\[general\_formula\_W\_Z\]), with one positive and two negative power laws. Again, the additive logarithmic contribution is absent but it acquires a multiplicative behavior. Notice the exponent $l$ in the logarithmic argument, which is required to provide correct fits for this particle. Moreover, $a_1$, $c_1$, $d_1$ and $d_2$ parameters appear to be independent of the WIMP mass. The rest of parameters, i.e., $b_1$, $n_1$, $c_2$, $p$, $q$ and $j$, are WIMP mass dependent and were determined in [@Dombriz_wimps_PRD]. The covered WIMP mass range for the top case was from $200$ to $10^5$ GeV. Nevertheless, at masses higher than 1000 GeV it was observed again [@Dombriz_wimps_PRD] that there is no significant change in the gamma-ray spectra. Consistency of this result was verified in [@Roberto_2011].
Some results: $\tau$ lepton and $t$ quark
=========================================
In order to illustrate the explained procedure, we present here some representative annihilation channels: $\tau$ lepton and $t$ quark.
For $\tau$ lepton channel the studied mass range was from 25 GeV to 50 TeV. The mass dependent parameters in expression (\[general\_formula\_1\]) are only $n_1$ and $p$ whereas $a_1$, $b_1$, $b_2$, $n_2$, $c_1$, $d_1$, $c_2$, $d_2$ and $q$ are mass independent. Figure 1 presents $\tau$ channel spectra for 1 and 50 TeV WIMP masses.
Finally, for $t$ quark channel the studied mass range was from 200 GeV to 10 TeV although the spectra are the same from 1 GeV onwards. The mass dependent parameters in expression (\[general\_formula\_t\]) are $b_1$, $n_1$, $c_2$, $p$, $q$ and $l$ whereas $a_1$, $c_1$, $d_1$ and $d_2$ are mass independent. Figure 2 presents $t$ channel spectra for two different WIMP masses, 500 and 1000 GeV.
{height=".235\textheight"} {height=".235\textheight"}
{height=".235\textheight"} {height=".235\textheight"}
Brane-world theory as an example
================================
It has been suggested that our universe could be a 3-dimensional brane where the SM fields live embedded in a D-dimesional space-time. In flexible braneworlds, in addition to the SM fields, new degrees of freedom appear on the brane associated to brane fluctuations, that is the branons. In brane-world models with low tension, branons appear to be massive and weakly interacting fields, so natural candidates to DM [@CDM; @BR; @ACDM]. Therefore, their annihilations by pairs may produce SM particles and gamma photons by the subsequent processes of hadronization and decay. Limits on the model parameters, the WIMPs mass $M$ and the tension scale $f$, are given both from collider experiments and indirect search of DM. In particular, the self-annihilation cross section of branons depends on the two parameters of the model [@CDM; @CDM_cross_sections]. In the case of heavy branons, neglecting three body annihilations and direct production of two photons, the main contribution to the photon flux comes from branon annihilation into $ZZ$ and $W^+ W^-$ (see Figure 1 in [@gammaflux]) according to the expression $$\begin{aligned}
\langle\sigma_{Z,W} v\rangle=\frac{M^2\sqrt{\,1-\left(\frac{m_{Z,W}}{M}\right)^2}
\left(4M^4-4M^2m_{Z,W}^2+3m_{Z,W}^4\right)}{64f^8\pi^2}.
\label{Cross_section_WZ}\end{aligned}$$ The contribution from heavy fermions, i.e. annihilation in $t\bar{t}$ channel, can be shown to be subdominant [@indirect]. Therefore, expression (\[Cross\_section\_WZ\]) represents the self-annihilation cross section to be considered in (\[eq:integrand\]) for the study of these theories.
The astrophysical part of (\[eq:integrand\]) depends as already mentioned on both the performed experiment and DM profile of the source. $<J>_{\Delta\Omega}$ value is approximately $10^{23}\,{\rm GeV^2 cm^{-5} sr^{-1}}$ for dSphs galaxy, but strongly dependent also from the distance of the source for a given DM profile. The technical details of the different experiments and the value of the background also affect the minimum detectable gamma ray flux. The minimum expected value of this flux as coming from a given source and instrument may be given by the following expression $$\frac{\Phi_\gamma\sqrt{\Delta\Omega A_{eff}t}}{\sqrt{\Phi_\gamma+\Phi_{Bg}}} \geq 5,
\label{minflu}$$ By integrating expression (\[eq:integrand\]) over the energy threshold of the selected device, an estimation of $N_\gamma<\sigma v>$ can be found and matched with the expected one [@gammaflux] depending on the theoretical model. This procedure allows to select the most promising target to be investigated with current ground-based or satellites experiments (MAGIC [@Mag11], EGRET [@EGRET], FERMI [@Fer]) or with a new generation of them (CTAs [@CTA]). Refer to [@gammaflux] for further details.
Conclusions
===========
We have presented the model-independent fitting functions for the photon spectra coming from WIMPs pair annihilation into Standard Model particle-antiparticle pairs for all the phenomenologically relevant channels. This analysis is model independent and therefore, provided a theoretical model our formulas make it possible to obtain the expected photon spectrum in a relatively simple way. Explicit calculations for all studied channels \[2\] are available at the websites [@Mathematica_code] and [@Fortran_code].
Acknowledgments
---------------
AdlCD acknowledges financial support from National Research Foundation (NRF, South Africa), MICINN (Spain) project numbers FIS 2008-01323, FPA 2008-00592 and MICINN Consolider-Ingenio MULTIDARK CSD2009-00064. AdlCD is particularly grateful to PHOTON11 organizing committee for inviting him to present these results. VG acknowledges financial support from MICINN (Spain) Consolider-Ingenio MULTIDARK CSD2009-00064.
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http://teorica.fis.ucm.es/$\sim$PaginaWeb/downloads.html
http://teorica.fis.ucm.es/$\sim$PaginaWeb/descargas/damasco.C
|
---
abstract: 'Matter-wave interferometry with atoms [@Cronin2009] and molecules [@Hornberger2012] has attracted a rapidly growing interest over the past two decades, both in demonstrations of fundamental quantum phenomena and in quantum-enhanced precision measurements. Such experiments exploit the non-classical superposition of two or more position and momentum states which are coherently split and rejoined to interfere [@Estermann1930; @Keith1988; @Kasevich1991; @Borde1989; @Rasel1995a; @Fray2004; @Mueller2008b; @Moskowitz1983; @Giltner1995]. Here, we present the experimental realization of a universal near-field interferometer built from three short-pulse single-photon ionization gratings [@Reiger2006; @Nimmrichter2011]. We observe quantum interference of fast molecular clusters, with a composite de Broglie wavelength as small as 275 fm. Optical ionization gratings are largely independent of the specific internal level structure and are therefore universally applicable to different kinds of nanoparticles, ranging from atoms to clusters, molecules and nanospheres. The interferometer is sensitive to fringe shifts as small as a few nanometers and yet robust against velocity-dependent phase shifts, since the gratings exist only for nanoseconds and form an interferometer in the time domain.'
author:
- Philipp Haslinger
- 'Nadine D[ö]{}rre'
- Philipp Geyer
- Jonas Rodewald
- Stefan Nimmrichter
- Markus Arndt
bibliography:
- 'otimaref.bib'
title: 'A universal matter-wave interferometer with optical ionization gratings in the time domain'
---
Recent progress in atom interferometry has been driven by the development of wide-angle beam splitters [@Chiow2011], large interferometer areas [@Lan2012] and long coherence times [@Mueller2008]. Most interferometers operate in a Mach-Zehnder [@Kasevich1991; @Keith1991], $\text{Ramsey-Bord\'{e}}$ [@Borde1994] or Talbot-Lau [@Clauser1994] configuration, some of them also in the time-domain [@Szriftgiser1996; @Cahn1997]. Here we ask how to generalize these achievements to atoms, molecules, clusters or nanoparticles - independent of their internal states.
Mechanical nanomasks [@Juffmann2012] could be considered as universal if it were not for their van der Waals attraction on the traversing matter waves, which induces sizable dispersive, that is, velocity-dependent, phase shifts even for gratings as thin as 10 nm.
Optical [@Mueller2008b; @Chiow2011] or measurement-induced [@Storey1992] gratings eliminate this effect, but most methods so far relied on closed transitions and required an individual light source for every specific kind of atom or molecule.
It is possible to circumvent this restriction by using the spatially periodic electric dipole potential in an off-resonant standing light wave. Its field then modulates the phase of the matter wave rather than the amplitude. This implies, however, that the spatial coherence of the incident matter wave needs to be prepared by other means before - such as by collimation, cooling [@Deng1999] or the addition of another absorptive (material) mask [@Hornberger2012].
Here, we demonstrate a new method for coherence experiments with a wide class of massive particles and show how a sequence of ionizing laser grating pulses [@Reiger2006] can form a generic matter-wave interferometer in the time-domain [@Nimmrichter2011].
{width="65.00000%"}
Figure \[fig1\] shows a schematic of the layout of our experiment, which we here realize specifically for clusters of anthracene (Ac) molecules. The molecules are evaporated in an Even-Lavie valve [@Even2000] that injects the organic vapor with a pulse width of about 30 s into the vacuum chamber. The adiabatic co-expansion with a noble gas cools the molecules and fosters the formation of organic clusters - here typically up to $\text{Ac}_{15}$.
The bunch of neutral nanoparticles passes a differential pumping stage, enters the interferometer chamber and flies in a short distance (0.1 - 4 mm) from the surface of a super-polished $\text{CaF}_{2}$ mirror before it reaches the laser ionization region of a time-of-flight mass spectrometer (ToF-MS) where it creates the signal peaks.
The pulsed beams of three synchronized $\text{F}_2$-excimer lasers ($\lambda$ = 157.63 nm) hit the mirror surface and the cluster beam under normal incidence with a variable pulse energy of 1 - 3 mJ and a duration of about 7 ns. The laser beams are separated in space by $\sim$ 20 mm along the cluster trajectory. Their mutual time delay is adjusted with an accuracy of a few nanoseconds. We choose the laser beam diameters ($\sim$ 1 mm $\times$ 10 mm rectangular flat top, extended along the cluster beam) to cover a wide particle bunch emitted by the source, whereas the detection laser beam is narrow enough to post-select only those clusters that have interacted with all three laser light pulses.
{width="\textwidth"}
All three laser gratings interact with the matter waves in two different ways [@Nimmrichter2011]: they imprint a periodic phase and, more importantly, they act as transmission gratings because the photon energy of $\sim$ 7.9 eV exceeds the ionization energy of the nanoclusters. Particles that traverse the antinodes of a laser grating ionize with high probability after absorption of one or more photons and a weak electric field removes them from the beam. Close to the nodes of the standing light waves the clusters remain neutral and move on in the interferometer. This process imprints a periodic modulation onto the matter-wave amplitude - as if the clusters had passed a mechanical nanomask.
A strong spatial localization inside the first laser grating is important for preparing a comb of emergent wavelets whose transverse coherence will cover a few antinodes in the second light grating further downstream. This is a prerequisite for interference to occur, that is, for the formation of a free-flying cluster density pattern at precisely defined moments in time, which is probed with nanosecond precision by the third ionizing standing wave.
The three laser pulses form a Talbot-Lau interferometer in the time domain, which exhibits transmission resonances when the delay between two subsequent pulses is close to the Talbot time $\text{T}_\text{m} = \text{md}^{2}\text{/h}$, with m the cluster mass and h Planck’s constant. In our setting the grating period d =$\lambda$/2 = 78.8 nm results in $\text{T}_\text{m}$ = 15 ns/amu. All particles see the same gratings at the same time irrespective of their velocity. Even though they may enclose different areas in real-space ($x-z$), they will accumulate the same phase and contribute constructively to the same interferogram for each given mass (Figure 1b).
We trace the emergent interference pattern in four different ways: its mass characteristics, its dependence on the pulse separation and pulse sequence asymmetry, and by visualizing its structure in position space.
![\[fig3\] **Interferometric resonance and timing precision.** Cluster self-imaging in a pulsed near-field interferometer is a resonant process with a short acceptance window for the matter waves to rephase. **(a)** Pulse sequence and **(b)** difference $\Delta \text{S}_\text{N}$ between the resonant and off-resonant signals detected at a mass of $\text{Ac}_7$ as a function of $\Delta$T. In our setup and for a pulse separation time T of 18.9 s interference occurs during a time window of 48 ns (FWHM). The error bars represent one standard deviation of statistical error (see Suppl. Inf. h). ](figure3.eps){width="\columnwidth"}
We start by monitoring the ToF-MS signal and toggle between a resonant and a non-resonant setting. In the resonant mode the delays $\text{t}_2-\text{t}_1$ = T, $\text{t}_3-\text{t}_2$ = T + $\Delta \text{T}$ between two subsequent laser pulses are equal, $\Delta$T = 0, and quantum interference is expected to modulate (enhance or reduce, depending on the phase) the transmission for the mass whose Talbot time matches the pulse separation T. In the off-resonant mode, the pulse delays are imbalanced by $\Delta$T = 200 ns and this tiny mismatch suffices to destroy the interferometric signal. We extract the interference contrast from the normalized difference $\Delta \text{S}_\text{N} = (\text{S}_\text{R} - \text{S}_\text{O}) / \text{S}_\text{O}$ between the resonant $\text{S}_\text{R}$ and the off-resonant signal $\text{S}_\text{O}$ and plot it as a function of mass in Figure \[fig2\]. The experimental mass spectra and $\Delta \text{S}_\text{N}$ bars (green) can be well understood by a quantum mechanical model (violet bars), as described in the Methods Section, and both are in marked discrepancy with a classical model (grey bars) [@Nimmrichter2011].
The role of the pulse separation T is demonstrated by changing the seed gas from argon to neon. Shifting the most probable jet velocity from 690 to 925 m$\text{s}^{-1}$ allows us to decrease T. The quantum model then predicts the highest contrast to occur at smaller masses, as confirmed by the experimental data in Figure 2b.
Figure \[fig3\] shows a clear resonance in $\Delta \text{S}_\text{N}$ as a function of the time imbalance $\Delta \text{T} \in$ \[-70, +70\] ns with a width determined by the transverse momentum distribution of the cluster beam [@Nimmrichter2011]. The momentum spread inferred from a Gaussian fit to the data in Figure \[fig3\] corresponds to a divergence angle along the grating of 2.1 mrad, in good agreement with the experimental settings.
In our set-up, the pulsed supersonic expansion determines the cluster velocity distribution and the pulsed mass detection post-selects its relative width to $\Delta$v/v $\simeq$ 3 $\%$. It is then justified to interpret the observations in position space: With the de Broglie wavelength given by $\lambda_{\text{dB}}$ = h/mv the mass distribution also represents a wavelength spectrum. The most prominent interference peak in Figure 2b at 1248 amu corresponds to the heptamer $\text{Ac}_7$ with $\lambda_{\text{dB}}$ $\simeq$ 345 fm, at v $\simeq$ 925 m$\text{s}^{-1}$. The highest mass peaks in the spectrum reach down to below $\lambda_{\text{dB}}\simeq$ 275 fm.
![\[fig4\] **$\Delta \text{S}_\text{N}$ as a function of the mirror displacement for different clusters.** The second grating laser beam was tilted by 5.1 $\pm$ 0.3 mrad in the direction of the molecular beam to stretch the effective grating period by about 0.013 per mille. This suffices to induce a fringe shift of half a grating period for molecules travelling around 1.5 mm distance from the mirror surface. The mirror height is varied to effectively shift the second grating with regard to the other two which allows us to scan the cluster interference pattern. We extract the periodicity for $\Delta \text{S}_\text{N}$ as a function of the mirror distance by fitting a damped sine curve to the experimental data. This periodicity corresponds to the expected effective period [@Nimmrichter2011] of the interferogram of 80 nm. The error bars represent one standard deviation of statistical error (see Suppl. Inf. h). ](figure4.eps){width="\columnwidth"}
Finally, we can also prove the formation of an interference pattern in real space by modifying the period of the central grating: While all laser beams had originally been set to normal incidence on the interferometer mirror - with an uncertainty of about 200 rad - we now explicitly tilt the central laser beam by 5.1 mrad along the cluster beam. The direction of the standing-light-wave grating remains defined by the orientation of the mirror surface, but an increasing tilt angle $\theta$ reduces the modulus of the wave vector perpendicular to the surface, $\text{k}_\text{p} \simeq \text{k}\cdot\cos \theta$. We can shift the interference pattern by half a grating period when the clusters pass the mirror surface at an average distance of 1.5 mm. We plot the fringe shift as a function of the separation between the beam and the mirror in Figure \[fig4\] and find a damped sinusoidal transmission curve for all clusters with the expected period. The overall damping results from the limited coherence of the laser system and the vertical extension of the Ac cluster beam.
All tests presented here confirm the successful experimental realization of an **o**ptical **t**ime-domain **i**onizing **ma**tter-wave (OTIMA) interferometer [@Hornberger2012; @Nimmrichter2011], which exploits pulsed ionization gratings. This versatile tool for quantum interferometry will be applicable to a large class of nanoparticles. Owing to the pulsed gratings, all phase shifts caused by constant external forces become velocity-independent and leave the contrast unaffected. The dispersive Coriolis shift [@Lan2012] can be well compensated by a suitable orientation of the interferometer, if needed.
The wide applicability and non-dispersive nature of pulsed ionization gratings make the OTIMA design particularly appealing for quantum experiments with highly complex particles, eventually even with nanoparticles at the length scale of the grating period. As high-mass interferometry requires coherence of the order of the Talbot time, practical mass limits are imposed by free fall in the gravitational field on Earth in combination with the limited coherence of vacuum ultraviolet lasers and the finite phase-space density of the available particle sources. However, none of them is fundamental. Even in the presence of thermal radiation at room temperature (particle and environment) and collisional decoherence at a background pressure of $10^{-9}$ mbar, the OTIMA design is predicted to enable new tests of quantum physics, such as tests of spontaneous localization, with particle masses around $10^6$ amu and beyond [@Nimmrichter2011b].
On the applied side, the OTIMA set-up is expected to improve the accuracy of molecule and cluster deflectometry because it ensures the same interaction (phase accumulation) time for all particles with the external fields [@Heer2011] and a position readout at the nanometre scale. Our interferometer concept therefore establishes also the basis for a new class of quantum-enhanced precision metrology experiments.
Methods {#methods .unnumbered}
=======
*Absorption and optical polarizability.* The central grating influences the propagation of the coherent matter wave by modulating both its amplitude and phase. It does this by removing particles from the anti-nodes of the standing light field and by imprinting a phase onto the matter wave in proportion to the clusters’ optical polarizability at 157 nm. In the first and third grating the phase modulation has no effect, since the clusters enter with random phases, and since the last grating merely acts as a transmission mask. Neither the absorption cross sections $\sigma_{157}$(N) nor the polarizabilities $\alpha_{157}$(N) are known, a priori, for each cluster of N molecules in the vacuum ultraviolet wavelength range. However, $\sigma_{157}$(N) enters the model only through the mean number of photons absorbed $\text{n}_0\left(\text{N}\right)$ in each grating which we can determine by monitoring the cluster loss rate. While this parameter influences the general shape of the interference curve as a function of mass, the polarizability may modify the predicted contrast of each individual cluster. We assume the polarizability and the absorption cross section to exhibit the same N-scaling as retrieved from our $\text{n}_0\left(\text{N}\right)$ measurements and we allow the polarizability to vary by $\pm$ 30 $\%$ (light violet confidence areas in Figure \[fig2\]) around the single-molecule value. We use the polarizability $\alpha_{157}$(1) = 25.4 $\times 10^{-30}\,\text{m}^3$ from Marchese et al. [@Marchese1977] and we extract an absorption cross section of $\sigma_{157}$(1) = 1.1 $\times 10^{-20}\,\text{m}^2$ from Malloci et al. [@Malloci2004].
This yields the quantum and classical theory curve in Figure \[fig2\]. Apart from the uncertain polarizability, the deviations from the experimental data may be attributed to a limited efficiency of single-photon ionization and contributions by fragmentation processes. While the absolute interference contrast is sensitive to a variety of different cluster properties which still wait to be extracted in combination with more refined cluster theory, the fringe shift will become valuable for precisely measuring the interplay between internal cluster properties and external forces.
acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge support by the Austrian science funds (FWF-Z149-N16 Wittgenstein and DK CoQuS W1210-2) as well as infrastructure funds by the Austrian ministry of science and research BMWF (IS725001). We thank Uzi Even and Ori Cheshnovsky for emphasizing the benefits of organic molecules with the Even-Lavie valve and Klaus Hornberger for collaborations on the modeling of the OTIMA interferometer. We thank Bernd von Issendorff for discussions on cluster sources.
Supplementary Information {#supplementary-information .unnumbered}
=========================
(a) Source {#a-source .unnumbered}
----------
We use an Even-Lavie (EL) valve to create a pulsed neutral molecular cluster beam. Anthracene (Ac) molecules are heated close to their melting point (491 K) in the valve and they are co-expanded into high vacuum with a supersonic noble gas jet (p $\simeq$ 1 - 10 bar). There they cool and condense to form clusters ranging from $\text{Ac}_1$ - $\text{Ac}_{15}$. The EL valve is operated at a repetition rate of 100 Hz and it is synchronized with the three vacuum ultraviolet (VUV) grating lasers and the detection laser.
(b) VUV Laser system {#b-vuv-laser-system .unnumbered}
--------------------
The gratings are generated by three synchronized GAM lasers, model EX50F, $\simeq$ 5 mJ, shot-to-shot energy stability $\simeq$ 5$\%$, coherence length $\simeq$ 1 cm. The grating transmission function depends on the laser energy which we monitor for every individual pulse via the photodiode P (in Figure \[fig1\]). All mass spectra are sorted according to a given laser energy and pulse delay $\Delta$T. The laser beam lines are evacuated and purged with dry nitrogen (6.0) at 1 mbar to avoid both absorption and laser-induced deposition of debris on the mirrors. The ionizing VUV laser at the TOF-MS is a Coherent Excistar $\text{F}_2$-laser (3 mJ) with an energy stability of better than 5$\%$.
The $\text{F}_2$ laser operates mainly at 157.63 nm, with an additional weak line at 157.52 nm [@Sansonetti2001]. Our own measurements on the GAM lasers confirm the specified coherence length of $\text{l}_c \simeq$ 1 cm which ensures a standing wave in a few millimeters distance to the mirror. The transverse coherence of each excimer laser is given by its output aperture and amounts to $\simeq$ 40 $\mu$m at the mirror.
The timing sensitivity of the OTIMA scheme at short pulse separation periods requires a precise monitoring of the laser jitter. The intrinsic short term jitter of all three GAM lasers is less than 7 ns (FWHM). They exhibit, however, long term instability of the order of 100 ns, which we measure and compensate. Our Coherent laser jitters by $\simeq$ 20 ns (FWHM). The timing of the grating laser pulses is recorded to post-select the interferograms according to their pulse-delays. All measurements were made with a maximal jitter smaller than 5 ns.
(c) VUV mirror {#c-vuv-mirror .unnumbered}
--------------
The dielectric interferometer mirror (Jenoptik, Germany) is made from VUV grade $\text{CaF}_2$ coated with a reflectivity of R $>$ 96$\%$ under normal incidence. The finite reflectivity allows us to monitor the position and shape of the laser beams via their scattering on the frosted backside of the mirror.
Moreover it causes a small running wave to add to the standing wave. The constant intensity offset would only slightly reduce the interference contrast if every cluster was always ionized by a single photon. Our own and independent measurements indicate a minimal spherical deformation across the two-inch mirror towards its edges up to 100 nm.
(d) Mass spectrometry {#d-mass-spectrometry .unnumbered}
---------------------
The time of flight mass spectrometer (ToF-MS, Kaesdorf Munich) is built as an orthogonal reflection MS with $\Delta$m/m = 1/3000. The relative mass spread across every individual multiplet is as small as 0.1 - 1 $\%$. A mass variation of $\pm$ 2 amu on m = 1400 amu gives rise to a variation of 30 ns on 20 s Talbot time. This leads only to a negligible reduction of the interference contrast.
(e) Number of absorbed photons {#e-number-of-absorbed-photons .unnumbered}
------------------------------
We chose anthracene as a test molecule because its ionization energy $\text{E}_\text{i}$ is smaller than the photon energy at 157.63 nm (7.9 eV) and the contrast is highest if the absorption of a single photon suffices to ionize the particle. If this condition is fulfilled for a certain cluster number N it will be generally met for all higher clusters too, since $\text{E}_\text{i}$ decreases with cluster size to approach the work function of the bulk. Photon absorption without subsequent ionization would diminish the interference contrast. The photoionization quantum yield [@Jochims1996] of anthracene at 7.9 eV is only 10 $\%$. Our data are compatible with the assumption that it is close to one for clusters composed of several molecules. Different structural isomers may respond differently to the incident light, but a full assessment of all optical properties for all cluster sizes is beyond the scope of this first demonstration of experimental OTIMA interferometry.
(f) Vacuum system {#f-vacuum-system .unnumbered}
-----------------
The source chamber is evacuated to $\text{p}_1$ = 1 $\times 10^{-5}$ mbar, the interference chamber to $\text{p}_2 = 2 \times 10^{-8}$ mbar and the optical beam line to $\text{p}_3$ = 1 mbar.
(g) Data Recording and Processing {#g-data-recording-and-processing .unnumbered}
---------------------------------
The TOF-MS voltage signal is recorded using a 10 bit digitizer card (Agilent Acquiris DC282) with 0.5 ns time resolution. We run the experiment with 100 Hz repetition rate. A data file for one mass spectrum has a size of 1 mega points. Data are post-processed in real time using a custom developed software solution. The software also records the laser timings and pulse energies.
(h) Figures {#h-figures .unnumbered}
-----------
**Figure \[fig2\]:** The TOF-MS data were averaged over about 28000 individual mass spectra for panel (a) and about 14500 spectra for panel (b). An overall TOF-MS background was subtracted, for all masses equally. The green columns in the upper panels of Figure 2a and 2b were generated by summing the mass spectra (bottom panels) over a mass region whose width is indicated by the width of the columns. It accounts for the majority of the isotopic spread of a given cluster peak. The experimental error bar was determined as follows: Since the experimental response to the incidence of an ion is a voltage peak whose amplitude changes both from shot to shot and with increasing ion mass, we chose to extract a measure for the true count rate from the observation of “no count” - a small discriminator threshold was set to distinguish between the presence or absence of ions - in every mass bin. Assuming a Poissonian distribution of the cluster counts one can then infer the average detected cluster number and its standard deviation ($\delta \text{S}_\text{O}$ and $\delta\text{S}_\text{R})$ from the probability of finding zero counts. The error of the normalized signal difference $\delta\left(\Delta \text{S}_\text{N}\right)$ is then computed using Gaussian error propagation. The data has been evaluated and plotted with Matlab R2010b and arranged using Adobe Illustrator CS5.\
**Figure \[fig3\]:** For this data set, TOF-MS data were averaged over 3300 - 3500 frames for each data point. The error bar was determined by the same procedure as in Figure 2. The data has been evaluated and plotted with Matlab R2010b and arranged using Adobe Illustrator CS5.\
**Figure \[fig4\]:** For this data set we averaged over roughly 25000 mass spectra for every step in mirror distance. Uncertainty bars were generated using the same procedure as in Figure 2 and 3. The data has been evaluated with Matlab R2010b, plotted using the Matplotlib package for Python and arranged using Adobe Illustrator CS5.
|
---
abstract: 'Federated learning (FL) is an emerging paradigm for distributed training of large-scale deep neural networks in which participants’ data remains on their own devices with only model updates being shared with a central server. However, the distributed nature of FL gives rise to new threats caused by potentially malicious participants. In this paper, we study targeted data poisoning attacks against FL systems in which a malicious subset of the participants aim to poison the global model by sending model updates derived from mislabeled data. We first demonstrate that such data poisoning attacks can cause substantial drops in classification accuracy and recall, even with a small percentage of malicious participants. We additionally show that the attacks can be targeted, i.e., they have a large negative impact only on classes that are under attack. We also study attack longevity in early/late round training, the impact of malicious participant availability, and the relationships between the two. Finally, we propose a defense strategy that can help identify malicious participants in FL to circumvent poisoning attacks, and demonstrate its effectiveness.'
author:
- Vale Tolpegin
- Stacey Truex
- Mehmet Emre Gursoy
- Ling Liu
bibliography:
- 'labelflip\_fl.bib'
title: |
Data Poisoning Attacks Against\
Federated Learning Systems
---
Introduction {#sec:intro}
============
Preliminaries and Attack Formulation {#sec:attack_eval}
====================================
Analysis of Label Flipping Attacks in FL {#sec:experiments}
========================================
Related Work {#sec:related_work}
============
Conclusion {#sec:conclusion}
==========
**Acknowledgements**. This research is partially sponsored by NSF CISE SaTC 1564097. The second author acknowledges an IBM PhD Fellowship Award and the support from the Enterprise AI, Systems & Solutions division led by Sandeep Gopisetty at IBM Almaden Research Center. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other funding agencies and companies mentioned above.
DNN Architectures and Configuration {#appendix:a}
===================================
|
---
abstract: 'Photometric calibration is currently the dominant source of systematic uncertainty in exploiting type Ia supernovae to determine the nature of the dark energy. We review our ongoing program to address this calibration challenge by performing measurements of both the instrumental response function and the optical transmission function of the atmosphere. A key aspect of this approach is to complement standard star observations by using NIST-calibrated photodiodes as a metrology foundation for optical flux measurements. We present our first attempt to assess photometric consistency between synthetic photometry and observations, by comparing predictions based on a NIST-diode-based determination of the PanSTARRS-1 instrumental response and empirical atmospheric transmission measurements, with fluxes we obtained from observing spectrophotometric standards.'
author:
- 'Christopher W. Stubbs$^1$, John L. Tonry$^2$'
bibliography:
- 'stubbs\_c.bib'
title: 'Addressing the Photometric Calibration Challenge: Explicit Determination of the Instrumental Response and Atmospheric Response Functions, and Tying it All Together.'
---
=1
Introduction and Motivation
===========================
The challenge of photometry is to extract knowledge of the location and flux distribution of astronomical sources, based on measurements of the 2 dimensional distribution of detected photons in a focal plane. Each pixel $i$ in the detector array sees a signal $S_i$ given by
$$S_i = \sum_{sources~j} \int \Phi_j(\lambda) R_i(\lambda) T(\lambda) G(\lambda) A_i ~ d\lambda,
\label{eq:psignal}$$
where the sum is taken over all sources (including the sky) that contribute to the flux in the pixel, $\Phi_j(\lambda)$ is the photon spectrum for source $j$, $R_i(\lambda)$ is the throughput of the pixel, including the transmission of the optics and the pixel’s quantum efficiency, $T(\lambda)$ is the optical transmission of the atmosphere, $G(\lambda)$ accounts for non-atmospheric extinction processes along the line of sight to the source (necessary to convert from top-of-the-atmosphere fluxes to the SED of the source), and $A_i$ is the effective aperture of the system for pixel $i$, essentially the wavelength-independent part of the instrumental response.
Photometric calibration uncertainties currently limit our ability to use type Ia supernovae to determine the nature of the dark energy [@SNLS11]. The essential tasks are 1) to establish a clear relationship between the zeropoints in the various passbands used (to avoid contamination of the Hubble diagram with systematic zeropoint errors), and 2) to map out the shapes of the passbands (to enable precise K corrections). The absolute overall zeropoint of the system is degenerate with the intrinsic brightness of SN Ia’s. Precise colors are what matter. However we do note that in splicing together a Hubble diagram from multiple surveys, it is important to understand the overall zeropoints of the relevant instrumental photometric systems.
There are a number of alternative approaches to establishing well-understood colors in a multiband photometric system, comprising some combination of the following:
1. Using terrestrial sources to establish a spectral scale for celestial sources. This is the approach taken by [@Hayes75a] and [@Hayes75b], that in effect uses Vega as a transfer standard. In this instance the radiometric properties of the terrestrial source is used as the fundamental system calibration.
2. Exploiting the theoretical understanding of stars to predict the SEDs of individual stars. In practice white dwarfs are the favored class of object. Observations of appropriate sources are then used to calibrate instruments, and the theoretical spectrum is the basis for the relative system calibration. This approach has been used for the HST calibrations described by [@AbsCal94; @Bohlin96] and presented at this meeting by Hunt. This approach is also being pursued by the SkyMapper project (as described by Bessell at this meeting). The status of theoretical spectral modeling is described by Rauch in his contribution to this meeting.
3. Statistical properties of stars can also be used to establish consistency across the sky, and to correct for various types of chromatic attenuation. This approach was pursued by [@SLR], and was used by [@dust11] to show that the commonly-used SFD [@SFD] extinction map ($G(\lambda)$ of equation 1) requires renormalization of 0.86 in $E(B-V)$ compared to SFD.
4. An alternative approach, described in [@ST06] and in the contribution by Cramer to this meeting, attempts to bring modern metrology methods to bear on the calibration challenge. This approach uses NIST-calibrated detectors [@NIST07] as the fundamental reference for establishing the system’s sensitivity function. Initial results from this approach were obtained on the CTIO 4 meter [@CTIOlasercal], and more recently on the PanSTARRS-1 system [@PSlasercal]. The experimental challenge in this approach is to obtain a reliable measurement of the instrumental response function that is traceable to SI standards.
We stress that these approaches can all be pursued in parallel, and the relative consistency we achieve can be used to understand, quantify, and overcome sources of systematic error.
Throughput Measurements of the PanSTARRS-1 Survey System
========================================================
Full-aperture system response function measurements
---------------------------------------------------
We have used a photodiode to monitor the flux emanating from a back-illuminated flat-field screen in the dome of the PanSTARRS-1 telescope to map out the full-aperture relative system response function. Details of this measurement are given in [@PSlasercal] and [@JT12]. The results are shown in Figure \[fig:filters\]. We project light from a tunable laser onto the flat-field screen in the dome. We measure the flux emanating from the screen, incident on the telescope pupil, with a calibrated photodiode. We then compare the flux detected by the instrument to the incident flux, as measured by the photodiode. Performing this measurement at a succession of wavelengths allows us to determine system throughput as a function of wavelength, using the calibrated photodiode as the fundamental reference. A philosophically similar full-aperture calibration system for the Dark Energy Survey is described in these proceedings by Marshall.
Collimated-Light Determination of Ghosting in the PanSTARRS-1 System
--------------------------------------------------------------------
During the processing of the throughput data described above, we became increasingly concerned about stray light, multipath effects, and ghosting in the optical train. The basic problem is that by imaging a uniform surface brightness screen we can’t distinguish the focussing light paths through the system (which is how we measure celestial sources) from other light paths. This is the underlying reason why dome flats, twilight flats and sky flats typically mutually disagree, and require an “illumination correction”. Especially at wavelengths where the filter transmission is low, we find a substantial amount of light scatters from the focal plane, up to the filter, and back down to the CCDs. This is a source of systematic error. The contribution by Regnault to this meeting describes the merits of ray tracing to model these ghosts, and the LSST calibration team is undertaking a similar exercise (see paper here by Lynne Jones).
In order to quantify the amount of ghosting in the system, we set up a collimated telescope that was fed by an optical fiber to send a beam of collimated light onto the primary. This controls the phase space distribution of rays entering the telescope much better than the light emanating from the flat field screen. Moreover, we can adjust the focus of the collimating telescope to control the size of the image on the PS-1 focal plane. Figure \[fig:ghosts\] shows an example image obtained with this configuration.
We used the ghost-light fraction as a function of wavelength to correct the flat-screen generated throughput data. This is described in more detail in [@JT12].
Atmospheric Transmission
========================
Our team’s review paper on the atmosphere [@PASPatmos] describes the variable components of atmospheric transmission that require attention in order to achieve improved photometric performance. Water vapor, ozone, clouds and aerosols are the primary concerns. Rayleigh scattering from $O_2, N_2$ and other well-mixed gases is essentially deterministic, are the molecular absorption line strengths from $O_2$. Although a number of groups are conducting or are planning to undertake explicit atmospheric transmission monitoring (see contributions to this conference by McGraw, the Texas A&M team, the ESO program, and Blake), it remains the case that our community does not yet know the angular and temporal correlation functions of these variable atmospheric attenuation processes. Max Fagin and Justin Albert’s presentations to this conference describe our group’s program to fly a set of laser diodes to characterize aerosol extinction, so we will focus here on our measurements of water vapor.
Water Vapor Measurements at the site, using Polaris
----------------------------------------------------
We have constructed an objective grating slitless imaging spectrograph (Shivvers [*et al. *]{}in prep). The configuration is very similar to the spectrophotometric system described by McGraw. We decided to point at the North Celestial Pole, so as to obtain data at fixed airmass over an extended period of time. The camera in this system uses a Pixis 1024BR deep-depletion detector and so it has low fringing. The dispersion and focal length are such that we can obtain a spectrum spanning 300 to 1000 nm, and the 50-50 objective transmission grating greatly suppresses second order contamination light. Figure \[fig:MODTRAN\] shows the comparison between one of our 1-d spectra and the MODTRAN model that has been tuned to obtain an equivalent width of water vapor absorption to match the observations. The instrument was installed on Haleakala in July 2011 and has been acquiring data intermittently ever since. Figure \[fig:EW\] shows the temporal evolution, over a one month period, of the equivalent widths (EW, in nm) of the $O_2$ and water vapor features. We obtain an excellent match to the MODTRAN prediction of the EW for $O_2$, and there is no evidence of this feature changing over the duration of the observations. Both of these facts indicate that we are obtaining meaningful and reliable measurements of absorption features. The water vapor attenuation does show considerable variation over this period, and we assess our fractional accuracy in PWV determination to be about 10%. This is adequate to obtain an overall precision of $< 1\%$ in the determination of transmission in the passbands.
MODTRAN, tweaked
----------------
For comparison with the standard star observations, we used the MODTRAN “Generic Tropical" model atmosphere, with the “Desert Extinction (Spring-Summer)" aerosol choice. No attenuation from clouds was included. The PWV at the base of the atmosphere was set to 0.65 cm, since this produced a good match between our Polaris observations on the night the standard stars were observed, and the MODTRAN prediction for the line of sight from the altitude of the summit of Haleakala, looking towards the celestial pole. As described below we ended up making an adjustment to the aerosol component, based on the airmass dependence we observed. We note that the other, deterministic, aspects of atmospheric transmission were not adjusted. We used the default ozone column. This amounted to a MODTRAN transmission model with three empirically adjusted parameters: PWV, aerosol optical depth, aerosol Angstrom exponent.
The Total PanSTARRS-1 System Throughput: Instrument and Atmosphere
==================================================================
Taking the ghosting-corrected relative system throughput function in conjunction with the MODTRAN atmosphere (with parameters adjusted based on our observed PWV), we obtained the PanSTARRS system throughput functions shown in Figure \[fig:throughput\]. We have adopted a convention of 1.2 airmasses as the definition of the PS throughput function [@JT12].
With the PS-1 system throughput function in hand, we have used measurements of HST spectrophotometric standards to compare our NIST-based flux calibration to the fluxes observed from these standards. Details of this are presented in [@JT12]. An important aspect of this comparison is to determine the atmospheric transmission.
For the bandpasses we integrated a set of power law SEDs against each of these model atmospheres for each bandpass and created an interpolation function for the extinction as a function of four variables: $z$ for airmass ($\sec \zeta$ where $\zeta$ is the zenith distance), $h$ for precipitable water vapor (PWV) (typically 0.65 cm at sea level), $a$ for “aerosol exponent” (nominally 1; we modify the Modtran aerosol component by applying this power to the transmission, thereby mostly affecting the aerosol amplitude), and $p$ for SED power law (we calculate the extinction for pure power law SEDs, where $p=+2$ for $f_{\nu}\sim\nu^{+2}$ corresponds to an O star with $(r{-}i)=-0.43$, $p=0$ for $f_{\nu}\sim\hbox{const}$ corresponds to an F star with $(r{-}i)=0.00$, and $p=-2$ for $f_{\nu}\sim\nu^{-2}$ corresponds to an K5 star with $(r{-}i)=+0.42$. The extinction $dm$ in magnitudes is parameterized the extinction as $$\ln dm = \ln C + Z\ln z + A\ln a + P p + \ln h(H_0 + H_1\ln z + H_2\ln h)
\label{eq:modterp}$$ The coefficients for each of the filters are given in Table \[tab:modterp\].
Filter $C$ $Z$ $A$ $P$ $H_0$ $H_1$ $H_2$ err
-------------- ------- ------- ------- ---------- ---------- ---------- ------- -----
$g_{\rm P1}$ 0.204 0.982 0.227 $ 0.021$ $ 0.001$ $-0.000$ 0.000 1.7
$r_{\rm P1}$ 0.123 0.975 0.283 $ 0.012$ $ 0.012$ $-0.000$ 0.005 2.0
$i_{\rm P1}$ 0.092 0.831 0.304 $ 0.005$ $ 0.125$ $-0.011$ 0.035 2.7
$z_{\rm P1}$ 0.060 0.878 0.375 $-0.004$ $ 0.330$ $-0.070$ 0.055 4.9
$y_{\rm P1}$ 0.154 0.680 0.145 $ 0.014$ $ 0.549$ $-0.084$ 0.024 3.5
: PanSTARRS extinction coefficients. The columns contain coefficients described above for each of the PanSTARRS bandpasses that interpolate the Modtran extinction calculations. The final column is the percentage scatter of these fits relative to the calculated values. Note that the saturation of the water lines means that the extinction is [*not*]{} proportional to $\sec\zeta$ ($Z\neq1$), particularly for $y_{\rm P1}$. This is consistent with the modeling results from MODTRAN.
\[tab:modterp\]
Standard Star Observations
==========================
MJD 55744 (UT 02 July 2011) was a photometric night during which we observed a substantial number of spectrophotometric standard stars from the STIS Calspec [@CalSpec01] tabulation: 1740346, KF01T5, KF06T2, KF08T3, LDS749B, P177D, and WD1657- 343. These were observed throughout the night at airmasses between 1 and 2.2 in all six filters and also with no filter in the beam. Each observation was repeated, and exposure times were chosen to stay well clear of any non-linearities but still permit good accuracy. Observations in $y_{\rm P1}$ of the fainter white dwarfs were curtailed at 100 sec duration, so their uncertainties are relatively large. In addition, Medium Deep Field 09 (which overlaps SDSS Stripe 82) was observed a dozen times in each of $g_{\rm P1}$ $r_{\rm P1}$ $i_{\rm P1}$ $z_{\rm P1}$ and $y_{\rm P1}$ offering the opportunity to tie the spectrophometric data to a well-observed Pan-STARRS1 field. All stars were placed on OTA 34 and cell 33, so their integration was on the same silicon and used the same amplifier for read out (gain measured to be 0.97 e$^-$/ADU). The observations were bias subtracted and flatfielded as part of the normal IPP processing, and the IPP fluxes (instrumental magnitudes) were then available for comparison with tabulated SEDs. The IPP performs an aperture correction and reports fluxes within a radius of 25 pixels (13 arcsec diameter). Observations of Polaris on MJD 55744 with the spectroscopic sky probe had a PWV indistinguishable from the long term mean of 0.65 cm.
Putting it all Together: Closing the Photometric Loop and Assessing Consistency
===============================================================================
As described in more detail in [@JT12], we compared the observed $g_{\rm P1}$ $r_{\rm P1}$ $i_{\rm P1}$ $z_{\rm P1}$ and $y_{\rm P1}$ fluxes for the HST spectrophotometric standards with the predictions obtained from using equation (1), using our in-dome determination of the instrumental response, the observationally adjusted MODTRAN model for the atmosphere, and the HST Calspec data for the source SEDs. We find that we need to apply a gently varying “tweak” to the system response function, with an rms of around 3-4%, in order to obtain consistency between the synthetic photometry using CalSpec SEDs and the on-sky observations.
Conclusions
===========
The results presented here comprise, to our knowledge, the first instance of a NIST-calibrated telescope response function and a MODTRAN atmospheric model generating synthetic photometry that is then compared with on-sky measurements of spectrophotometric standards. We obtain agreement at the 5% level, except for the $y_{\rm P1}$ band where the discrepancy is 10%. We consider this to be encouraging, since it’s our first attempt to “close” this photometric loop. Our objectives for future work include:
1. Obtaining more precise on-site determination of atmospheric aerosols.
2. Improving the in-dome calibration technique, and our corrections for optical multipath effects.
3. Identifying the origin of the “tweak” we need to apply to obtain overall consistency between the NIST-based calibration and the expectations from stellar SEDs.
We are grateful to the US National Science Foundation, under grants AST-0443378, AST-0507475 and AST-1009749, and AST-0551161 (awarded for LSST development). We are also grateful to the LSST Corporation, Harvard University, and the US Department of Energy Office of Science for their support of the continuing development of these techniques under grant DE-SC0007881 and NIST under grant 70NANB8H8007. We thank the following colleagues for their contributions and collaboration on this overall effort: Justin Albert (U Victoria), Tim Axelrod (U Arizona) Ralph Bohlin (STSCI), Steve Brown (NIST), Yorke Brown (Dartmouth), David Burke (SLAC), Ken Chambers (UH), Claire Cramer (NIST), Susana Deustua (STSCI), Peter Doherty (Harvard), Maxwell Fagin (Dartmouth), Will High (Univ. Chicago), Keith Lykke (NIST), John McGraw (UNM), Gautham Narayan (Harvard), Abi Saha (NOAO), Isaac Shivvers (Berkeley), Chris Smith, CTIO), Amali Vaz (Harvard), John Woodward (NIST), and Peter Zimmer (UNM).
We also thank the conference organizers for their hard work in arranging a most interesting and informative meeting.
Discussion {#discussion .unnumbered}
==========
Q: (S. Deustua) What about flatfields? How reliable is your flat/illumination system in that a source placed on any pixel gives the same result? You have a very wide (3 deg) field.
Q: (N. Regnault) It is difficult to apply a photometric calibration obtained from a flat-field-like illumination to point source photometry. Do you have specific plans to bridge this gap?
A: We agree that it’s difficult to distinguish the focusing light paths from the stray and scattered light that lands on the focal plane, when the illumination comes from a diffuse flat field screen. As described above, we have used a collimated beam to attempt to disentangle these illumination paths, and to determine the response function at one location on the CCD array. The task of tying together the system sensitivity across the array is a distinct but related problem. A number of groups are working to develop various calibration devices that control the phase space distribution of the rays (in angle and position) that are sent into the telescope pupil from a monitored calibration source. We think that making an “illumination correction” is an essential ingredient in reducing and assessing sources of systematic error. This is best obtained on the sky, either by rastering stable sources across the focal plane, or through the “ubercal” procedure that uses the multiple observations per field from a wide-field multi-epoch survey. As a community we are still developing an overall methodology for obtaining the best possible accuracy and precision, and this meeting has been very useful in helping move this forward.
|
---
abstract: |
Based directly on the microscopic lattice dynamics, a simple high temperature expansion can be devised for non-equilibrium steady states. We apply this technique to investigate the disordered phase and the phase diagram for a driven bilayer lattice gas at half filling. Our approximation captures the phases first observed in simulations, provides estimates for the transition lines, and allows us to compute signature observables of non-equilibrium dynamics, namely, particle and energy currents. Its focus on non-universal quantities offers a useful analytic complement to field-theoretic approaches.\
**[KEY WORDS]{}: Non-equilibrium steady states; driven lattice gases; high temperature series expansion.**
address: |
$^1$Department of Physics and Engineering,\
Washington and Lee University, Lexington, VA 24450;\
$^2$Center for Stochastic Processes in Science and Engineering,\
Physics Department, Virginia Tech, Blacksburg, VA 24061-0435, USA.
author:
- 'I. Mazilu$^1$ and B. Schmittmann$^2$'
date: 'January 22, 2003 '
title: High temperature expansion for a driven bilayer system
---
epsf.sty
Introduction
============
Many-particle systems in a state of thermal equilibrium are the exception, rather than the rule. Physical reality is overwhelmingly in a far-from-equilibrium state. Examples range from living cells and weather patterns to ripples on water and sand. As we leave the framework of standard Gibbs ensemble theory for equilibrium systems, we have to search for new avenues and tools, seeking to understand and classify non-equilibrium behavior. As a first step along this road, the study of the simplest generalizations of equilibrium systems, i.e., [*non-equilibrium steady states*]{} (NESS), has been particularly fruitful [@SZ-rev; @other-revs]. Progress has relied predominantly on simulations, mean-field theory and renormalization group analyses for simple model systems. A class of models which exhibit especially interesting behavior are driven diffusive systems. Microscopically, these are lattice gases, consisting of one or more species of particles and holes, whose densities are conserved. An external driving force, combined with suitable boundary conditions, maintains a NESS. In the simplest case [@KLS], a uniform bias, or drive, $E$, is imposed on an Ising lattice gas such that a nonzero steady-state mass current is induced. This model differs significantly from the usual Ising model: it displays generic long-range correlations [@KLS; @ZWLV; @GLMS], and belongs to a non-equilibrium universality class [@crit] with upper critical dimension $d_{c}=5$. The ordered phase is phase-separated into two strips of high vs low density aligned with the bias. In contrast to equilibrium, bulk and interfacial properties are inextricably intertwined here [@lowT].
To avoid the complications due to the presence of interfaces, a bilayer structure was suggested [@KKM]: in the two-dimensional case, a second lattice was introduced, allowing for particle-hole exchanges between each site and its mirror image. This bilayer system is half filled with particles, and both layers are driven in the same direction. In the absence of any energetic couplings between the two layers, it was hoped that typical ordered configurations would show [*homogeneous*]{} densities on each layer, one almost full and the other nearly empty. Remarkably, however, this expectation proved too naive: Monte Carlo simulations [@2l-early] showed a sequence of[* two*]{} phase transitions, as the temperature is lowered: the first transition takes the system from a disordered (D) phase to a strip-like (S) structure showing phase-separation [*within each layer*]{}, with interfaces parallel to the drive and ‘on top of’ one another. The anticipated “full-empty” (FE) phase, with uniform densities on both layers, only emerges after a second transition which occurs at a lower temperature. Once an interaction $J$, of either sign, between nearest neighbors on different layers is introduced, the full phase diagram in ($J,E$) space can be mapped out [@HZS; @CW], using Monte Carlo simulations. As one might expect, the S (FE) phase dominates for attractive (repulsive) cross-layer coupling $J$. Remarkably, however, there is a small but finite region where the S-phase prevails even though the cross-layer coupling is weakly repulsive (cf. Fig. 1). The presence of this domain puts the two transitions, observed for $J=0$, into perspective. We note for completeness that universal properties along the lines of continuous transitions have been analyzed in [@TSZ] with the help of renormalized field theory.
To provide additional motivation for the study of layered structures, we note that multilayer models have a long history in equilibrium statistical mechanics[@ballentine; @binder; @hansen]. On the theoretical side, they allow for the study of dimensional crossover [@dim-cross]; on the more applied side, they provide natural models for the analysis of intercalated systems [@intercalation], interacting solid surfaces or thin films [ferrenberg]{}. Since intercalated systems are often driven by chemical gradients or electric fields, to speed the diffusion of foreign atoms into the host material, it is quite natural to study driven layered structures.
Simulations relie, of course, on discrete lattice models. In contrast, field theories operate in the continuum, and thus, all discrete degrees of freedom have to be coarse-grained before these powerful techniques can be applied. In the process, non-universal information is lost, such as, e.g., the location of transition lines in the phase diagram. It is therefore desirable to identify a second analytic approach which is based directly on the microscopic model and thus complements both, simulations and continuum theories. Fortunately, high temperature expansion techniques [@HTS-revs] can be generalized to interacting driven lattice gases [@ZWLV; @SZ; @LZS]. For the single-layer case, two-point correlation functions can be computed approximately [@ZWLV; @SZ] and display the expected power law decays in the steady state. With some care, the approximate location of order-disorder transitions can be extracted and compared to simulation results [@SZ]. Given the nature of the approximation, [*quantitative*]{} accuracy cannot be expected, but the [*qualitative*]{} agreement of data and approximation is remarkably good.
While the high temperature expansion is quite successful for the usual driven lattice gas, it is not clear to what extent it is capable of capturing the main features of other driven systems. This motivates the work presented in this paper, namely, the analysis of the bilayer system with this technique. Within a first-order approximation, we compute the two-point correlation functions and several related quantities, such as the particle current and the energy flux through the system. We extract the approximate location of the continuous transition lines and compare our results to the Monte Carlo data. As in the single-layer case, the qualitative features of the transition lines are reproduced as well as can be expected. Some limitations of the method will be discussed.
This paper is organized as follows. We first introduce the bilayer model.After a brief summary of the high temperature expansion, we derive the closed set of equations satisfied by the two-point functions. We then obtain the solutions and extract the transition lines. Next, we show how the mass and energy currents through the system can be expressed in terms of pair correlations. We conclude with some comments and open questions.
The bilayer model
=================
A variant of the driven Ising model [@KLS], the model consists of two square lattices, one stacked above the other, resulting in a bilayer structure of size $L^{2}\times 2$. Each lattice site $\vec{r}\equiv (x,y,z)$, with $x,y=1,2,...,L$ and $z=0,1$, carries a spin variable $s(\vec{r})=\pm 1
$. Often, we also use lattice gas language, mapping spins into particles or holes, via $s(\vec{r})\equiv 2n(\vec{r})-1$. The local occupation variable $%
n(\vec{r})$ takes the values $1$ or $0$, indicating whether a particle is present or not. The total magnetization, $\sum_{\vec{r}}s(\vec{r})$, is fixed at zero so that the Ising critical point can be accessed. Within each layer, nearest-neighbor spins interact through a ferromagnetic exchange coupling $J_{0}>0$; in contrast, the cross-layer interaction $J$, which couples spins $s(x,y,0)$ and $s(x,y,1)$, can take both signs. These choices are motivated by the physics of intercalated systems [@intercalation]. Thus, the Hamiltonian of the system can be written in the form $$H=-J_{0}\sum_{z}\sum_{nn}s(x,y,z)s(x^{\prime },y^{\prime
},z)-2J\sum_{x,y}s(x,y,0)s(x,y,1) \label{H}$$where $\sum_{nn}$ denotes the sum over all nearest-neighbor pairs $(x,y,z)$ and $(x^{\prime },y^{\prime },z)$ within the same plane. A heat bath at temperature $T$ is coupled to the system, in order to model thermal fluctuations. We use fully periodic boundary conditions in all directions; hence the factor of $2$ in front of the cross-layer coupling $J$.
In the absence of the drive, particles hop to empty nearest-neighbor sites according to the usual Metropolis [@Metropolis] rates, $\min \left\{
1,\exp \left( -\beta \Delta H\right) \right\} $, where $\Delta H$ is the energy difference due to the jump. Respecting the conservation of density, the phase diagram of this system is easily found. At high temperatures, a disordered phase persists, characterized by correlations which fall off exponentially. At a critical temperature $T_{c}(J)$, a continuous transition occurs into the S (FE) phase for $J>0$ ($J<0$). At $J=0$, the critical temperature takes the Onsager value [@Onsager] $%
T_{c}(0)=2.269...J_{0}/k_{B}$. For finite $J$, $T_{c}(J)$ is even in $J$, due to a simple gauge symmetry, and increases monotonically with $\left|
J\right| $. For $J\rightarrow \pm \infty $, nearest-neighbor spin pairs, with the partners located on different layers, combine into dimers who couple to neighboring dimers with strength $2J_{0}$. As a result, the critical temperature approaches the limit $T_{c}(\pm \infty )=2T_{c}(0)$. The line $J=0$, $T<T_{c}(0)$ is a line of first-order transitions between the S and FE phases. It ends in a bicritical point at $J=0$, $T=T_{c}(0)$.
To drive the system out of equilibrium, we apply a bias (an “electric” field) $\vec{E}$ along the positive $x$-axis. The contents of two sites, $%
\vec{r}$ and $\vec{r}+\hat{a}$, separated by a (unit) lattice vector $ \hat{a}$, are exchanged according to the rate $$c(\vec{r},\vec{r}+\hat{a};\left\{ s\right\} )=\min \left\{ 1,\exp \left[
-\beta \Delta H+\beta \,\hat{a} \cdot \vec{E}\,(n(\vec{r})-n(\vec{r}+\hat{a}))%
\right] \right\} \label{rates}$$ The argument $\left\{ s\right\} $ reminds us that the rate depends on a local neighborhood of the central pair. Due to $E$, particle hops against the drive become unfavorable. In conjunction with periodic boundary conditions in the $x$- and $y$-directions, the system settles into a non-equilibrium steady state with a net particle current.
The phase diagram, resulting from Monte Carlo simulations at $J_{0}=1$ and infinite $E$, is shown in Fig. 1. The same phases and transitions are found, but the bicritical point and its attached first order line are shifted to higher $T$ and into the $J<0$ region. Thus, the S phase is observed to be stable in a finite window of negative interlayer coupling, so that two transitions must occur along the $J=0$ axis. This discovery represents the most unexpected new characteristic of this driven diffusive system. We also note the decrease of the critical temperatures for very large $\left|
J\right| $. In a recent paper [@CW], this phase diagram was extended to include [*unequal intra-layer*]{} attractive couplings. In this case, the bicritical point is shifted even further into the negative region of $J$ as the coupling transverse to the bias increases.
We now turn to the analysis of this model in terms of a high temperature expansion.
= 0.7= 0.7
High temperature expansion
==========================
The dynamics underlying the Monte Carlo simulations is easily expressed via a master equation. The latter provides a convenient starting point for a high temperature expansion. For simplicity, we take the thermodynamic limit within each plane, i.e., $L\rightarrow \infty $. Following [@ZWLV], we first derive the equations of motion for the two-point functions. By virtue of the familiar hierarchy, they are coupled to the three-point functions; however, we will argue that these are negligible (while non-zero, they are numerically rather small), so as to arrive at a closed system of equations for the two-point correlations. Temperature appears in these equations through the rates, via the combinations $\beta J$, $\beta J_{0}$, and $\beta
E$. To preserve the non-equilibrium nature of our dynamics, we expand in $%
\beta J$ and $\beta J_{0}$, keeping $\beta E$ finite. Technically, this requires that $E$ always dominates the energetic contribution, i.e., $%
E>\Delta H$ for all jumps along $E$. To first order, a linear, ${\em %
inhomogeneous}$ system of equations results, which can be solved exactly [@SZ] and forms the basis of our analysis.
The equations of motion and their solution.
-------------------------------------------
Before turning to any detailed calculations, let us introduce the key quantities. The [*two-point correlation function*]{} is defined as: $$G(\vec{r}-\vec{r}\text{ }^{\prime })=\left\langle s(\vec{r})s(\vec{r}\text{ }%
^{\prime })\right\rangle \label{CF}$$ where $\left\langle \cdot \right\rangle $ denotes the configurational average. Due to translation invariance, $G$ depends only on the difference of the two vectors. Moreover, $G$ is invariant under reflection of one or several lattice directions; e.g., $G(x,y,z)=G(-x,y,z)$, etc. The correlation function at the origin is obviously unity, $G(\vec{0})=\left\langle s^{2}(%
\vec{r})\right\rangle =1$. We also introduce the Fourier transform of $G$, i.e., the[** **]{}[*structure factor*]{}: $$S(k,p,q)\equiv \sum_{z=0,1}\sum_{x,y=-\infty }^{\infty
}G(x,y,z)e^{-i(kx+py+qz)} \label{SF}$$ Since we take the thermodynamic limit $L\rightarrow \infty $, the wave vectors $k$ and $p$ are continuous, but restricted to the first Brillouin zone $[-\pi ,\pi ]$, while $q$ is discrete, taking only the two values $0$ and $\pi $. For completeness, we also give the inverse transform, $$\begin{aligned}
G(x,y,z) &=&\frac{1}{2(2\pi )^{2}}\sum_{q=0,\pi }\int_{-\pi }^{+\pi
}dk\int_{-\pi }^{+\pi }dpS(k,p,q)e^{i(kx+py+qz)} \nonumber \\
&\equiv &\int S(k,p,q)e^{i(kx+py+qz)} \label{FT}\end{aligned}$$ where the second line just defines some simplified notation.
To set up the high temperature expansion, we first define the actual expansion parameters of our theory, namely $$\begin{aligned}
K_{0} &\equiv &\beta J_{0} \nonumber \\
K &\equiv &\beta J \label{Ks}\end{aligned}$$For $K=K_{0}=0$, the steady-state distribution is exactly known [spitzer]{} to be uniform for all $E$: $P^{\ast }\propto 1$, so that we are expanding about a well-defined zeroth order solution. The correlation functions and structure factors for this limit are trivial, namely, $G(\vec{r%
})=\delta _{\vec{r},\vec{0}}$ where $\delta $ denotes the Kronecker symbol, and $S(k,p,q)=1$. Returning to the interacting case, we note that $G(\vec{r})
$, for $\vec{r}\neq \vec{0}$, is already of first order in the small parameter. Similarly, we can write the structure factor as a sum of two terms. The first term is just the zeroth order solution, while the second, $%
\tilde{S}$, carries the information about the interactions, $$S(k,p,q)=1+\tilde{S}(k,p,q) \label{S}$$so that we can recast $G(\vec{r})$, for $\vec{r}\neq \vec{0}$, in the form $$G(x,y,z)=\int \widetilde{S}(k,p,q)e^{i(kx+py+qz)}\text{\ for }x,y,z\neq 0
\label{G}$$
The exact equations of motion for $G$ are easily derived from the master equation [@ZWLV]: $$%TCIMACRO{\dfrac{d}{dt}}%
%BeginExpansion
{\displaystyle{d \over dt}}%
%EndExpansion
\left\langle s(\vec{r})s(\vec{r}\text{ }^{\prime })\right\rangle =\sum_{\vec{%
x},\vec{x}^{\prime }}\left\langle s(\vec{r})s(\vec{r}\text{ }^{\prime })%
\left[ s(\vec{x})s(\vec{x}\text{ }^{\prime })-1\right] c\left( \vec{x},\vec{x%
}\text{ }^{\prime };\left\{ s\right\} \right) \right\rangle \label{eom}$$ Here, the sum runs over[* nearest-neighbor*]{} pairs ($\vec{x},\vec{x}$ $%
^{\prime }$) such that $\vec{x}\in \left\{ \vec{r},\vec{r}\text{ }^{\prime
}\right\} $ but $\vec{x}$ $^{\prime }\notin \left\{ \vec{r},\vec{r}\text{ }%
^{\prime }\right\} $. Stationary correlations are obtained by setting the left hand side to zero. Clearly, jumps along and against all three lattice directions will contribute to the right hand side of Eq. (\[eom\]).
To proceed, let us write the jump rates in a form which makes their dependence on the spin configuration $\left\{ s\right\} $ explicit, so that the configurational averages in Eq. (\[eom\]) can be performed. For [*infinite*]{} drive, a particle jumps along the field with rate unity, but never against it, so that the transition rates [*parallel to the field*]{} can be written as: $$c_{\Vert }^{\infty }\left( \vec{r},\vec{r}+\hat{x};\left\{ s\right\} \right)
=%
%TCIMACRO{\dfrac{1}{4}}%
%BeginExpansion
{\displaystyle{1 \over 4}}%
%EndExpansion
\left[ s(\vec{r})-s(\vec{r}+\hat{x})+2\right] \label{c_par_Einf}$$ Here, $\hat{x}$ is a unit vector in the positive $x$-direction. In the case of [*finite*]{} drive, our restriction $E>\Delta H$ ensures that jumps along $E$ still occur with unit rate, while those against $E$ are suppressed by a factor of $\exp \left[ -\beta \left( \Delta H+E\right) %
\right] $. Defining $$\varepsilon \equiv e^{-\beta E} \label{eps}$$ Eq. (\[c\_par\_Einf\]) must be amended to $$c_{\Vert }\left( \vec{r},\vec{r}+\hat{x};\left\{ s\right\} \right) =%
%TCIMACRO{\dfrac{1}{4}}%
%BeginExpansion
{\displaystyle{1 \over 4}}%
%EndExpansion
\left[ s(\vec{r})-s(\vec{r}+\hat{x})+2\right] +%
%TCIMACRO{\dfrac{\varepsilon }{4}}%
%BeginExpansion
{\displaystyle{\varepsilon \over 4}}%
%EndExpansion
\left[ s(\vec{r}+\hat{x})-s(\vec{r})+2\right] \exp (-\beta \Delta H)
\label{c_par}$$ Transverse to the field we have two jump rates, corresponding to the two transverse directions ($y$ and $z$). Both of these are regulated by the energy difference due to a jump: $$c_{\bot }\left( \vec{r},\vec{r}+\hat{a};\left\{ s\right\} \right) =\min
\left\{ 1,\exp \left( -\beta \Delta H\right) \right\} \label{c_perp}$$ We are now ready to expand the rates in powers of $K$ and $K_{0}$ while keeping $\varepsilon $ finite: $$\begin{aligned}
c_{\Vert }\left( \vec{r},\vec{r}+\hat{x};\left\{ s\right\} \right) &=&%
%TCIMACRO{\dfrac{1}{4}}%
%BeginExpansion
{\displaystyle{1 \over 4}}%
%EndExpansion
\left[ s(\vec{r})-s(\vec{r}+\hat{x})+2\right] +%
%TCIMACRO{\dfrac{\varepsilon }{4}}%
%BeginExpansion
{\displaystyle{\varepsilon \over 4}}%
%EndExpansion
\left[ s(\vec{r}+\hat{x})-s(\vec{r})+2\right] \left( 1-\beta \Delta H\right)
+O(\beta ^{2}) \label{c_par_exp} \\
c_{\bot }\left( \vec{r},\vec{r}+\hat{a};\left\{ s\right\} \right) &=&1+\beta
c_{2}\left( \vec{r},\vec{r}+\hat{a};\left\{ s\right\} \right) +O(\beta ^{2})
\label{c_perp_exp}\end{aligned}$$ with $$c_{2}\left( \vec{r},\vec{r}+\hat{a};\left\{ s\right\} \right) =-\frac{1}{2}%
(\Delta H+\left| \Delta H\right| )$$
Given these simple forms for the rates, we can now derive the equations of motion satisfied by the pair correlations directly from Eq. (\[eom\]), following [@ZWLV]. A few details are outlined in the Appendix. Keeping only corrections to first order in $K$, $K_{0}$ and neglecting three-point correlations, we obtain a [*closed set of linear equations*]{} for $G(x,y,z)$.[** **]{}Since the dynamics is restricted to nearest-neighbor processes, it is not surprising that the equations involve an anisotropic lattice Laplacian acting on $G(x,y,z)$. For $x,y,z$ near the origin, the Laplacian may include the origin and will thus generate inhomogeneities in the system of equations. The detailed form depends on the chosen boundary conditions, and, of course, on the three parameters $K$, $K_{0}$, and $\varepsilon $. Below, we show the set of equations for fully periodic boundary conditions. The first three equations result from nearest neighbors of the origin, $\vec{%
r}=(1,0,0)$, $(0,1,0)$, and $(0,0,1)$: $$\begin{aligned}
\partial _{t}G(1,0,0) &=&(1+\varepsilon
)[G(2,0,0)-G(1,0,0)]+4[G(1,1,0)-G(1,0,0)] \nonumber \\
&&+4[G(1,0,1)-G(1,0,0)]+2\varepsilon K_{0}+8K_{0} \nonumber \\
\partial _{t}G(0,1,0) &=&2(1+\varepsilon
)[G(1,1,0)-G(0,1,0)]+2[G(0,2,0)-G(0,1,0)] \nonumber \\
&&+4[G(0,1,1)-G(0,1,0)]+4\varepsilon K_{0}+6K_{0} \label{G-10} \\
\partial _{t}G(0,0,1) &=&2(1+\varepsilon
)[G(1,0,1)-G(0,0,1)]+4[G(0,1,1)-G(0,0,1)] \nonumber \\
&&+8K+8\varepsilon K \nonumber\end{aligned}$$ By virtue of invariance under reflections, these equations also hold for the other nearest neighbors $\vec{r}=(-1,0,0)$, $(0,-1,0)$, and $%
(0,0,-1) $. The following three equations arise from the next-nearest neighbor sites, $\vec{r}=(1,1,0)$, $(0,1,1)$, and $(1,0,1)$, and their reflections: $$\begin{aligned}
\partial _{t}G(1,1,0) &=&(1+\varepsilon
)[G(2,1,0)+G(0,1,0)-2G(1,1,0)]+2[G(1,2,0)+G(1,0,0) \nonumber \\
&&-2G(1,1,0)]+\newline
4[G(1,1,1)-G(1,1,0)]-2K_{0}-2\varepsilon K_{0}\newline
\nonumber \\
\partial _{t}G(0,1,1) &=&2(1+\varepsilon
)[G(1,1,1)-G(0,1,1)]+2[G(0,2,1)+G(0,0,1)-2G(0,1,1)] \nonumber \\
&&+\newline
4[G(0,1,0)-G(0,1,1)]-4[K_{0}+K]\newline
\label{G-11} \\
\partial _{t}G(1,0,1) &=&(1+\varepsilon
)[G(2,0,1)+G(0,0,1)-2G(1,0,1)]+4[G(1,1,1)-G(1,0,1)] \nonumber \\
&&+\text{\newline
}4[G(1,0,0)-G(1,0,1)]-4\varepsilon K-4K_{0} \nonumber\end{aligned}$$ Increasing the separation of the participating sites further, to $\vec{r}%
=(2,0,0)$ and $(0,2,0)$, we obtain: $$\begin{aligned}
\partial _{t}G(2,0,0) &=&(1+\varepsilon
)[G(3,0,0)+G(1,0,0)-2G(2,0,0)]+4[G(2,1,0)-G(2,0,0)] \nonumber \\
&&+4[G(2,0,1)-G(2,0,0)]-2\varepsilon K_{0} \nonumber \\
\partial _{t}G(0,2,0) &=&2(1+\varepsilon
)[G(1,2,0)-G(0,2,0)]+2[G(0,3,0)+G(0,1,0)-2G(0,2,0)] \label{G-20} \\
&&+4[G(0,2,1)-G(0,2,0)]\newline
-2K_{0} \nonumber\end{aligned}$$ And finally, all $G$’s with $\left| x\right| +\left| y\right| +\left|
z\right| >2$ satisfy homogeneous equations: $$\begin{aligned}
\partial _{t}G(i,j,k) &=&(1+\varepsilon )[G(i+1,j,k)+G(i-1,j,k)-2G(i,j,k)]
\nonumber \\
&&+2[G(i,j+1,k)+G(i,j-1,k)-2G(i,j,k)]+\newline
4[G(i,j,k-1)-G(i,j,k)] \label{G-rest}\end{aligned}$$ The last equation contains the full anisotropic lattice Laplacian, acting on $G(i,j,k)$, without any inhomogeneities being generated. We note, for further reference, that the right hand sides of Eqns (\[G-10\]-\[G-rest\]) contain contributions from exchanges along and against the three lattice directions. Starting from Eq. (\[eom\]), it is of course easy to keep track of terms originating in transverse vs parallel jumps. Below, this distinction will become important when we turn to energy currents.
To solve this system, we closely follow the method presented in [@SZ]. Returning to Eq. (\[S\]), we need to focus only on $\tilde{S}$, since this quantity carries the information about the interactions. Recalling Eq. ([G]{}), we first express $G$ through its Fourier transform $\tilde{S}$, exploiting translation invariance and linearity. Then, we invoke the completeness of complex exponentials to project out an equivalent set of (algebraic) equations for $\tilde{S}$.
To follow through with this program, we first define the anisotropic lattice Laplacian in Fourier space, $$\delta (k,p,q)\equiv 2(1+\varepsilon )(1-\cos k)+4(1-\cos p)+4(1-\cos q)
\label{Lap}$$and second, introduce the three (as yet unknown) quantities: $$\begin{aligned}
I_{1} &\equiv &\int \tilde{S}(1-\cos k) \nonumber \\
I_{2} &\equiv &\int \tilde{S}(1-\cos p) \label{Is} \\
I_{3} &\equiv &\int \tilde{S}(1-\cos q) \nonumber\end{aligned}$$With these definitions, the system can be expressed in terms of $\tilde{S}$, resulting in: $$\begin{aligned}
2\varepsilon K_{0}+8K_{0} &=&\int \tilde{S}\delta \exp (ik)+(1+\varepsilon
)I_{1} \nonumber \\
4\varepsilon K_{0}+6K_{0} &=&\int \tilde{S}\delta \exp (ip)+2I_{2} \nonumber
\\
8\varepsilon K\newline
+8K &=&\int \tilde{S}\delta \exp (iq)+4I_{3} \nonumber \\
-2\varepsilon K_{0}-2K_{0} &=&\int \tilde{S}\delta \exp (i(k+p)) \nonumber
\\
-4(K_{0}+K) &=&\int \tilde{S}\delta \exp (i(p+q)) \\
-4\varepsilon K-4K_{0} &=&\int \tilde{S}\delta \exp (i(k+q)) \nonumber \\
-2\varepsilon K_{0} &=&\int \tilde{S}\delta \exp (2ik) \nonumber \\
-2K_{0} &=&\int \tilde{S}\delta \exp (2ip) \nonumber \\
0 &=&\int \tilde{S}\delta \exp (i(kx+py+qz))\text{ for }\left| x\right|
+\left| y\right| +\left| z\right| >2 \nonumber\end{aligned}$$To proceed, we treat $I_{1},I_{2},I_{3}$ for the time being as simple coefficients and move them to the left-hand side. Finally, we need one additional equation for $x=y=z=0$, which is easily obtained: $$\int \tilde{S}\delta =\int \tilde{S}[2(1+\varepsilon )(1-\cos k)+4(1-\cos
p)+4(1-\cos q)]=2(1+\varepsilon )I_{1}+4I_{2}+4I_{3}$$Now, we are ready to invoke the completeness relation for complex exponentials, namely, $$\sum_{x,y,z}\exp [i(kx+py+qz)]=2(2\pi )^{2}\delta (k)\delta (p)\delta _{q,0}$$which allows us to solve for $\tilde{S}$: $$\tilde{S}(k,p,q)=%
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)} }%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}
%EndExpansion
\label{S-tilde}$$where $$\begin{aligned}
L(k,p,q) &\equiv &2(1+\varepsilon )\left( 1-\cos k\right) I_{1}+4\left(
1-\cos p\right) I_{2}+4\left( 1-\cos q\right) I_{3} \nonumber \\
&&+\left( 2\varepsilon K_{0}+8K_{0}\right) 2\cos k+\left( 4\varepsilon
K_{0}+6K_{0}\right) 2\cos p \nonumber \\
&&+\left( 8\varepsilon K\newline
+8K\right) \cos q-\left( 2\varepsilon K_{0}+2K_{0}\right) 4\cos k\cos p
\label{L} \\
&&-\left( 4\varepsilon K+4K_{0}\right) 2\cos k\cos q-4(K_{0}+K)2\cos p\cos q
\nonumber \\
&&-4\varepsilon K_{0}\cos 2k-4K_{0}\cos 2p \nonumber\end{aligned}$$However, Eq. (\[S-tilde\]) is not yet a fully explicit solution for $%
\tilde{S}$, due to the appearance of the three integrals $I_{1}$, $I_{2}$, and $I_{3}$ in $L$. To determine these three coefficients, we need three linearly independent equations. One of these equations is given by the value of $G$ at the origin, $1=G(0,0,0)=\int (1+\tilde{S})$, and the remaining two can be obtained directly from the definitions of $I_{1}$ and $I_{3}$ in Eq. (\[Is\]): $$\begin{aligned}
0 &=&\int
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)} }%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}
%EndExpansion
\nonumber \\
0 &=&-I_{1}+\int
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)}}%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}%
%EndExpansion
(1-\cos k) \label{matrix} \\
0 &=&-I_{3}+\int
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)}}%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}%
%EndExpansion
(1-\cos q) \nonumber\end{aligned}$$After inserting Eq. (\[L\]) for $L$, this leads to a set of three inhomogeneous, linear equations for the three unknowns $I_{1}$, $I_{2}$, and $I_{3}$, which are easily solved. Since the details of the associated matrix inversion are straightforward but tedious, we relegate a few details to the Appendix. We just note the following overall features: ([*i*]{}) All three coefficients are functions of $K$, $K_{0}$, and $\varepsilon $; ([*ii*]{}) for the whole range of fields $\varepsilon $ and for $K_{0}=1$ and $K=\pm 1$ (attractive and repulsive inter-layer interactions), $I_{1}$ and $I_{2}$ are negative, while $I_{3}$ is positive for $K=-1$ and negative for $K=+1$.
This concludes the calculation of the structure factor. To summarize, we find $$S(k,p,q)=1+%
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)}}%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}%
%EndExpansion
+O(K^{2},K_{0}^{2},KK_{0}) \label{S-final}$$ Even at the lowest nontrivial order, this solution carries a significant amount of information about the phase diagram of our system. In particular, we can extract an approximate shape of the critical lines, as we will show in the following.
The critical lines.
-------------------
The location of a continuous phase transition is marked by the divergence of a suitably chosen structure factor, as a function of the external control parameters. For example, we can locate the order-disorder phase transition of the usual, two-dimensional Ising model by seeking those values of temperature (at zero magnetic field) for which the structure factor, $S(\vec{%
k})$, diverges. In the absence of a conservation law, the only singularity occurs at the Onsager temperature if $\vec{k}=0$, indicating that the system orders into a spatially homogeneous state. For a lattice gas, however, $S(%
\vec{0})$ is fixed by the conservation law, and we need to seek the onset of phase [*separation*]{}, i.e., the emergence of macroscopic spatial inhomogeneities in the system. In this case, singular behavior occurs in $%
\lim_{\vec{k}\rightarrow 0}S(\vec{k})$, provided the system is half-filled and tuned to the Onsager temperature.
For the bilayer system, we need to locate, and distinguish, [*two types*]{} of continuous transitions, namely, from disorder (D) into the strip (S) and the full-empty (FE) phases, respectively. Since the D-S transition is marked by the appearance of phase-separated strips in each layer, aligned with the driving force, it can be located by seeking singularities in $%
\lim_{p\rightarrow 0}S(0,p,0)$. In contrast, the D-FE transition exhibits homogeneous, but opposite magnetizations in the two planes, so that it can be found by considering $S(0,0,\pi )$. In fact, these two structure factors were precisely the order parameters chosen in the MC studies [@HZS].
Yet, another subtlety must be considered: in a typical high temperature expansion such as ours, only a finite number of terms can be computed. Hence, any perturbative result for the structure factor must be finite, and instead, the radius of convergence of the expansion must be estimated. Even this is not practical here, since we have only two terms of the series. To circumvent these restrictions [@SZ], we extract the singularity by looking for [*zeros*]{} of $S^{-1}$, to first order in $K$ and $K_{0}$.
Starting from Eq. (\[S-final\]), we obtain $$S^{-1}(k,p,q)=1-%
%TCIMACRO{\dfrac{L(k,p,q)}{\delta (k,p,q)}}%
%BeginExpansion
{\displaystyle{L(k,p,q) \over \delta (k,p,q)}}%
%EndExpansion
+O(K^{2},K_{0}^{2},KK_{0}) \label{S-1}$$and seek to locate the zeros of $\lim_{p\rightarrow 0}S^{-1}(0,p,0)$ for the D-S transition, and of $S^{-1}(0,0,\pi )$ for the D-FE transition. Of course, we should ensure that these are the first zeros which are encountered upon lowering the temperature. Therefore, we consider, more generally, the behavior of $S^{-1}(k,p,q)$ at small $k,p$ and fixed $q$. The denominator of Eq. (\[S-1\]), being the lattice Laplacian, is positive definite: $$\lim_{k,p\rightarrow 0}\delta (k,p,q)=\ (1+\varepsilon
)k^{2}+2p^{2}+4(1-\cos q)+O(k^{4},p^{4},k^{2}p^{2})$$and vanishes only at the origin. Similarly, we obtain $$\begin{aligned}
\lim_{k,p\rightarrow 0}L(k,p,q) &=&16K_{0}\left( 1-\cos q\right) +4\left(
1-\cos q\right) I_{3} \\
&&+k^{2}\left[ (1+\varepsilon )I_{1}+10\varepsilon K_{0}-4K_{0}+\left(
4\varepsilon K+4K_{0}\right) \cos q\right] \nonumber \\
&&+p^{2}\left[ 2I_{2}+6K_{0}+(4K_{0}+4K)\cos q\right]
+O(k^{4},p^{4},k^{2}p^{2}) \nonumber\end{aligned}$$We note, briefly, that the anisotropic momentum dependence of numerator and denominator leads to power law correlations in the $x$- and $y$-directions [@ZWLV; @SZ; @AGMA]. Combining our results so far, it is apparent that the zeros of $S^{-1}$ are identical to those of $\delta -L$ in Eq. (\[S-1\]). To simplify notation, we write $$\lim_{k,p\rightarrow 0}\left[ \delta (k,p,q)-L(k,p,q)\right] \equiv \tau
_{\Vert }(q)k^{2}+2\tau _{\bot }(q)p^{2}+4\tau _{z}(1-\cos
q)+O(k^{4},p^{4},k^{2}p^{2})$$and read off $$\begin{aligned}
\tau _{\Vert }(q) &=&(1+\varepsilon )\left( 1-I_{1}\right) -10\varepsilon
K_{0}+4K_{0}-\left( 4\varepsilon K+4K_{0}\right) \cos q \nonumber \\
\tau _{\bot }(q) &=&1-I_{2}-3K_{0}-(2K_{0}+2K)\cos q \label{tau} \\
\tau _{z} &=&1-I_{3}-4K_{0} \nonumber\end{aligned}$$In a field-theoretic context [@TSZ], these quantities play the role of diffusion coefficients: $\tau _{\Vert }$ and $\tau _{\bot }$ control the in-plane diffusion in the parallel and transverse directions, respectively, while $\tau _{z}$ controls the cross-plane hopping.
For high temperatures, i.e., small values of $K_{0}=\beta J_{0}$ and $%
K=\beta J$, all three $\tau $-coefficients are positive. Seeking zeros of these expressions, as $K_{0}$ and $K$ increase, we need to consider the two cases $q=0$ and $q=\pi $ separately. For $q=0$, we find that $\tau _{\bot
}(0)$ has a single zero at a critical $\beta _{c}^{S}$, for given $J_{0}$, $%
J $ and $\varepsilon $. At these parameter values, $\tau _{\Vert }(0)$ and $%
\tau _{z}$ remain positive. Similarly, for $q=\pi $, the coefficient $\tau
_{z}$ is the one which vanishes first as $\beta $ increases, reaching zero at a critical $\beta _{c}^{FE}$. Converting into temperatures, we obtain two functions, $T_{c}^{S}(J_{0},J,\varepsilon )$ and $T_{c}^{FE}(J_{0},J,%
\varepsilon )$, and we need to identify the larger of the two: If $\max %
\left[ T_{c}^{S}(J_{0},J,\varepsilon ),T_{c}^{FE}(J_{0},J,\varepsilon )%
\right] =T_{c}^{S}(J_{0},J,\varepsilon )$, the order-disorder transition is of the D-S type. Otherwise, if $\max \left[ T_{c}^{S}(J_{0},J,\varepsilon
),T_{c}^{FE}(J_{0},J,\varepsilon )\right] =T_{c}^{FE}(J_{0},J,\varepsilon )$, the FE phase is selected upon crossing criticality.
While the two critical lines, $T_{c}^{S}(J_{0},J,\varepsilon )$ and $%
T_{c}^{FE}(J_{0},J,\varepsilon )$, can in principle be found analytically, the details are not particularly illuminating. Instead, we present a range of numerical results below. For example, for infinite $E$ ($\varepsilon =0$), we obtain $$\begin{aligned}
k_{B}T_{c}^{S}(J_{0},J,0) &=&4.39J_{0}+2.11J \label{eps=0} \\
k_{B}T_{c}^{FE}(J_{0},J,0) &=&4.14J_{0}\ -1.36J \nonumber\end{aligned}$$For finite $E$ with $\varepsilon =\exp (-\beta E)=0.5$, all coefficients decrease and we find $$\begin{aligned}
k_{B}T_{c}^{S}(J_{0},J,0.5) &=&4.15J_{0}+2.03J \label{eps=0.5} \\
k_{B}T_{c}^{FE}(J_{0},J,0.5) &=&4.05J_{0}\ -1.70J \nonumber\end{aligned}$$In each case, the bicritical point is defined through the solution of $%
T_{c}^{S}(J_{0},J,\varepsilon )=T_{c}^{FE}(J_{0},J,\varepsilon )$. For comparison, we also quote the equilibrium ($E=0$) results (see the Appendix for details): $$\begin{aligned}
k_{B}T_{c}^{S}(J_{0},J,1) &=&4J_{0}+2J \label{equ} \\
k_{B}T_{c}^{FE}(J_{0},J,1) &=&4J_{0}-2J \nonumber\end{aligned}$$which exhibit the expected $J\rightarrow -J$ symmetry.
Recalling that the MC simulations were performed at fixed, positive in-plane coupling $J_{0}$, we need to consider only the dependence on the cross-plane coupling $J$ which may take either sign. All of our results show that, for positive $J$, the D-S transition dominates while, for [*sufficiently negative*]{} $J$, a D-FE transition is found. In the following, we discuss the non-equilibrium ($E\neq 0$) results in more detail.
Fig. 2 shows the critical lines for two typical values of the parameters, $%
\varepsilon =0.5$ and $J_{0}=1$. Being the result of a first order approximation, the critical lines must of course be linear in $J$. Therefore, quantitative agreement with the simulation data cannot be expected; nevertheless, several important features are reproduced: [*the existence of two ordered phases* ]{}and the[* shift of the bicritical point*]{} to higher values of $T$ and negative $J$. As a result, the S phase survives for small, negative $J$, despite being energetically unfavorable. This phenomenon can be explained qualitatively [@HZS] by noting that [*long-range negative*]{} correlations transverse to $E$ dominate the ordering process for positive $J$, and this mechanism continues to be effective for a small region of negative $J$. For large and negative $J$, the disordered state orders into the full-empty (FE) phase, characterized by the planes being mainly full or empty. Finally, we comment on the dependence of the critical lines, specifically $%
T_{c}^{S}(1,1,\varepsilon )$ and $T_{c}^{FE}(1,-1,\varepsilon )$, on the field parameter $\varepsilon =\exp \left( -\beta E\right) $, shown in Fig. 3. For $E=0$, both temperatures are equal, by virtue of the $J\rightarrow -J$ symmetry of the equilibrium system. As the field becomes stronger, the critical temperature of the D-S transition increases, in contrast to the critical temperature of the D-FE transition which decreases. This behavior agrees qualitatively with the trend observed in the simulations [@HZS; @CH].
= 0.7= 0.7
There are several other quantities of physical interest which are immediately related to the two-point correlations, such as the steady-state particle and energy currents. To extract these, we first discuss the inverse Fourier transform of the structure factor, focusing specifically on the nearest-neighbor correlations.
= 0.7= 0.7
Related physical observables.
-----------------------------
[*Nearest-neighbor correlations.* ]{}These are easily found from our solution for the structure factor, Eq. (\[S-final\]). For example, the nearest-neighbor correlation in the field direction is given by: $$G(1,0,0)=\int \tilde{S}(k,p,q)\cos k+O(K^{2},K_{0}^{2},KK_{0})$$Since $\int \tilde{S}=0$ by virtue of $G(0,0,0)=1$, we obtain $$G(1,0,0)=-\int \tilde{S}(k,p,q)\left( 1-\cos k\right)
+O(K^{2},K_{0}^{2},KK_{0})=-I_{1}$$and similarly, $$\begin{aligned}
G(0,1,0) &=&-I_{2} \\
G(0,0,1) &=&-I_{3} \nonumber\end{aligned}$$These three integrals are already known since they were required for the discussion of the critical lines. Specifically, for $\varepsilon =0.5$ we find, neglecting corrections of $O(K^{2},K_{0}^{2},KK_{0})$:$$\begin{aligned}
G(1,0,0) &=&0.949K_{0}+0.030K \nonumber \\
G(0,1,0) &=&0.849K_{0}-0.034K \\
G(0,0,1) &=&-0.055K_{0}+1.702K \nonumber\end{aligned}$$For reference, we also quote the first order results for the equilibrium ($%
E=0$) correlations: $$\begin{aligned}
G^{eq}(1,0,0) &=&G^{eq}(0,1,0)=K_{0} \\
G^{eq}(0,0,1) &=&2K \nonumber\end{aligned}$$
In the following graphs (Fig. 4a-c), we show the drive dependence of the three [*nearest-neighbor*]{} correlation functions, at $K_{0}=1$ and $K=\pm
1 $, to illustrate their behavior in two typical domains (attractive and repulsive cross-layer coupling). Of course, these values of $K$ and $K_{0}$ are not “small”, but in a linear approximation they just serve to fix a scale. Consistent with the interpretation of the drive as an additional noise which tends to break bonds, all correlations are reduced compared to their equilibrium value. Further, as the field is switched on, the $%
J\rightarrow -J$ symmetry of the equilibrium system is broken, and the correlations for repulsive and attractive cross-layer coupling differ from one another. The details of how they differ provides some insight into the ordered phases which will eventually emerge.
The first plot (Fig. 4a) shows the correlation function for a nearest-neighbor bond within a given plane, aligned with the drive direction. It is interesting to note that the correlations for repulsive cross-layer coupling are more strongly suppressed than their counterparts for attractive $J$. This feature becomes more transparent when we consider nearest-neighbor correlations transverse to the drive, but still within the same plane (Fig. 4b). For attractive cross-layer coupling, we note that $%
G(1,0,0)$ is considerably enhanced over $G(0,1,0)$, while the two correlations are roughly equal in the repulsive case. This indicates a tendency to form droplets of correlated spins which are elongated in the field direction for $J=+1$ while remaining approximately isotropic for $J=-1$, hinting at the nature of the associated ordered phases (strip-like vs uniform within each layer). This picture is completed when we consider the cross-plane correlations $G(0,0,1)$ (Fig. 4c): These are positive in the attractive, and negative in the repulsive case, demonstrating the tendency towards equal vs opposite local magnetizations on the two layers. Given that we have performed only a first order calculation, the results really carry a remarkable amount of information about the system. Encouraged by these observations, we now consider two other quantities, namely, the particle and energy currents.
epsf
= 1.= 1.
(a)
= 1.= 1.
(b)
= 1.= 1.
(c)
[*The particle current.* ]{}Due to the bias in conjunction with periodic boundary conditions, the bilayer system carries a net particle current, $%
\left\langle j\right\rangle $. Since only nearest-neighbor exchanges are possible, this current is proportional to the density (number per site) of available particle-hole pairs in the field direction. The transition rate $%
c_{\Vert }$ along this direction, given in Eq. (\[c\_par\]), then counts the fraction of these pairs which will actually exchange per unit time. Specifically, in configuration $\{s\}$, the particle current can be written as $$j(\{s\})=\frac{1}{2L^{2}}\sum_{\vec{r}}\frac{s(\vec{r})-s(\vec{r}+\hat{x}%
)}{2}c_{\Vert }(\vec{r},\vec{r}+\hat{x};\{s\})\newline$$For infinite $E$, this expression simplifies considerably, since jumps against the field will be completely suppressed.
After a few straightforward algebraic manipulations, the [*average current*]{} can be expressed through the pair correlations along the field direction. To [*first order*]{} in $K$ and $K_{0}$, we obtain$$\left\langle j\right\rangle =%
%TCIMACRO{\dfrac{1}{4}}%
%BeginExpansion
{\displaystyle{1 \over 4}}%
%EndExpansion
(1-\varepsilon )[1-G(1,0,0)]+O(K^{2},K_{0}^{2},K_{0}K)$$which shows that it is non-zero only if $E\neq 0$. Further, it takes its maximum value at infinite temperature and is reduced by (attractive) nearest-neighbor interactions. The graph (Fig. 5) shows the field-dependence of this current, for $K_{0}=1$ and $K=\pm 1$. Since nearest-neighbor correlations along the field are much larger for positive $J$, indicating a predominance of particle-particle or hole-hole pairs, the current is reduced relative to the repulsive case.
= 0.7= 0.7
[*Energy currents.* ]{}Another interesting quantity associated with driven dynamics is the change in configurational [*energy*]{} during one Monte Carlo step. In the steady state, by definition, the average configurational energy is of course constant. However, particle-hole exchanges [*parallel*]{} to the field direction tend to increase the energy, since the drive can easily break bonds. In contrast, exchanges [*transverse*]{} to $E$ are purely energetically driven and hence, prefer to satisfy bonds so that the energy decreases [@SZ-rev]. In summary, we have $$\left\langle
%TCIMACRO{\dfrac{dH}{dt}}%
%BeginExpansion
{\displaystyle{dH \over dt}}%
%EndExpansion
\right\rangle _{\Vert }=-\left\langle
%TCIMACRO{\dfrac{dH}{dt}}%
%BeginExpansion
{\displaystyle{dH \over dt}}%
%EndExpansion
\right\rangle _{\perp }>0$$Even if a particle current were absent, the presence of energy currents would signal the[* non-equilibrium* ]{}steady state.
Since the configurational energy involves only nearest-neighbor bonds, it is obvious that only the time evolution of nearest-neighbor correlations plays a role in these two fluxes. Specifically, we have $$L^{-2}\left\langle
%TCIMACRO{\dfrac{dH}{dt}}%
%BeginExpansion
{\displaystyle{dH \over dt}}%
%EndExpansion
\right\rangle _{\Vert }=-J_{0}\newline
\left( \frac{\partial }{\partial t}\right) _{\Vert }\,\left[
G(1,0,0)+G(0,1,0)\right] -2J\left( \frac{\partial }{\partial t}\right)
_{\Vert }\,G(0,0,1)$$where the subscript on the time derivatives reminds us to select only those processes which are due to parallel exchanges alone. These can be easily identified from the terms contributing to Eq. (\[eom\]) or (\[G-10\]). Of course, there is an analogous equation for $\left\langle
dH/dt\right\rangle _{\perp }$. Collecting the relevant contributions and multiplying both sides by the inverse temperature $\beta $ to express everything in terms of $K_{0}$ and $K$, we find:
$$\begin{aligned}
L^{-2}\left\langle
%TCIMACRO{\dfrac{d\beta H}{dt}}%
%BeginExpansion
{\displaystyle{d\beta H \over dt}}%
%EndExpansion
\right\rangle _{\Vert } &=&-K_{0}\left\{ (1+\varepsilon )\left[
G(2,0,0)-G(1,0,0)\right] +2(1+\varepsilon )\left[ G(1,1,0)-G(0,1,0)\right]
+6\varepsilon K_{0}\right\} \nonumber \\
&&-2K\left\{ 2(1+\varepsilon )\left[ G(1,0,1)-G(0,0,1)\right] +8\varepsilon
K\right\}\end{aligned}$$
The correlation functions spanning next- and next-next nearest neighbors which appear here can again be determined from our solution for the structure factors (see Appendix). The result, at $K_{0}=1$ and $K=\pm 1$, is shown in Fig. 6 as a function of $\varepsilon $. As expected, this flux is always non-negative and monotonically increasing as a function of $E$. We note that the current for $K=-1$ is slightly larger than its counterpart for $K=+1$. Since it is a complicated function of the couplings and several correlations, we cannot offer a simple intuitive explanation of this property.
= 0.7= 0.7
Concluding remarks
==================
Based directly on the microscopic lattice dynamics, the high temperature series provides us with a simple analytic tool which complements field theoretic approaches. Even in a first order approximation, it is remarkable how many features of the MC results are – at least qualitatively – reproduced. To summarize our results briefly, we derive, and solve, a set of equations for the stationary pair correlation functions and their Fourier transforms, the equal-time structure factors. By matching the series expansion of the latter with the expected critical singularity, we find two critical lines, separating the disordered phase from the strip phase (S) and the full-empty phase (FE), respectively. We also observe the shift of the bicritical point which marks the juncture of these two lines, in very good qualitative agreement with the simulations. To illustrate the non-equilibrium character of the steady state, we compute the particle current and the energy flux through the system. The particle current is determined by the nearest-neighbor correlations in the field direction, and takes its maximum value in the absence of interactions. Our findings for the energy current confirm intuitive expectations: parallel exchanges tend to lower, while transverse exchanges tend to increase, the configurational energy.
A brief comment on boundary conditions is in order. Even though it is quite natural to use periodic boundary conditions in all lattice directions, it is not unreasonable to consider other choices, especially in the $z$-direction. To recall, periodicity in $z$ implies that the site ($x,y,0$) is connected to the site ($x,y,1$) via [*two*]{} bonds which enter into [*both*]{} the energetics [*and*]{} and the dynamics (i.e., there are two channels for a particle to move from one layer to the other). Alternately, we can choose open boundary conditions in $z$ and consider only a single energetic bond and single dynamical channel between these two sites. Mixtures of these two cases can also be constructed: i.e., imposing periodic boundary conditions on the energetics, but allowing only a single channel for particle moves, or vice versa. The first (second) “mixed” case is reducible to the case of open (periodic) boundary conditions, with $J$ replaced by $2J$ ($J/2$). Even though details are not presented here, we did, in fact, compute the critical lines for different cross-plane boundary conditions. The main conclusions of our study, namely, the existence of the two continuous phase transitions and the shift of the bicritical point, hold for all of these variations.
The high temperature expansion presented here has two shortcomings. First, our results provide no insight into the first-order transitions between the FE and S phases which were observed in the simulations. As in all high temperature series, the first singularity which is encountered as $T$ is lowered determines the radius of convergence. A low-temperature approach would be necessary to capture transitions between ordered phases. Second, our series is currently limited to just one nontrivial term. In order to compute the second order correction to the pair correlations, we would need to know the full stationary distribution, $P^{\ast }$, to first order. Writing the stationary master equation in the form $0=LP^{\ast }$ where $L$ is the linear operator (“Liouvillean”) defined by the transition rates, this requires the full inverse of $L$, to zeroth order. Finding this inverse is a highly nontrivial (and as yet unsolved) problem.
Inspite of these drawbacks, the high temperature expansion is one of the few analytic tools which provide insight into non-equilibrium steady states. It is conceptually and mathematically straightforward, and – at least at the qualitative level – surprisingly reliable. Since it is based directly on the microscopic lattice dynamics, it still carries information about nonuniversal properties which would be lost upon taking a continuum limit. It is therefore a valuable complement to both simulations and field theoretic methods.
[**Acknowledgements.**]{} We thank U.C. Täuber and R.K.P. Zia for fruitful discussions. Financial support from the NSF through the Division of Materials Research is gratefully acknowledged.
Equations for the two-point correlations.
-----------------------------------------
To illustrate the general procedure, we provide a few details here [ZWLV]{}. As an example, we choose the two-point correlation $G(1,1,0)$. We start with the equation of motion for the pair correlations:
$${\displaystyle{d \over dt}}\left\langle s(\vec{r})s(\vec{r}\,^{\prime })\right\rangle =\sum_{\vec{x},\vec{x}^{\prime }}\left\langle s(\vec{r})s(\vec{r}\,^{\prime })\left[ s(\vec{x})s(\vec{x}\,^{\prime })-1\right] \,c\left( \vec{x},\vec{x}\,^{\prime
};\left\{ s\right\} \right) \right\rangle$$where the sum runs over nearest-neighbor pairs ($\vec{x},\vec{x}\,^{\prime }$) such that $\vec{x}\in \{\vec{r},\vec{r}\,^{\prime }\}$ but $\vec{x}\,^{\prime }\notin \{\vec{r},\vec{r}\,^{\prime }\}$. To obtain the equation satisfied by $G(1,1,0)$, we choose, e.g. $\vec{r}\equiv (0,0,0)$ and $\vec{r}\,^{\prime }\equiv (1,1,0)$. The two participating spins have a total of $8$ distinct nearest neighbors: $6$ of these lie in the $z=0$ plane, and $2$ are found on the $z=1$ plane. One possible ($\vec{x},\vec{x}\,^{\prime }$) pair is, for example, the pair $\vec{x}\equiv \vec{r}$ and $\vec{x}\,^{\prime
}\equiv (1,0,0)$. The corresponding exchange occurs along the field direction, and hence, we must use the rate $c_{\Vert }$ from Eq. (\[c\_par\]). Considering [*only*]{} the contribution due to this particular pair of sites, we obtain $$\begin{aligned}
\partial _{t}G(1,1,0) &=&{\displaystyle{1 \over 4}}\left\langle s(0,0,0)s(1,1,0)\left[ s(0,0,0)s(1,0,0)-1\right] \right. \\
&&\times \left. \left\{ \left[ s(0,0,0)-s(1,0,0)+2\right] +\varepsilon \left[
s(1,0,0)-s(0,0,0)+2\right] \exp (-\beta \Delta H)\right\} \right\rangle +...\end{aligned}$$Here, $\ ...$ stands for the contributions due to all other possible ($\vec{x},\vec{x}\,^{\prime }$) pairs which can be handled in an analogous manner. After multiplying out a few terms and neglecting $3$-point functions, the expression above simplifies to $$\begin{aligned}
\partial _{t}G(1,1,0) &=&{\displaystyle{1 \over 2}}\left[ G(0,1,0)-G(1,1,0)\right] \\
&&+{\displaystyle{\varepsilon \over 4}}\left\langle \left[ s(1,0,0)s(1,1,0)-s(0,0,0)s(1,1,0)\right] \left[
s(1,0,0)-s(0,0,0)+2\right] \exp (-\beta \Delta H)\right\rangle +...\end{aligned}$$Note that we have replaced $\left\langle s(1,0,0)s(1,1,0)\right\rangle $ by the corresponding correlation function, $G(0,1,0)$. Next, we expand $\exp
(-\beta \Delta H)$ in powers of $K$ and $K_{0}$, according to Eq. ([c\_par\_exp]{}). The zeroth order contribution is easily accounted for, leaving us with the $O(\beta )$ correction: $$\begin{aligned}
\partial _{t}G(1,1,0) &=&{\displaystyle{1 \over 2}}\left( 1+\varepsilon \right) \left[ G(0,1,0)-G(1,1,0)\right] \\
&&-\beta
{\displaystyle{\varepsilon \over 4}}\left\langle \left[ s(1,0,0)s(1,1,0)-s(0,0,0)s(1,1,0)\right] \left[
s(1,0,0)-s(0,0,0)+2\right] \left( \Delta H\right) \right\rangle +...\end{aligned}$$The change in energy, $\Delta H$, involves the nearest-neighbor spins of the selected pair. Since these terms are already explicitly first order in $\beta $, they are averaged using the zeroth order approximation to the steady state solution. The latter is uniform so that the averages are trivial. Collecting, we find that the contribution of this particular exchange to $G(1,1,0)$ is $$\partial _{t}G(1,1,0)={\displaystyle{1 \over 2}}\left( 1+\varepsilon \right) \left[ G(0,1,0)-G(1,1,0)\right] -\varepsilon
K_{0}+...$$Including all the other ($\vec{x},\vec{x}\,^{\prime }$) pairs, we arrive at the complete equation: $$\begin{aligned}
\partial _{t}G(1,1,0) &=&(1+\varepsilon
)[G(2,1,0)+G(0,1,0)-2G(1,1,0)]+2[G(1,2,0)+G(1,0,0) \nonumber \\
&&-2G(1,1,0)]+\newline
4[G(1,1,1)-G(1,1,0)]-2K_{0}-2\varepsilon K_{0}\newline
\nonumber\end{aligned}$$Of course, this procedure is easily extended to any other two-point function. Moreover, it is straightforward to track which terms in the equations arise from parallel, and which from transverse, exchanges. This distinction is crucial for the computation of the energy fluxes.
Matrix inversion.
-----------------
We seek to invert the system of equations (\[matrix\]) for the three unknown expressions $I_{1}$, $I_{2}$, and $I_{3}$. We follow the method first outlined in [@SZ]. With a little algebra, it becomes apparent that the coefficients and inhomogeneities in these equations involve integrals of the form $$\begin{aligned}
Q_{lmn}(\varepsilon )\equiv \int \frac{(1-\cos k)^{l}(1-\cos p)^{m}(1-\cos
q)^{n}}{\delta } \nonumber\end{aligned}$$ with non-negative integer $l,m,n$ and $l+m+n\leq 3$. The task of determining these integrals is simplified by a series of relations, namely, $$\begin{aligned}
1 &=&\int
{\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{100}+4Q_{010}+4Q_{001} \\
1 &=&\int (1-\cos k){\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{200}+4Q_{110}+4Q_{101} \\
1 &=&\int (1-\cos p){\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{110}+4Q_{020}+4Q_{011} \\
{{\frac{3}{2}}} &=&\int (1-\cos k)^{2}{\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{300}+4Q_{210}+4Q_{201} \\
{{\frac{3}{2}}} &=&\int (1-\cos p)^{2}{\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{120}+4Q_{030}+4Q_{021} \\
1 &=&\int (1-\cos k)(1-\cos p){\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{210}+4Q_{120}+4Q_{111} \\
1 &=&\int (1-\cos q){\displaystyle{\delta \over \delta }}=2(1+\varepsilon )Q_{101}+4Q_{011}+4Q_{002}\end{aligned}$$The computation of the remaining integrals, while tedious, is completely straightforward. Once these coefficients are known, Eqns (\[matrix\]) can be inverted.
The equilibrium solution
------------------------
Since our expansion presumed $E>\Delta H$, we may not simply set $\varepsilon =1$ in our equations of motion for the two-point correlations. Instead, one should rederive the whole set carefully, noting the absence of the driving field. Of course, this is trivial, since only the first order term in the expansion of the equilibrium (Boltzmann) distribution is required here. In this order, only nearest-neighbor correlations can be nonzero, so that the only non-vanishing $G$’s are $$\begin{aligned}
G^{eq}(0,0,0) &=&1 \\
G^{eq}(\pm 1,0,0) &=&G^{eq}(0,\pm 1,0)=K_{0} \\
G^{eq}(0,0,1) &=&2K\end{aligned}$$Performing the Fourier transform to structure factors and exploiting the boundary conditions in the $z$-direction, we find immediately that $$\begin{aligned}
S(k,p,q) &=&\sum_{z=0,1}\sum_{x,y=-\infty }^{\infty }G(x,y,z)e^{-i(kx+py+qz)}
\\
&=&1+2K_{0}\left( \cos k+\cos p\right) +2K\cos q\end{aligned}$$resulting in $$\begin{aligned}
\lim_{k,p\rightarrow 0}S^{-1}(k,p,q) &=&1-2K_{0}\left( 2-\frac{1}{2}k^{2}-\frac{1}{2}p^{2}\right) -2K\cos q+O(k^{4},p^{4}) \\
&\equiv &\tau (q)+O(k^{2},p^{2})\end{aligned}$$with $$\tau (q)=1-4K_{0}-2K\cos q$$For $q=0$, this vanishes at $$\ k_{B}T_{c}^{S}=4J_{0}+2J$$and for $q=\pi $, the zero shifts to $$\ k_{B}T_{c}^{FE}=4J_{0}-2J$$Thus, D-S transitions are observed for $J>0$, and D-FE transitions dominate for $J<0$. The bicritical point is located at $k_{B}T_{c}(J=0)=4J_{0}$.
Other correlations near the origin.
-----------------------------------
These are required to compute the energy fluxes along the parallel and transverse directions. Specifically, we need the following correlation functions: $G(1,1,0)$, $G(1,0,1)$, $G(0,1,1)$, $G(2,0,0)$, and $G(0,2,0)$. We want to write these correlation functions in terms of the already calculated integrals $I_{1},I_{2},I_{3}$ and also in terms of the set of $Q_{lmn}$ integrals defined earlier.
We start with the definition for $G(2,0,0)$ and substitute the expression for the structure factor: $$\begin{aligned}
\ G(2,0,0) &=&\int \tilde{S}\exp (2ik)=2\int \tilde{S}(1-\cos k)^{2}-4I_{1}\newline
\\
&=&2\left\{ 2(1+\varepsilon )I_{1}Q_{200}+4\varepsilon
(5K_{0}+2K)Q_{300} \right. \\
&& + \left. 4(I_{2}+5K_{0}+2K)Q_{210}+4\left( I_{3}+4K_{0}\right)
Q_{201} \right. \\
&& - \left. 8K_{0}(1+\varepsilon )Q_{310}-8\left( K+K_{0}\right) Q_{211} \right.
\\
&& - \left. 8\left( \varepsilon K+K_{0}\right) Q_{301}-8\varepsilon
K_{0}Q_{400}-8K_{0}Q_{220}-2I_{1}\right\}\end{aligned}$$and similarly, $$\begin{aligned}
G(0,2,0) &=&2\int \tilde{S}(1-\cos p)^{2}-4I_{2}\newline
\\
&=&2\left\{ 2(1+\varepsilon )I_{1}Q_{020}+4\varepsilon
(5K_{0}+2K)Q_{120} \right. \\
&& + \left. 4(I_{2}+5K_{0}+2K)Q_{030}+4\left( I_{3}+4K_{0}\right)
Q_{021} \right. \\
&& - \left. 8K_{0}(1+\varepsilon )Q_{130}-8\left( K+K_{0}\right) Q_{031} \right.
\\
&& - \left. 8\left( \varepsilon K+K_{0}\right) Q_{121}-8\varepsilon
K_{0}Q_{220}-8K_{0}Q_{040}-2I_{2}\right\}\end{aligned}$$The remaining correlation functions follow in the same way: $$\begin{aligned}
G(1,1,0) &=&\int \tilde{S}(1-\cos k)(1-\cos p)-I_{1}-I_{2}\newline
\ \\
&=&\left\{ 2(1+\varepsilon )I_{1}Q_{110}+4\varepsilon
(5K_{0}+2K)Q_{210}\right. \\
&&\left. +4(I_{2}+5K_{0}+2K)Q_{120}+4\left( I_{3}+4K_{0}\right)
Q_{111}\right. \\
&&\left. -8K_{0}(1+\varepsilon )Q_{220}-8\left( K+K_{0}\right) Q_{121}\right.
\\
&&\left. -8\left( \varepsilon K+K_{0}\right) Q_{211}-8\varepsilon
K_{0}Q_{310}-8K_{0}Q_{130}-I_{1}-I_{2}\right\}\end{aligned}$$$$\begin{aligned}
G(1,0,1) &=&\int \tilde{S}(1-\cos k)(1-\cos q)-I_{1}-I_{3}\newline
\ \\
&=&\left\{ 2(1+\varepsilon )I_{1}Q_{101}+4\varepsilon
(5K_{0}+2K)Q_{201}\right. \\
&&\left. +4(I_{2}+5K_{0}+2K)Q_{111}+4\left( I_{3}+4K_{0}\right)
Q_{102}\right. \\
&&\left. -8K_{0}(1+\varepsilon )Q_{211}-8\left( K+K_{0}\right) Q_{112}\right.
\\
&&\left. -8\left( \varepsilon K+K_{0}\right) Q_{202}-8\varepsilon
K_{0}Q_{301}\right. \\
&&\left. -8K_{0}Q_{121}-I_{1}-I_{3}\right\}\end{aligned}$$$$\begin{aligned}
\ G(0,1,1) &=&\int \tilde{S}(1-\cos p)(1-\cos q)-I_{2}-I_{3} \\
&=&\left\{ 2(1+\varepsilon )I_{1}Q_{011}+4\varepsilon
(5K_{0}+2K)Q_{111}\right. \\
&&\left. +4(I_{2}+5K_{0}+2K)Q_{021}+4\left( I_{3}+4K_{0}\right)
Q_{012}\right. \\
&&\left. -8K_{0}(1+\varepsilon )Q_{121}-8\left( K+K_{0}\right) Q_{022}\right.
\\
&&\left. -8\left( \varepsilon K+K_{0}\right) Q_{112}-8\varepsilon
K_{0}Q_{211}\right. \\
&&\left. -8K_{0}Q_{031}-I_{2}-I_{3}\right\}\end{aligned}$$After the additional $Q$-integrals have been determined, the energy currents are easily found.
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---
abstract: 'In the quickest change detection problem in which both nuisance and critical changes may occur, the objective is to detect the critical change as quickly as possible without raising an alarm when either there is no change or a nuisance change has occurred. A window-limited sequential change detection procedure based on the generalized likelihood ratio test statistic is proposed. A recursive update scheme for the proposed test statistic is developed and is shown to be asymptotically optimal under mild technical conditions. In the scenario where the post-change distribution belongs to a parametrized family, a generalized stopping time and a lower bound on its average run length are derived. The proposed stopping rule is compared with the stopping time and the naive 2-stage procedure that detects the nuisance or critical change using separate CuSum stopping procedures for the nuisance and critical changes. Simulations demonstrate that the proposed rule outperforms the stopping time and the 2-stage procedure, and experiments on a real dataset on bearing failure verify the performance of the proposed stopping time.'
author:
- 'Tze Siong Lau, and Wee Peng Tay, [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'StringDefinitions.bib'
- 'refs.bib'
title: Quickest Change Detection in the Presence of a Nuisance Change
---
Quickest change detection, nuisance change, Generalized Likelihood Ratio Test (GLRT), average run length, average detection delay
Introduction {#sec:intro}
============
The problem of detecting a change in the statistical properties of a signal with the shortest possible delay after the change is known as quickest change detection (QCD). Given a sequence of independent and identically distributed (i.i.d.) observations $\{x_t:t\in \mathbb{N}\}$ with distribution $f$ up to an unknown change point $\nu$ and i.i.d. with distribution $g\neq f $ after $\nu$, we aim to detect this change as quickly as possible while maintaining a false alarm constraint. Detecting for a change has applications in many areas, including manufacturing quality control[@woodall2004using; @lai1995sequential], fraud detection[@bolton02], cognitive radio[@lai2008quickest], network surveillance[@sequeira02; @tartakovsky2006novel; @LuoTayLen:J16], structural health monitoring[@sohn00], spam detection[@xie12; @JiTayVar:J17; @TanJiTay:J18], bioinformatics[@muggeo10], power system line outage detection[@banerjee2014power], and sensor networks[@coppin1996digital; @Hong2004; @YanZhoTay:J18].
For the non-Bayesian formulation of QCD, the change-point is assumed to be unknown but deterministic. When both the pre- and post-change distributions are known, Page [@page54] developed the Cumulative Sum Control Chart (CuSum) for quickest change detection. Lorden[@lorden71] proved that the CuSum test has asymptotically optimal worst-case average detection delay as the false alarm rate goes to zero. Moustakides [@moustakides86] later established that the CuSum test is exactly optimal under Lorden’s optimality criterion. Later, Lai showed in [@lai98] that the CuSum test is asymptotically optimal under Pollak’s criterion[@pollak85], as the false alarm rate goes to zero. For the case where the post-change distribution is unknown, Lorden[@lorden71] showed that the CuSum is asymptotically optimal for the case of finite multiple post-change distributions. Other methods were also proposed for the case when the post-change distribution is unknown to a certain degree [@siegmund95; @lai98; @banerjee15; @lau17; @lau2017optimal; @lau19]. We refer the reader to [@tartakovsky2014sequential; @veeravalli2013quickest; @poor2009quickest] and the references therein for an overview of the QCD problem.
In many practical applications, the signal of interest may undergo different types of change. However, only a subset of these changes may be of interest to the user. One example is the problem of bearing failure detection using accelerometer readings[@smith2015rolling]. During normal operations, the bearings are driven at two different activity levels, idle or active. In a typical bearing failure detection scenario, the bearing is initially driven at the idle state. A change to the active state results in a change in the statistical properties of the accelerometer readings. However, this change is not of interest to us and is called a nuisance change. We are only interested in the change arising from the failure of the bearing, which is known as a critical change. Furthermore, the statistical properties of the observations obtained when the bearing is faulty depend on the activity level that it is driven at. The traditional QCD framework does not allow us to distinguish between critical and nuisance changes. Furthermore, due to the nuisance change, the observations are no longer i.i.d. either in the pre-change or post-change regime, depending on when the nuisance change occurs. In this paper, we investigate the non-Bayesian formulation of the QCD problem under a nuisance change, and propose a window-limited stopping time that ignores the nuisance change but detects the critical change as quickly as possible.
Related Work
------------
Existing works in QCD that consider the problem where observations are not generated i.i.d. before and after the change-point can be categorized into three main categories. In the first category, the papers[@fuh2003; @fuh2004asymptotic; @fuh2015quickest] consider the problem where the pre-change distribution and the post-change distribution are modeled as hidden Markov models (HMMs). In [@fuh2003], the authors proved the asymptotic optimality of the CuSum procedure for the HMM signal model in the sense of Lorden. In [@fuh2004asymptotic], the authors developed the Shiryayev-Roberts-Pollak (SRP) rule for the HMM signal model and proved its optimality in the sense of Pollak. The authors of [@fuh2015quickest] consider the problem where the vector parameter of a two-state HMM changes at some unknown time. The second category of papers[@moustakides1998quickest; @tartakovsky2005general] considers a QCD problem which relaxes the i.i.d. assumption. In [@moustakides1998quickest], the authors established the optimality of CuSum and the Shiryayev-Roberts stopping rule in the class of random processes with likelihood ratios that satisfy certain independence and stationary conditions. The class of random processes includes Markov chains, AR processes, and processes evolving on a circle. In [@tartakovsky2005general], the authors considered the Bayesian QCD problem where conditions on the asymptotic behavior of the likelihood process are assumed. Unlike all the aforementioned papers, the signal model in our QCD problem with nuisance change cannot be modeled by an HMM, and the likelihood ratios generated by our signal model are non-stationary. In the third category, the papers [@Krishnamurthy12; @guepie2012sequential; @ebrahimzadeh2015sequential; @Moustakides16; @zou2018quickest; @heydari2018quickest] consider QCD of transient changes, where the change is either not persistent or multiple changes occur throughout the monitoring process. Unlike our QCD problem which allows some changes to be considered nuisance, all changes are considered critical in the aforementioned papers.
Our Contributions {#subsec:contributions}
-----------------
In this paper, we consider the non-Bayesian QCD problem where both nuisance and critical changes may occur, and our objective is to detect the critical change as quickly as possible while ignoring the nuisance change. Our goal is to develop a sequential algorithm with computational complexity that increases linearly with the number of samples observed. Our main contributions are as follows:
1. We formulate the QCD problem with a nuisance change and propose a window-limited simplified (W-SGLR) stopping time.
2. We derive a lower bound for the to a false alarm, and the asymptotic upper bound of the for our proposed test.
3. We prove the asymptotic optimality of the W-SGLR stopping time under mild technical assumptions.
4. We provide simulation and experimental results that verify the theoretical guarantees of our proposed test and also illustrate the performance of our proposed test on a real dataset.
A preliminary version of this work was presented in [@lau2018quickest; @lau2019quickest]. To the best of our knowledge, there are no existing works that consider the QCD problem for a signal that may undergo a nuisance change.
The rest of this paper is organized as follows. In Section \[sec:problem\], we present our signal model and problem formulation. We propose the W-SGLR stopping time and derive the theoretical properties of our test statistics in Section \[sec:teststats\]. In Section \[sec:discussion\], we discuss a modification of the proposed stopping time when the post-change distribution belongs to a parametrized family. We present numerical simulations and experiments on a real dataset to illustrate the performance of our proposed stopping time in Section \[sec:numerical\]. We conclude in Section \[sec:conclusion\].
*Notations:* The operator $\E_f$ denotes mathematical expectation the probability density (pdf) $f$, and $X \sim f$ means that the random variable $X$ has distribution with pdf $f$. If the nuisance change point is at $\nu_n$, and the critical change point is at $\nu_c$, we let $\P{\nu_n,\nu_c}$ and $\E_{\nu_n,\nu_c}$ be the probability measure and mathematical expectation, respectively. The Gaussian distribution with mean $\mu$ and variance $\sigma^2$ is denoted as $\calN(\mu,\sigma^2)$. Convergence in $\P$-probability is denoted as $\xrightarrow{\ \P\ }$. We use $\indicatore{E}$ as the indicator function of the set $E$, and $\KLD{\cdot}{\cdot}$ to denote the divergence. We use $\Nat$, $\mathbb{R}$ and $\mathbb{R}_{>0}$ to denote the set of positive integers, real numbers and positive real numbers, respectively.
Problem formulation {#sec:problem}
===================
In many applications, the statistical distribution of the observed signal may undergo different changes over time. For example, in the application of fault detection in motor bearings [@smith2015rolling], we aim to raise an alarm as soon as possible after a bearing fault has occurred (critical change). This is done by monitoring the accelerometer readings from the motor to detect any changes in the signal statistics. However, the accelerometer readings are also affected by non-critical or nuisance changes like variation in the motor-load of the bearing. It would be undesirable if we declare that a fault has taken place whenever the motor-load changes. This motivates a need to define a change-point model that allows both critical and nuisances changes and to develop change detection techniques that can effectively ignore nuisance changes while efficiently identifying critical changes.
In this paper, we assume that the signals observed, $X_1, X_2, \ldots$, may undergo two types of change: a critical change at $\nu_c \geq 0$ and a nuisance change at $\nu_n \geq 0$. Both the critical and nuisance change points are unknown *a priori*. We are interested in detecting the critical change while the nuisance change is not of interest. Let $f,f_n,g,g_n$ be probability distributions. At each time $t$, we let $h_{\nu_n,\nu_c,t}$ to be the distribution that generates the observation $X_t$ when the nuisance change point is at $\nu_n$ and the critical change point is at $\nu_c$: $$\begin{aligned}
\label{eqn:definition_of_h}
h_{\nu_n,\nu_c,t}=\begin{cases}
f\quad \text{if $t<\min\{\nu_c,\nu_n\}$,} \\
f_n\quad \text{if $\nu_n\leq t<\nu_c$,} \\
g\quad \text{if $\nu_c\leq t<\nu_n$,} \\
g_n\quad \text{if $\max\{\nu_c,\nu_n\}\leq t$.} \\
\end{cases}
\end{aligned}$$ Thus, in our model (cf. \[fig:h\_model\]), $f$ is the pre-change distribution.If $\nu_c < \nu_n < \infty$, the signal distribution first changes to $g$ at $\nu_c$ and then to $g_n$ at $\nu_n$. If $\nu_n < \nu_c < \infty$, the distribution first changes to $f_n$ at $\nu_n$ and then to $g_n$ at $\nu_c$. If $\nu_n=\nu_c$, then the distribution changes from $f$ to $g_n$ at the common change point.
[.25]{} ![Evolution of the signal distribution in our model.[]{data-label="fig:h_model"}](signal_model_a "fig:"){width="0.8\linewidth"}
[.25]{} ![Evolution of the signal distribution in our model.[]{data-label="fig:h_model"}](signal_model_b "fig:"){width="0.8\linewidth"}
The sequence of observations $X_1,X_2,\ldots$ is a sequence of random variables satisfying $X_t \sim h_{\nu_n,\nu_c,t}$ where $\{X_t\}_{t\in\mathbb{N}}$ are mutually independent given $\nu_n,\nu_c$. The quickest change detection problem is to detect the critical change $\nu_c$ through observing $X_1,X_2,\ldots,$ as quickly as possible while ignoring the nuisance change and keeping the false alarm rate low. In our signal model, the nuisance change also changes the distribution that generates the observations after the critical change point. This creates a dependence between the nuisance change point and the distribution after the critical change point. Our formulation is different from assuming composite pre-change and post-change distribution families[@mei2003asymptotically] since the nuisance change leads to non-stationarity in the distribution of $X_t$ before or after the critical change, depending on whether the nuisance change occurs before or after the critical change, respectively.
In a typical sequential change detection procedure, at each time $t$, a test statistic $S(t)$ is computed based on the currently available observations $X_1,\ldots,X_t$, and the observer decides that a change has occurred at a stopping time $
\tau(b)=\inf\{t:S(t)\geq b\}.
$
In the traditional QCD framework[@lorden71], the rate of false alarms is quantified by the mean time between false alarms. Since the nuisance change-point affects the distributions generating the signal, this quantity varies with the nuisance change point. In this paper, we consider the worst-possible rate as the nuisance change point varies by considering the smallest mean time between false alarms for all possible nuisance change points. A similar generalization can be made to quantify the detection delay by taking the largest detection delay over all possible nuisance change points.
Mathematically, our QCD problem can be formulated as a minimax problem similar to Lorden’s formulation[@lorden71], where we seek a stopping time that minimizes the WADD subject to an constraint: $$\label{eqn:lorden_formulation}
\begin{aligned}
& \min_\tau & & {\text{WADD}}(\tau) \\
& \st & & {\text{ARL}}(\tau)\geq\gamma,
\end{aligned}$$ where $\gamma$ is a predefined threshold, $\tau$ is a stopping time the filtration $\{\sigma(X_1,X_2,\ldots,X_t) :\ t\geq 0\}$, $$\begin{aligned}
&{\text{WADD}}(\tau) =\sup_{\nu_n\in\mathbf{N}\cup\{\infty\}}{\text{WADD}}_{\nu_n}(\tau),\label{eqn:WADD}\\
&{\text{WADD}}_{\nu_n}(\tau) = \sup_{\nu_c\geq 1}\esssup \E{\nu_n,\nu_c}[(\tau-\nu_c+1)^+]{X_1,\ldots,X_{\nu_c-1}},\\
&{\text{ARL}}(\tau) =\inf_{\nu_n\in\mathbb{N}\cup\{\infty\}}\E{\nu_n,\infty}[\tau],
\end{aligned}$$ and $\esssup$ is the essential supremum operator. In the next section, we propose a stopping time for .
A closely related topic is [@bojdecki1980probability; @pollak2013shewhart; @moustakides2014multiple; @poor18FMA] where the change only occurs for a finite period of time and the objective is to detect if such a change has occurred within a predefined window or not instead of detecting the change as quickly as possible. There are two widely adopted methods for the TCD problem, the window-limited CuSum stopping time [@guepie2012sequential] and the stopping time[@poor18FMA]. The stopping time has been shown to perform well for the TCD problem, and we will use the FMA stopping time as a comparison in \[sec:numerical\]. When $\nu_c < \nu_n$, our system model can be seen to be a generalization of the TCD problem variant where one seeks to detect the transient change as quickly as possible by letting $g_n = f$. In \[eqn:WSGLR-test-statistic\] below, we propose a test statistic and stopping time for \[eqn:lorden\_formulation\]. By setting $g_n=f$ and $f_n=f$ in our test statistic, our proposed stopping time reduces to the window-limited CuSum stopping time with pre-change distribution $f$ and post-change distribution $g$.
Test Statistic for QCD with Nuisance Change {#sec:teststats}
===========================================
In this section, we derive a test-statistic and stopping time for QCD under a nuisance change. Suppose that we observe the sequence $X_1, X_2,\ldots$ and know *a priori* that the nuisance change does not take place (i.e., $\nu_n=\infty$), then Page’s CuSum test statistic[@page54] given as $
S_{\text{CuSum}}(t)=\max_{1\leq k\leq t+1}\sum_{i=k}^t \log\tfrac{g(X_i)}{f(X_i)},
$ can be used and we declare that a critical change has taken place at $$\begin{aligned}
\label{eqn:rpt_sprt}
\tau_{\text{CuSum}}(b) & =\inf\left\{t\ :\ S_{\text{CuSum}}(t)\geq b\right\} \\
& =\inf\left\{t\ :\ \max_{1\leq k\leq t+1}\sum_{i=k}^t \log \tfrac{g(X_i)}{f(X_i)}\geq b\right\},
\end{aligned}$$ where $b$ is a pre-determined threshold. The CuSum test statistics has a convenient recursion $
S_{\text{CuSum}}(t+1)=\max\left\{S_{\text{CuSum}}(t)+\log\tfrac{g(X_t)}{f(X_t)} ,0\right\},
$ which allows the CuSum stopping time to be implemented efficiently.
If the nuisance change takes places at a time $\nu_n<\infty$ and $\nu_n$ is known, a modification of Page’s test statistic gives the following: $$\begin{aligned}
S_{\text{CuSum}}(t) & =\max_{1\leq k\leq t+1}\sum_{i=k}^t \log\tfrac{h_{\nu_n,1,i}(X_i)}{h_{\nu_n,\infty,i}(X_i)}, \\
\tau_{\text{CuSum}}(b) & =\inf\left\{t\ :\ \max_{1\leq k\leq t+1}\sum_{i=k}^t \log\tfrac{h_{\nu_n,1,i}(X_i)}{h_{\nu_n,\infty,i}(X_i)}\geq b\right\},
\end{aligned}$$ where $h_{\nu_n,1,i}(x)$ and $h_{\nu_n,\infty,i}(x)$ are as defined in and are the probability distributions corresponding to the cases where the critical change has already occurred or will never occur, respectively. Similar to the case where $\nu_n=\infty$, the CuSum test statistics admits a convenient recursion for efficient implementation. Furthermore, for both the cases mentioned above, $\tau_{\text{CuSum}}$ was shown to be asymptotically optimal by [@lorden71].
A naive approach is to utilize four variants of $\tau_{\text{CuSum}}$, one for detecting for a change in each of the cases: from $f$ to $f_n$, from $f$ to $g$, from $f$ to $g_n$, and from $f_n$ to $g_n$. In the first stage, we monitor for changes from $f$ to either $f_n$, $g$ or $g_n$. If a change to $f_n$ is detected, then we monitor for a change from $f_n$ to $g_n$. The difficulty in such an approach is that any false alarm or miss detection in the first stage propagates to the second stage. We demonstrate that such an approach is suboptimal in \[subsec:numerical\_results\_sim\_signals\].
In our problem formulation, the nuisance change-point $\nu_n$ is unknown. Replacing $\nu_n$ with its maximum likelihood estimator in both the numerator and denominator, we obtain the following test statistic and stopping time: $$\begin{aligned}
\Lambda_{\text{GLR}}(k,t) & =\frac{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,1,i}(X_i)}{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,\infty,i}(X_i)}\nn
& =\frac{\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}g(X_i)\prod_{i=j}^{t}g_n(X_i)}{\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}f(X_i)\prod_{i=j}^{t}f_n(X_i)}, \label{Lambda_GLR} \\
S_{\text{GLR}}(t)
& =\max_{1\leq k\leq t+1} \log\Lambda_{\text{GLR}}(k,t),\\
\tau_{\text{GLR}}(b) & =\inf\left\{t\ :\ S_{\text{GLR}}(t)\geq b\right\}. \label{tau_GLR}
\end{aligned}$$ From our simulations in \[subsec:numerical\_results\_sim\_signals\], it turns out that \[tau\_GLR\] does not achieve the best trade-off between and to false alarm over a wide range of threshold values $b$. Furthermore, its ARL is challenging to characterize theoretically since the GLR test statistic $\Lambda_{\text{GLR}}(k,t)$ is not a likelihood ratio and standard techniques in the QCD literature (e.g., Theorem 6.16 of [@poor2009quickest]) cannot be used to analyze its ARL. This is a critical problem for practical applications that require us to pre-determine a suitable threshold $b$ to achieve a desired ARL.
To develop a stopping time with ARL that can be characterized theoretically, we simplify the maximum likelihood estimation in the numerator of \[Lambda\_GLR\] to be the maximum of only two cases $j=k$ and $j=t+1$. This gives us the Simplified GLR (SGLR) test statistic and stopping time as follows: $$\begin{aligned}
\Lambda_{\text{SGLR}}(k,t) & =\frac{\max\left\{\prod_{i=k}^{t}g(X_i),\prod_{i=k}^{t}g_n(X_i)\right\}}{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,\infty,i}(X_i)} \label{eqn:SGLR_teststats} \\
S_{\text{SGLR}}(t) & =\max_{1\leq k\leq t+1} \log\Lambda_{\text{SGLR}}(k,t), \\
\tau_{\text{SGLR}}(b) & =\inf\left\{t\ :\ S_{\text{SGLR}}(t)\geq b\right\}. \label{tau_SGLR}
\end{aligned}$$
Unlike the CuSum test statistic, the SGLR test statistic does not have a convenient recursion. Any implementation of the SGLR stopping time would require computational resources that increases with the number of samples observed. The requirement on computational resources would be a significant limitation for many practical applications. To limit the computational resources required,in the same spirit as [@lai98], we propose the Window-Limited SGLR (W-SGLR) test statistic and stopping time as follows: $$\begin{aligned}
S_{\text{W-SGLR}}(t) & =\max_{t-m_b\leq k\leq t+1} \log\Lambda_{\text{SGLR}}(k,t), \label{eqn:WSGLR-test-statistic} \\
{\tau_{\text{W-SGLR}}}(b) & =\inf\left\{t\ :\ S_{\text{W-SGLR}}(t)\geq b\right\},\label{tau_WSGLR}
\end{aligned}$$ where the window size $m_b$ is chosen such that $$\begin{aligned}
\label{mb}
\liminf_{b\to\infty} \frac{m_b}{b}>I^{-1} \text{ and } \log m_b=o(b),
\end{aligned}$$ with $$\begin{aligned}
\begin{split}
I=\min\left\{\KLD{g}{f},\ \KLD{g}{f_n},\ \KLD{g_n}{f},\ \KLD{g_n}{f_n}\right\}, \label{I}
\end{split}
\end{aligned}$$ and $o(b)$ denoting a term that goes to zero as $b\to\infty$. Window-limited test statistics were first introduced by [@willsky1976generalized]. The paper [@lai98] further discussed their properties and the choice of window size and thresholds. We make the following assumption.
\[assumpt:moments\] The first four moments of $\log\tfrac{f_n(X)}{f(X)}$ both $g$ and $g_n$ are finite, and $\rho_g, \rho_{g_n}\neq 0$, where we define $ \rho_{g}=\E{g}[\log \tfrac{f_n(X)}{f(X)}]$, $\sigma_{g}^2=\E{g}[\left(\log \tfrac{f_n(X)}{f(X)}-\rho_g\right)^2]$, $\omega_{g}^4=\E{g}[\left(\log \tfrac{f_n(X)}{f(X)}-\rho_g\right)^4]$, $\rho_{g_n}=\E{g_n}[\log \tfrac{f_n(X)}{f(X)}]$, $\sigma_{g_n}^2=\E{g_n}[\left(\log \tfrac{f_n(X)}{f(X)}-\rho_{g_n}\right)^2]$ and $\omega_{g_n}^4=\E{g_n}[\left(\log \tfrac{f_n(X)}{f(X)}-\rho_{g_n}\right)^4]$.
In \[thm:main\_result\] of \[subsec:properties\_stoptime\], we show that the proposed ${\tau_{\text{W-SGLR}}}(b)$ is asymptotically optimal as $b\to\infty$ under \[assumpt:moments\] and an additional technical assumption. To do that, we first analyze the asymptotic properties of ${\tau_{\text{W-SGLR}}}$. We let $$\begin{aligned}
\Lambda(k,t) & =\tfrac{\prod_{i=k}^t g(X_i)}{\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}f(X_i)\prod_{i=j}^{t}f_n(X_i)},\label{Lambda} \\
\Lambda_n(k,t) & =\tfrac{\prod_{i=k}^t g_n(X_i)}{\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}f(X_i)\prod_{i=j}^{t}f_n(X_i)},\label{Lambdan}
\end{aligned}$$ and study their properties in \[subsec:properties\_stats\]. Then, using the relationships $$\begin{aligned}
\Lambda_{\text{SGLR}}(k,t) & =\max\left\{\Lambda(k,t),\Lambda_n(k,t)\right\}, \\
\tau_{\text{SGLR}}(b) & =\min\{\tau(b),\tau_n(b)\}, \label{tau_SGLR2} \\
\tau_{\text{W-SGLR}}(b) & =\min\{\widetilde{\tau}(b),\widetilde{\tau_n}(b)\}, \label{tau_WSGLR2}
\end{aligned}$$ where $$\begin{aligned}
\begin{split}\tau(b) & =\inf\{t\ :\ \max_{k\leq t}\log\Lambda(k,t)\geq b \},\\
\tau_n(b)&=\inf\{t\ :\ \max_{k\leq t}\log\Lambda_n(k,t)\geq b \}, \\
\widetilde{\tau}(b) & =\inf\{t\ :\ \max_{t-m_b \leq k\leq t}\log\Lambda(k,t)\geq b \},\\
\quad\widetilde{\tau}_n(b)&=\inf\{t\ :\ \max_{t-m_b \leq k\leq t}\log\Lambda_n(k,t)\geq b \},\end{split}\label{tau_taun}
\end{aligned}$$ we finally show the asymptotic optimality of ${\tau_{\text{W-SGLR}}}$ under mild technical conditions in \[subsec:properties\_stoptime\].
Log Likelihood Ratio Growth Rates {#subsec:properties_stats}
---------------------------------
In this subsection, we derive properties of $\Lambda$ and $\Lambda_n$ as defined in \[Lambda,Lambdan\], respectively. The stopping times $\tau_{\text{SGLR}}(b)$ and $\tau_{\text{W-SGLR}}(b)$ are defined by the first time the test statistics $S_{\text{SGLR}}$ and $S_{\text{W-SGLR}}$ cross the threshold $b$ respectively. The rates of growth, $\tfrac{1}{t-k+1}\log\Lambda(k,t)$ and $\tfrac{1}{t-k+1}\log\Lambda_n(k,t)$, allow us to understand the detection delay of these stopping times. We show that these rates of growth converge in probability as $t\to\infty$. In particular, the limit that the rate of growth converges to depends on the sign of $\rho_{g}$ and $\rho_{g_n}$.
As the nuisance change point is unknown, the denominator of both $\Lambda$ and $\Lambda_n$ contains a maximization of the likelihood $
\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}f(X_i)\prod_{i=j}^{t}f_n(X_i).
$ If the first moment $\rho_g<0$, the distribution $g$ is closer to the distribution $f$ as compared to $f_n$ in the divergence sense. When the critical change point is at $\nu_c=1$ and no nuisance change has taken place, we expect the denominator to approach $\prod_{i=k}^{t}f(X_i)$. Thus, our statistic $\Lambda(k,t)$ can be approximated by $\prod_{i=k}^t\tfrac{g(X_i)}{f(X_i)}$. A similar argument can be made for $\Lambda_n$ when $\nu_n=\nu_c=1$. This observation is made precise in the following two propositions.
\[prop:converge\_to\_post\_nui\_LR\] Suppose that \[assumpt:moments\] holds, and $\rho_{g}<0$. For any $\nu_c\leq k<\infty$ and $\epsilon>0$, we have $$\begin{aligned}
& \lim_{t\to\infty}\P{\infty,\nu_c}(\left|\tfrac{\log \Lambda(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log \tfrac{g(X_i)}{f(X_i)}\right|\geq\epsilon)=0, \\
& \lim_{t\to\infty}\P{\infty,\nu_c}(\left|\tfrac{\log\Lambda_n(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log \tfrac{g_n(X_i)}{f(X_i)}\right|\geq\epsilon)=0.
\end{aligned}$$
See \[sec:AppProp1\].
\[prop:converge\_to\_pre\_nui\_LR\] Suppose that \[assumpt:moments\] holds, and $\rho_{g_n}>0$. For any $\nu_c\leq k<\infty$, $\nu_n<\infty$, and $\epsilon>0$, we have $$\begin{aligned}
& \lim_{t\to\infty}\P{\nu_n,\nu_c}(\left|\tfrac{\log\Lambda(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log \tfrac{g(X_i)}{f_n(X_i)}\right|\geq\epsilon)=0, \\
& \lim_{t\to\infty}\P{\nu_n,\nu_c}(\left|\tfrac{\log\Lambda_n(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log \tfrac{g_n(X_i)}{f_n(X_i)}\right|\geq\epsilon)=0.
\end{aligned}$$
See \[sec:AppProp1\].
Using \[prop:converge\_to\_post\_nui\_LR,prop:converge\_to\_pre\_nui\_LR\] together with the weak law of large numbers, we obtain the following result.
\[thm:convergence\_in\_probability\_special\] Suppose that \[assumpt:moments\] holds, $\rho_{g}=\KLD{g}{f}-\KLD{g}{f_n}<0$, and $\rho_{g_n}=\KLD{g_n}{f}-\KLD{g_n}{f_n}>0$. For any $\nu_c\leq k<\infty$, $$\begin{aligned}
\tfrac{\log \Lambda(k,t)}{t-k+1} & \xrightarrow{\P{\infty,\nu_c}} \KLD{g}{f}, \\
\tfrac{\log \Lambda_n(k,t)}{t-k+1} & \xrightarrow{\P{\infty,\nu_c}} \KLD{g}{f}-\KLD{g}{g_n},
\end{aligned}$$ as $t\to\infty$. Furthermore, for any $\nu_n < \infty$, $$\begin{aligned}
\tfrac{\log \Lambda(k,t)}{t-k+1} & \xrightarrow{\P{\nu_n,\nu_c}} \KLD{g_n}{f_n}-\KLD{g_n}{g}, \\
\tfrac{\log \Lambda_n(k,t)}{t-k+1} & \xrightarrow{\P{\nu_n,\nu_c}} \KLD{g_n}{f_n},
\end{aligned}$$ as $t\to\infty$.
In \[thm:convergence\_in\_probability\_special\], we have assumed that $\rho_{g}<0$ and $\rho_{g_n}>0$. If we vary the signs of $\rho_{g}$ and $\rho_{g_n}$, a similar argument to that provided in \[thm:convergence\_in\_probability\_special\] gives us the following result.
\[thm:convergence\_in\_probability\_general\] Suppose that \[assumpt:moments\] holds. For any $\nu_c\leq k<\infty$ and $\nu_n<\infty$, we have the following convergences in probability as $t\to\infty$ shown in the table below.
---------- -------------- ------------------------------------ --------------------------------------
$\rho_g$ $\rho_{g_n}$ $\tfrac{\log \Lambda(k,t)}{t-k+1}$ $\tfrac{\log \Lambda_n(k,t)}{t-k+1}$
>0 >0 $\KLD{g}{f_n}$ $\KLD{g_n}{f_n}$
>0 <0 $\KLD{g}{f_n}$ $\KLD{g_n}{f}$
<0 >0 $\KLD{g}{f}$ $\KLD{g_n}{f_n}$
<0 <0 $\KLD{g}{f}$ $\KLD{g_n}{f}$
---------- -------------- ------------------------------------ --------------------------------------
gives us the average rate of growth of the statistics $\log \Lambda(k,t)$ and $\log\Lambda_n(k,t)$. Since $I$ in \[I\] is the minimum of the growth rates in \[thm:convergence\_in\_probability\_general\], we see that the average growth rate of these statistics is at least $I$ regardless of the signs of $\rho_g$ and $\rho_{g_n}$. This suggests that the WADD of $\tau_{\text{W-SGLR}}(b)$ grows linearly with respect to $b$ with a gradient bounded above by $I$. This observation is made precise in the next subsection.
Conditions for Asymptotic Optimality {#subsec:properties_stoptime}
------------------------------------
In this subsection, we establish the asymptotic - trade-off under \[assumpt:moments\] and provide a sufficient condition for $\tau_{\text{W-SGLR}}$ to be asymptotically optimal. In particular, we show that $\tau_{\text{W-SGLR}}$ is asymptotically optimal if in addition to \[assumpt:moments\], the following assumption holds.
\[assumpt:kldiv\] The KL divergences $\KLD{g}{f}$, $\KLD{g}{f_n}$, $\KLD{g_n}{f}$ and $\KLD{g_n}{f_n}$ satisfy $$\begin{aligned}
\KLD{g}{f_n}>\min\left\{\KLD{g}{f},\ \KLD{g_n}{f},\ \KLD{g_n}{f_n}\right\}.
\end{aligned}$$
\[assumpt:kldiv\] essentially says that $g$ cannot be too similar to $f_n$, which makes intuitive sense as otherwise it is difficult to distinguish the critical change from the nuisance change (see \[fig:h\_model\]). A sufficient condition for \[assumpt:kldiv\] is $\rho_g <0$ as assumed in \[thm:convergence\_in\_probability\_special\]. For example, in the problem of spectrum sensing in cognitive radio[@poor08qcdspectrum], we are often interested in detecting a variance change of a signal generated by independent Gaussian distributions. Furthermore, in many signal processing applications, a change in mean may be due to sensor drift as a result of long duration monitoring. This change in mean is usually not of interest and interferes with the actual signal processing task[@Wang17drift; @li2015Driftdetection; @Takruri07drift]. A typical signal model of this type is given by $f=\calN(\mu_0,\sigma_0^2)$, $f_n=\calN(\mu_1,\sigma_0^2)$, $g=\calN(\mu_0,\sigma_1^2)$, and $g_n=\calN(\mu_1,\sigma_1^2)$ with $\mu_0,\mu_1\in\mathbb{R}$, $\mu_0\neq\mu_1$, $\sigma_0,\sigma_1\in\mathbb{R}_{>0}$. While \[assumpt:kldiv\] may seem artificial, it is shown in \[lem:sufficient\_conditions\_exponential\_family\] that this model satisfies $\rho_g <0$ and hence \[assumpt:kldiv\]. In this case, the W-SGLR stopping time achieves asymptotic optimality.
We use techniques from the proof of Theorem 6.16 in [@poor2009quickest] to obtain a lower bound for the ARL of $\tau_{\text{SGLR}}$ in \[tau\_SGLR\]. Since ${\tau_{\text{W-SGLR}}}\geq \tau_{\text{SGLR}}$, the same lower bound also applies for the ARL of $\tau_{\text{W-SGLR}}$ in \[tau\_WSGLR\]. In the previous subsection, we have shown that the rate of growth of the statistics $\Lambda$ and $\Lambda_n$ converge to constants as $t\to\infty$. This means that, asymptotically, $\Lambda$ and $\Lambda_n$ grow linearly $t$. Heuristically, this implies that the WADD of the stopping times, $\tau(b)$ and $\tau_n(b)$, grow linearly the threshold $b$ in \[tau\_taun\].
derives an upper-bound for the probability of a false alarm for a stopping time related to ${\tau_{\text{W-SGLR}}}(b)$. Following Theorem 6.16 in [@poor2009quickest], this upper bound then yields a lower bound for the ARL of $\tau_{\text{W-SGLR}}(b)$ in \[thm:main\_result\].
\[lem:fa\_prob\] For $b>0$, let $\eta^k,\eta^k_n$ be stopping times defined by $$\begin{aligned}
\eta^k & =\inf\{t\geq k :\ \log\Lambda(k,t)\geq b\}, \\
\eta^k_n & =\inf\{t\geq k :\ \log \Lambda_n(k,t)\geq b\},
\end{aligned}$$ so that $\tau(b)=\inf_{k \geq 1} \eta^k$ and $\tau_n(b)=\inf_{k \geq 1} \eta^k_n$ in \[tau\_taun\]. For any $\nu_n\in\mathbb{N}\cup\{\infty\}$, we have $$\begin{aligned}
\P{\nu_n,\infty}(\eta^1<\infty)\leq e^{-b}\quad \text{and}\quad\P{\nu_n,\infty}(\eta_n^1<\infty)\leq e^{-b},
\end{aligned}$$ and $$\begin{aligned}
\label{ARL_tauWSGLR}
\E{\nu_n,\infty}[{\tau_{\text{W-SGLR}}}(b)] \geq \E{\nu_n,\infty}[{\tau_{\text{SGLR}}}(b)] \geq \ofrac{2}e^b.
\end{aligned}$$
See \[sec:AppLem1\].
The next lemma checks that our proposed stopping time satisfies the assumption required in [@lai98] to relate the asymptotic upper-bound for the WADD to the threshold $b$ in \[prop:ADD\].
\[lem:limsup\_assumption\] Suppose that \[assumpt:moments\] holds. For any $\delta>0$, we have
(i) \[eqn:limsup:it1\]$\displaystyle\lim_{t\to\infty}\quad\ \ \sup_{{\nu_n\in\mathbb{N}, 1\leq \nu_c \leq k}} \P{\nu_n,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)=0$, and
(ii) \[eqn:limsup:it2\]$\displaystyle\lim_{t\to\infty}\sup_{1\leq \nu_c \leq k} \P{\infty,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)=0$.
See \[sec:AppLem2\].
\[prop:ADD\] Suppose that \[assumpt:moments\] holds. There exists a $B$ such that for all $b \geq B$, we have
(i) \[prop:ADD:it1\] $\displaystyle\sup_{\nu_n,\nu_c\geq1}\esssup \E{\nu_n,\nu_c}[(\widetilde{\tau}_n(b)-\nu_c+1)^+]{X_1,\ldots,X_{\nu_c-1}}\\\leq (I^{-1}+o(1))b$, and
(ii) \[prop:ADD:it2\] $\displaystyle\sup_{\nu_c\geq1}\esssup \E{\infty,\nu_c}[(\widetilde{\tau}(b)-\nu_c+1)^+]{X_1,\ldots,X_{\nu_c-1}}\leq (I^{-1}+o(1))b$.
See \[sec:AppProp3\].
Finally, we show the asymptotic optimality of ${\tau_{\text{W-SGLR}}}$ in the following result.
\[thm:main\_result\] Suppose that \[assumpt:moments\] holds. For any $b>0$, $$\begin{aligned}
{\text{ARL}}({\tau_{\text{W-SGLR}}}(b)) & \geq \tfrac{1}{2}e^b, \\
\text{WADD}({\tau_{\text{W-SGLR}}}(b)) & \leq(I^{-1}+o(1))b, \label{WSGLR_opt}
\end{aligned}$$ where $o(1)$ is a term going to zero as $b\to\infty$. Furthermore, if \[assumpt:kldiv\] holds, then the stopping time ${\tau_{\text{W-SGLR}}}(b)$ is asymptotically optimal for the problem as $b\to\infty$.
See \[sec:AppThm1\].
In \[thm:main\_result\], we have shown that ${\tau_{\text{W-SGLR}}}$ is asymptotically optimal under \[assumpt:moments\] and \[assumpt:kldiv\]. In the next lemma, we derive sufficient conditions for \[assumpt:kldiv\] when $f,f_n,g,g_n$ belong to an exponential family.
\[lem:sufficient\_conditions\_exponential\_family\] Suppose that $f$, $f_n$, $g$, $g_n\in\{\phi :\ \phi(x)=h(x)\exp\left(\sum_{i=1}^s B_i(\theta)T_i(x)-A(\theta)\right)\}$, an exponential family of distributions on $\mathbb{R}^N$ with parameters $\theta=\theta_f,\theta_{f_n},\theta_{g},\theta_{g_n},$ respectively. Here, $T_i\in\mathbb{R}^N\times\mathbb{R}$ and $A, B_i\in\mathbb{R}^M\times\mathbb{R}$ for $i=1,\ldots,s$. If any of the following inequalities hold: $$\begin{aligned}
&A(\theta_{f_n})-A(\theta_{f})-\sum_{i=1}^s\left(B_i(\theta_{f_n})-B_i(\theta_{f})\right)\E{g}[T_i(X)]>0, \label{eqn:condition_for_exp_fam}\\
\begin{split}
&A(\theta_{f_n})-A(\theta_{g})-\sum_{i=1}^s\left(B_i(\theta_{f_n})-B_i(\theta_{g})\right)\E{g}[T_i(X)]\\
&>A(\theta_{f})-A(\theta_{g_n})-\sum_{i=1}^s\left(B_i(\theta_{f})-B_i(\theta_{g_n})\right)\E{g_n}[T_i(X)],
\end{split}\label{eqn:condition_for_exp_fam_1}\\
\begin{split}
&A(\theta_{f_n})-A(\theta_{g})-\sum_{i=1}^s\left(B_i(\theta_{f_n})-B_i(\theta_{g})\right)\E{g}[T_i(X)]\\
&>A(\theta_{f_n})-A(\theta_{g_n})-\sum_{i=1}^s\left(B_i(\theta_{f_n})-B_i(\theta_{g_n})\right)\E{g_n}[T_i(X)],
\end{split}\label{eqn:condition_for_exp_fam_2}
\end{aligned}$$ then \[assumpt:kldiv\] holds.
In particular, if $f=\calN(\mu_0,\sigma_0^2), f_n=\calN(\mu_1,\sigma_0^2),g=\calN(\mu_0,\sigma_1^2)$, and $g_n=\calN(\mu_1,\sigma_1^2)$ with $\mu_0,\mu_1\in\mathbb{R}$, $\mu_0\neq\mu_1$, $\sigma_0,\sigma_1\in\mathbb{R}_{>0}$, and $\sigma_0\neq \sigma_1$, \[assumpt:kldiv\] holds.
To show that \[eqn:condition\_for\_exp\_fam\] implies \[assumpt:kldiv\], we rearrange the terms on the of \[eqn:condition\_for\_exp\_fam\] to obtain $$\begin{aligned}
\E{g}[\log\tfrac{f_n(X)}{f(X)}]=\E{g}[\log\tfrac{ h(X)\exp\left(\sum_{i=1}^{s}B_i(\theta_{f_n})T_i(X)-A(\theta_{f_n})\right)}{h(X)\exp\left(\sum_{i=1}^{s}B_i(\theta_{f})T_i(X)-A(\theta_{f})\right)}]< 0.\label{eqn:exp_fam_proof_1}
\end{aligned}$$ This implies that $$\begin{aligned}
\KLD{g}{f}< \KLD{g}{f_n},\label{eqn:exp_fam_proof_2}
\end{aligned}$$ and hence \[assumpt:kldiv\] holds. A similar argument shows that \[eqn:condition\_for\_exp\_fam\_1,eqn:condition\_for\_exp\_fam\_2\] imply $\KLD{g_n}{f}<\KLD{g}{f_n}$ and $\KLD{g_n}{f_n}<\KLD{g}{f_n}$, respectively.
If $f=\calN(\mu_0,\sigma_0^2),\ f_n=\calN(\mu_1,\sigma_0^2),\ g=\calN(\mu_0,\sigma_1^2)$, and $g_n=\calN(\mu_1,\sigma_1^2)$ with $\mu_0,\mu_1\in\mathbb{R}$, $\mu_0\neq\mu_1$, $\sigma_0,\sigma_1\in\mathbb{R}_{>0}$, and $\sigma_0\neq \sigma_1$, we can define $\theta_f=[\mu_0,\sigma_0^2],\ \theta_{f_n}=[\mu_1,\sigma_0^2],\ \theta_g=[\mu_0,\sigma_1^2],\ \theta_{g_n}=[\mu_1,\sigma_1^2]$, with the functions $B_1(\mu,\sigma^2)=\mu/\sigma^2$, $B_2(\mu,\sigma^2)=\tfrac{-1}{2\sigma^2}$, $T_1[X]=X$, $T_2[X]=X^2$ and $A(\mu,\sigma^2)=\tfrac{\mu^2}{2\sigma^2}+\log \sigma$. The of \[eqn:condition\_for\_exp\_fam\] becomes $\tfrac{\mu_1^2-\mu_0^2}{\sigma_0^2}-\left(\tfrac{{\mu_1-\mu_0}}{\sigma_0}\right)\mu_0$. Simplifying, the of \[eqn:condition\_for\_exp\_fam\] becomes $\tfrac{(\mu_1-\mu_0)^2}{2\sigma^2_0}$. Thus, for any $\mu_1\neq\mu_0$ and $\sigma_0,\sigma_1\in\mathbb{R}_{>0}$, the inequality \[eqn:condition\_for\_exp\_fam\] holds. The proof is now complete.
Parametrized Families of Post-Change Distributions {#sec:discussion}
==================================================
In many applications, the post-change distribution $g$ and nuisance post-change distribution $g_n$ may contain unknown parameters. In this section, we modify $\tau$ and $\tau_n$ in \[tau\_taun\] to obtain a -based stopping time $\widehat{\tau}_{\text{W-SGLR}}$ for the following signal model: Let $\Theta\subseteq \Real^d$ be a set with non-empty interior and $X_1,X_2,\ldots$ be a sequence of independent random variables satisfying: $
X_t \sim h_{\nu_n,\nu_c,\theta,\theta_n,t}
$ where $$\begin{aligned}
h_{\nu_n,\nu_c,\theta,\theta_n,t}(\cdot)=\begin{cases}
f(\cdot) & \text{if $t<\min\{\nu_c,\nu_n\}$,} \\
f_n(\cdot) & \text{if $\nu_n\leq t<\nu_c$,} \\
g(\cdot;\theta) & \text{if $\nu_c\leq t<\nu_n$,} \\
g_n(\cdot;\theta_n) & \text{if $\max\{\nu_c,\nu_n\}\leq t$,} \\
\end{cases}
\end{aligned}$$ and $\theta,\theta_n\in\text{Int}(\Theta)$, the interior of $\Theta$. We derive a lower bound for the ARL of $\widehat{\tau}_{\text{W-SGLR}}$ under the following assumption.
\[assup:compact2nddiff\] $\Theta$ is a compact $d$-dimensional sub-manifold of $\mathbb{R}^d$. The pdfs of the post-change distributions $g(\cdot;\theta)$ and nuisance post-change distribution $g_n(\cdot;\theta)$ are twice continuously differentiable $\theta$.
A commonly used method to handle unknown parameters is to replace the likelihood ratio $\Lambda(k,t)$ with the generalized likelihood ratio. We define the generalized W-SGLR test statistic $\widehat{S}_{\text{W-SGLR}}$ as $$\begin{aligned}
\widehat{\Lambda}_{\text{W-SGLR}}(k,t,\theta) & =\tfrac{\max\left\{\prod_{i=k}^{t}g(X_i;\theta),\prod_{i=k}^{t}g_n(X_i;\theta)\right\}}{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,\infty,\theta,\theta,i}(X_i)},\nonumber \\
\widehat{S}_{\text{W-SGLR}}(t) & =\max_{t-m_{b}+1\leq k\leq t-m_{b}'}\max_{\theta\in\Theta}\log\widehat{\Lambda}(k,t,\theta),\label{eqn:generalised_wsglr}
\end{aligned}$$ where the minimal delay $m_b'$ is required to prevent difficulties of under-determination when performing maximum likelihood estimation of the parameter $\theta$. While \[eqn:generalised\_wsglr\] is commonly used, the maximization over $\Theta$ make it difficult to theoretically quantify the of the stopping time $\inf\{t:\ \widehat{S}_{\text{W-SGLR}}(t) \geq b\}$. To work around this problem, we modify the stopping times $\tau$ and $\tau_n$ as follows. Let $\lambda_{\max}(A)$ denote the largest eigenvalue of the symmetric matrix $A$. Fix $m_b'\geq 0$. We let $$\begin{aligned}
& \widehat{\Lambda}(k,t,\theta)=\tfrac{\prod_{i=k}^{t}g(X_i;\theta)}{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,\infty,\theta,\theta,i}(X_i)},\\
& \widehat{\tau}(b)=\min_{m_{b}'\leq l\leq m_{b}} \widehat{\eta}_l(b), \label{eqn:modified_SGLR1}\\
\begin{split}
&\widehat{\eta}_l(b)=\inf\bigg\{t\geq l:\ \log\widehat{\Lambda}(t-l+1,t,\widehat{\theta})\geq b,\\
&\quad \sup_{\|\theta-\widehat{\theta}\|<1/\sqrt{b}}\lambda_{\max}\left(-\nabla^2\log\widehat{\Lambda}(t-l+1,t,\theta)\right)\leq b,\\
&\quad\quad\text{and} \ \widehat{\theta}=\argmax_\theta \log\widehat{\Lambda}(t-l+1,t,\theta)\in \text{Int}(\Theta) \bigg\},
\end{split}\\
& \widehat{\Lambda}_n(k,t,\theta)=\tfrac{\prod_{i=k}^{t}g_n(X_i;\theta)}{\max_{k\leq j\leq t+1}\prod_{i=k}^th_{j,\infty,\theta,\theta,i}(X_i)},\\
& \widehat{\tau}_n(b)=\min_{m_{b}'\leq l\leq m_{b}} \widehat{\eta}_{n,l}(b)\label{eqn:modified_SGLR2},\\
\begin{split}
&\widehat{\eta}_{n,l}(b)=\inf\bigg\{t\geq l:\ \log\widehat{\Lambda}_n(t-l+1,t,\widehat{\theta}_n)\geq b,\\
&\sup_{\|\theta-\widehat{\theta}_n\|<1/\sqrt{b}}\lambda_{\max}\left(-\nabla^2\log\widehat{\Lambda}_n(t-l+1,t,\theta)\right)\leq b,\\
&\quad\text{and}\ \widehat{\theta}_n=\argmax_\theta \log\widehat{\Lambda}_n(t-l+1,t,\theta)\in \text{Int}(\Theta) \bigg\}.
\end{split}
\end{aligned}$$ We define the generalized W-SGLR stopping time as $$\begin{aligned}
\widehat{\tau}_{\text{W-SGLR}}(b)=\min\{\widehat{\tau}(b),\widehat{\tau}_n(b)\}.
\end{aligned}$$ Note that $\widehat{\tau}_{\text{W-SGLR}}$ is a modification of ${\tau_{\text{W-SGLR}}}$ with additional conditions required for stopping.
The paper [@willsky1976generalized] first introduced window-limited generalized detection rules. We compute the false alarm probability of $\widehat{\eta}_l$ and $\widehat{\eta}_{n,l}$ . We then use this false alarm probability to obtain a lower bound for the ARL of $\widehat{\tau}$ and $\widehat{\tau}_n$ in Proposition \[prop:generalised\_arl\].
\[lem:error\_prob\_glrt\] Suppose that \[assup:compact2nddiff\] holds. Given any $0<\delta< 1$, there exists $b_\delta>0$ such that $\P{\nu_n,\infty}(\widehat{\eta}_k<\infty)\leq \exp\left(-(1-\delta)b\right)$ and $\P{\nu_n,\infty}(\widehat{\eta}_{n,l}<\infty)\leq \exp\left(-(1-\delta)b\right)$ for any $\nu_n\in\mathbb{N}\cup\{\infty\}$ and $b\geq b_\delta$.
As the proof is similar to Lemma 2 in [@lai98], we omit it here and refer the reader to the extended version in [@lau2019quickestarxiv].
\[prop:generalised\_arl\] Suppose that \[assup:compact2nddiff\] holds. For any $0< \delta< 1$, there exists $b_\delta>0$ such that for all $b\geq b_\delta$ and $\nu_n\in\mathbb{N}\cup\{\infty\}$, we have $$\begin{aligned}
{\text{ARL}}(\widehat{\tau}_{\text{W-SGLR}}(b))=\inf_{\nu_n}\E_{\nu_n,\infty}[\widehat{\tau}_{\text{W-SGLR}}(b)]\geq \tfrac{1}{2}e^{(1-\delta)b}.
\end{aligned}$$
Fix $0< \delta<1$. By Lemma \[lem:error\_prob\_glrt\], there exists $b_\delta\geq 0$ such that $
\P{\nu_n,\infty}(\min\{\widehat{\eta}_1,\widehat{\eta}_{n,1}\}<\infty)\leq 2\exp(-(1-\delta)b)
$ for all $b\geq b_\delta$. Applying results from Theorem 6.16 in [@poor2009quickest], we obtain $$\begin{aligned}
\E_{\nu_n,\infty}[\widehat{\tau}_{\text{W-SGLR}}]\geq \tfrac{1}{2}e^{(1-\delta)b}
\end{aligned}$$ for all $b\geq b_\delta$. Taking infimum over $\nu_n$, we have $
{\text{ARL}}(\widehat{\tau}_{\text{W-SGLR}}(b))\geq \tfrac{1}{2}e^{(1-\delta)b},
$ and the proof is complete.
Numerical Results {#sec:numerical}
=================
In this section, we first illustrate the performance of the proposed W-SGLR stopping time under the assumption that the distributions $f,f_n,g$ and $g_n$ are known. Next, we illustrate the performance of the proposed generalised W-SGLR stopping time when $g$ and $g_n$ belongs to a parametrized family of distributions. Finally, we evaluate the performance of the proposed W-SGLR stopping time on real data from the Case Western Reserve University Bearing Dataset[@smith2015rolling].
W-SGLR on Synthetic Data Satisfying \[assumpt:kldiv\] {#subsec:numerical_results_sim_signals}
-----------------------------------------------------
![W-SGLR test statistics $S_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(2,10)$ and $I=3.34$.[]{data-label="fig:satassumpt"}](simple_nuisance_mean_nui1500_crit1000.eps){width="0.85\columnwidth"}
![W-SGLR test statistics $S_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(2,10)$ and $I=3.34$.[]{data-label="fig:satassumpt"}](simple_nuisance_mean_nui1000_crit1500.eps){width="0.85\columnwidth"}
In our first set of simulations, we let $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, and $g_n=\calN(2,10)$ where the critical change is a change in variance and the nuisance change is a change in mean (see example after \[assumpt:kldiv\] for motivation). We ran the simulations with two change-point configurations to illustrate the behaviour of the W-SGLR test statistic for different window-sizes. In , we set $\nu_{c}=1000,\nu_n=1500$, while in , we set $\nu_{c}=1500,\nu_n=1000$. In and , the test statistic $S_{\text{W-SGLR}}(t)$ remains low before the critical change-point and grows linearly with the gradient of at least $I=3.34$, as described in \[I\], after the critical change-point. This trend continues until the test statistic approximately achieves the value of $m_b I$. From our choice of $m_b$ in \[mb\], we see that $m_b I > b$ for $b$ large, i.e., our ${\tau_{\text{W-SGLR}}}$ is able to detect the critical change given sufficient delay for every choice of $b$ sufficiently large. However, choosing a larger $m_b$ is more resistant to outlier noise. For example, in , when $m_b=1024$, we note that the test statistic continues to grow linearly with the gradient $I$ even after the nuisance change point. The trade-off is the increase in memory requirement and computational complexity. In , we note that the test statistic continues to remain low during the period between the nuisance and the critical change point. This demonstrates that ${\tau_{\text{W-SGLR}}}$ is oblivious to the nuisance change.
![Comparison of the trade-off performance for $\tau_{\text{W-SGLR}}$, $\tau_{\text{GLRT}}$, $\tau_{\text{FMA}}$ and $\tau_{\text{2-stage}}$.[]{data-label="fig:comparisonwithtwostage"}](comparison_with_two_stage_with_more_mb){width="0.85\columnwidth"}
Next, we compare ${\tau_{\text{W-SGLR}}}$, the GLRT stopping time $\tau_{\text{GLRT}}$ developed in [@lau2018quickest], the finite-moving average (FMA) stopping time $\tau_{\text{FMA}}$ and a naive 2-stage CuSum stopping time denoted as $\tau_{\text{2-stage}}$. Following ideas from the literature[@poor18FMA], the FMA stopping time $\tau_{\text{FMA}}$ is constructed by replacing the maximum in the test statistic \[eqn:WSGLR-test-statistic\] with a sum across the entire window, i.e. setting $k=t-m_b$. It should be noted that while the stopping time has been shown to perform well for the TCD problem, there are no guarantees that it will perform as well for the QCD problem. The naive stopping time $\tau_{\text{2-stage}}$ is constructed from stopping times based on the CuSum stopping time described in with $
\tau_{\{p\to q\}}(b)=\inf\left\{t\ :\ \max_{1\leq k\leq t+1}\prod_{i=k}^t \tfrac{p(x_i)}{q(x_i)}>e^b\right\}
$ for any pair of pdfs $p$ with $q\neq p$. We consider four stopping times: $\tau_{f\to f_n}(b_n)$, $\tau_{f\to g}(b_c)$, $\tau_{f\to g_n}(b_c)$, and $\tau_{f_n\to g_n}(b_c)$, where the threshold for declaring a critical change is $b_c$ and the threshold for declaring a nuisance change is $b_n$. In the first stage, we apply the stopping times $\tau_{f\to g}(b_c)$, $\tau_{f\to g_n}(b_c)$ and $\tau_{f\to f_n}(b_n)$ to the observations. If $\tau_{f\to g}(b_c)$ or $\tau_{f\to g_n}(b_c)$ stops the process before $\tau_{f\to f_n}(b_n)$, we declare that a critical change has occurred and set $\tau_{\text{2-stage}}=\min\{\tau_{f\to g}(b_c),\tau_{f\to g_n}(b_c)\}$. Otherwise, we apply $\tau_{f_n\to g_n}(b_c)$ to the rest of the observations after the stopping time $\tau_{f\to g_n}(b_n)$ and set $\tau_{\text{2-stage}}=\tau_{f_n\to g_n}(b_c)$.
In our simulations, our signal is generated using $f=\mathcal{N}(0,1)$, $g=\mathcal{N}(0.5,1)$, $f_n=\mathcal{N}(0,2)$, $g_n=\mathcal{N}(0.5,2)$. Here, the critical change is a change in mean from $0$ to $0.5$, and the nuisance change is a change in variance from $1$ to $2$. We generate a signal of length $2^{16}=65536$ and independently select the nuisance change point and critical change point with uniform probability on the $2^{16}$ possible data points. A total of $2^{12}=4096$ signals are generated. We compare the trade-off between the and the empirical of the proposed ${\tau_{\text{W-SGLR}}}$, $\tau_{\text{GLR}}$, $\tau_{\text{FMA}}$ and $\tau_{\text{2-stage}}$ in . We observe that our proposed ${\tau_{\text{W-SGLR}}}$ achieves a lower as compared to $\tau_{\text{FMA}}$, $\tau_{\text{GLR}}$ and $\tau_{\text{2-stage}}$ for large empirical .
![W-SGLR test statistics $S_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(5,20)$ and $I=3.34$. []{data-label="fig:rho_negative"}](simple_nuisance_mean_nui1500_crit1000_rho_g_negative.eps){width="0.85\columnwidth"}
![W-SGLR test statistics $S_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(5,20)$ and $I=3.34$. []{data-label="fig:rho_negative"}](simple_nuisance_mean_nui1000_crit1500_rho_g_negative.eps){width="0.85\columnwidth"}
In the next set of simulations, we let $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, and $g_n=\calN(5,20)$ where unlike the first set of simulations, the change in mean before and after the critical change point differs, and the change in variance before and after the nuisance change point differs. In , we set $\nu_{c}=1000 < \nu_n=1500$. We see that the test statistic $S_{\text{W-SGLR}}(t)$ remains low before the critical change-point and grows linearly with a gradient of at least $I=3.34$ (cf. \[I\]), after the critical change-point. This trend continues until the nuisance change point $\nu_n$ where rate of growth changes to $\E_g[\log\frac{g_n}{f_n}]>I$ until it approximately achieves the value of $m_b \E_g[\log\frac{g_n}{f_n}]$. While the growth of the test statistic after the change-point is not linear, the observation that the overall rate of growth from the critical change point is at least $I$ is consistent with \[lem:limsup\_assumption\]. In , we set $\nu_{c}=1500 > \nu_n=1000$ and note that the test statistic continues to remain low during the period between the nuisance and the critical change points. This demonstrates that ${\tau_{\text{W-SGLR}}}$ is oblivious to the nuisance change prior to the critical change-point.
W-SGLR on Synthetic Data Violating \[assumpt:kldiv\] {#subsec:numerical_results_sim_signals_no_assumpt_2}
----------------------------------------------------
![Comparison of the trade-off performance for $\tau_{\text{W-SGLR}}$, $\tau_{\text{2-stage}}$ and $\tau_{\text{FMA}}$ when \[assumpt:kldiv\] is violated with $f=\calN(0,1)$, $f_n=\calN(2,5)$, $g=\calN(3,10)$ and $g_n=\calN(5,10)$.[]{data-label="fig:non_asump_2_ARL_ADD"}](ADD_ARL_rhog_positive_nui_crit.eps){width="0.85\columnwidth"}
\
![Comparison of the trade-off performance for $\tau_{\text{W-SGLR}}$, $\tau_{\text{2-stage}}$ and $\tau_{\text{FMA}}$ when \[assumpt:kldiv\] is violated with $f=\calN(0,1)$, $f_n=\calN(2,5)$, $g=\calN(3,10)$ and $g_n=\calN(5,10)$.[]{data-label="fig:non_asump_2_ARL_ADD"}](ADD_ARL_rhog_positive_crit_nui.eps){width="0.85\columnwidth"}
When \[assumpt:kldiv\] is violated, \[thm:main\_result\] still provides the asymptotic trade-off between the ARL and the WADD. However, the asymptotic optimality of the W-SGLR stopping is not guaranteed. Here, we provide discussions and numerical simulations that suggests that the W-SGLR stopping time out-performs the two-stage stopping time and FMA stopping time with respect to \[eqn:lorden\_formulation\].
If \[assumpt:kldiv\] is violated, from \[thm:main\_result\], we have $$\begin{aligned}
\text{WADD}({\tau_{\text{W-SGLR}}}(b)) & \leq\left(\KLD{g}{f_n}^{-1}+o(1)\right)b.
\end{aligned}$$ This worst-case performance of the W-SGLR stopping time is achieved when $\nu_n=\infty$. For the rest of this discussion, we let $\nu_n=\infty$ to compare the two-stage stopping time with our proposed W-SGLR stopping time under this worst-case scenario.
Since $\KLD{g}{f}\geq\KLD{g}{f_n}$, the CuSum $$\max_{1\leq k\leq t+1}\sum_{i=k}^t\log\frac{f_n(X_i)}{f(X_i)}$$ associated with the stopping time $\tau_{f\to f_n}$ experiences a positive drift when $\nu_c\leq t<\nu_n$. Thus, for any finite threshold $b_n$ and sufficiently large $b_c$, the two-stage stopping time declares that a nuisance change has taken place and transits into the second stage after the critical change point. The CuSum associated with the stopping time $\tau_{f_n\to g_n}$ in the second stage is expected to grow at a rate of $\E{g}[\log \frac{g_n}{f_n}]$ for when $\nu_c\leq t<\nu_n$. In contrast, the W-SGLR test statistic, from \[thm:convergence\_in\_probability\_general\], is expected to grow at a rate of $\E{g}[\log \frac{g}{f_n}]\geq \E{g}[\log \frac{g_n}{f_n}]$. Heuristically, this means that, when $\nu_n=\infty$, the ${\text{WADD}}(\tau_{\text{2-stage}})\geq {\text{WADD}}(\tau_{\text{W-SGLR}})$ as the ${\text{ARL}}\to\infty$. It should also be noted that it is possible that the stopping time $\tau_{\text{2-stage}}$ fails completely when $\E{g}[\log \frac{g_n}{f_n}]$ is negative and $\nu_n=\infty$.
In \[fig:nui4000\_crit2000\_ARL\_ADD,fig:nui2000\_crit4000\_ARL\_ADD\], we compare the trade-off between the ADD and the ARL of the different stopping times when \[assumpt:kldiv\] is violated under the cases $(\nu_n,\nu_c)=(2000,4000)$ and $(\nu_n,\nu_c)=(4000,2000)$, respectively. To estimate the empirical ARL, the stopping times are applied to a set of $4096$ signals each of length $2^{16}$ with nuisance change point independently selected with uniform probability on the $2^{16}$ possible data points. To compute the corresponding ADD for the stopping times, they are applied to a set of $4096$ signals of length $4500$. It can be seen from both \[fig:nui4000\_crit2000\_ARL\_ADD,fig:nui2000\_crit4000\_ARL\_ADD\] that the W-SGLR stopping time achieves a lower ADD as compared to both $\tau_{\text{2-stage}}$ and $\tau_{\text{FMA}}$ for large empirical . Consistent with our intuition, it can be seen that $\tau_{\text{W-SGLR}}$ significantly outperforms $\tau_{\text{2-stage}}$ when $\nu_c<\nu_n$.
Parametrized Post-Change Distributions {#subsec:parameterized}
--------------------------------------
![The generalized W-SGLR test statistics $\widehat{S}_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(2,10)$ and $I=3.34$.[]{data-label="fig:composite_nuisance_mean"}](composite_nuisance_mean_nui666_crit333.eps){width="0.85\columnwidth"}
![The generalized W-SGLR test statistics $\widehat{S}_{\text{W-SGLR}}(t)$ with $f=\calN(0,1)$, $f_n=\calN(2,1)$, $g=\calN(0,10)$, $g_n=\calN(2,10)$ and $I=3.34$.[]{data-label="fig:composite_nuisance_mean"}](composite_nuisance_mean_nui333_crit666.eps){width="0.85\columnwidth"}
In this set of simulations, we let $f=\calN(0,1)$, the critical change to be a change in variance where $g=\calN(0,\theta^2)$, and the nuisance change to be a change in the mean where $f_n=\calN(2,1)$ and $g_n=\calN(2,\theta_n^2)$. The parameters $\theta=\theta_n=10$ are unknown to the change detection algorithms. This corresponds to the case where the transmission power is unknown in the problem of spectrum sensing[@poor08qcdspectrum]. We ran the simulations with two change-point configurations to demonstrate the behavior of the generalized W-SGLR test statistic used in $\widehat{\tau}_{\text{W-SGLR}}$ as described in and for window-sizes $m_b=16,32,64,128,256,512$ and $1024$. In , we set the critical change point to be $\nu_{c}=333$ and nuisance change point to be $\nu_n=666$. It can be observed that our proposed generalized W-SGLR test statistic remains low during the pre-change regime, increases in the post-change regime and continue to increase in the nuisance post-change regime when the window is sufficiently large. This demonstrates that our stopping time $\widehat{\tau}_{\text{W-SGLR}}$ is able to detect the critical change even in the nuisance critical change region. In , we set the critical change point to be $\nu_{c}=666$ and nuisance change point to be $\nu_n=333$. We see that our stopping time $\widehat{\tau}_\text{W-SGLR}$ is effective in detecting critical changes while ignoring the nuisance change in pre-change regime for window sizes as small as $m_b=16$. In practice, we can use graphs like \[fig:composite\_nuisance\_mean\_nui666\_crit333\] and \[fig:composite\_nuisance\_mean\_nui333\_crit666\] to compare if the increase in the test-statistic after the critical change is discernible from the test-statistic in the pre-change regime. This would provide assistance in determining if the choice window-size is suitable.
Next, we compare the generalized W-SGLR stopping time with the W-SGLR stopping time. In our simulations, our signal is generated using the following distributions $f=\mathcal{N}(0,1)$, $f_n=\mathcal{N}(0,2)$, $g=\mathcal{N}(\theta,1)$, $g_n=\mathcal{N}(\theta_n,2)$. Here we set $\theta=\theta_n=2$ and assume that the condition that $\widehat{\theta}\in\text{Int}(\Theta)$ is always satisfied. We generate a signal of length $2^{16}=65,536$ and independently select the nuisance change point and critical change point with uniform probability on the $2^{16}$ possible data points. A total of $2^{12}=4096$ signals are generated. We compare the trade-off between the and the of the proposed W-SGLR stopping time when $\theta$ and $\theta_n$ are known against the generalized W-SGLR stopping time when $\theta$ and $\theta_n$ are unknown in . We observe that the generalized W-SGLR stopping time has a higher as compared to the W-SGLR stopping time. Our experiments suggest that the difference in is bounded as the becomes large.
![Comparison of trade-off performance for the proposed stopping time $\widehat{\tau}_{\text{W-SGLR}}$ and $\tau_{\text{W-SGLR}}$.[]{data-label="fig:comparisonwithcomposite"}](comparison_with_composite){width="0.85\columnwidth"}
Real Data {#subsec:RealData}
---------
In this subsection, we test our proposed stopping time $\tau_{\text{W-SGLR}}$ on the Case Western Reserve University Bearing Dataset [@smith2015rolling]. The dataset is collected from experiments conducted using an electric motor with accelerometer data measured at locations near to and remote from the motor bearings. Samples were collected at 12 KHz. We pre-process the signal by de-trending the signal using a first order finite difference: for each signal sample time $t$, let $
X_t = Y_{t} - Y_{t-1},
$ where $Y_t$ is the observed raw signal sample at time $t$.
![Plots of the pdfs of the observed de-trended signal in different scenarios.[]{data-label="fig:pdf_of_different_scenario"}](pdf_of_different_scenario.eps){width="0.85\columnwidth"}
We consider signals $X_t$ obtained at a motor load of 1hp and 2hp with normal bearings and also faulty bearings with a 0.007-inch fault diameter. We assume that the critical change would be the transition from a normal to faulty bearing, and a nuisance change would be a change in the motor load. We use the first 12,000 samples as training data to build a model for each of the following scenarios: normal bearings under a motor load of 1hp, normal bearings under a motor load of 2hp, faulty bearings under a motor load of 1hp, and faulty bearings under a motor load of 2hp. shows the learned distributions of the de-trended signals observed in each scenario.
![Examples of the W-SGLR test statistic with $m_b=1024$.[]{data-label="fig:real_example"}](real_example_cri_nui.eps){width="0.85\columnwidth"}
![Examples of the W-SGLR test statistic with $m_b=1024$.[]{data-label="fig:real_example"}](real_example_nui_cri.eps){width="0.85\columnwidth"}
There are two challenges faced in testing our proposed stopping time on real data: (i) we lack theoretical results for the of the 2-stage stopping times for the selection of appropriate thresholds for comparison and (ii) real run-to-failure data is difficult to obtain. We divide the remaining samples into 3 disjoint sets to address the above challenges.
For the first set, we create a training set of 1000 signals each with length 36,000 with a randomly selected nuisance change point $\nu_n$ for each signal such that there is a period of $\nu_n-1$ samples for a normal bearing under a motor load of 1hp, and a period of $36,000-\nu_n+1$ samples for a normal bearing under a motor load of 2hp. We select appropriate thresholds for each of the stopping times so that the empirical ARL varies between 1200 and 18,000.
The next two sets are testing sets. We create 1000 signals of length 3600 each with (i) a period of 1200 samples for a normal bearing under a motor load of 1hp, which transitions to (ii) a period of 1200 samples for a normal bearing under a motor load of 2hp, which finally transitions to (iii) a period of 1200 samples for a faulty bearing under a motor load of 2hp.
Similarly, we create 1000 signals for the scenario where a normal bearing under a motor load of 1hp transitions to a faulty bearing under a motor load of 1hp and finally a faulty bearing under a motor load of 2hp.
Finally, we apply the selected thresholds obtained from the first training set to the two testing sets to compute the stopping times’ empirical ADD performance. The window size of $m_b=8$ for the FMA stopping time is selected to minimize its empirical ADD on the test set. For this dataset, if $m_b$ is chosen to be $128$ or $1024$, the empirical ADD of the FMA stopping time becomes much larger compared to the empirical ADD of the W-SGLR and 2-stage stopping times. Thus, we only present the ARL-ADD trade-off for $m_b=8$. In \[fig:real\_example\_cri\_nui,fig:real\_example\_nui\_cri\], we present some examples of the performance of the W-SGLR test statistic. It can be seen that in both cases, the test-statistic remains low before the bearing failure and quickly rises after the bearing fails even as the motor load changes.
In \[fig:real\_example\_add\_vs\_arl\_vs\_2stage\_nui\_crit\] and \[fig:real\_example\_add\_vs\_arl\_vs\_2stage\_crit\_nui\], we present the trade-off between the empirical ADD and ARL for the proposed W-SGLR stopping time with $m_b=1024$, the 2-stage stopping times with different thresholds $b_n$ and the FMA stopping time. It can be seen that our proposed stopping time $\tau_{\text{W-SGLR}}$ achieves better ADD-ARL trade-off compared to the other stopping times. However, as the KL divergences $\KLD{g_n}{f_n},\KLD{g}{f_n},\KLD{g_n}{f},\KLD{g}{f}$ are large, the empirical ADD for all the algorithms remains low across the range of ADD tested. In this case, the reduction in empirical ADD is small, between $1$ to $4$ samples, over the range of ARLs tested. In terms of computational complexity, up till sample $t$, the W-SGLR stopping time requires $O(m_bt)$ operations[@lau2019quickest], which is slightly more than both the two-stage stopping time and the FMA stopping time, both of which require $O(t)$ operations. Thus, for applications that have limited computational resources and large differences in their pre and post-change distributions, we may want to consider using the FMA or the 2-stage stopping time as the degradation in performance is small.
![Trade-off between the empirical ADD and ARL.[]{data-label="fig:real_example_add_vs_arl_vs_2stage"}](ADD_vs_ARL_twostage_vs_WSGLR_nui_crit.eps){width="0.85\columnwidth"}
\
![Trade-off between the empirical ADD and ARL.[]{data-label="fig:real_example_add_vs_arl_vs_2stage"}](ADD_vs_ARL_twostage_vs_WSGLR_crit_nui.eps){width="0.85\columnwidth"}
Discussions and Conclusions {#sec:conclusion}
===========================
We have studied the non-Bayesian QCD problem where the signal may be subjected to a nuisance change. We proposed the W-SGLR stopping time that quickly detects the critical change while ignoring the nuisance change. The limited window size ensures that the W-SGLR stopping time does not require increasing computational resources as more samples are observed. We also derived the stopping time’s asymptotic behavior and showed that it is asymptotically optimal under mild technical assumptions. A generalized W-SGLR stopping time is also proposed for the case where the critical and nuisance post-change distributions are unknown but belong to a parametrized family. Numerical simulations and experiments on a real dataset demonstrated that the W-SGLR stopping time achieves better ADD-ARL trade-off than various other competing stopping times.
In this paper, we have assumed that if both the critical and nuisance changes occur, the eventual distribution that generates the signal is the same, regardless of which change comes first. A more general model would be to allow the eventual distribution to depend on the order of the change points. An easy generalization of the W-SGLR stopping would be to include all the different eventual distributions into the numerator of \[eqn:SGLR\_teststats\]. The asymptotic trade-off between the WADD and ARL can be derived using similar techniques in \[subsec:properties\_stats\]. However, deriving the conditions for asymptotic optimality of this stopping time is more complicated and would be a possible direction for future research.
Another possible future research direction is to consider a modification of the W-SGLR stopping time for the TCD problem under the possibility of a nuisance change. As the performance metrics of the TCD problem are different from the QCD problem, its asymptotic trade-off between the worst-case false alarms and missed detection within a specified window needs to be studied. Also, as the FMA performs well in the TCD problem, it will be interesting to consider if the FMA stopping time can be adapted to solve our QCD problem.
Proof of Propositions \[prop:converge\_to\_post\_nui\_LR\] and \[prop:converge\_to\_pre\_nui\_LR\]
==================================================================================================
We start off with some notation definitions. Let $L_i=\log\tfrac{f_n(X_i)}{f(X_i)}$. For any $N\geq0$, let $
L_{i,>N}=L_i\mathbf{1}_{\{|L_i|>N\}}\ \text{and}\ L_{i,\leq N}=L_i\mathbf{1}_{\{|L_i|\leq N\}}.
$ For any $k,t\in\mathbb{N}$ such that $k\leq t$, we define the following averages: $$\begin{aligned}
\begin{split}&\overline{L^{k:t}}=\tfrac{1}{t-k+1}\sum_{i=k}^tL_i,\ \overline{L^{k:t}_{>N}}=\tfrac{1}{t-k+1}\sum_{i=k}^tL_{i,>N},\\
&\quad\quad\quad\overline{L^{k:t}_{\leq N}}=\tfrac{1}{t-k+1}\sum_{i=k}^tL_{i,\leq N}\end{split}
\end{aligned}$$ We have $$\begin{aligned}
\label{eqn:mean_truncation}
\overline{L^{k:t}}=\overline{L^{k:t}_{>N}}+\overline{L^{k:t}_{\leq N}}.
\end{aligned}$$ For the case where $k>t$, we let $\overline{L^{k:t}}=\overline{L^{k:t}_{>N}}=\overline{L^{k:t}_{\leq N}}=0$. Finally, we define the random variable $$\begin{aligned}
V_{k,t}=\arg\max_{k\leq j\leq t+1}\prod_{i=k}^{j-1}f(X_i)\prod_{i=j}^{t}f_n(X_i).
\end{aligned}$$
An outline of the proof of Propositions \[prop:converge\_to\_post\_nui\_LR\] and \[prop:converge\_to\_pre\_nui\_LR\] is as follows. Lemma \[lem:no\_nui\_change\_gap\_small\] and Lemma \[lem:truncation\_error\_bound\] provide the results required for controlling the error bound in Proposition \[prop:converge\_to\_post\_nui\_LR\]. Similarly, Lemma \[lem:nui\_change\_gap\_small\] and Lemma \[lem:truncation\_error\_bound\] provide the results required for controlling the error bound in Proposition \[prop:converge\_to\_pre\_nui\_LR\]. The Lemmas \[lem:no\_nui\_change\_gap\_small\], \[lem:nui\_change\_gap\_small\] and \[lem:truncation\_error\_bound\] require that decay in the tail probabilities of the average log-likelihood ratio $\tfrac{1}{n}\sum_{i=k}^{k+n-1}\log\tfrac{f_n(X_i)}{f(X_i)}$ to be at most $O(n^{-2})$, which is shown in \[lem:fourth\_order\_tail\_bound\].
\[lem:fourth\_order\_tail\_bound\] For any $\nu_c,\nu_n,k,l,n\in\mathbb{N}$ such that $\nu_c\leq k$ and $\max\{\nu_n,\nu_c\}\leq l$, and $\epsilon>0$, we have $$\begin{aligned}
\P{\infty,\nu_c}(\left|\overline{L^{k:k+n-1}}-\rho_{g}\right|\geq \epsilon) & \leq \tfrac{K_{g}}{\epsilon^4 n^2}, \label{4order1} \\
\P{\nu_n,\nu_c}(\left|\overline{L^{l:l+n-1}}-\rho_{g_n}\right|\geq \epsilon) & \leq \tfrac{K_{g_n}}{\epsilon^4 n^2}, \label{4order2}
\end{aligned}$$ where $K_g=\omega_g^4+\tfrac{3}{2}\sigma_g^4$ and $K_{g_n}=\omega_{g_n}^4+\tfrac{3}{2}\sigma_{g_n}^4$.
As the proof is elementary, we omit it here and refer the reader to the extended version in [@lau2019quickestarxiv].
We first prove the first inequality \[4order1\]. It can be shown that $$\begin{aligned}
\E{\infty,\nu_c}[\left(\overline{L^{k:k+n-1}}-\rho_{g}\right)^4]=\tfrac{\omega_g^4+\tfrac{3}{2}(n-1)\sigma_g^4}{n^3},
\end{aligned}$$ and an application of Markov’s inequality yields $$\begin{aligned}
\P{\infty,\nu_c}(\left|\overline{L^{k:k+n-1}}-\rho_g\right|\geq \epsilon)
& \leq \ofrac{\epsilon^4}\E{\infty,\nu_c}[\left(\overline{L^{k:k+n-1}}-\rho_{g}\right)^4] \\
& = \tfrac{\omega_g^4+\tfrac{3}{2}(n-1)\sigma_g^4}{\epsilon^4n^3} \\
& \leq\tfrac{K_g}{\epsilon^4n^2}.
\end{aligned}$$ The proof of the second inequality is similar. The proof is now complete.
From \[lem:fourth\_order\_tail\_bound\], for any $\nu_c \leq k \leq v \leq t < \infty$, we have for $\rho_g < 0$, $$\begin{aligned}
& \P{\infty,\nu_c}(V_{k,t}=v) \nn
& \leq\P{\infty,\nu_c}(\left|\overline{L^{v:t}}-\rho_g\right|\geq |\rho_g|)\nn
& \leq\tfrac{K_{g}}{|\rho_g|^4 (t-v+1)^2}.\label{Vkt_bound}
\end{aligned}$$ Similarly, for $\max\{\nu_c,\nu_n\} \leq k < v \leq t+1 <\infty$, and $\rho_{g_n}>0$, we have $$\begin{aligned}
\P{\nu_n,\nu_c}(V_{k,t}=v) & \leq\tfrac{K_{g}}{|\rho_{g_n}|^4 (v-k)^2}.\label{Vktn_bound}
\end{aligned}$$
For the next two lemmas, we use bounds on the tail probability of $\overline{L^{k:t}_{>N_{g}}}$ to derive asymptotic properties of the random variable $V_{k,t}$ under the distributions $\P{\nu_n,\nu_c}$ and $\P{\infty,\nu_c}$ for any $\nu_n,\nu_c\in\mathbb{N}$.
\[lem:no\_nui\_change\_gap\_small\] For any $0<c< 1$, $\nu_c\leq k<\infty$, and $\rho_g < 0$, we have $$\begin{aligned}
\label{eqn:no_nui_change_gap_small}
\lim_{t\to\infty}\P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}> c)=0.
\end{aligned}$$
We have $$\begin{aligned}
& \P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}> c)\nn
& \leq\sum_{v=k}^{\floor{(1-c)(t+1)+ck}}\P{\infty,\nu_c}(V_{k,t}=v)\nn
& \leq\sum_{v=k}^{\floor{(1-c)(t+1)+ck}}\tfrac{K_{g}}{|\rho_g|^4 (t-v+1)^2}\label{eqn:use_fourth_moment_tail_bound} \\
& \leq\left(\floor{(1-c)(t+1)+ck}-k+1\right)\tfrac{K_{g}}{|\rho_g|^4 (t-k+1)^2}\to 0, \nonumber
\end{aligned}$$ as $t\to\infty$. The inequality follows from \[Vkt\_bound\]. The proof is now complete.
\[lem:nui\_change\_gap\_small\] [For any $0<c< 1$ and $\nu_c\leq k<\infty$, let $k'=\max\{k,\nu_n\}$. If $\rho_{g_n}>0$, we have $$\begin{aligned}
\label{eqn:nui_change_gap_small}
{\lim_{t\to\infty}\P{\nu_n,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}> c)=0}.
\end{aligned}$$ ]{}
For $t>\tfrac{k-k'}{c}+k-1$, we have $c(t-k+1)+k'>k$ and $$\begin{aligned}
& \P{\nu_n,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}> c)\nn
& =\sum_{v=\ceil{c(t-k+1)+k'}}^{t+1}\P{\nu_n,\nu_c}(V_{k,t}=v)\nn
& \leq\sum_{v=\ceil{c(t-k+1)+k'}}^{t+1}\tfrac{K_{g_n}}{|\rho_{g_n}|^4 (v-k)^2}\label{eqn:nui_use_fourth_moment_sum_bound} \\
& \leq\left(t+1-\ceil{c(t-k+1)+k'}+1\right)\tfrac{K_{g_n}}{|\rho_{g_n}|^4 (\ceil{c(t-k+1)}+k'-k)^2}\to 0, \nonumber
\end{aligned}$$ as $t\to\infty$, where \[eqn:nui\_use\_fourth\_moment\_sum\_bound\] follows from \[Vktn\_bound\]. The proof is now complete.
\[lem:truncation\_error\_bound\] Suppose that $\rho_{g}<0$ and $\rho_{g_n}>0$. For any $\epsilon,\delta>0$ and $k\geq \max\{\nu_n,\nu_c\}$, there exist $N_{g_n},N_g\in\mathbb{N}$ such that for any $t\geq k$, $$\begin{aligned}
\P{\infty,\nu_c}(\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|\geq \epsilon) & \leq\delta,\label{eqn:truncate_error_1} \\
\P{\nu_n,\nu_c}(\left|\overline{L^{k:V_{k,t}-1}_{>N_{g_n}}}\right|\geq \epsilon) & \leq\delta.\label{eqn:truncate_error_2}
\end{aligned}$$
Given any $\epsilon,\delta>0$, since the fourth moment of $\log\tfrac{f_n(X)}{f(X)}$ exists, by the monotone convergence theorem, there exists $N_g$ and $N_{g_n}$ such that $$\begin{aligned}
\E{g_n}[\left|\log \tfrac{f_n(X)}{f(X)}\right|^4\mathbf{1}_{\left\{\left|\log \tfrac{f_n(X)}{f(X)}\right|>N_{g_n}\right\}}]\leq\tfrac{\epsilon^4\delta^4}{M^4},\label{LNgn} \\
\E{g}[\left|\log \tfrac{f_n(X)}{f(X)}\right|^4\mathbf{1}_{\left\{\left|\log \tfrac{f_n(X)}{f(X)}\right|>N_{g}\right\}}]\leq\tfrac{\epsilon^4\delta^4}{M^4},
\end{aligned}$$ where $M=\sum_{v=1}^{\infty}\left(\tfrac{K_{g}}{|\rho_g|^4 v^2}\right)^{\tfrac{3}{4}} < \infty$. Applying Markov’s inequality, we obtain $$\begin{aligned}
\label{eqn:tallprobability_control}
\P{\infty,\nu_c}(\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|\geq \epsilon)\leq\ofrac{\epsilon}\E{\infty,\nu_c}[\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|].
\end{aligned}$$
Next, we derive an upper bound for $\E{\infty,\nu_c}[\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|]$. For any $v \leq t$, we have $$\begin{aligned}
\E{\infty,\nu_c}[\left|\overline{L^{v:t}_{>N_{g}}}\right|^4] & =\E{\infty,\nu_c}[\left|\tfrac{1}{t-v+1}\sum_{i=v}^t L_{i,> N_g}\right|^4]\nn
& \leq\tfrac{1}{t-v+1}\sum_{i=v}^t\E{\infty,\nu_c}[\left|L_{i,>N_g}\right|^4]\label{jensen} \\
& \leq \tfrac{\epsilon^4\delta^4}{M^4},\label{bddLvt}
\end{aligned}$$ where \[jensen\] follows from Jensen’s inequality, and \[bddLvt\] follows from \[LNgn\]. We obtain $$\begin{aligned}
\E{\infty,\nu_c}[\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|] & =\sum_{v=k}^{t+1}\E{\infty,\nu_c}[\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|\indicator{V_{k,t}=v}]\nn
& =\sum_{v=k}^{t}\E{\infty,\nu_c}[\left|\overline{L^{v:t}_{>N_{g}}}\right|\indicator{V_{k,t}=v}]\label{eqn:reduced_sum_1} \\
& \leq\sum_{v=k}^{t}\E{\infty,\nu_c}[\left|\overline{L^{v:t}_{>N_{g}}}\right|^4]^{\ofrac{4}}\P{\infty,\nu_c}(V_{k,t}=v)^{\tfrac{3}{4}}\label{holder} \\
& \leq\tfrac{\epsilon\delta}{M}\sum_{v=k}^{t}(\P{\infty,\nu_c}(V_{k,t}=v))^{\tfrac{3}{4}}, \label{bdd1} \\
& \leq \tfrac{\epsilon\delta}{M}\sum_{v=k}^{t}\left(\tfrac{K_{g}}{|\rho_g|^4 (t-v+1)^2}\right)^{\tfrac{3}{4}}\label{bdd2} \\
& \leq \epsilon\delta. \label{ELNg_bdd}
\end{aligned}$$ where is because $\overline{L^{t+1:t}_{>N_{g}}}=0$, \[holder\] follows from Hölder’s inequality, \[bdd1\] from \[bddLvt\], \[bdd2\] from \[Vkt\_bound\], and \[ELNg\_bdd\] from the definition of $M$. From , we have $
\P{\infty,\nu_c}(\left|\overline{L^{V_{k,t}:t}_{>N_{g}}}\right|\geq \epsilon)\leq \tfrac{\epsilon\delta}{\epsilon} = \delta,
$ and \[eqn:truncate\_error\_1\] is proved. The proof of is similar and the lemma is proved.
Proof of Proposition \[prop:converge\_to\_post\_nui\_LR\]
---------------------------------------------------------
It suffices to show that for any $\epsilon,\delta>0$, there exists $T$ such that for all $t\geq T$ we have $$\begin{aligned}
\label{eqn:prop_3_1}
\P{\infty,\nu_c}(\left|\tfrac{\log \Lambda(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log \tfrac{g(X_i)}{f(X_i)}\right|\geq\epsilon)\leq\delta.
\end{aligned}$$ For any $N\geq0$ and $c>0$, the left-hand side of becomes $$\begin{aligned}
& \P{\infty,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=V_{k,t}}^t\log L_i \right|\geq\epsilon)\nn
\begin{split}& \leq \P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}\left| \overline{L^{V_{k,t}:t}_{>N}}\right|\geq\tfrac{\epsilon}{2})\\&\quad+ \P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}\left| \overline{L^{V_{k,t}:t}_{\leq N}}\right|\geq\tfrac{\epsilon}{2})\end{split}\nn
\begin{split}& \leq \P{\infty,\nu_c}(\left| \overline{L^{V_{k,t}:t}_{>N}}\right|\geq\tfrac{\epsilon}{2})\\&\quad+ \P{\infty,\nu_c}(\left\{\tfrac{t-V_{k,t}+1}{t-k+1}\left| \overline{L^{V_{k,t}:t}_{\leq N}}\right|\geq\tfrac{\epsilon}{2}\right\}\bigcap\left\{\tfrac{t-V_{k,t}+1}{t-k+1} \leq c\right\})\end{split}\nn
& + \P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}> c). \label{eqn:prop_3_1_bdd}
\end{aligned}$$ From \[lem:truncation\_error\_bound\], there exists $N$ such that $\P{\infty,\nu_c}(\left| \overline{L^{V_{k,t}:t}_{>N}}\right|\geq\tfrac{\epsilon}{2})\geq\tfrac{\delta}{2}$. Next, by choosing $c=\tfrac{\epsilon}{4N}$, we have $
\P{\infty,\nu_c}(\left\{\tfrac{t-V_{k,t}+1}{t-k+1}\left| \overline{L^{V_{k,t}:t}_{\leq N}}\right|\geq\tfrac{\epsilon}{2}\right\}\bigcap\left\{\tfrac{t-V_{k,t}+1}{t-k+1} \leq c\right\})=0.
$ Finally, from Lemma \[lem:no\_nui\_change\_gap\_small\], there exists $T$ such that for all $t\geq T$, we have $
\P{\infty,\nu_c}(\tfrac{t-V_{k,t}+1}{t-k+1}> c)\leq\tfrac{\delta}{2}.
$ The right-hand side of \[eqn:prop\_3\_1\_bdd\] is then upper bounded by $\delta$, and the proof is complete.
Proof of Proposition \[prop:converge\_to\_pre\_nui\_LR\]
--------------------------------------------------------
It suffices to show that for any $\epsilon,\delta>0$, there exists $T$ such that for all $t\geq T$ we have $$\begin{aligned}
\label{eqn:prop_4_1}
\P{\nu_n,\nu_c}(\left|\tfrac{\log \Lambda(k,t)}{t-k+1}-\tfrac{1}{t-k+1}\sum_{i=k}^t\log
\tfrac{g(X_i)}{f_n(X_i)}\right|\geq\epsilon)\leq\delta.\end{aligned}$$ Let $k'=\max\{k,\nu_n\}$. The left-hand side of can be written as $$\begin{aligned}
& \P{\nu_n,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=k}^{V_{k,t}-1}\log L_i\right|\geq\epsilon)\nonumber \\
& \leq\P{\nu_n,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=k}^{k'-1}\log L_i \right|\geq\tfrac{\epsilon}{2})\label{eqn:prop_4_pre_nui} \\
& \quad+\P{\nu_n,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=k'}^{V_{k,t}-1}\log L_i \right|\geq\tfrac{\epsilon}{2}).\label{eqn:prop_4_post_nui}\end{aligned}$$ Applying Markov’s inequality to \[eqn:prop\_4\_pre\_nui\], there exists $T_1$ such that for all $t\geq T_1$, we have $$\begin{aligned}
& \P{\nu_n,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=k}^{k'-1}\log L_i \right|\geq\tfrac{\epsilon}{2})\nonumber \\
& \leq \frac{2}{\epsilon(t-k+1)}\E{\nu_n,\nu_c}[\left|\sum_{i=k}^{k'-1}\log L_i \right|]
<\tfrac{\delta}{3}.\nonumber
\end{aligned}$$ For any $N\geq 0$ and $c\geq 0$, \[eqn:prop\_4\_post\_nui\] becomes $$\begin{aligned}
& \P{\nu_n,\nu_c}(\tfrac{1}{t-k+1}\left|\sum_{i=k'}^{V_{k,t}-1}\log L_i \right|\geq\tfrac{\epsilon}{2})\nonumber \\
\begin{split}&\leq \P{\nu_n,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}\left| \overline{L^{k':V_{k,t}-1}_{>N}}\right|\geq\tfrac{\epsilon}{4})\\&\quad+ \P{\nu_n,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}\left| \overline{L^{k':V_{k,t}-1}_{\leq N}}\right|\geq\tfrac{\epsilon}{4})\end{split}\nonumber \\
\begin{split}&\leq \P{\nu_n,\nu_c}(\left| \overline{L^{k':V_{k,t}}_{>N}}\right|\geq\tfrac{\epsilon}{4})\\&\quad+ \P{\nu_n,\nu_c}(\left\{\tfrac{V_{k,t}-k'}{t-k+1}\left| \overline{L^{V_{k,t}:t}_{\leq N}}\right|\geq\tfrac{\epsilon}{2}\right\}\bigcap\left\{\tfrac{V_{k,t}-k'}{t-k+1} \leq c\right\})\end{split}\nonumber \\
& + \P{\nu_n,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}> c). \label{eqn:prop_4_post_nui_bdd}
\end{aligned}$$
From \[lem:truncation\_error\_bound\], there exists $N$ such that $\P{\nu_n,\nu_c}(\left| \overline{L^{k':V_{k,t}-1}_{>N}}\right|>\tfrac{\epsilon}{4})<\tfrac{\delta}{3}$. Next, by choosing $c=\tfrac{\epsilon}{4N}$, we have $\P{\nu_n,\nu_c}(\left\{\tfrac{V_{k,t}-k'}{t-k+1}\left| \overline{L^{k':V_{k,t}-1}_{\leq N}}\right|>\tfrac{\epsilon}{4}\right\}\bigcap\left\{\tfrac{V_{k,t}-k'}{t-k+1} \leq c\right\})=0.$ Finally, from \[lem:no\_nui\_change\_gap\_small\], there exists $T_2$ such that for all $t\geq T_2$, we have $
\P{\infty,\nu_c}(\tfrac{V_{k,t}-k'}{t-k+1}> c)<\tfrac{\delta}{3}.$ The right-hand side of \[eqn:prop\_4\_post\_nui\_bdd\] is then upper bounded by $\delta$, and the proof is complete.
In this section, we discuss results that lead to the proof of \[thm:main\_result\]. The first result relates the ARL of $\tau_{\text{W-SGLR}}(b)$ in \[tau\_WSGLR\] to the threshold $b$. Lemma \[lem:fa\_prob\] derives an upper-bound for the probability of a false alarm for a related stopping time. Using techniques from [@poor2009quickest], the upper bound in \[lem:fa\_prob\] can be used to obtain a lower bound for the ARL of $\tau_{\text{W-SGLR}}(b)$. The next result relates the WADD of $\tau_{\text{W-SGLR}}(b)$ to the threshold $b$. Lemma \[lem:limsup\_assumption\] checks that the stopping time proposed satisfies the assumption required in [@lai98]. Using \[lem:limsup\_assumption\], we use techniques similar to [@lai98] to relate the asymptotic upper-bound for the WADD to the threshold $b$ in Proposition \[prop:ADD\].
Proof of Lemma \[lem:fa\_prob\]
===============================
For any $b>0$, we have $$\begin{aligned}
& \P{\nu_n,\infty}(\eta^1<\infty) \\
& =\sum_{k=1}^\infty\P{\nu_n,\infty}(\eta^1=k) \\
& =\sum_{k=1}^\infty\int_{\{\eta^1=k\}}\prod_{i=1}^k h_{\nu_n,\infty,i}(x_i)\ \ud \mathbf{x}_{1:k} \\
& \leq \sum_{k=1}^\infty\int_{\{\eta^1=k\}}e^{-b}\Lambda(1,k)\prod_{i=1}^k h_{\nu_n,\infty,i}(x_i)\ \ud \mathbf{x}_{1:k} \\
\begin{split}& = e^{-b}\sum_{k=1}^\infty\int_{\{\eta^1=k\}}\tfrac{\prod_{i=1}^k g(x_i)}{\max_{1\leq j\leq k+1}\prod_{i=1}^{j-1}f(x_i)\prod_{i=j}^{k}f_n(x_i)}\\&\quad\quad\quad\quad\quad\times\prod_{i=1}^k h_{\nu_n,\infty,i}(x_i)\ \ud \mathbf{x}_{1:k} \end{split}\\
& \leq e^{-b}\sum_{k=1}^\infty\int_{\{\eta^1=k\}}\prod_{i=1}^k g(x_i)\ \ud \mathbf{x}_{1:k} \\
& \leq e^{-b}\P{\infty,1}(\eta^1<\infty)\leq e^{-b}.
\end{aligned}$$ The proof that $\P{\nu_n,\infty}(\eta^1_n<\infty)\leq e^{-b}$ is similar. We then have $
\P{\nu_n,\infty}(\min\{\eta^1,\eta^1_n\}<\infty)\leq 2e^{-b}.
$ Since ${\tau_{\text{SGLR}}}=\min\{\tau(b),\tau_n(b)\}=\inf_{k\geq1}\min\{\eta^k,\eta^k_n\}$, applying [@poor2009quickest Theorem 6.16], we obtain $\E{\nu_n,\infty}[{\tau_{\text{SGLR}}}(b)]\geq \tfrac{1}{2}e^b$ and \[ARL\_tauWSGLR\] follows from ${\tau_{\text{W-SGLR}}}\geq{\tau_{\text{SGLR}}}$. The proof is now complete.
Proof of Lemma \[lem:limsup\_assumption\]
=========================================
It suffices to show that for any $\epsilon>0$, there exists $T$ such that for all $t\geq T$ we have $$\begin{aligned}
\sup_{\mathclap{\nu_n\in\mathbb{N},1\leq \nu_c \leq k}}\P{\nu_n,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)\leq\epsilon.\label{eqn:limsup_assumption_epsilon_delta}
\end{aligned}$$ The set over which the supremum in \[eqn:limsup\_assumption\_epsilon\_delta\] is taken can be divided into two subsets: $A_1=\{(\nu_n,\nu_c,k)\ :\ \max\{\nu_c,\nu_n\}\leq k\leq t\}$ and $A_2=\{(\nu_n,\nu_c,k)\ :\ \nu_c\leq k<\nu_n\leq t\}$. We have $$\begin{aligned}
\begin{split}&\sup_{A_1}\P{\nu_n,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)\\
& =\P{1,1}(\tfrac{1}{t}\log \Lambda(1,t)-I\leq -\delta)
\leq \tfrac{\epsilon}{2},\end{split}\label{eqn:reduction_of_supremum_1}
\end{aligned}$$ where the last inequality follows from \[thm:convergence\_in\_probability\_general\] for $t$ sufficiently large.
If $\nu_c\leq k<\nu_n\leq t$, we obtain $$\begin{aligned}
& \P{\nu_n,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)\nn
& =\P{\nu_n-k+1,1}(\tfrac{1}{t}\log \Lambda(1,t)-I\leq -\delta) \nn
& \leq \P{\nu_n-k+1,1}(\tfrac{\nu_n-k}{t}\left(\tfrac{\log \Lambda(1,\nu_n-k)}{\nu_n-k}-I\right)\leq -\tfrac{\delta}{2})\nonumber \\
& \quad+\P{\nu_n-k+1,1}(\tfrac{t-(\nu_n-k)}{t}\left(\tfrac{\log \Lambda(\nu_n-k+1,t)}{t-(\nu_n-k)}-I\right)\leq -\tfrac{\delta}{2})\nonumber \\
& \leq \P{\infty,1}(\tfrac{\nu_n-k}{t}\left(\tfrac{\log \Lambda(1,\nu_n-k)}{\nu_n-k}-I\right)\leq -\tfrac{\delta}{2})\label{eqn:case2_infty1} \\
& \quad+\P{1,1}(\tfrac{t-(\nu_n-k)}{t}\left(\tfrac{\log \Lambda(1,t-(\nu_n-k))}{t-(\nu_n-k)}-I\right)\leq -\tfrac{\delta}{2}).\label{eqn:case2_11}
\end{aligned}$$ From \[thm:convergence\_in\_probability\_general\], there exists $N_1$ such that for all $n\geq N_1$, we have $$\begin{aligned}
\P{\infty,1}(\tfrac{1}{n}\log \Lambda(1,n)-I\leq -\tfrac{\delta}{2})\leq\tfrac{\epsilon}{2},\label{eqn:conv_in_prop_bound_infty1} \\
\P{1,1}(\tfrac{1}{n}\log \Lambda(1,n)-I\leq -\tfrac{\delta}{2})\leq\tfrac{\epsilon}{2}.\label{eqn:conv_in_prop_bound_11}
\end{aligned}$$ From Markov’s inequality and \[assumpt:moments\], there exists $N_2$ such that for all $1\leq n< N_1$ and $m\geq N_2$, we have $$\begin{aligned}
\begin{split}&\P{\infty,1}(\tfrac{n}{m}\left(\tfrac{1}{n}\log \Lambda(1,n)-I\right)\leq -\tfrac{\delta}{2})\\
&\quad\quad\quad\quad \leq\tfrac{2n}{m\delta}\E{\infty,1}[\left|\tfrac{1}{n}\log \Lambda(1,n)-I\right|]
\leq \tfrac{\epsilon}{2},\end{split}\label{eqn:markov_bound_infty1} \\
\begin{split}&\P{1,1}(\tfrac{n}{m}\left(\tfrac{1}{n}\log \Lambda(1,n)-I\right)\leq -\tfrac{\delta}{2})\\
&\quad\quad\quad\quad \leq\tfrac{2n}{m\delta}\E{1,1}[\left|\tfrac{1}{n}\log \Lambda(1,n)-I\right|]
\leq \tfrac{\epsilon}{2}.\end{split}\label{eqn:markov_bound_11}
\end{aligned}$$
Next, we show that for any $t\geq T=2N_1+N_2$, both \[eqn:case2\_infty1\] and \[eqn:case2\_11\] are bounded by $\epsilon/2$. There are three possible cases:
1. $\nu_n-k\geq N_1$ and $t-(\nu_n-k)\geq N_1$,
2. $\nu_n-k< N_1$ and $t-(\nu_n-k)\geq N_1$,
3. $\nu_n-k\geq N_1$ and $t-(\nu_n-k)< N_1$.
Applying \[eqn:conv\_in\_prop\_bound\_infty1\] and \[eqn:conv\_in\_prop\_bound\_11\] in the first case, \[eqn:conv\_in\_prop\_bound\_11\] and \[eqn:markov\_bound\_infty1\] in the second case, and \[eqn:conv\_in\_prop\_bound\_infty1\] and \[eqn:markov\_bound\_11\] in the thrid case to \[eqn:case2\_infty1\] and \[eqn:case2\_11\], respectively, we obtain $
\sup_{A_2} \P{\nu_n,\nu_c}(\tfrac{1}{t}\log \Lambda(k,k+t-1)-I\leq -\delta)\leq\epsilon.
$ The proof for \[eqn:limsup:it2\] is similar and proof is now complete.
Proof of Proposition \[prop:ADD\]
=================================
From \[mb\], there exists $\gamma>0$ such that $m_b \geq (1+\gamma) b/I$ for all $b$ sufficiently large. For any $0<\epsilon < \gamma/(1+\gamma)$, let $n_b=\ceil{\tfrac{b}{(1-\epsilon)I}}$ and $\delta=\epsilon I$. There exists $b_1>0$ such that $n_b(I-\delta) \geq b$ for all $b\geq b_1$. From \[lem:limsup\_assumption\], by choosing $b_1$ sufficiently large, we have for all $b\geq b_1$, $$\begin{aligned}
& \sup_{1\leq \nu_c\leq k}\P{\nu_n,\nu_c}(\log\Lambda(k,k+n_b-1)<b)\nn
& \leq \sup_{\mathclap{1\leq \nu_c\leq k}}\P{\nu_n,\nu_c}(\log\Lambda(k,k+n_b-1)\leq n_b(I-\delta))\leq \epsilon.\label{eqn:boundnb}
\end{aligned}$$ Let $b_2 \geq b_1$ be such that $I/b_2 \leq 1+\gamma - (1-\epsilon)^{-1}$. Then, for $b\geq b_2$, we have $$\begin{aligned}
\tfrac{m_b}{n_b}
& \geq \tfrac{bI^{-1}(1+\gamma)}{\tfrac{bI^{-1}}{1-\epsilon}+1} \\
& \geq \tfrac{1+\gamma}{(1-\epsilon)^{-1}+I/b}\geq 1.
\end{aligned}$$
For any $k,\nu_n,\nu_c\geq 1$, we then have $$\begin{aligned}
& \esssup \P{\nu_n,\nu_c}(\widetilde{\tau}_n(b)-\nu_c+1 > kn_b){X_1,\ldots,X_{\nu_c-1}} \\
\begin{split}& = \esssup \mathbb{P}_{\nu_n,\nu_c}\left(\max_{t-m_{b}\leq k'\leq t}\log\Lambda(k',t)<b\right.\\& \quad\quad\quad\quad\quad\quad\quad\text{for all $t\leq \nu_c+kn_b-1$}\ \bigg|
X_1,\ldots,X_{\nu_c-1}\bigg)\end{split} \\
\begin{split}& \leq \esssup \mathbb{P}_{\nu_n,\nu_c}(\log\Lambda(\nu_c+(j-1)n_b,\nu_c+jn_b-1)<b\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{for all $1\leq j \leq k$}\ |\ X_1,\ldots,X_{\nu_c-1}) \end{split} \\
& =\prod_{j=1}^k \P{\nu_n,\nu_c}(\log\Lambda(\nu_c+(j-1)n_b,\nu_c+jn_b-1)<b)\leq \epsilon^k,
\end{aligned}$$ where the last equality follows from independence and the last inequality from \[eqn:boundnb\]. Therefore, for any $b\geq b_2$, we have $$\begin{aligned}
& \sup_{{\nu_n,\nu_c\geq 1}}\esssup \E{\nu_n,\nu_c}[(\widetilde{\tau}_n(b)-\nu_c+1)^+]{X_1,\ldots,X_{\nu_c-1}} \\
& =\sup_{{\nu_n,\nu_c\geq 1}}\esssup \sum_{i=0}^\infty\P{\nu_n,\nu_c}((\widetilde{\tau}_n(b)-\nu_c+1)>i){X_1,\ldots,X_{\nu_c-1}} \\
& \leq\sum_{k=0}^\infty n_b \sup_{{\nu_n,\nu_c\geq 1}}\esssup \P{\nu_n,\nu_c}(\widetilde{\tau}_n(b)-\nu_c+1>kn_b){X_1,\ldots,X_{\nu_c-1}} \\
& \leq \tfrac{n_b}{1-\epsilon}
\leq \tfrac{b}{I(1-\epsilon)^2}+\tfrac{1}{1-\epsilon}=b\left(I^{-1}+\tfrac{2\epsilon-\epsilon^2}{I(1-\epsilon)^2}+\tfrac{1}{b(1-\epsilon)}\right),
\end{aligned}$$ which yields \[prop:ADD:it1\]. The proof for \[prop:ADD:it2\] is similar and the proposition is proved.
Proof of Theorem \[thm:main\_result\]
=====================================
From \[lem:fa\_prob\], taking infimum on both sides of \[ARL\_tauWSGLR\], we obtain $
{\text{ARL}}(\tau_{\text{W-SGLR}}(b))=\inf_{\nu_n\in\mathbb{N}\cup\{\infty\}}\E{\nu_n,\infty}[\tau_{\text{W-SGLR}}(b)]\geq \tfrac{1}{2}e^b.
$ Since $\tau_{\text{W-SGLR}}\leq\widetilde{\tau}$ and $\tau_{\text{W-SGLR}}\leq\widetilde{\tau}_n$, by \[prop:ADD\], we have $
\text{WADD}(\tau_{\text{W-SGLR}}(b))\leq(I^{-1}+o(1))b
$ as $b\to\infty$.
To see that $\tau_{\text{W-SGLR}}(b)$ is asymptotically optimal when \[assumpt:kldiv\] is satisfied, let $C_{\gamma}=\{\tau\ :\ {\text{ARL}}(\tau)\geq \gamma\}$ be the set of stopping times satisfying ${\text{ARL}}(\tau)\geq \gamma$. By expanding ${\text{WADD}}(\tau)$ using \[eqn:WADD\], we obtain $$\begin{aligned}
\inf_{\tau\in C_{\gamma}}{\text{WADD}}(\tau)&=\inf_{\tau\in C_{\gamma}}\sup_{\nu_n\in\mathbb{N}\cup\{\infty\}}{\text{WADD}}_{\nu_n}(\tau)\nn
&\geq \sup_{\nu_n\in\mathbb{N}\cup\{\infty\}}\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau)\label{eqn:min-max_ineq}\\
&\geq \sup_{\nu_n\in\{0,\nu_c,\infty\}}\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau),\label{eqn:supinf}
\end{aligned}$$ where \[eqn:min-max\_ineq\] is due to the min-max inequality[@boyd2004convex]. For each of the cases $\nu_n\in\{0,\nu_c,\infty\}$, by Theorem 6.17 in [@poor2009quickest], we have $$\begin{aligned}
\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau) & \geq\left(\KLD{g_n}{f_n}^{-1}+o(1)\right)b, \quad \text{when $\nu_n=0$,} \\
\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau) & \geq\left(\KLD{g_n}{f}^{-1}+o(1)\right)b, \quad \text{when $\nu_n=\nu_c$,} \\
\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau) & \geq\left(\KLD{g}{f}^{-1}+o(1)\right)b, \quad \text{when $\nu_n=\infty$.}
\end{aligned}$$ Since \[assumpt:kldiv\] is satisfied, we have $$\begin{aligned}
I=\min\left\{\KLD{g}{f},\ \KLD{g_n}{f},\ \KLD{g_n}{f_n}\right\}.\label{eqn:I}
\end{aligned}$$ Therefore, from \[eqn:supinf\] and \[eqn:I\], we obtain $$\begin{aligned}
\inf_{\tau\in C_{\gamma}}{\text{WADD}}_{\nu_n}(\tau)
& \geq \Big(\max\big\{\KLD{g_n}{f_n}^{-1},\ \KLD{g_n}{f}^{-1}, \\
& \qquad\qquad \KLD{g}{f}^{-1}\big\} + o(1) \Big)b \\
& = \left(I^{-1}+o(1)\right)b,
\end{aligned}$$ and the proof is now complete.
Proof of Lemma \[lem:error\_prob\_glrt\] {#sec:App_Lemma_error_prob}
========================================
We use techniques is similar to [@lai98] to prove Lemma \[lem:error\_prob\_glrt\]. To analyze the probability $\P{\nu_n,\infty}(\widehat{\eta}_k<\infty)$, we use a change-of-measure argument. For any $0<\delta<1$, choose $b_\delta\geq0$ so that for any $b\geq b_\delta$ $$\begin{aligned}
|\Theta| \pi^{-\tfrac{d}{2}}{\Gamma\left(\tfrac{d}{2}+1\right)}b^{\tfrac{d}{2}} \exp\left(-b\right) \leq \exp\left(-(1-\delta)b\right),
\end{aligned}$$ where $|\Theta|$ is the volume or Lebesgue measure of $\Theta\subset\Real^d$ and $\Gamma(\cdot)$ is the gamma function. From Kolmogorov’s Consistency Theorem, there is a probability measure $G_\theta$ for the stochastic process $(X_i)_{i\geq k}$ under which the pdf of each $X_i$ is $g(\cdot;\theta)$. Define a measure $H(\cdot)=\int_{\Theta}G_\theta(\cdot)\ \ud \theta$. Since $\Theta$ is compact in $\mathbb{R}^d$, the measure $H$ is finite. For each $t\geq k$, the Radon-Nikodym derivative of the law of $(X_k,X_{k+1},\ldots,X_t)$ under $H$ $\P{\nu_n,\infty}$ is $$\begin{aligned}
R_t=\int_{\Theta}\exp\left(\sum_{i=k}^t\log \tfrac{g(X_i;\theta)}{h_{\nu_n,\infty,\theta,\theta_n,i}(X_i)}\right)\ \ud\theta,
\end{aligned}$$ which follows from Fubini’s Theorem. By Wald’s likelihood ratio identity, $$\begin{aligned}
\P{\nu_n\infty}(\widehat{\eta}_k<\infty) & =\int_{\{\widehat{\eta}_k<\infty\}}R_{\widehat{\eta}_k}^{-1}\ \ud H\nn
& =\int_{\Theta}\left\{\int_{\{\widehat{\eta}_k<\infty\}}R_{\widehat{\eta}_k}^{-1}\ \ud G_\theta\right\}\ \ud \theta. \label{eqn:wald_llr_identity}
\end{aligned}$$ Suppose $\widehat{\eta}_k=t$. Since $\widehat{\theta}=\argmax_\theta \log\widehat{\Lambda}(k,t,\theta)\in\text{Int}(\Theta)$, from Taylor series, there exists $\theta^*\in\Theta$ such that $$\begin{aligned}
\sum_{i=k}^t \log g(x_i;\theta)=\sum_{i=k}^t \log g(x_i;\hat{\theta})+\tfrac{1}{2}(\theta-\hat{\theta})^T\left[\nabla^2\sum_{i=k}^t \log g(x_i,\theta^*)\right](\theta-\hat{\theta}).
\end{aligned}$$ Thus, for $\|\theta-\widehat{\theta}\|<1/\sqrt{b}$, we have $$\begin{aligned}
\log\widehat{\Lambda}(k,t,\hat{\theta})-\log\widehat{\Lambda}(k,t,\theta)
& =-\tfrac{1}{2}(\theta-\hat{\theta})^T\left[\nabla^2 \log\widehat{\Lambda}(k,t,\theta^*)\right](\theta-\hat{\theta}) \\
& \leq\tfrac{1}{2}\|\theta-\widehat{\theta}\|^2\lambda_{\max}\left(-\nabla^2\log\widehat{\Lambda}(k,t,\theta)\right)\\
&\leq \ofrac{2}\|\theta-\widehat{\theta}\|^2 b,
\end{aligned}$$ where the last inequality follows from $\sup_{\|\theta-\widehat{\theta}\|<1/\sqrt{b}}\lambda_{\max}\left(-\nabla^2\log\widehat{\Lambda}(k,t,\theta)\right)\leq b$. We obtain $$\begin{aligned}
\tfrac{R_t}{\widehat{\Lambda}(k,t,\widehat{\theta})} & =\int_{\Theta} \exp\left(-\left[\log\widehat{\Lambda}(k,t,\widehat{\theta})-\sum_{i=k}^t \log \tfrac{g(X_i,\theta)}{h_{\nu_n,\infty,\theta,\theta_n,i}(X_i)} \right]\right)\ \ud \theta \\
& \geq\int_{\Theta} \exp\left(-\left[\log\widehat{\Lambda}(k,t,\widehat{\theta})-\log \widehat{\Lambda}(k,t,\theta) \right]\right)\ \ud \theta \\
& \geq\int_{\|\theta-\widehat{\theta}\|\leq 1/\sqrt{b}} \exp\left(-\ofrac{2}\|\theta-\widehat{\theta}\|^2b\right)\ \ud \theta\\
& \geq\int_{\|\theta-\widehat{\theta}\|\leq 1/\sqrt{b}} 1\ \ud \theta \\
& \geq\tfrac{\pi^{\tfrac{d}{2}}}{\Gamma(\tfrac{d}{2}+1)}b^{-\tfrac{d}{2}}
\end{aligned}$$
Therefore, we have $$\begin{aligned}
R_t & \geq \tfrac{\pi^{\tfrac{d}{2}}}{\Gamma(\tfrac{d}{2}+1)}b^{-\tfrac{d}{2}}\widehat{\Lambda}(k,t,\widehat{\theta}) \\
& \geq \tfrac{\pi^{\tfrac{d}{2}}}{\Gamma(\tfrac{d}{2}+1)}b^{-\tfrac{d}{2}} \exp\left(b\right).
\end{aligned}$$ This yields the upper bound $$\begin{aligned}
R_t^{-1} & \leq \pi^{-\tfrac{d}{2}}{\Gamma(\tfrac{d}{2}+1)}b^{\tfrac{d}{2}} \exp\left(-b\right).
\end{aligned}$$ Applying this upper bound to , we obtain $$\begin{aligned}
\P{\nu_n,\infty}(\widehat{\eta}_k<\infty)
& \leq \int_{\Theta}\left\{ \pi^{-\tfrac{d}{2}}{\Gamma\left(\tfrac{d}{2}+1\right)}b^{\tfrac{d}{2}} \exp\left(-b\right)\int_{\{\widehat{\eta}_k<\infty\}}\ \ud G_\theta\right\}\ \ud \theta \\
& \leq \int_{\Theta}\left\{ \pi^{-\tfrac{d}{2}}{\Gamma\left(\tfrac{d}{2}+1\right)}b^{\tfrac{d}{2}} \exp\left(-b\right) \right\}\ \ud \theta \\
& = |\Theta| \pi^{-\tfrac{d}{2}}{\Gamma\left(\tfrac{d}{2}+1\right)}b^{\tfrac{d}{2}} \exp\left(-b\right) \\
& \leq \exp\left(-(1-\delta)b\right),
\end{aligned}$$ for all $b\geq b_\delta$. The proof that $\P{\nu_n,\infty}(\widehat{\eta}_{n,l}<\infty)\leq \exp\left(-(1-\delta)b\right)$ is similar, and the lemma is proved.
[^1]: This research is supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant 2017-T1-001-059 (RG20/17) and Tier 2 grant MOE2018-T2-2-019.
[^2]: T. S. Lau and W. P. Tay are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: TLAU001@e.ntu.edu.sg, wptay@ntu.edu.sg).
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author:
- |
[^1]\
Dipartimento di Fisica [*Ettore Pancini*]{}, Università di Napoli [*Federico II*]{}, and INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy\
E-mail:
title: Dark Matter scenarios at IceCube
---
Introduction
============
The recent discovery of a diffuse neutrino flux at the TeV–PeV range by the IceCube collaboration [@IceCube] has ushered us into a new era for astroparticle physics. Due to the very feeble interaction with the other Standard Model particles, neutrinos represent the best messenger for observing and studying the cosmos. Indeed, the observation of extraterrestrial neutrinos provides an important tool that can be adopted to examine the acceleration mechanisms of hadronic cosmic-rays and the properties of both galactic and extragalactic astrophysical environments. Moreover, the IceCube Neutrino Telescope has measured the most energetic neutrinos, offering us the possibility to explore the neutrino physics at energies where phenomena beyond the Standard Model can be relevant.\
The origin of the diffuse neutrino flux is still unknown. In the last few years, the scientific community has proposed a variety of astrophysical sources (extragalactic Supernovae and Hypernovae remnants [@Chakraborty:2015sta], blazars [@Kalashev:2014vya], and gamma-ray bursts [@Waxman:1997ti]) as potential candidates for the IceCube observations. In general, under the reasonable assumption of a correlation with the spectrum of hadronic cosmic-rays, one expects that the differential neutrino flux has a power-law behavior $$\frac{{\rm d}\phi^{\rm Astro}}{{\rm d}E_\nu {\rm d}\Omega} = \phi^{\rm Astro}_0 \left( \frac{E_\nu}{100~{\rm TeV}} \right)^{-\gamma}\,,
\label{eq:astro}$$ where $\gamma$ is called [*spectral index*]{} and $\phi^{\rm Astro}_0$ is the normalization of the neutrino flux at 100 TeV. The astrophysical neutrino flux is assumed to be isotropic and it is characterized by an equal flavour ratio $\left(1:1:1\right)$ at the Earth. While the standard Fermi acceleration mechanism at shock fronts implies $\gamma = 2.0$ at first order, the spectral index can assume larger values depending on the astrophysical source considered. Indeed, in case of neutrinos produced by hadronuclear $p$–$p$ interactions then $\gamma \lesssim 2.2$ [@ppsources] (see also Ref. [@Murase:2013rfa]), while for photohadronic $p$–$\gamma$ interactions the spectral index has to be larger than $2.3$ [@Winter:2013cla].\
The IceCube observations of different data samples have revealed a preference for a soft power-law spectrum and the IceCube combined analysis has provided the best-fit $\gamma_{\rm best}=2.50\pm0.09$ [@IceCube]. However, very recently, the analysis on the 6-year up-going muon neutrinos shows that at high energy $\left(E_\nu \geq 100\,{\rm TeV}\right)$ the best-fit spectral index is $\gamma = 2.13$, value that is disfavoured by 3.3$\sigma$ with respect to $\gamma_{\rm best}$. As also stated by the IceCube collaboration, this tension may indicate the presence of a second galactic component in the diffuse neutrino flux.\
The tension with the assumption of a single power-law is further strengthened by considering the [*multi-messenger*]{} analyses. Indeed, the contributions of different astrophysical sources to the IceCube spectrum are strongly constrained if one assumes a correlation between the diffuse neutrino flux and the isotropic diffuse gamma-ray background measured by Fermi-LAT [@Ackermann:2014usa]. For instance, it has been pointed out that the contribution of star-forming galaxies ($p$–$p$ sources) has to be smaller than $\sim 30\%$ at $100$ TeV and $\sim 60\%$ at $1$ PeV [@Bechtol:2015uqb] (see also Ref. [@Murase:2015xka]). On the other hand, gamma-ray bursts [@Aartsen:2014aqy] and blazars [@blazarMulti] ($p$–$\gamma$ sources) can only provide a contribution of $\sim1\%$ and $\sim20\%$ to the IceCube neutrino spectrum, respectively. Therefore, the multi-messenger analyses suggest that a hard neutrino spectrum is preferred with respect to a soft one.
![\[fig:1\]Residuals in the number of neutrino events (2-year MESE) with respect to a single astrophysical power-law with spectral index 2.0, in the southern and northern hemispheres (see Ref. [@Chianese:2016kpu]).](res_s20p.pdf "fig:"){width="42.00000%"} 3.mm ![\[fig:1\]Residuals in the number of neutrino events (2-year MESE) with respect to a single astrophysical power-law with spectral index 2.0, in the southern and northern hemispheres (see Ref. [@Chianese:2016kpu]).](res_n20p.pdf "fig:"){width="42.00000%"}
\
As discussed in Ref.s [@Chianese:2016kpu; @Chianese:2016opp], once a hard neutrino spectrum $E_\nu^{-2.0}$ is considered according to the multi-messenger analyses, the IceCube data show an excess in the 10–100 TeV range ([*low-energy excess*]{}) with a maximum local statistical significance of 2.3$\sigma$ (2.0$\sigma$) in the 2-year MESE (4-year HESE) data (see Fig. \[fig:1\] for the residuals in the 2-year MESE data). However, as the spectral index increases, the excess moves towards PeV energies ([*high-energy excess*]{}) [@Boucenna:2015tra].\
All the above considerations lead to the conclusion that the IceCube measurements are explained in terms of a [*two-component*]{} neutrino flux. In particular, we have studied in detail the intriguing two-component scenario where, in addition to the neutrino background, one component is related to astrophysical sources and one is originated from Dark Matter [@Chianese:2016kpu; @Chianese:2016opp; @Boucenna:2015tra] (see also Ref. [@Chen:2014gxa]). In this case, the total differential neutrino flux takes the following expression $$\frac{{\rm d}\phi}{{\rm d}E_\nu {\rm d}\Omega} = \frac{{\rm d}\phi^{\rm Astro}}{{\rm d}E_\nu {\rm d}\Omega} + \frac{{\rm d}\phi^{\rm DM}}{{\rm d}E_\nu {\rm d}\Omega}\,.
\label{eq:tot_flux}$$ Here, the first term is given by the astrophysical power-law of Eq. (\[eq:astro\]). The Dark Matter second term (see Ref. [@Chianese:2016kpu] for its detailed expression) depends on the particular interaction with the Standard Model particles (Dark Matter particles decaying or annihilating into leptonic or hadronic final states) and on the halo density distribution of our galaxy.
Two-component Neutrino Flux and Dark Matter
===========================================
![\[fig:2\]Best-fit of the two-component hypothesis for the IceCube diffuse neutrino flux in case of 2-year MESE (left panel) and 3-year HESE (right panel) data (see Ref.s [@Chianese:2016kpu] and [@Boucenna:2015tra], respectively). The Dark Matter contribution is respectively obtained by considering Dark Matter decaying into tau leptons (low-energy excess) and leptophilic three-body decays at PeV energies (high-energy excess).](pMESE.pdf "fig:"){width="40.00000%"} 3.mm ![\[fig:2\]Best-fit of the two-component hypothesis for the IceCube diffuse neutrino flux in case of 2-year MESE (left panel) and 3-year HESE (right panel) data (see Ref.s [@Chianese:2016kpu] and [@Boucenna:2015tra], respectively). The Dark Matter contribution is respectively obtained by considering Dark Matter decaying into tau leptons (low-energy excess) and leptophilic three-body decays at PeV energies (high-energy excess).](pHESE.pdf "fig:"){width="40.00000%"}
The proposed two-component interpretation of the IceCube data is depicted in left and right panels of Fig. \[fig:2\] for the low-energy excess [@Chianese:2016kpu; @Chianese:2016opp] and the high-energy one [@Boucenna:2015tra], respectively.[^2] The low-energy excess is explained in terms of the decay channel ${\rm DM} \to \tau^+\tau^-$, while the excess at high energy is fitted by the leptophilic model predicting ${\rm DM} \to \ell^+\ell^-\nu$. In both cases, the Navarro-Frenk-White halo density distribution has been considered. The low-energy excess has been characterized by analyzing the 2-year MESE data, since they have a lower energy threshold with respect to the HESE ones (see the different energy scale in the two plots).\
Regarding the PeV neutrinos, the leptophilic model proposed in Ref. [@Boucenna:2015tra] has two peculiar characteristics: the neutrino spectrum is spread differently from the case of two-body decays and it is peaked in a particular energy range due to the absence of quarks in the final states. Such a model is also able to account for the Dark Matter production in the early Universe through a freeze-in mechanism [@Chianese:2016smc]. It is worth observing that considering the unrealistic behavior $E_\nu^{-3.0}$ of the astrophysical component is equivalent to assume a power-law with $\gamma=2.0$ exponentially suppressed for $E_\nu \geq 100$ TeV (neutrino spectrum expected for extragalactic Supernovae remnants [@Chakraborty:2015sta]).
Conclusions
===========
The tension of the IceCube observations with the assumption of a single power-law flux indicates the presence of a second contribution to the diffuse neutrino flux. Therefore, we have examined the case where this second component is related to Dark Matter. A likelihood-ratio statistical test has shown that two-component scenario of Eq. (\[eq:tot\_flux\]) is favoured by 2$\sigma$–4$\sigma$ with respect to a single power-law [@Chianese:2016kpu]. The statistical significance depends on the Dark Matter model and the slope of the astrophysical contribution. In general, the leptophilic decaying Dark Matter models are preferred by both IceCube and Fermi-LAT measurements, while the other cases are disfavoured by multi-messenger studies. Better statistics could confirm the presence of an excess in the neutrino spectrum and this would potentially shed light on some of the deepest mysteries in contemporary physics: the nature of Dark Matter.
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[^1]: I thank the organizers of the Neutrino Oscillation Workshop and the conveners of my session. I acknowledge the financial support by the Instituto Nazionale di Fisica Nucleare I.S. TASP and the PRIN 2012 “Theoretical Astroparticle Physics" of the Italian Ministero dell’Istruzione, Università e Ricerca.
[^2]: See references cited in Ref.s [@Chianese:2016kpu; @Chianese:2016opp; @Boucenna:2015tra] for other Dark Matter interpretations of IceCube measurements.
|
---
abstract: 'A virtual chemical spectrophotometer for the simultaneous analysis of nickel (Ni) and cobalt (Co) was developed based on an artificial neural network (ANN). The developed ANN correlates the respective concentrations of Co and Ni given the absorbance profile of a Co-Ni mixture based on the Beer’s Law. The virtual chemical spectrometer was trained using a 3-layer jump connection neural network model (NNM) with 126 input nodes corresponding to the 126 absorbance readings from 350 nm to 600 nm, 70 nodes in the hidden layer using a logistic activation function, and 2 nodes in the output layer with a logistic function. Test result shows that the NNM has correlation coefficients of 0.9953 and 0.9922 when predicting \[Co\] and \[Ni\], respectively. We observed, however, that the NNM has a duality property and that there exists a real-world practical application in solving the dual problem: Predict the Co-Ni mixture’s absorbance profile given \[Co\] and \[Ni\]. It turns out that the dual problem is much harder to solve because the intended output has a much bigger cardinality than that of the input. Thus, we trained the dual ANN, a 3-layer jump connection nets with 2 input nodes corresponding to \[Co\] and \[Ni\], 70-logistic-activated nodes in the hidden layer, and 126 output nodes corresponding to the 126 absorbance readings from 250 nm to 600 nm. Test result shows that the dual NNM has correlation coefficients that range from 0.9050 through 0.9980 at 356 nm through 578 nm with the maximum coefficient observed at 480 nm. This means that the dual ANN can be used to predict the absorbance profile given the respective Co-Ni concentrations which can be of importance in creating academic models for a virtual chemical spectrophotometer.'
author:
- |
Jaderick P. Pabico\
\
\
Jose Rene L. Micor\
\
\
Elmer Rico E. Mojica\
\
\
bibliography:
- 'paper.bib'
title: A Neural Prototype for a Virtual Chemical Spectrophotometer
---
Introduction {#intro}
============
An artificial neural network (ANN) is a system loosely based or modeled on the human brain. It can goes by many names, such as natural intelligent system, connectionism, neuron computing, parallel distributed processing, machine learning algorithms and artificial neural networks [@ref1]. The ANN is able to obtain or attain information and offer or present models even when the information and data are complex, noise contaminated, nonlinear and incomplete [@ref2; @ref3; @ref4; @ref5].
Artificial neural networks (ANNs) are powerful tools, exceptionally suited for a diverse tasks in information processing, such as recognizing patterns, generalizing, analyzing non-linear multivariate data and other things [@ref4; @ref5]. Currently, the most commonly used ANN type is a multi-layer feed forward network, which is trained by the back propagation (BP) algorithm. The application of ANNs to data is claimed to constitute so-called “soft models,” since the models have an ability to learn and extract $X$–$Y$ relationships from the presentation of a set of training samples. Their flexibility has been a decisive asset compared with parametric techniques that require the assumption of a specific hard model form. In addition, ANNs avoid the time-consuming and possibly expensive task of hard model identification [@ref4]. There are many applications of ANNs particularly in fields like medicine [@ref6], engineering, chemistry, physics, agriculture, music, economy and management, archeology, as well as industry [@ref7; @ref8; @ref9; @ref10; @ref11; @ref12; @ref13; @ref14]. In chemistry, ANNs have the ability to tackle the problem of complex relationships among variables that cannot be accomplished by more traditional methods. The ANN modeling method has found extensive application in the field of simultaneous determination of several species in a given sample. This method makes it possible to eliminate or reduce the effect of the analyte-analyte interaction, the multi-step process and any other unknown non-linearity in systems [@ref15]. Another advantage of ANN is its anti-jamming, anti-noise and robust nonlinear transfer ability which in a proper model would results in lower calibration errors and prediction errors [@ref16].
Nickel (II) and cobalt (II) are metals that appear together in many real samples, both natural and artificial. Several techniques such as atomic absorption, atomic fluorescence, X-ray fluorescence, voltammetric and spectrophotometric methods have been used for the determination of these ions in different samples. Among the most widely used analytical methods are those based on the UV\_visible spectrophotometric techniques due to the resulting experimental rapidly, simplicity and the wide application [@ref17].
Simultaneous determination of trace amounts of metals in environmental samples is still a challenging analytical problem because of the sensitivity and specificity required in environmental monitoring and regulations. Recently, spectrophotometric methods based on ANNs have found increasing applications for multicomponent determination. This method is effective because they can improve the performance and application of the analytical method with the use of simultaneous analysis of several spectra. There are several studies that reported on the simultaneous analysis of metals using ANN. An example of this is the study on the simultaneous spectrophotometric determination of Co(II) and Ni(II) based on formation of their complexes with EDTA complexes with pyrolidine and carbon disulfide [@ref17]. It uses ANN to analyze the mixture spectra of the complex solutions formed. In this paper, mixture of untreated Ni(II) and Co (II) were prepared and the concentration of each component was determined based on the absorbance profile of the standard solutions.
Methodology
===========
Experimental
------------
All chemicals were of analytical reagent grade and deionized water was used throughout the experiment. Stock standard solution (0.50 M) of Ni(II) and Co(II) were prepared by dissolving appropriate amounts of nickel nitrate (Sigma) and cobalt nitrate (Sigma), respectively in 25 mL volumetric flasks and diluted to the mark with deionized water.
Procedure
---------
Sample solutions ranging from 0.02 to 0.10 M of Ni(II) and Co(II) were prepared in 5 mL volumetric flask by taking a required volume of the stock solution and then diluted to the mark with deionized water. A mixture was also prepared ranging with either one of the component containing a concentration that ranged from 0.025 to 0.10 M.
Instrumentation
---------------
Quartz cuvettes (1 cm$^-2$) were used for all spectroscopic experiments and all measurements were performed at room temperature (25 $\pm$ 1C). All absorbance measurements were carried out on a Hewlett Packard 8452A diode array spectrophotometer with a 1 nm spectral bandpass. For each concentration of each metal and the mixture, the spectrum was scanned in the wavelength of 350–600 nm.
Training Data and Randomization
-------------------------------
The data collected from the spectrophotometer totaled 6,000 records. To facilitate input into the ANN, the absorbance data were normalized to values between 0 and 1. A randomization function was created that fed randomized data into the ANN during the training and testing phases.
ANN Optimization by Artificial Chemistry
----------------------------------------
Artificial Chemistry (AChem) is a computational paradigm for search, optimization, and machine learning. In this paradigm, the artificial molecules represent machine or data and the interactions among these molecules are driven by an algorithm. The duality of the molecule to represent either a machine (or operator) or data (or operand) enables a molecule to process other molecules or be processed. This dualism property of molecules enables one to implicitly define a constructive computational procedure using the dynamics of chemical reaction as a metaphor to solve complex optimization problems [@pabico03]. The capabilities of AChem to simultaneously find solutions to different problems makes it a potent solution to finding the best ANN structure for this problem.
An AChem system was setup to simultaneously find the optimal ANN structure. The AChem system ran for 10,000 simulation cycles when 80% of the molecules already encode the same ANN structure. Each AChem molecule encodes the number of hidden layers, the number of neurons on each hidden layer, a binary digit that flags whether the structure will use a jump-connection or not, the learning rate, and the momentum value. The number of neurons in the input and output layers are fixed at 2 and 126, respectively. The 2 input neurons correspond to the \[Co\] and \[Ni\] while the 126 output neurons correspond to the absorbance readings from 250 nm to 600 nm. Specialized reaction rules were devised such that the collision of the two molecules, as well as the collision of a molecule with the artificial reaction tank, will create more molecules. A reactor algorithm was also devised to filter out molecules that encode better solutions [@pabico07].
Each encoded ANN structure on AChem’s molecule was evaluated by running a feed-forward, back-propagation learning algorithm over the training data using the accompanying encoded learning rate and momentum. The ANN used 4,200 records as the training set. A test set, disjoint from the training set, was extracted containing 600 records. The ANN training was stopped when the network error over the test set has not improved after 100 epochs. The trained ANN was when the network error over the test set was at the minimum. A validation set, disjoint from both the training and test sets, was extracted containing 1,200 records. The trained ANN was run over the validation set and its classification rate recorded. The recorded classification rate becomes the molecule’s evaluation.
Results and Discussion
======================
The absorbance profile of the solutions containing nickel, cobalt and both nickel and cobalt were obtained. Figures \[fig1\]-\[fig3\] showed the increase of absorbance at increasing concentration of the given solution. The increase in absorbance is directly proportional to the increase in concentration. Ni solution has an absorbance maximum at around 394 nm while the cobalt solution had an absorbance maximum at around 510 nm. These maxima was also observed in the absorbance profile of the solution containing both metals. Although overlapping of the spectrum was observed, the absorbance maxima is still proportional to the concentration of the metal. With this result, a direct relationship correlating the concentration of metal in the solution with the absorbance was obtained and used to develop an artificial neural network which can predict the concentration of the components in a given solution.
The AChem optimization routine found the optimal ANN structure as a 3-layer jump connection nets with 2 input nodes corresponding to \[Co\] and \[Ni\], 70-logistic-activated nodes in the hidden layer, and 126 output nodes corresponding to the 126 absorbance readings from 250 nm to 600 nm. Test result shows that the dual NNM has correlation coefficients that range from 0.9050 through 0.9980 at 356 nm through 578 nm with the maximum coefficient observed at 480 nm. This means that the dual ANN can be used to predict the absorbance profile given the respective Co-Ni concentrations which can be of importance in creating academic models for a virtual chemical spectrophotometer.
|
=1
amstex.tex epsf
1.5
**SOME SPECULATIONS ABOUT BLACK HOLE**
**ENTROPY IN STRING THEORY**
By Leonard Susskind[^1]
Physics Department
Stanford University
Stanford, CA 94305-4060
and
Department of Physics and Astronomy
Rutgers University
Piscataway, NJ 08855-0849
There are some puzzles concerning the entropy of black holes which I would like to consider from the point of view of string theory. First of all the meaning of the Bekenstein entropy $$S_B = {1\over 4} \, {\text{Area}\over 4G\hbar}\tag 1.1$$ has always been mysterious. Entropy, as generally understood, has to do with the counting of configurations of some set of degrees of freedom. What the degrees of freedom of the horizon are and why they give entropy of order $1/\hbar$ has remained obscure.[^2] The second puzzle concerns the higher order quantum corrections to the entropy. G ’tHooft has emphasized[@1] that conventional quantum fields contribute an ultraviolet divergence to $S$ which blows up at the horizon. This is despite the fact that no curvature or other invariant signal becomes large. The ultraviolet divergent entropy is proportional to the area and although down by a factor of $\hbar$ from $S_B$ it is infinitely larger. It has not been properly appreciated except by ’t Hooft that this ultraviolet divergence of $S$ is the same problem as Hawking’s information paradox and that any theory which naturally produces a finite entropy will also solve this problem. From the perspective of a distant observer nothing ever reaches the horizon. Instead all matter settles into layers which eternally sink toward it. If the entropy of matter near the horizon is infinite, indefinite amounts of information can be stored arbitrarily close by. This information can not be emitted until the horizon shrinks to quantum mechanical proportions and perhaps not even then. By contrast, a theory in which the information storage capacity is finite has no choice but to reemit information as the horizon shrinks. In a previous paper[@2] I showed that in string theory, from the outside observers vantage point, the substance of infalling strings not only never reaches the horizon but never entirely sinks past the stretched horizon. Therefore it seems appropriate to ask whether string theory leads to a finite entropy. The last puzzle concerns the connection between the spectrum of black holes and that of unperturbed strings. In both cases that level density increases rapidly with mass. Furthermore, most of the spectrum of strings must actually be black holes since they lie within their Schwarzschild radii. Nevertheless I do not know of any speculation that the two spectra may really be the same.[@3] In fact at first sight such a suggestion seems nonsensical. The level density of black holes grows like $\exp 4\pi M^2$ while that of strings is exponential in the first power of the mass. We shall see that this argument is wrong and that the two spectra, when properly interpreted, could easily be the same. I will begin by outlining the procedures involved in constructing the thermodynamics of Rindler space. The euclidean continuation is ordinary flat space in cylindrical polar coordinates $$ds^2 = r^2 d\theta^2 +dr^2 + dx^2_\bot \tag2.1$$ where the angular variable $\theta$ is the euclidean time, $r$ is the radical variable (not to be confused with the Schwarzschild coordinate) and $X_\bot$ is the 2 space parallel to the horizon. The horizon itself is the surface $r=0$. The Rindler hamiltonian $H_R$ is the generator of $\theta$-rotations. $$H_R = {\partial\over \partial\theta}\tag2.2$$ The partition function from which thermodynamics is derived is $$Z = Tr \exp\{-\beta H_R\}\tag2.3$$For physical applications $\beta$ should be set equal to 2$\pi$. For the purposes of thermodynamic analysis it must be left as a free variable. Note that varying $\beta$ is equivalent to introducing a conical singularity at the horizon with an angular deficit $2\pi-\beta$. The free energy $F(\beta)$ is given by $$F = {1\over \beta} \log Z\tag 2.4$$ and the entropy by $$S = {\partial F\over \partial T_R}\tag2.5$$ where the Rindler temperature is $T_R = 1/\beta$. Eq. 2.5 is the reason we need to be able to vary $\beta$ away from $2\pi$. Consider first the thermodynamics of a free scalar field. The partition function can be carried out as a sum over first quantized closed path particle trajectories in a well known manner. The only new ingredient is the conical singularity at $r = 0$. It is easy to see that paths which do not wind around $r=0$ contribute no $\beta$ dependence to the free energy. This means that they can be dropped when calculating the entropy. Thus we find that the entropy is the first order variation with respect to $\beta$ of the sum of paths which wind one or more times around the horizon. This principle extends to interacting field theories described in terms of networks of paths forming Feynman diagrams. Only those networks which topologically encircle $r= 0$ contribute to entropy. It is clear that the entropy divergences found in conventional quantum field theory are due to very small loops near $r = 0$. Now I want to study black hole entropy using an analogous method. Consider the euclidean continuation of the Schwarzschild metric in which time is periodic. The geometry near $r = 0$ is identical to Rindler space. Far from the horizon 2.1 is modified to $$\align
ds ^2 &= 16 m^2 G^2 d\theta^2 + dr^2\\
&+ \text{transverse metric}\\
&= - dt^2 + dr^2 +\text{transverse}
\tag2.6
\endalign$$ where $$t = - 4 \imath m G\theta\tag2.7$$ is the Schwarzshild time coordinate. The ordinary energy (mass) of a black hole is conjugate to Schwarzshild time and the conventional temperature is defined in terms of this energy. The energy and Schwarzshild temperature are given in terms of the mass by $$\align
S &= 4\pi M^2 G\\
Ts &= {1\over 8\pi MG}
\tag2.8
\endalign$$ In transforming energy from Schwarzshild to Rindler coordinates care must be taken to insure that the Rindler energy $ E_R$ is conjugate to $\imath\theta$. We assume $M$ is conjugate to $t$. Thus $$\align
[E_R(M), \imath\theta] &= \imath\\
[M, t] &= \imath
\tag2.9
\endalign$$ Using 2.7 and the usual properties of commitators gives $$E_R = 2M^2 G\tag2.10$$ It is interesting that both entropy and Rindler energy are extensive functions of the horizon area ($A = 16\pi M^2 G^2$). $$\align
E_R &= {A\over 8\pi G} = 4\pi 2M^2G\\
S &= {A\over 4G} = 4\pi M^2G\tag 2.11
\endalign$$ To find the Rindler temperature we use the first law of thermodynamics $$dE_R = T_R dS\tag 2.12$$ which gives $$T_R = {1\over 2\pi}\tag 2.13$$ To compute from first principles directly in Rindler coordinates we must calculate the free energy of a euclidean black hole with an angle deficit of ($2\pi - \beta$). Again this introduces a conical singularity at the horizon. The lowest order contribution to $F$ is order $1/\hbar$ and is given terms of the classical action. $$F = {1\over \beta} \, \text{Action}\tag 2.14$$ The action consists of three distinct terms. The first is proportional to the integrated Ricci scalar which vanishes by virtue of the Einstein equations. The second involves the extrinsic curvature at the boundary at large $r$. This term is proportional to $\beta$ and does not contribute to the entropy. The third contribution is due to the curvature delta function at the conical singularity. It is proportional to the horizon area and to the deficit angle $2\pi- \beta$. Explicit calculation gives $$F = {\beta- 2\pi\over \beta} \, {\text{Area}\over 8\pi G}\tag2.15$$ and $$S = {\partial F\over \partial T_R} = {\text{Area}\over 4G}\tag2.16$$ Evaluating the partition function in a conical background provides a general framework for calculating black hole entropy including the classical Bekenstein term and quantum corrections from both matter and gravitational field. To apply it to string theory requires formulating the theory in backgrounds with small but arbitrary angle deficits. This has not been done. I will therefore restrict my remarks to certain general features. The free energy is given as a sum over world sheet configurations of arbitrary genus. By an argument similar to that used in field theory, the only nonvanishing contributions to the entropy come from world sheets which in some way wrap around or touch the singularity at the horizon. For example a torus can surround the horizon as in fig(1) This configuration describes the contribution to the entropy of a free closed string. This is seen by slicing the figure at some fixed euclidean time such as $\theta = 0$ (See fig 2) Another configuration in which the horizon intersects the torus is shown in fig 3 In this case the instantaneous configurations involve strings with their ends frozen on the horizon as in fig 4 Such configurations must be included in the space of states of the black hole. The genus $k$ surfaces contribute with a coefficient $G^{k-1}$. Evidently the Bekenstein entropy corresponds to a genus zero surface with the topology of a sphere as in fig 5 These surfaces describe the evolution of a single string with ends on the horizon. In Minkowski space the endpoints of the string can not move because of the infinite time dilation at the horizon but the rest of the string is free to wiggle. This solves the puzzle of the origin of $S_B$ in String theory. One might wonder why the zero genus contributions do not vanish as they usually do in string theory. The reason involves both the conical singularity at the origin and the fact that the deficit angle at infinity does not vanish. The presence of these genus zero contributions very near the horizon is almost certainly related to a similar effect found by Atick and Witten[@4] in high temperature string theory and large $N$ $QCD$. We next turn to the question of the finiteness of the higher genus quantum corrections. In the absence of precise tools for quantizing strings on singular spaces I can only quote some circumstantial evidence for finiteness. However, before doing so we must take care of a trivial divergence associated with the infinite volume at spatial infinity. If the black hole is in thermal equilibrium with its environment, the nonvanishing temperature will cause a volume divergence in all extensive thermodynamical variables. This can be easily overcome by passing to the limit of infinite black hole mass. The resulting geometry is flat Rindler space with a deficit angle. In Rindler space-times of dimension greater than two the thermodynamical variables are infrared finite. The entropy per unit area is therefore well defined. As I have mentioned in sect 2 the ordinary field theoretic divergences in entropy arise from paths of vanishing length which encircle the horizon. In string theory the analogue of a path of given proper time is a torus with complex modular parameter $\tau$. It is well known that the integration region over $\tau$ which could potentially cause ultraviolet divergences should be excised because it corresponds to an infinite overcounting of geometrically identical tori. There is no mechanism for generating ultraviolet divergences if the relevant loop integrals are anything like other string amplitudes. The finiteness of string theoretic loops is due to the extreme paucity of degrees of freedom at short distances. This lack of short distance structure is seen in several ways. 1) Atick and Witten have shown[@4] that string theory behaves more softly at high temperatures than any possible continuum quantum field theory. Roughly speaking the high temperature thermodynamics is consistent with a lattice theory in which the spacing is finite. Similar evidence comes from the work of Klebanov and Susskind[@5] who show that exact string amplitudes can be derived from a space-time lattice theory with non vanishing spacing. 2) High energy scattering amplitudes at fixed angle are the traditional method of uncovering short distance structure. Gross and Mende[@6] have shown that such amplitudes vanish like gaussian functions of momentum transfer. 3) It appears to be impossible to force the dimensions of compact space dimensions to be smaller than a certain size of order $\ell_s$.[@7] 4) Following the progress of a string falling toward a horizon, an external observer fails to see the object Lorentz contract.[@2] There appears to be a minimum longitudinal size that strings can occupy. Furthermore for non vanishing coupling there is a bound to the number of strings that can pass through a small region without inducing violent interactions. All these facts point to a common conclusion. When we attempt to localize strings or parts of strings in distances much smaller than $\ell_s$ we discover a complete lack of local degrees of freedom. This strongly suggests that higher genus contributions to black hole entropy is finite and that to an external observer, indefinite quantities of information can not collect arbitrarily near the event horizon. What is desperately needed is a computation to confirm this. I turn now to a radically different way to estimate the entropy of a black hole. When strings fall on to a horizon an external observer sees them spread out and eventually fill the stretched horizon.[@2] One can regard this phenomenon as a melting of strings as they encounter Hagedorn temperature conditions[@4] at a distance $\sim\ell_s$ from the event horizon. The entropy of single string states is so large that strings on the horizon will tend to form a single string when the Hagedorn temperature is approached. The implication is that all black hole states are in one to one correspondence with single string states. Now it has been observed in the past that the high mass-low angular momentum states of string theory must be black holes since they lie within their Schwarzshild radii. I would like to make the heretical suggestion that the spectrum of black holes and the spectrum of single string states are identical.[@3] Furthermore this provides us with a direct way to estimate the number of levels and therefore the entropy. Before we can actually compare the spectra we must deal with somewhat trivial but numerically very important effect. The classical gravitational field outside the stretched horizon is not a low order effect. We must imagine removing the effects of this field before we try to compare low order string theory with black hole physics. The main effect which must be accounted for is the large red shift of clock rates that takes place between the stringy stretched horizon and an observer at infinity. In other words a rescaling of all energy levels of the black hole should be done. The stretched horizon is the place where the local Unruh temperature becomes hot enough for stringy effects to become important, i.e. the Hagedom temperature. This means a distance $\sim \ell_s =
(\alpha^1)^{-{1\over 2}}$ from the event horizon at $ r = 0$. At this distance, the proper time of a fiducial observer is given by $$\tau = \imath\theta\ell_s\tag4.1$$ where $\imath\theta$ is Rindler time. All quantities with units of energy should be rescaled by dividing the corresponding Rindler quantities by $\ell_s$. The resulting quantities would also be appropriate for an observer at infinity if the effects of the classical gravitational field could be removed. Thus we define the stretched horizon energy and temperature of the black hole to be $$\align
E(S.H.) &= {2 M^2 G\over \ell_s}\\
T(S.H.) &= 1/2 \pi \ell_s\tag 4.2
\endalign$$ Another way to think of the relation between $E(S.H.)$ and $M$ is that the long range field outside the stretched horizon renormalizes the mass from its “bare” value $E(SH)$ to its renormalized value $M$. Therefore I would like to suggest that a black hole of mass $M$ should be identified with a string state of mass $E(S.H.)$. We can estimate the entropy of a black hole by counting the levels of fundamental strings. The number of states at mass level $r = E$ satisfies $$\log N(E) \sim E\ell_s\tag4.3$$ Using 4.2 we find a black hole of mass $M$ has a level density satisfying $$\log N(M) \sim M^2G\tag 4.4$$ which says that the entropy is of order $S_B$. It is therefore not inconsistent to suppose that a correspondence exists between black holes and fundamental string states. Vafa has pointed out that this correspondence may extend to extreme charged black holes. In this case the mass renormalization due to the gravitational field energy exactly cancels the electromagnetic energy so that string states of mass $M$ should be compared with black holes of mass $M$. If, for example, we consider charge arising from winding modes then the minimum mass of a string of charge $Q$ is proportional to $Q$. This corresponds exactly to the extremal black hole. If the view of black hole entropy in this section is correct then there can be little doubt that the quantum corrections to $S_B$ are finite. These quantum corrections would result from the finite shifting of levels due to higher genus world sheets. Finally I would like to mention an observable effect of the string theory on black hole evaporation. In the usual picture the final evaporation process takes place at planckian temperatures. The last radiated particles would carry energy of order the Planck mass. To best appreciate the difference that string theory makes, it is helpful to pretend that the string coupling $g^2 = G/\ell^2_s$ is extremely small so that the Planck and string scales are extremely well separated. Let us consider the radius of an average excited string state of mass M, ignoring all higher order effects including the long range gravitational field. Ignoring quantum fluctuations of the string the mean radius of the average configuration of mass M can be shown to be $$R_{ST} = \sqrt{M\ell^3_s}\tag 4.5$$ Comparing this with the Schwarzschild radius, $R_{SCH} \sim MG =
Mg^2\ell^2_s$, we see the two are equal when $$M = M_a = (\ell_sg^4)^{-1}\tag 4.6$$ Thus for $g \ll 1$ there is a large range of masses for which the conventional string configuration is larger than $R_{SCH}$ and no black hole behavior should occur. On the otherhand an evaporating black hole should behave conventionally until the red shift factor at the stretched horizon is of order unity. At this point there is no large red shift factor and strings should behave like strings. The mass at this point is $$M = M_b = {1 \over g^2\ell_s}\tag 4.7$$ Several things happen at this point. The first is that the area of the black hole horizon has become equal to $\ell_s^2$. Second, the Hawking temperature has reached the Hagedorn temperature. Third, the conventional mass M and the stretched horizon energy E(S.H.) become equal. Finally at this point the Bekenstein entropy $\sim M^2G^2$ becomes of order the string entropy $M\ell_s$. Between $M_a$ and $M_b$ we have two descriptions, ie. string and black hole, both of which should apply but which disagree. The resolution of this inconsistency is that in this region there are two configurations. The first is metastable and describes a conventional string with radius larger than $R_{SCH}$. The second, with larger entropy is stable and consists of a black hole with the string gravitationally collapsed to the stretched horizon. Below mass $M_b$ the string entropy exceeds the black hole entropy so that the black hole becomes thermodynamically unstable. Accordingly when a black hole reaches the Hagedorn temperature it “inflates” to a string whose size exceeds that of a black hole by the factor $${R_{ST} \over R_{SCH}} \sim {1 \over g}\tag 4.8$$ Thereafter it decays like a weakly coupled string. In particular the momenta of the emitted particles never exceeds the Hagedorn temperature.
**Acknowledgments.**
This paper was written while the author was visiting Rutgers University. He is grateful not only for the hospitality, but for many exciting discussions with the members of the theory group and other visitors. In particular C. Vata pointed out the significance of extremal charged states in string theory. The observations about the final stages of evaporation arose in a conversation with Steve Shenker about possible experimental signatures. As always discussions with Tom Banks, Nathan Seiberg and Mike Douglas were very valuable and fun. Helpful discussions with Ed Witten and Curt Callan are also appreciated.
\[\]1 G. ’t Hooft Private Communication
\[\]2 L. Susskind Strings, Black Holes and Lorentz Contraction Stanford University preprint SU-ITP-93-21 hep-th/9308139
\[\]3 ’t Hooft has had the long standing belief that black holes are the extrapolation of elementary particles to high mass. See for example G. ’t Hooft Nucl. Phys. B 335 1988 138
\[\]4 J. Attick and E. Witten Nucl. Phys. B 310 1988 291
\[\]5 J. Polchinski Comm. Math Phys. 104 1986 37
\[\]6 I. Klebanov and L. Susskind Continuum strings from discrete field theories Nucl Phys. B 309 1988 175
\[\]7 D.J. Gross and P. Mende Nucl Phys. B 303 1988 407
\[\]8 K. Huang and S. Weinberg Phys REv Lett 25 1970 895 S. Fubini and G. Veneziano, Nuovo cimento 64 A 1969 1640
[^1]: susskinddormouse.stanford.edu
[^2]: Hereafter $\hbar$ will be set to 1
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abstract: 'We present [*Rossi X-ray Timing Explorer*]{} observations of the X-ray pulsar SMC X-1. The source is highly variable on short time scales ($<$ 1 h), exhibiting apparent X-ray flares occupying a significant fraction ($\sim$3 %) of the total observing time, with a recurrence time of $\sim$100 s. The flares seem to occur over all binary orbital phases, and correlate with the overall variability in the light curve. We find a total of 323 discrete flares which have a mean full width half maximum of $\sim$18 s. The detailed properties of SMC X-1 do not vary significantly between the flares and the normal state, suggesting that the flare may be an extension of the normal state persistent emission with increased accretion rates. The flares resemble Type II X-ray bursts from GRO J1744–28. We discuss the origin of the SMC X-1 flares in terms of a viscous instability near the inner edge of the accretion disk around a weakly magnetized X-ray pulsar, and find this is consistent with the interpretation that SMC X-1 is in fact an intermediate-stage source like GRO J1744–28.'
author:
- 'Dae-Sik Moon, Stephen S. Eikenberry, & Ira M. Wasserman'
title: 'SMC X-1 As An Intermediate-Stage Flaring X-ray Pulsar'
---
Introduction
============
Neutron star X-ray binaries are generally categorized into two groups: low mass X-ray binaries (LMXBs) and X-ray pulsars. The surface magnetic field of the central neutron star in an LMXB is thought to be $\sim$10$^8$ G. The mass accretion with this magnetic field is most likely spherical, so that no significant inhomogeneity in the X-ray emission over the neutron star surface is expected – i.e., no persistent coherent pulsations are observed. The strong magnetic field, $\sim$10$^{12}$ G, of an X-ray pulsar, on the other hand, can funnel the accretion matter onto the magnetic pole, which makes the central neutron star appear as a pulsar.
Of particular interest are the so-called “intermediate-stage sources” speculated to lie between the LMXBs and X-ray pulsars, including “the Rapid Burster" (MXB 1730–355), GRO J1744–28 (“the Bursting Pulsar"), and SAX J1808.4–3658 (“the accreting millisecond pulsar") (Lewin et al. 1976; Fishman et al. 1995; in ’t Zand et al. 1998). Both the Rapid Burster and GRO J1744–28 exhibit Type II X-ray bursts; however, only the former shows Type I X-ray bursts while only the latter has apparent coherent pulsations (Lewin et al. 1996). SAX J1808.4–3658, on the other hand, shows both Type I bursts and coherent pulsations (Wijnands & van der Klis 1998; Chakarbarty & Morgan 1998), but not Type II bursts. The magnetic field strengths of these sources have been inferred to be $\sim$10$^{8-11}$ G, between those of LMXBs and X-ray pulsars.
Another possible intermediate-stage source is the X-ray pulsar SMC X-1, which has similar properties to GRO J1744–28, including its fast spin period ($\sim$0.72 s for SMC X-1; $\sim$0.47 s for GRO J1744–28), steady spin-up, and inferred magnetic field ($\sim$10$^{11}$ G ) (Bildsten & Brown 1997; Li & van den Heuvel 1997). In addition, once SMC X-1 was observed with an X-ray burst that resembles Type II bursts (Angelini, Stella, & White 1991). It may be possible, therefore, that SMC X-1 and GRO J1744–28 belong to a distinctive group of X-ray binaries, “bursting pulsars", which show both coherent pulsations and Type II X-ray bursts (Li & van den Heuvel 1997). To investigate this important possibility, we analyze all publicly available RXTE data for SMC X-1, searching for phenomena that may be related to X-ray bursts. We report that SMC X-1 in fact exhibits active flares resembling Type II bursts from GRO J1744–28.
Data Analysis and a Flare Search
================================
We analyzed all publicly available RXTE Proportional Counter Array observations toward SMC X-1. Photon arrival times from the Good Xenon mode were transformed to the solar system barycenter using the JPL DE400 ephemeris. The Very Large Event Models were used to subtract backgrounds, and only the Standard 2 data obtained from the top xenon layers of Proportional Counter Units 0, 1, and 4 were considered for spectral analysis. Both the data within 30 minutes after passages through South Atlantic Anomaly and/or with a high ($>$ 0.1) electron ratio were ignored.
The data toward SMC X-1 are occasionally contaminated by outbursts of the nearby ($\sim$27$'$ away) transient source XTE J0111.2–7317 (Chakrabarty et al. 1998). The contamination was easily identified by the source’s $\sim$31-s pulsations, with powers reach up to $>$ 5000 in Leahy-normalized PDSs. We searched for weak contamination in PDSs as narrow statistically significant ($>$ 10 in Leahy-nomalized PDSs) peaks at $\sim$0.032 Hz (and/or at its harmonic frequencies). We excluded any data contaminated by XTE J0111.2–7317 from our analyses, as well as data containing dip-like features (e.g., Her X-1; Moon & Eikenberry 2001) or close to the eclipse (i.e., binary orbital phase between 0.9 and 0.1) of SMC X-1. We obtained a total of 150 data segments with an average length of $\sim$1360 s, giving a net observational duration of $\sim$204 ks. For spectral analysis, we considered only the spectrum between 2.5–25 keV range due to a PCA responsivity problem (R. Remillard 2001, private communication), and assumed a systematic uncertainty of 1 %.
In our analysis, we define “flares” to be the part of a light curve that has three or more consecutive data bins with photon counts larger than a threshold value of 3$\sigma$ Poisson noise above the mean photon count in a given light curve. We analyzed all the 150 light curves as follows searching for the flares. First, we binned each light curve to 4-s time resolution and made a flare list. We extended the search with 2- and 8-s time resolutions, excluding the flares already found with different time resolutions. We found a total of 323 flares, and fit them a Gaussian function to estimate width and peak intensity. We obtained $\sim$18 s and $\sim$15 s for the mean and rms standard deviation of FWHM; the reduced $\chi^2$ (= $\chi^2_{\nu}$) of the fits range from 0.5 to 1.9. The total duration above half-maximum is $\sim$5.8 ks, indicating that SMC X-1 spends $\sim$3 % of its time on flares.
Flare Examples and Correlations with Other Parameters
=====================================================
The rms variability of the 150 light curves ranges between 6 and 16 %, with a mean of $\sim$11 %. Figure 1 presents three light curves (and their PDSs) with very low (7 %), average (11 %), and very high (16 %) rms variability as examples representing three different variability levels. The mean photon count rates are 120.5 $\pm$ 8.5, 116.1 $\pm$ 12.9, and 146.4 $\pm$ 23.3, for Figure 1a, 1b, and 1c, respectively. No flare is found in Figure 1a, which has a low rms variability; three flares are found Figure 1b, which has an average rms variability. Strong flaring activity is well illustrated in Figure 1c, which has a high rms variability, with several apparent flares lasting for a few tens of seconds. At the flare peak, the photon count rate rises up to $\sim$2.5 times of that outside the flares. The inset shows a Gaussian fit to the intense flare at $t$ $\simeq$ 900 s, with an FWHM estimated to be $\sim$23 s. Leahy-normalized PDSs of the three light curves (Figure 1d, 1e, and 1f) show strong QPO-like peaks around 10 mHz, as well as the peaks caused by the source’s coherent pulsations at $\nu$ $\simeq$ 1.4 Hz (and their harmonics), indicating the existence of $\sim$100-s aperiodic variability independent of the flaring activity. The fractional rms (FRMS) amplitudes in the 2–50 mHz range are 7 $\pm$ 1, 9 $\pm$ 1, and 14 $\pm$ 1 % for 1d, 1e, and 1f, respectively.
Figure 2 presents the pulse profiles of the three light curves, all showing the double-peaked, smooth profile typical of SMC X-1 (e.g., Levine et al. 1993). They have the same pulsed fractions of $\sim$39 %, with their second peaks at phase 0.56 with respect to the first ones at phase 0. The ratio of the second peak to the first peak is $\sim$0.78 for Figure 2a, while it is $\sim$0.96 for 2b and 2c. No significant variation in the pulse peak ratio has been found within a given light curve, regardless whether it is obtained inside or outside flares.
Figure 3 compares the phase-averaged softness ratio, defined to be the ratio of the soft X-ray (2–5 keV) photon count rates to those of the hard X-ray (5–13 keV), of the three light curves in Figure 1 with the total photon count rates at 2–25 keV with 4-s resolution. The average softness ratio of the three light curves is invariant: 0.47 $\pm$ 0.03, 0.47 $\pm$ 0.02 , and 0.46 $\pm$ 0.04 for Figure 1a, 1b, and 1c, respectively. This contrasts with the softness ratio of the LMC X-4 flares (Moon, Eikenberry, & Wasserman 2002), which shows a strong linear correlation with the total photon count rate (inset in Figure 3).
We examined the spectral invariance of SMC X-1 over the flaring activity indicated by the constant softness ratio distribution more thoroughly via fitting the 32-s spectra obtained from the largest peak in each of the three light curves (i.e., the peak at $t$ $\simeq$ 900 s in Figure 1a, at $t$ $\simeq$ 1050 s in 1b, and at $t$ $\simeq$ 1110 s in 1c) to a model spectrum. The model spectrum consists of a power-law component with a high-energy cutoff (for non-thermal magnetospheric emission) and a Gaussian component (for iron line emission), together with a component for photoelectric absorption by intervening interstellar matter. We fixed the central energy of the Gaussian component to be 6.7 keV based on previous results [@aet91]. Figure 4 compares the observed spectra with the best-fit model spectra, and Table 1 lists the best-fit parameters and $\chi^2_{\nu}$ of the fits. Although all parameters are poorly constrained, the power-law index does not change significantly over the flaring activity – consistent with the softness ratio distribution (Figure 3). The difference in $N_{\rm H}$ may be caused by the motion of the precessing, tilted accretion disk suggested to be responsible for the super-orbital motion of SMC X-1, but the low energy limit of the spectral fits (i.e., 2.5 keV) makes it difficult to constrain $N_{\rm H}$ propertly, because the photoelectric absorption by intervening interstellar matter is expected to be most significant in the soft energy band.
In order to perform statistical analyses, we calculated the flare fraction, which we define to be the ratio of the integrated time that SMC X-1 is flaring (i.e., within FWHMs of the Gaussian fits) to the total observing time of a given parameter, and investigate its correlation with the parameter. Figure 5 shows the distribution of the flare fraction as functions of the binary orbital phase (5a), rms variability of the light curve (5b), and the pulse peak ratio (5c). Some important results are worth noticing: the flare fraction (1) is larger than 1.4 % over all the orbital phases (with its minimum at the orbital phase of $\sim$0.35), (2) shows a significant variation from phase to phase, and (3) increases with the rms variability of the light curve, as well as with the pulse peak ratio. The correlation between flare fraction and rms variability identified in Figure 5b remains very similar when we use the flare-subtracted rms variability.
Discussion
==========
SMC X-1 shows active flares that occupy $\sim$3 % of the total observing time. The flares seem to occur over all binary orbital phases, and the flaring activity is proportional to the rms variability of the light curve (Figure 5). Except for the small change in the pulse peak ratio (Figure 2 and 5), no significant change is found along with the flaring activity. All these suggest that the SMC X-1 flares may be simple extensions of the persistent emission of a normal state with increased accretion rates, but without significant changes in the geometry of the accretion flows and the magnetosphere. Under the hypothesis that the double peaks in the pulse profile (Figure 2) are due to the two magnetic poles of SMC X-1, one simple explanation for the change in the pulse peak ratio may be that the increase in the accretion rate onto the fainter pole is higher than that onto the brighter pole during the flares. The invariant pulse peak ratio in a given light curve (regardless of flares), together with the correlation between the flare-subtracted rms variability and the flare fraction, on the other hand, may indicate the existence of variability whose time scale is longer than the flare recurrence time, $\sim$100 s.
The SMC X-1 flares differ from Type I X-ray bursts from LMXBs for various reasons, including the shape of profiles and spectral properties. While the profile of Type I bursts shows an abrupt increase with an exponential decay in most cases, the SMC X-1 flares have symmetric Gaussian shapes. The X-ray spectrum of the SMC X-1 flare is far from thermal spectrum of Type I bursts (although the thermal spectrum of an X-ray pulsar is not well constrained, so it is not completely excluded that the SMC X-1 flare spectrum is thermal). In addition, the SMC X-1 spectrum does not show any apparent variation within a flare, while Type I burst shows spectral cooling as the burst continues. The changes in the softness ratio and pulse profile of the SMC X-1 flares differ from those of the LMC X-4 flares (Figure 3; Moon et al. 2002), which is one of the most well known and regular flaring X-ray binaries.
The SMC X-1 flares recall the Type II X-ray bursts found in GRO J1744–28 mainly due to the spectral invarince over the flaring activity. In fact, the burst spectrum of GRO J1744–28 was found to be very similar to that of SMC X-1, with a similar photon index of $\sim$1.2 and e-folding energy of $\sim$14 keV (Lewin et al. 1996). One difference is that the SMC X-1 flares lack the post-flare dip which often follows the Type II bursts from GRO J1744–28. The rather gradual rise of the SMC X-1 flares, however, may be responsible for it via offering sufficient time to replenish the material in the accretion disk. This is consistent with the interpretation that the SMC X-1 flares are simple extensions of a normal state with increased accretion rates, which is probably the most compelling argument for attributing the origin of the SMC X-1 flares to an accretion disk instability. Given the difference between GRO J1744–28 and SMC X-1 (i.e., a transient low-mass system versus a persistent high-mass one), we consider that the magnetic field strength, suggested to be comparable for the two sources, is critical to understanding the Type II bursts from GRO J1744–28 and the X-ray flares from SMC X-1.
We note that SMC X-1 may be capable of experiencing a viscous instability, namely the Lightman-Eardley instability (Lightman & Eardley 1974), due to its $\sim$10$^{29}$ G cm$^3$ magnetic moment [@lv97]. The radiation pressure in this case is comparable to the gas pressure around the inner disk radius (i.e., the magnetospheric radius), resulting in a viscous instability with slightly increased mass accretion (e.g., Cannizzo 1996, 1997). Because the instability develops near the inner edge of the accretion disk, it has an advantage in explaining bursts/flares with a short recurrence time. The viscosity parameter ($\alpha$) of the classical $\alpha$-disk (Shakura & Sunyaev 1973) at the transition radius between the “inner region" and the “middle region" around an 1.4 $M_{\odot}$ neutron star is $\alpha \simeq 216 \; t_{\rm vis}^{-3/2} \dot M_{17}$, where $t_{\rm vis}$ is the viscous time scale in seconds and $\dot M_{17}$ is the mass accretion rate in units of 10$^{17}$ g s$^{-1}$. For a typical value $\dot M_{17}$ $\simeq$ 20 for SMC X-1 (e.g., Wojdowski et al. 2000), the viscosity parameter is $\alpha$ $\simeq$ 0.14 and 4.4 when $t_{\rm vis}$ is 1000 and 100 s, respectively. The value 4.4 for $t_{\rm vis}$ = 100 s is somewhat larger than generally expected, $\alpha$ $\lesssim$ 1. However, even larger values ($\alpha$ $\simeq$ 10–100) were obtained for the dwarf nova HT Cas, possibly due to the patch nature of the accretion disk (Vrielmann, Hessman, & Horne 2002). Alternately, the strong magnetic field of SMC X-1 may help increase the viscosity parameter (R. Lovelace 2002, private communication). We need detailed numerical studies to investigate this scenario more thoroughly.
Summary and Conclusions
=======================
Through the analysis of the all publicly available RXTE data toward the X-ray pulsar SMC X-1, we find that the source is highly variable on short time scales ($<$ 1 h), and that Gaussian flares occur over all orbital phases with a $\sim$100 s recurrence time scale. The flares occupy $\sim$3 % of the total observing time, and the flaring activity is proportional to the overall variability of the source. While the pulse peak ratio shows a small systematic change along with the flaring activity, the PDSs, pulse profiles, softness ratios, and X-ray spectra during the flares are very similar to those outside the flares, indicating that the flares are probably extensions of a normal state just with increased accretion rates. This supports the interpretation that the SMC X-1 flares have their origin in an accretion disk instability and the suggestion that it may belong to a distinctive group of “bursting pulsars" with the “bursting pulsar" GRO J1744–28, owing to its $\sim$10$^{11}$ G surface magnetic field. A viscous instability near the inner-edge of the accretion disk might be responsible for the SMC X-1 flares, although detailed studies on this scenario need to be done in the future.
We would like to thank the anonymous referee for the useful comments and suggestions. D.-S. M. acknowledges Akiko Shirakawa, Dong Lai, and Richard Lovelace for their comments. This research has made use of data obtained from the [*High Energy Astrophysics Science Archive Research Center*]{} provided by NASA’s Goddard Space Flight Center. D.-S. M. is supported by NSF grant AST-9986898. S. S. E. is supported in part by an NSF Faculty Early Careeer Development award (NSF-9983830). I. M. W. is supported by the NASA grant NAG-5-8356.
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[lrrr]{} $N_{\rm H}$ (10$^{22}$ cm$^{-2}$) & 3.2(0.9) & 1.6(1.5) & 2.2(0.9)\
$\alpha$ & 1.6(0.3) & 1.3(0.5) & 1.6(0.3)\
$E_{\rm c}$ (keV) & 17.2(7.1) & 6.9(3.2) & 14.3(5.2)\
$E_{\rm f}$ (keV) & 7.8(6.9) & 12.6(10.2) & 14.0(10.6)\
$\chi^2_{\nu}$ & 0.81 & 1.3 & 0.81\
Flux (10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$) & 1.7 & 2.0 & 2.8\
|
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abstract: 'The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g. in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.'
author:
- Andreas Haier and Ilya Molchanov
title: 'Multivariate risk measures in the non-convex setting'
---
Introduction {#sec:introduction}
============
Multivariate financial positions (portfolios) are usually described by vectors in Euclidean space. However, if one aims to take into account possible exchanges between the components of the portfolio, it is necessary to consider the whole set of points in space that may be attained from the original position by allowed exchanges. In other words, considering a multiasset portfolio is indispensable from specifying which transactions may be applied to its components. For instance, if all components of the portfolio $C=(C^{(1)},\dots,C^{(d)})$ represent cash amounts in the same currency and transfers between the components are unrestricted with short-selling permitted, then the attainable positions are all random vectors such that the sum of their components equals the sum of components of $C$. By allowing disposal of assets (e.g., in the form of consumption), we arrive at the half-space $$\Big\{{{\mbf x}}\in{{\mathbb R}}^d:\; \sum_{i=1}^d x^{(i)}\leq \sum_{i=1}^d C^{(i)}\Big\}.$$ In this case and also in the presence of transaction costs not influenced by $C$, the attainable positions are points from $C+{{\mbf K}}$, where ${{\mbf K}}$ is the set of portfolios available at price zero, see [@kab:saf09]. In other situations, possible attainable positions may depend on $C$ in a non-linear way, for instance, when components represent capitals of members of a group and admissible transfers satisfy further restrictions, e.g., requiring that they do not cause insolvency of an otherwise solvent agent, see [@haier:mol:sch15].
In view of the above reasons, it is natural to represent multiasset portfolios as random closed sets. Recall that a *random closed set* ${{\mbf X}}$ is a measurable map from a probability space $(\Omega,{\mathfrak{F}},{\mathbf{P}})$ to the space of closed sets in ${{\mathbb R}}^d$ equipped with the $\sigma$-algebra generated by the Fell topology. In other words, the measurability of ${{\mbf X}}$ means that $\{\omega:\; {{\mbf X}}(\omega)\cap K\neq\emptyset\}\in{\mathfrak{F}}$ for all compact sets $K$ in ${{\mathbb R}}^d$, see [@mo1 Sec. 1.1.1].
A random closed set ${{\mbf X}}$ is said to be *lower* if almost all its realisations are lower sets, that is, for almost all $\omega$, $x\in{{\mbf X}}(\omega)$ and $y\leq x$ coordinatewisely imply that $y\in{{\mbf X}}(\omega)$. A random closed set is said to be *convex* if almost all its realisations are convex. If ${{\mbf X}}$ is a random closed set, then its closed convex hull $\operatorname{\overline{conv}}({{\mbf X}})$ is also a random closed set, see [@mo1 Th. 1.3.25].
For $p\in[1,\infty]$, denote by ${\boldsymbol{L}^{p}}({{\mbf X}})$ the family of $p$-integrable (essentially bounded if $p=\infty$) random vectors $\xi$ such that $\xi\in{{\mbf X}}$ a.s.; such random vectors are called *$p$-integrable selections* of ${{\mbf X}}$. Furthermore, ${\boldsymbol{L}^{0}}({{\mbf X}})$ is the family of all selections of ${{\mbf X}}$; this family is not empty if ${{\mbf X}}$ is a.s. non-empty, see [@mo1 Th. 1.4.1]. A random closed set ${{\mbf X}}$ is called *$p$-integrable* if it admits at least one $p$-integrable selection; it is called *$p$-integrably bounded* if $$\|{{\mbf X}}\|=\sup\{\|x\|:\; x\in{{\mbf X}}\}$$ is a $p$-integrable random variable for $p\in[1,\infty)$. The random closed set ${{\mbf X}}$ is said to be essentially bounded if $\|{{\mbf X}}\|$ is a.s. bounded by a constant.
If ${{\mbf X}}$ is integrable (that is, $1$-integrable), its *selection expectation* is defined by $$\label{eq:5}
{{\mathbf E}}{{\mbf X}}=\operatorname{cl}\{{{\mathbf E}}\xi:\; \xi\in{\boldsymbol{L}^{1}}({{\mbf X}})\},$$ where $\operatorname{cl}(\cdot)$ denotes the topological closure in ${{\mathbb R}}^d$. The closed *Minkowski sum* $${{\mbf X}}+{{\mbf Y}}=\operatorname{cl}\{x+y:\; x\in{{\mbf X}},y\in{{\mbf Y}}\}$$ of two random closed sets ${{\mbf X}}$ and ${{\mbf Y}}$ is also a random closed set. Note that $$-{{\mbf X}}=\{-x:\; x\in{{\mbf X}}\}$$ denotes the reflection of ${{\mbf X}}$ with respect to the origin; this is not the inverse operation to the addition. We refer to [@mo1] for further material concerning random closed sets.
The paper is organised as follows. In Section \[sec:select-risk-meas\] we introduce the *selection risk measure* of possibly non-convex random lower closed sets, thereby generalising the setting of [@haier:mol:sch15] and [@cas:mol14]. Due to the non-convexity, it is not possible to assess the risk by working with half-spaces containing the portfolio, as it is the case in [@ham:hey10; @ham:hey:rud11]. In Section \[sec:fixed-points-expect\] we discuss two basic set-valued risk measures, one based on considering the fixed points of set-valued portfolio, the other is given by the selection expectation of $-{{\mbf X}}$. These two cases correspond to taking the negative essential infimum and the negative expectation as the underlying numerical risk measures. Section \[sec:conv-law-invar\] explores the cases when the selection risk measure takes convex values and is law invariant. The important case of fixed transaction costs is considered in Section \[sec:fixed-trans-costs\]. Finally, Section \[sec:finite-trans-sets\] deals with the case of only a finite set of admissible transactions.
Selection risk measure of non-convex portfolios {#sec:select-risk-meas}
===============================================
Definition {#sec:definition}
----------
Fix $p\in\{0\}\cup [1,\infty]$ and a vector ${\mathbf{r}}(\xi)=({r}_1(\xi^{(1)}),\dots,{r}_d(\xi^{(d)}))$ of monetary ${\boldsymbol{L}^{p}}$-risk measures applied to components of a $p$-integrable random vector $\xi=(\xi^{(1)},\dots,\xi^{(d)})$. We refer to [@delb12] and [@foel:sch04] for the facts concerning risk measures for random variables. Assume that ${\mathbf{r}}(0)=0$ and that all components of ${\mathbf{r}}$ are finite on $p$-integrable random variables. When saying that ${\mathbf{r}}$ is coherent or convex, we mean that all its components are coherent or convex. The convexity or coherency properties will be imposed only when necessary and will be explicitly mentioned.
In many cases below, we consider the following basic numerical risk measures.
1. The negative essential infimum ${r}(\xi)=-\operatorname{essinf}\xi$, which is an ${\boldsymbol{L}^{\infty}}$-risk measure.
2. The negative expectation ${r}(\xi)=-{{\mathbf E}}\xi$, an ${\boldsymbol{L}^{1}}$-risk measure.
3. The Average Value-at-Risk (or Expected Shortfall in the non-atomic case) $${r}(\xi)=- \frac{1}{\alpha} \int_0^{\alpha} F_\xi^{-1}(t) dt.$$ at level $\alpha\in(0,1]$ for $\xi\in{\boldsymbol{L}^{1}}({{\mathbb R}})$, where $F_\xi$ is the cumulative distribution function of $\xi$ and $F_\xi^{-1}$ is the quantile function.
4. The distortion risk measure $$\label{eq:77}
{r}(\xi)=-\int_0^1 F_{\xi}^{-1}(t)d\tilde{g}(t)$$ for $\xi\in{\boldsymbol{L}^{p}}({{\mathbb R}})$, where $g:[0,1]\mapsto [0,1]$ is a (concave) distortion function, $\tilde{g}(t)=1-g(1-t)$ is the dual distortion function, and $p$ is chosen to ensure that the integral is finite.
The *selection risk measure* of a $p$-integrable lower random closed set ${{\mbf X}}$ is defined as $$\label{eq:1}
{\mathsf{R}}({{\mbf X}})=\operatorname{cl}\bigcup_{\xi\in{\boldsymbol{L}^{p}}({{\mbf X}})} ({\mathbf{r}}(\xi)+{{\mathbb R}}_+^d),$$ where the union is taken over all $p$-integrable selections of ${{\mbf X}}$. Thus, $x\in{\mathsf{R}}({{\mbf X}})$ if and only if $\liminf {\mathbf{r}}(\xi_n)\leq x$ for $\xi_n\in{\boldsymbol{L}^{p}}({{\mbf X}})$, $n\geq1$. The inequalities between vectors are always coordinatewise and the lower limit is also taken coordinatewisely. The selection risk measure takes values being upper sets, and can be seen as the primal representation of ${\mathsf{R}}({{\mbf X}})$. A dual representation is not feasible without imposing convexity on ${{\mbf X}}$.
A random set ${{\mbf X}}$ is said to be acceptable if $0\in{\mathsf{R}}({{\mbf X}})$. In other words, ${{\mbf X}}$ is acceptable if ${{\mbf X}}$ contains a sequence of selections whose risk converges to zero. The monetary property of ${\mathbf{r}}$ yields that ${\mathsf{R}}({{\mbf X}})$ is the set of all $x\in{{\mathbb R}}^d$ such that ${{\mbf X}}+x$ is acceptable, that is, $${\mathsf{R}}({{\mbf X}})=\{x:\; {\mathsf{R}}({{\mbf X}}+x)\ni 0\}.$$
Properties of the selection risk measure {#sec:prop-select-risk}
----------------------------------------
The selection risk measure was introduced in [@cas:mol14] for convex ${{\mbf X}}$ and coherent ${\mathbf{r}}$. Some of its properties for non-convex ${{\mbf X}}$ and general monetary ${\mathbf{r}}$ are easy-to-show replica of those known in the convex coherent setting adopted in [@cas:mol14].
\[thr:general\] The selection risk measure satisfies the following properties for $p$-integrable random lower closed sets ${{\mbf X}}$ and ${{\mbf Y}}$.
i) Monotonicity, that is, ${\mathsf{R}}({{\mbf X}})\subseteq {\mathsf{R}}({{\mbf Y}})$ if ${{\mbf X}}\subseteq {{\mbf Y}}$ a.s.
ii) Cash-invariance, that is, ${\mathsf{R}}({{\mbf X}}+a)={\mathsf{R}}({{\mbf X}})-a$ for all deterministic $a\in{{\mathbb R}}^d$.
iii) If ${\mathbf{r}}$ is homogeneous, then ${\mathsf{R}}$ is homogeneous, that is, ${\mathsf{R}}(c{{\mbf X}})=c{\mathsf{R}}({{\mbf X}})$ for all deterministic $c>0$.
iv) If ${\mathbf{r}}$ is convex, then ${\mathsf{R}}$ is convex, that is, $$\label{eq:3}
{\mathsf{R}}(\lambda{{\mbf X}}+(1-\lambda){{\mbf Y}})
\supseteq \lambda{\mathsf{R}}({{\mbf X}})+(1-\lambda){\mathsf{R}}({{\mbf Y}})$$ for all deterministic $\lambda\in[0,1]$.
We prove only the convexity, the rest is straightforward. All elements of the set on the right-hand side of are coordinatewisely larger than or equal to $$\liminf \big(\lambda{\mathbf{r}}(\xi_n)+(1-\lambda){\mathbf{r}}(\eta_n)\big)$$ for $\xi_n\in{\boldsymbol{L}^{p}}({{\mbf X}})$ and $\eta_n\in{\boldsymbol{L}^{p}}({{\mbf Y}})$, $n\geq1$. Then it suffices to note that this convex combination of risks of $\xi$ and $\eta$ dominates ${\mathbf{r}}(\lambda\xi_n+(1-\lambda)\eta_n)$, which is an element of the left-hand side of .
The monotonicity property of ${\mathbf{r}}$ yields that ${\mathsf{R}}(C+{{\mathbb R}}_-^d)={\mathbf{r}}(C)+{{\mathbb R}}_+^d$ for $C\in{\boldsymbol{L}^{p}}({{\mathbb R}}^d)$. The selection risk measure is said to be *coherent* if it is homogeneous and convex; this is the case if ${\mathbf{r}}$ has all coherent components. If ${\mathbf{r}}$ is coherent, $C$ is a $p$-integrable random vector, and ${{\mbf X}}$ is a $p$-integrable random lower closed set, then $$\label{eq:10}
{\mathsf{R}}(C+{{\mbf X}})\supseteq {\mathbf{r}}(C)+{\mathsf{R}}({{\mbf X}}).$$ This is easily seen from choosing $\lambda=1/2$, ${{\mbf Y}}=C+{{\mathbb R}}_-^d$, and using the homogeneity of ${\mathbf{r}}$. Note that the equality in is not guaranteed even if ${{\mbf X}}$ is a deterministic set. Still, in this case, it provides a useful acceptability condition: $C+{{\mbf X}}$ is acceptable if ${\mathbf{r}}(C)+{\mathsf{R}}({{\mbf X}})\ni 0$.
A general set-valued function (not necessarily constructed using selections) defined for $p$-integrable random sets is said to be monotonic, cash invariant, homogeneous or convex if it satisfies the corresponding properties from Theorem \[thr:general\]. The set-valued (selection) risk measure is called *law invariant* if its values on identically distributed random sets coincide.
Choice of selections {#sec:choice-selections}
--------------------
The definition of the selection risk measure involves taking union over all $p$-integrable selections of ${{\mbf X}}$. This family may be very rich even for simple random closed sets. In the following, we discuss general approaches suitable to reduce the family of selections needed to determine the selection risk measure.
With a lower closed set $F$ we associate the set $\partial^+F$ of its *Pareto optimal* points, that is, the set of points $x\in F$ such that $y\geq x$ for $y\in F$ is only possible if $y=x$. If ${{\mbf X}}$ is a random lower closed convex set, then the set $\partial^+{{\mbf X}}$ of Pareto optimal points of ${{\mbf X}}$ is a random closed set, see [@haier:mol:sch15 Lemma 3.1]. In the non-convex case, the cited result establishes that $\partial^+{{\mbf X}}$ is graph measurable, so that its closure $\operatorname{cl}\partial^+{{\mbf X}}$ is a random closed set, see [@lep:mol17 Prop. 2.6]. If $\partial^+{{\mbf X}}$ is closed and $p$-integrable, then it is possible to reduce the union in to selections of $\partial^+{{\mbf X}}$.
A lower random closed set ${{\mbf X}}$ is said to be *quasi-bounded* if $\partial^+{{\mbf X}}$ is essentially bounded; ${{\mbf X}}$ is $p$-integrably quasi-bounded if $\|\partial^+{{\mbf X}}\|$ is $p$-integrable.
Consider $$\label{eq:8}
{{\mbf X}}=F_1\cup\cdots \cup F_m,$$ where $F_1,\dots,F_m$ are deterministic lower *convex closed cones*. For the following result, assume that ${\mathbf{r}}$ is convex law invariant, and the probability space is non-atomic. In this case, ${\mathbf{r}}$ satisfies the dilatation monotonicity property, that is, ${\mathbf{r}}(\xi)$ dominates coordinatewisely the risk of a conditional expectation of $\xi$, see [@foel:sch04 Cor. 4.59] and [@leit04].
\[prop:det-set\] If ${{\mbf X}}$ is a deterministic set given by , then it is possible to reduce the union in to selections $\xi=\sum_{i=1}^m x_i{\mathbf{1}}_{A_i}$ for deterministic $x_i\in F_i$, $i=1,\dots,m$, and partitions ${\mathcal{A}}=\{A_1,\dots,A_m\}$ of the probability space.
Consider $\xi=\sum \eta_i{\mathbf{1}}_{A_i}$ for $\eta_i\in{\boldsymbol{L}^{p}}(F_i)$, $i=1,\dots,m$. By the dilatation monotonicity, ${\mathbf{r}}(\xi)$ dominates the risk of the conditional expectation of $\xi$ given ${\mathcal{A}}$. Thus, it is possible to replace $\eta_i$ by its conditional expectation, which is also a point in $F_i$.
In the convex setting, if ${{\mbf X}}$ is the sum of $C$ and a convex closed set $F$, then the union in can be reduced to the selections that are measurable with respect to the $\sigma$-algebra generated by $C$.
Fixed points and the expectation {#sec:fixed-points-expect}
================================
For a random closed set ${{\mbf X}}$, $$F_{{\mbf X}}=\{x:\; {{\mathbf{P}}\{x\in{{\mbf X}}\}}=1\}$$ denotes the set of its *fixed points*. The set $F_{{\mbf X}}$ is a lower closed set if ${{\mbf X}}$ is a lower closed set, it is convex if ${{\mbf X}}$ is convex.
\[prop:ess-inf\] Let ${{\mbf X}}$ be a $p$-integrable random lower closed set. For the selection risk measure generated by any monetary risk measure ${\mathbf{r}}$, we have $$\label{eq:6}
-F_{{\mbf X}}\subseteq {\mathsf{R}}({{\mbf X}}).$$ If all components of ${\mathbf{r}}$ are the negative of the essential infimum, then ${\mathsf{R}}({{\mbf X}})$ equals the set of fixed points of $-{{\mbf X}}$.
By taking constant selections $\xi=x\in F_{{\mbf X}}$ in and using the fact that ${\mathbf{r}}(x)=-x$, we see that holds.
If ${\boldsymbol{L}^{\infty}}({{\mbf X}})\neq\emptyset$, then $F_{{\mbf X}}\neq\emptyset$, since ${{\mbf X}}$ is a lower set. Choosing ${\mathbf{r}}$ with all components being negative of the essential infima, it is easily seen that ${{\mbf X}}$ is acceptable if it has a selection with all a.s. non-negative components. In this case, $0\in{{\mbf X}}$ a.s., whence $0\in F_{{\mbf X}}$. Note also that $F_{-{{\mbf X}}}=-F_{{\mbf X}}$.
The set of fixed points is a coherent selection risk measure, which is law invariant and not necessarily convex-valued.
\[ex:fx-non-convex\] The convex hull of $F_{{\mbf X}}$ is a (possibly, strict) subset of the set of fixed points of $\operatorname{\overline{conv}}({{\mbf X}})$. Let ${{\mbf X}}$ be a random set in ${{\mathbb R}}^2$ which equally likely take values $\{(-a,a),(a,-a)\}+{{\mathbb R}}_-^2$ and $\{(-b,b),(b,-b)\}+{{\mathbb R}}_-^2$ for $0<a<b$. Then $F_{{\mbf X}}=\{(-b,a),(a,-b)\}+{{\mathbb R}}_-^2$, while the set of fixed points of $\operatorname{\overline{conv}}({{\mbf X}})$ is the sum of the segment with end-points $(-a,a),(a,-a)$ and ${{\mathbb R}}_-^2$.
The set of fixed points appears also in the following context. Let $\Omega=\{\omega_1,\dots,\omega_n\}$ be a finite probability space, and let all components of ${\mathbf{r}}$ be the Average Value-at-Risk at level $\alpha\leq{\mathbf{P}}(\{\omega_i\})$, $i=1,\dots,n$. Then $${\mathsf{R}}({{\mbf X}})=-F_{{\mbf X}}=-\bigcap_{i=1}^{n} {{\mbf X}}(\omega_i).$$ Indeed, since ${\mathbf{P}}(\{\omega_i\})\geq \alpha$ for all $i$, we have ${r}(\xi^{(j)})=-\min\{\xi^{(j)}(\omega_i),i=1,\dots,n\}$ for any $\xi\in{\boldsymbol{L}^{1}}({{\mbf X}})$. Because each ${{\mbf X}}(\omega_i)$ is a lower set, we have $-{\mathbf{r}}(\xi)\in {{\mbf X}}(\omega_i)$ for all $i$. To show the reverse inclusion, assume that $x\in F_{{\mbf X}}$. Then $\xi=x$ is a deterministic selection of ${{\mbf X}}$, whence $-x={\mathbf{r}}(\xi)\in
{\mathsf{R}}({{\mbf X}})$.
If $p=1$ and ${\mathbf{r}}(\xi)=-{{\mathbf E}}\xi$ is the negative expectation of $\xi$, then ${\mathsf{R}}({{\mbf X}})$ becomes the selection expectation of $(-{{\mbf X}})$. Note that ${\mathsf{R}}({{\mbf X}})=-{{\mathbf E}}{{\mbf X}}$ is a coherent selection risk measure, which is law invariant on convex random sets, but may be not law invariant on non-convex ones. Indeed, if the non-convex deterministic set $F$ is considered a random closed set defined on the trivial probability space, then ${{\mathbf E}}F=F$, while ${{\mathbf E}}F=\operatorname{\overline{conv}}(F)$ if the underlying probability space is non-atomic, see [@mo1 Th. 2.1.26].
It might be tempting to define a set-valued risk measure by taking intersection of expected random sets with respect to varying probability measures. This would correspond to the construction of a convex function by taking the supremum of linear ones. However, taking expectation results in convex values for the risk measure if the probability space is non-atomic; otherwise, it depends on the atomic structure of the space. Furthermore, even in the convex setting, such a construction might not correspond to the existence of an acceptable selection from ${{\mbf X}}$, as the following remark shows.
For any family ${\mathcal{Z}}\subset{\boldsymbol{L}^{q}}({{\mathbb R}}_+^d)$ such that ${{\mathbf E}}\zeta=1$ for all $\zeta\in{\mathcal{Z}}$, define $$\label{eq:4}
{\mathsf{R}}_{\mathcal{Z}}({{\mbf X}})=\bigcap_{\zeta\in{\mathcal{Z}}} {{\mathbf E}}(-\zeta{{\mbf X}}),$$ where $\zeta{{\mbf X}}=\{\zeta x:\; x\in{{\mbf X}}\}$. Note that we use vector notation, e.g., ${{\mathbf E}}\zeta=1$ means that all components of $\zeta$ have mean $1$, and $$\zeta\xi=(\zeta^{(1)}\xi^{(1)},\dots,\zeta^{(d)}\xi^{(d)})$$ is the coordinatewise product of $\zeta$ and $\xi$. The so defined ${\mathsf{R}}_{\mathcal{Z}}(\cdot)$ satisfies all properties from Theorem \[thr:general\]. However, ${\mathsf{R}}_{\mathcal{Z}}$ in general is not a selection risk measure. Indeed, by letting ${{\mbf X}}=\xi+{{\mathbb R}}_-^d$, we see that the corresponding coherent vector-valued risk measure is given by $${\mathbf{r}}(\xi)=\sup_{\zeta\in{\mathcal{Z}}} {{\mathbf E}}(-\zeta\xi),\quad
\xi\in{\boldsymbol{L}^{p}}({{\mathbb R}}^d).$$ Assume that $\partial^+{{\mbf X}}$ is $p$-integrably bounded, so that ${{\mathbf E}}(\zeta{{\mbf X}})$ is closed for all $\zeta\in{\boldsymbol{L}^{q}}$. Then $0\in{\mathsf{R}}_{\mathcal{Z}}({{\mbf X}})$ if and only if $0\in{{\mathbf E}}(-\zeta{{\mbf X}})$ for all $\zeta\in{\mathcal{Z}}$, equivalently, for each $\zeta\in{\mathcal{Z}}$ there is $\xi_\zeta\in{\boldsymbol{L}^{p}}(\partial^+{{\mbf X}})$ such that ${{\mathbf E}}(-\zeta\xi_\zeta)\leq
0$. Since these selections $\xi_\zeta$ may be different for different $\zeta$, we cannot infer that ${{\mbf X}}$ is acceptable with respect to a selection risk measure. Indeed, the acceptability of ${{\mbf X}}$ requires the existence of a *single* selection $\xi\in{\boldsymbol{L}^{p}}({{\mbf X}})$ such that ${{\mathbf E}}(-\zeta\xi)\leq 0$ for all $\zeta$. Thus, ${\mathsf{R}}_{\mathcal{Z}}$ is an example of a coherent set-valued risk measure, which, however, is not necessarily a selection one. The acceptability of ${{\mbf X}}$ under ${\mathsf{R}}_{\mathcal{Z}}$ does not guarantee the existence of an acceptable selection of ${{\mbf X}}$. Furthermore, this risk measure does not distinguish between ${{\mbf X}}$ and its convex hull.
Convexity and law invariance {#sec:conv-law-invar}
============================
The monotonicity property yields that ${\mathsf{R}}({{\mbf X}})$ is a subset of ${\mathsf{R}}(\operatorname{\overline{conv}}({{\mbf X}}))$. It is well known that the selection expectation of an integrable random closed set is convex if the underlying probability space is non-atomic, see [@mo1 Th. 2.1.26]. This result follows from Lyapunov’s theorem on ranges of vector-valued measures. The same holds for selection risk measures of convex random sets, if the underlying risk measure ${\mathbf{r}}$ is convex, see [@cas:mol14 Th. 3.4]. This is however not the case for non-convex arguments, see Example \[ex:fx-non-convex\] and Section \[sec:fixed-trans-costs-1\].
Still, in some cases ${\mathsf{R}}({{\mbf X}})$ is convex even for non-convex ${{\mbf X}}$. Assume that $p\in[1,\infty]$, and the components of ${\mathbf{r}}=({r}_1,\dots,{r}_d)$ are $\sigma({\boldsymbol{L}^{p}},{\boldsymbol{L}^{q}})$-lower semicontinuous convex risk measures, so that $$\label{eq:2}
{r}_i(\xi)=\sup_{\zeta\in{\boldsymbol{L}^{q}}({{\mathbb R}}_+), {{\mathbf E}}\zeta=1}
\Big({{\mathbf E}}(-\zeta\xi)-\alpha_i(\zeta)\Big), \quad \xi\in{\boldsymbol{L}^{p}}({{\mathbb R}}),
\quad i=1,\dots,d,$$ where $\alpha_i:{\boldsymbol{L}^{q}}({{\mathbb R}}_+)\mapsto(-\infty,\infty]$, $i=1,\dots,d$, are the penalty functions corresponding to the components of ${\mathbf{r}}$. The following result generalises Lyapunov’s theorem in the sublinear setting, see also [@sag09].
\[thr:convexify\] Let $(\Omega,{\mathfrak{F}},{\mathbf{P}})$ be a non-atomic probability space, and let the components of ${\mathbf{r}}$ admit representation with $\alpha_1(\zeta),\dots,\alpha_d(\zeta)$ being all infinite unless $\zeta$ belongs to a finite family from ${\boldsymbol{L}^{q}}({{\mathbb R}})$. Then ${\mathsf{R}}({{\mbf X}})$ is convex.
We need to show that for two selections $\xi',\xi''\in{\boldsymbol{L}^{p}}({{\mbf X}})$ and $\lambda \in [0,1]$, there is $\xi\in{\boldsymbol{L}^{p}}({{\mbf X}})$ such that ${\mathbf{r}}(\xi)\leq\lambda{\mathbf{r}}(\xi')+(1-\lambda){\mathbf{r}}(\xi'')$. In view of the convexity of ${\mathbf{r}}$, it suffices to ensure that $${\mathbf{r}}(\xi)\leq {\mathbf{r}}(\lambda\xi'+(1-\lambda)\xi'').$$
Assume that all components of $\alpha(\zeta)$ are infinite for $\zeta$ outside a finite set ${\mathcal{Z}}=\{\zeta_1,\dots,\zeta_m\}$. Consider the mapping which assigns to each measurable subset $A\subseteq \Omega$ the vector $$\upsilon(A)=({{\mathbf E}}(-{\mathbf{1}}_A\zeta_1\xi'),\dots,{{\mathbf E}}(-{\mathbf{1}}_A\zeta_m\xi'),
{{\mathbf E}}(-{\mathbf{1}}_A\zeta_1\xi''),\dots,{{\mathbf E}}(-{\mathbf{1}}_A\zeta_m\xi''))\in {{\mathbb R}}^{2md}.$$ It is easily verified that this map is a vector-valued measure. By Lyapunov’s theorem, its image is closed convex, hence there is a measurable subset $A\subseteq \Omega$ such that $$\upsilon(A)=\lambda
\upsilon(\Omega)+(1-\lambda)\upsilon(\emptyset)
=\lambda\upsilon(\Omega).$$ Then ${{\mathbf E}}(-{\mathbf{1}}_A\zeta_i\xi')=\lambda{{\mathbf E}}(-\zeta_i\xi')$ and ${{\mathbf E}}(-{\mathbf{1}}_A\zeta_i\xi'')=\lambda{{\mathbf E}}(-\zeta_i\xi'')$ for all $i$. Hence, $${{\mathbf E}}(-\lambda\zeta_i\xi'-(1-\lambda)\zeta_i\xi'')
={{\mathbf E}}(-\zeta_i(\xi''+{\mathbf{1}}_A(\xi'-\xi'')))={{\mathbf E}}(-\zeta_i\xi),$$ where $\xi=\xi'{\mathbf{1}}_A+\xi''{\mathbf{1}}_{A^c}$ is a selection of ${{\mbf X}}$. Therefore, $${{\mathbf E}}(-\zeta_i\xi)-\alpha(\zeta_i)
={{\mathbf E}}(-\zeta_i(\lambda\xi'+(1-\lambda)\xi''))-\alpha(\zeta_i)
\leq {\mathbf{r}}(\lambda\xi'+(1-\lambda)\xi'')$$ for all $i$, so $\xi$ is indeed the required selection.
For a *deterministic* lower closed set $F$, the selection risk measure ${\mathsf{R}}(F)$ is not always equal to $(-F)$. For instance, this is not the case in the framework of Theorem \[thr:convexify\], or in the context of fixed transaction costs in Section \[sec:fixed-trans-costs-1\]. The set $F$ is said to be ${\mathbf{r}}$-convex (or risk-convex for ${\mathbf{r}}$), if with any $x_1,x_2\in F$ and any $A\subseteq \Omega$ we also have $-{\mathbf{r}}({\mathbf{1}}_A x_1+
{\mathbf{1}}_{A^c} x_2)\in F$. Then $F$ is ${\mathbf{r}}$-convex if and only if ${\mathsf{R}}(F)=-F$. It is easy to see that the intersection of risk convex sets is also risk convex. If ${\mathbf{r}}$ is the negative expectation and the probability space is non-atomic, the risk convexity corresponds to the usual notion of convexity. If ${\mathbf{r}}$ is the negative essential infimum, each lower set is risk convex.
Consider ${{\mbf X}}=\{\xi,\eta\}+{{\mathbb R}}_-^d$. Then ${\mathsf{R}}({{\mbf X}})$ is convex if and only if, for each $t\in(0,1)$, there exists $A\in{\mathfrak{F}}$ such that $$t{\mathbf{r}}(\xi)+(1-t){\mathbf{r}}(\eta)\geq
{\mathbf{r}}(\xi{\mathbf{1}}_A+\eta{\mathbf{1}}_{A^c}).$$
The families of selections of random sets are not necessarily law invariant, i.e. they can differ for two random sets having the same distribution, see [@mo1 Sec. 1.4.1]. This could result in the selection risk measure ${\mathsf{R}}$ not being law invariant. Still, the law invariance of ${\mathbf{r}}$ yields the law invariance of the selection risk measure for convex ${{\mbf X}}$, see [@cas:mol14]. Below we consider the case of a possibly non-convex ${{\mbf X}}$.
The risk measure ${\mathbf{r}}$ is said to be *Lebesgue continuous* if it is continuous on a.s. convergent uniformly $p$-integrably bounded sequences of random vectors.
\[thr:law-inv\] Assume that the probability space is non-atomic and that ${\mathbf{r}}$ is a Lebesgue continuous risk measure. Then the selection risk measure ${\mathsf{R}}({{\mbf X}})$ is law invariant on $p$-integrably quasi-bounded portfolios.
Let ${{\mbf X}}$ and ${{\mbf X}}'$ share the same distribution, so that the corresponding closures ${{\mbf Y}}=\operatorname{cl}\partial^+{{\mbf X}}$ and ${{\mbf Y}}'=\operatorname{cl}\partial^+{{\mbf X}}'$ of their Pareto optimal points are $p$-integrably bounded and share the same distribution. By the Lebesgue property and the $p$-integrable boundedness of ${{\mbf Y}}$ and ${{\mbf Y}}'$, it is possible to take the union in over $p$-integrable selections of ${{\mbf Y}}$ and ${{\mbf Y}}'$ respectively.
Let $x\in{\mathbf{r}}(\xi)+{{\mathbb R}}_+^d$ for $\xi\in{\boldsymbol{L}^{p}}({{\mbf Y}})$. Since the weak closures of ${\boldsymbol{L}^{0}}({{\mbf Y}})$ and ${\boldsymbol{L}^{0}}({{\mbf Y}}')$ coincide (see [@mo1 Th. 1.4.3]), there is a sequence $\eta_n\in{\boldsymbol{L}^{p}}({{\mbf Y}}')$ converging weakly to $\xi$. Then $\|\eta_n\|\leq \|{{\mbf Y}}'\|$, and the latter random variable is integrable. Thus, $\{\eta_n,n\geq1\}$ is relatively compact in ${\boldsymbol{L}^{1}}({{\mathbb R}}^d)$. By passing to a subsequence, it is possible to assume that $\eta_{n_k}\to\xi$ almost surely.
The Lebesgue continuity property yields that ${\mathbf{r}}(\eta_{n_k})\to {\mathbf{r}}(\xi)$. Thus, ${\mathbf{r}}(\xi)\in{\mathsf{R}}({{\mbf Y}}')$, since the latter set is closed. Finally, $x\in{\mathsf{R}}({{\mbf X}}')$, since the latter set is upper.
It is known that each ${\boldsymbol{L}^{p}}$-risk measure with finite values and $p\in[1,\infty)$ is Lebesgue continuous, see [@kain:rues09]. For $p=\infty$, [@jouin:sch:touz06 Thms. 2.4, 5.2] provide equivalent formulations of the Lebesgue continuity property for convex risk measures. We give below another criterion.
\[prop:leb-p\] Assume that ${\mathbf{r}}$ is a coherent ${\boldsymbol{L}^{\infty}}$-risk measure such that $${\mathbf{r}}(\xi)=\sup_{\zeta\in{\mathcal{Z}}} {{\mathbf E}}(-\zeta\xi),$$ where ${\mathcal{Z}}$ is a uniformly integrable subset of ${\boldsymbol{L}^{1}}({{\mathbb R}}_+^d)$. Then ${\mathbf{r}}$ is Lebesgue continuous.
Assume that $\xi_n\to\xi$ a.s. and $\|\xi_n\|\leq c$ a.s. for all $n$ and $c>0$. By Egorov’s theorem, for each ${\varepsilon}>0$, there is an event $A$ of probability at most ${\varepsilon}$ such that $\xi_n\to\xi$ uniformly on the complement $A^c$ of $A$.
Using the fact that the absolute value of the difference of two suprema is bounded by the suprema of the absolute values of the differences, we have $$\|{\mathbf{r}}(\xi_n)-{\mathbf{r}}(\xi)\|
\leq \sup_{\zeta\in{\mathcal{Z}}} \|{{\mathbf E}}(-\zeta(\xi_n-\xi)\|
\leq \sup_{\zeta\in{\mathcal{Z}}}{{\mathbf E}}\|\zeta\| \sup_{\omega\notin
A} \|\xi_n(\omega)-\xi(\omega)\|+2c\|{{\mathbf E}}(-\zeta{\mathbf{1}}_A)\|.$$ The first term on the right-hand side converges to zero by the uniform convergence on $A^c$, while the second converges to zero by the uniform integrability of ${\mathcal{Z}}$.
Fixed transaction costs {#sec:fixed-trans-costs}
=======================
Bounds on the selection risk measure {#sec:general-properties}
------------------------------------
Assume that the components of $C$ represent the same currency and transfers are not restricted, but whenever a transfer is made, a fixed cost ${\varkappa}>0$ is incurred. If $C$ is the capital position, then the corresponding set of attainable positions is given by ${{\mbf X}}=C+{I}_{\varkappa}$ with non-convex set $${I}_{\varkappa}={{\mathbb R}}_-^d\cup H_{-{\varkappa}}$$ of portfolios available at price zero, where $$H_t=\{x\in{{\mathbb R}}^d:\; \sum_{i=1}^d x_i\leq t\}, \quad t\in{{\mathbb R}}.$$ The following bounds for the selection risk measure of $C+{I}_{\varkappa}$ are straightforward.
\[prop:bounds\] We have $$\label{eq:7}
({\mathbf{r}}(C)-{I}_{\varkappa})\cup {\mathsf{R}}(C+H_{-{\varkappa}})
\; \subseteq\; {\mathsf{R}}(C+{I}_{\varkappa})\;\subseteq\; {\mathsf{R}}(C+H_0).$$
The first inclusion follows from the fact that $C+x$ is a selection of $C+{I}_{\varkappa}$ for all deterministic $x\in{I}_{\varkappa}$ and that $H_{-{\varkappa}}\subset{I}_{\varkappa}$. The second inclusion holds, since ${I}_{\varkappa}\subset H_0$.
\[ex:genC\] The inclusion on the left-hand side of can be strict. Let $d=2$, and let $C$ be $(-1,1)$ with probability $\alpha$ and $(0,0)$ otherwise. For any $0\leq \beta \leq \alpha$, we can define a selection $\xi\in{I}_{\varkappa}$ such that $C+\xi$ equals $(-{\varkappa},0)$ with probability $\beta$, $(-1,1)$ with probability $\alpha-\beta$, and $(0,0)$ with probability $1-\alpha$. Taking the risk measure of such selections shows that ${\mathsf{R}}(C+{I}_{\varkappa})$ contains all points on the segments with end-points $(1,0)$ and $({\varkappa},0)$.
The following result provides rather simple bounds on the selection risk measure in case of fixed transaction costs.
i) If ${\varkappa}_1\leq {\varkappa}_2$ and $C_1\geq C_2$ componentwisely, then $${\mathsf{R}}(C_1+{I}_{{\varkappa}_1})\supseteq
{\mathsf{R}}(C_2+{I}_{{\varkappa}_2}).$$
ii) If ${\mathbf{r}}$ is subadditive, then $${\mathsf{R}}(C_1+C_2+{I}_{\varkappa})\supseteq {\mathsf{R}}(C_1+{I}_{{\varkappa}_1}) +
{\mathsf{R}}(C_2+{I}_{{\varkappa}_2})$$ whenever ${\varkappa}\leq \min({\varkappa}_1,{\varkappa}_2)$.
i\) Note that ${I}_{{\varkappa}_1}\supseteq {I}_{{\varkappa}_2}$ for ${\varkappa}_1
\leq {\varkappa}_2$, and $$C_1+{I}_{{\varkappa}_1}\supseteq C_1+{I}_{{\varkappa}_2}\supseteq C_2+{I}_{{\varkappa}_2}.$$
ii\) follows from ${I}_{{\varkappa}_1}+{I}_{{\varkappa}_2} \subseteq
{I}_{\varkappa}$ and the monotonicity of the selection risk measure.
The following result identifies the selection risk measure of $C+H_t$ in some cases in terms of the risk of the total payoff $$D=C^{(1)}+\cdots+C^{(d)}.$$ If ${\mathbf{r}}$ is coherent with all identical components, it is easy to see that $C+H_t$ is acceptable if and only if $D$ is acceptable. The following result concerns the case, when all but one components of ${\mathbf{r}}$ are identical.
\[prop:Ht\]
i) If all the components of ${\mathbf{r}}$ are identical convex risk measures ${r}$, then $${\mathsf{R}}(C+H_t)=-H_{t-d{r}(D/d)}.$$
ii) If one of the components of ${\mathbf{r}}$ is the negative essential infimum and all others are identical convex risk measures ${r}$, then $${\mathsf{R}}(C+H_t)=-H_{t-(d-1){r}(\frac{D}{d-1})}.$$
iii) If one of components of ${\mathbf{r}}$ is the negative expectation and all others are identical convex risk measures ${r}$ such that ${r}(\xi)\geq -{{\mathbf E}}\xi$ for all $\xi\in{\boldsymbol{L}^{1}}({{\mathbb R}})$, then $${\mathsf{R}}(C+H_t)=-H_{t-{{\mathbf E}}D}.$$
By cash-invariance, it is possible to asssume that $t=0$. The statement i) is shown in [@haier:mol:sch15 Th. 5.1].
ii\) Assume that the first component of ${\mathbf{r}}$ is the negative essential infimum. Note that $0\in{\mathsf{R}}(C+H_0)$ if and only if there is a selection $\xi$ such that $\sum_{i=1}^{d} \xi^{(i)}\leq 0$, $C^{(1)}+\xi^{(1)}\geq 0$ a.s. and ${r}(C^{(i)}+\xi^{(i)})\leq 0$ for $i=2,\dots, d$. By convexity and monotonicity of ${r}$, $$\begin{aligned}
\label{eq:diagonal}
{r}\big(\frac{D}{d-1}\big)
={r}\Big(\frac{1}{d-1}\sum_{i=1}^{d} C^{(i)}\Big)
&\leq{r}\Big(\frac{1}{d-1}\sum_{i=2}^{d} (C^{(i)}+\xi^{(i)})\Big)\\
&\leq \sum_{i=2}^{d} \frac{1}{d-1}{r}(C^{(i)}+\xi^{(i)}).
\end{aligned}$$ Hence, if ${r}(C^{(i)}+\xi^{(i)})\leq 0$ for all $i=2,\dots,d$, then $0\in
-H_{-(d-1){r}(\frac{D}{d-1})}$. On the other hand, if ${r}(\frac{D}{d-1})\leq 0$, then letting $\xi^{(1)}=-C^{(1)}$ and $\xi^{(i)}=-C^{(i)}+D/(d-1)$, $i=2,\dots,d$, yields a selection $\xi$ of $C+H_0$ such that ${\mathbf{r}}(C+\xi)\leq 0$.
iii\) If $0\in {\mathsf{R}}(C+H_0)$, then there is $\xi$ such that ${{\mathbf E}}(C^{(1)}+\xi^{(1)})\geq 0$ and ${r}(C^{(i)}+\xi^{(i)})\leq 0$, $i=2,\dots,d$. Denote $\eta=D-C^{(1)}-\xi^{(1)}$. Since $\sum_{i=2}^{d}\xi^{(i)}\leq -\xi^{(1)}$, we have $$\sum_{i=2}^{d} (C^{(i)}+\xi^{(i)})
= D-C^{(1)}+ \sum_{i=2}^{d} \xi^{(i)}
\leq \eta.$$ Thus, $$\label{eq:diagonal2}
{r}\big(\eta/(d-1)\big)
\leq \frac{1}{d-1}
{r}\big(\sum_{i=2}^{d} (C^{(i)}+\xi^{(i)})\big)
\leq \frac{1}{d-1}\sum_{i=2}^{d} {r}(C^{(i)}+\xi^{(i)}) \leq 0.$$ Note that ${{\mathbf E}}(C^{(1)}+\xi^{(1)})\geq 0$ is equivalent to ${{\mathbf E}}\eta\leq
{{\mathbf E}}D$. Inequality yields that $-{{\mathbf E}}\eta\leq{r}(\eta)\leq 0$. Therefore, $0\leq {{\mathbf E}}\eta\leq {{\mathbf E}}D$ as desired.
If ${{\mathbf E}}D\geq 0$, define a selection of $C+H_0$ by letting $\xi^{(1)}=-C^{(1)}+D$ and $\xi^{(i)}=-C^{(i)}$, $i=2,\dots,d$. Then ${{\mathbf E}}(C^{(1)}+\xi^{(1)})\geq 0$ and $C^{(i)}+\xi^{(i)}=0$ for $i=2,\ldots,d$, whence $0\in {\mathsf{R}}(C+H_0)$.
Fixed transaction costs in case $C=0$ {#sec:fixed-trans-costs-1}
-------------------------------------
If $C=0$, then the portfolio ${{\mbf X}}=C+{I}_{\varkappa}={I}_{\varkappa}$ is deterministic. However, in the non-convex case, ${\mathsf{R}}({I}_{\varkappa})$ may be a strict superset of $(-{I}_{\varkappa})$. For instance, this happens in the context of Theorem \[thr:convexify\] when ${\mathsf{R}}({I}_{\varkappa})=-H_0=\operatorname{\overline{conv}}(-{I}_{\varkappa})$.
In the following assume that ${\mathbf{r}}$ is a coherent risk measure and $d=2$. By Proposition \[prop:det-set\], it suffices to consider selections $\xi=(x,y){\mathbf{1}}_{A}$ satisfying $x+y=-{\varkappa}$ with $(x,y)\notin{{\mathbb R}}_-^2$. If $x\geq 0$ and so $y\leq 0$, then $${\mathbf{r}}(\xi)=(x{r}_1({\mathbf{1}}_A),-y{r}_2(-{\mathbf{1}}_A)).$$ If $x<0$, then $${\mathbf{r}}(\xi)=(-x{r}_1(-{\mathbf{1}}_A),y{r}_2({\mathbf{1}}_A)).$$ Thus, the risk of ${I}_{\varkappa}$ is determined by the set $$\begin{aligned}
B_{\mathbf{r}}=\{({r}_1({\mathbf{1}}_A),{r}_2(-{\mathbf{1}}_A)):\; A\in{\mathfrak{F}}\},\end{aligned}$$ where ${\mathbf{P}}(A)=\beta$ varies between $0$ and $1$. Then $$\label{eq:88}
{\mathsf{R}}({{\mbf X}})=\bigcup_{t\geq0, (b^{(1)},b^{(2)})\in B_{\mathbf{r}}}
\big\{(tb^{(1)},(-{\varkappa}-t)b^{(2)}),
(tb^{(2)},(t-{\varkappa})b^{(1)})\big\}+{{\mathbb R}}_+^2.$$
\[ex:c-zero\] Let $d=2$, and let the both components of ${\mathbf{r}}=({r},{r})$ be the Average Value-at-Risk at level $\alpha$. If ${\mathbf{P}}(A)=\beta$, then $$({r}({\mathbf{1}}_A),{r}(-{\mathbf{1}}_A))=
\begin{cases}
(0,\beta/\alpha),& \beta\leq \min\big(\alpha,1-\alpha),\\
(0,1),& \alpha<\beta\leq 1-\alpha,\;
\alpha\leq 1/2,\\
(-1+(1-\beta)/\alpha,\beta/\alpha),&
1-\alpha<\beta\leq \alpha,\;
\alpha> 1/2,\\
(-1+(1-\beta)/\alpha,1),& \max(\alpha,1-\alpha)<\beta\leq1.
\end{cases}$$ Thus, if $\alpha\leq 1/2$, then $B_{\mathbf{r}}$ is the union of two segments $[(0,0),(0,1)]$ and $[(0,1),(-1,1)]$ and it does not depend on $\alpha$. In this case, yields that ${\mathsf{R}}({I}_{\varkappa})=-{I}_{\varkappa}$.
Assume now that $\alpha>1/2$. Then $B_{\mathbf{r}}$ is the line that joins the points $(0,0)$, $(0,1/\alpha-1)$, $(1/\alpha-2,1)$ and $(-1,1)$. Only the middle segment differs from the case $\alpha\leq
1/2$. If $t>0$, then the points $$\{(tb^{(1)},(t-{\varkappa})b^{(2)}):\;
(b^{(1)},b^{(2)})\in B_{\mathbf{r}}\}$$ constitute the segment with the end-points $(0,(t-{\varkappa})(1/\alpha-1))$ and $(t(1/\alpha-2),t-{\varkappa})$. A calculation of the lower envelope of these segments yields that $$\begin{gathered}
{\mathsf{R}}({I}_{\varkappa})=\Big\{(-x,y):\; x\geq 0,
y\geq\min\Big({\varkappa}+x,\big(\sqrt{x}+\sqrt{{\varkappa}(1/\alpha-1)}\big)^2\Big)\Big\}\\
\bigcup
\Big\{(x,-y):\; y\geq 0,
x\geq\min\Big({\varkappa}+y,\big(\sqrt{y}+\sqrt{{\varkappa}(1/\alpha-1)}\big)^2\Big)\Big\}.
\end{gathered}$$ Figure \[fig:1\] shows the risk of ${I}_{\varkappa}$ for $\alpha=0.75$. This set increases as $\alpha$ grows and becomes $\operatorname{\overline{conv}}(-{I}_{\varkappa})$ if $\alpha=1$.
Finite sets of admissible transactions {#sec:finite-trans-sets}
======================================
We consider another special case when the selection risk measure of a non-convex set can be calculated explicitly. Assume that possible transactions are restricted to belong to a finite deterministic set $M$ in ${{\mathbb R}}^d$, that is, $${{\mbf X}}=C+M+{{\mathbb R}}_-^d.$$ Let ${\mathbf{r}}$ have all components ${r}$ being the distortion risk measure with distortion function $g$. Since the analytical calculation of ${\mathsf{R}}({{\mbf X}})$ is not feasible, it is possible to use to arrive at the bound $${\mathsf{R}}(C+M+{{\mathbb R}}_-^d)\supseteq {\mathbf{r}}(C)+{\mathsf{R}}(M+{{\mathbb R}}_-^d).$$ In the following we determine the last term on the right-hand side in dimension $d=2$.
\[ex:two-point\] Consider the case of a two-point set $M$. By translating, it is always possible to assume that $0\in M$. If $M$ consists of two points $(0,0)$ and $(x,y)$ with $xy<0$, then ${\mathsf{R}}({{\mbf X}})$ is determined by the set of values ${\mathbf{r}}((x,y){\mathbf{1}}_A)$ for all $A\in{\mathfrak{F}}$. Without loss of generality assume that $x>0$ and $y<0$. Since ${r}({\mathbf{1}}_A)=-g(\beta)$ and ${r}(-{\mathbf{1}}_A)=1-g(1-\beta)=\tilde{g}(\beta)$ if ${\mathbf{P}}(A)=\beta$, we have $${\mathsf{R}}(M+{{\mathbb R}}_-^2)=\bigcup_{\beta\in[0,1]}
\Big(-g(\beta)x,(g(1-\beta)-1)y\Big)+{{\mathbb R}}_+^2.$$
Let $M=\{(x_1,y_1),(x_2,y_2),(x_3,y_3)\}$ consist of three points, and assume that $x_1<x_2=0<x_3$ and $y_1>y_2=0>y_3$. In this case, possible selections can be either two-points-selections of two of these three points (in this case the risk is calculated as in Example \[ex:two-point\]), and three point selection attaining all three points with positive probabilities $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1+\alpha_2+\alpha_3=1$. The risk of the three-point selection can be directly calculated, so that $${\mathsf{R}}(M+{{\mathbb R}}_-^2)=\bigcup_{\alpha_1+\alpha_3\leq 1,\alpha_1,\alpha_3\geq0}
\Big(-x_1\tilde{g}(\alpha_1)-x_3g(\alpha_3),
-y_1g(\alpha_1)-y_3\tilde{g}(\alpha_3)\Big)+{{\mathbb R}}_+^2.$$
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---
author:
- |
Ryosuke Kurihara$^1$ [^1], Keisuke Mitsumoto$^1$, Mitsuhiro Akatsu$^2$, Yuichi Nemoto$^{1, 5}$,\
Terutaka Goto$^{1, 5}$ [^2], Yoshiaki Kobayashi$^{3, 5}$, and Masatoshi Sato$^{3, 4, 5}$ [^3]
title: Critical Slowing Down of Quadrupole and Hexadecapole Orderings in Iron Pnictide Superconductor
---
Introduction
============
Since the discovery of superconductivity in iron pnictide compounds with the formula LaFeAs(O$_{1-x}$F$_x$) with a high transition temperature of $T_\mathrm{SC} = 26$ K by Hosono and his coworkers in 2008 [@Kamihara], many researchers have been involved in the search for various iron-based composites showing high superconducting transition temperatures and the inherent mechanism of the superconductivity. Among the many iron-based compounds, the family of LaFeAsO with the 1111-type ZrCuSiAs structure shows transition temperatures as high as $T_\mathrm{SC} \sim 56$ K [@Ren]. The family of BaFe$_2$As$_2$ with the 122-type ThCr$_2$Si$_2$ structure exhibits superconductivity with $T_\mathrm{SC} \sim 38$ K upon the substitution of barium for potassium [@Rotter1]. The other family of LiFeAs with the 111-type PbFCl structure shows superconductivity with $T_\mathrm{SC} \sim 18$ K [@Tapp], and FeSe with the 11-type PbO structure shows superconductivity with $T_\mathrm{SC} \sim 10$ K [@Hsu]. These compounds have a common lattice structure with a two-dimensional square of iron layers, which gives the electronic band structure due to 3$d$ orbitals of Fe$^{2+}$ ions favorable for achieving superconductivity with high transition temperatures.
The 122-type compounds of BaFe$_2$As$_2$ showing the superconductivity under either chemical doping or applying pressure have received particular attention [@Ni1; @Drotziger; @Fukazawa], because high-quality single crystals with a fair size are available for experiments. The end material of BaFe$_2$As$_2$ exhibits a structural phase transition from the tetragonal phase with space group $D_{4h}^{17}$ $(I4/mmm)$ to the orthorhombic phase with space group $D_{2h}^{23}$ $(Fmmm)$ simultaneously accompanied by antiferro-magnetic ordering with a stripe-type spin structure at transition temperatures of $T_\mathrm{s} = T_\mathrm{N} = 140$ K [@Rotter1; @Rotter2; @Huang]. The chemical doping by substituting Co ions with 3$d^7$ electrons for Fe ions with $3d^6$ electrons in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ reduces both the structural and antiferro-magnetic transition temperatures while slightly splitting the two transition temperatures so that $T_\mathrm{s} > T_\mathrm{N}$ [@Ni2; @Chu; @Lester; @Nandi]. The Co-ion doped compounds with $x > 0.03$ reveal the superconductivity below the successive structural and magnetic transitions. With increasing the Co dopant concentration $x$ to the quantum critical point (QCP) of $x_\mathrm{QCP} = 0.061$, the structural and antiferro-magnetic orderings disappear and the superconducting phase manifests itself. Upon further doping over $x_\mathrm{QCP}$, the optimized superconducting transition temperature of as high as $T_\mathrm{SC} = 23$ K emerges in the compound with $x = 0.071$.
In order to clarify the interplay of the structural transition to the superconductivity in the iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, the softening of the elastic constant $C_{66}$ as a precursor of the structural transition has been intensively investigated. By using resonance ultrasonic spectroscopy, Fernandes *et al*. first showed the softening of $C_{66}$ in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ for the end material and the over-doped compound with $x = 0.08$, the latter exhibiting a superconducting transition at $T_\mathrm{SC} = 16$ K, and proposed that the softening of $C_{66}$ is caused by spin nematic fluctuations of the Fe $3d^6$ electrons [@Fernandes]. By using the ultrasonic pulse-echo method, Goto *et al*. showed the softening of $C_{66}$ by $21\%$ for $x = 0.071$ from 300 K down to the optimized superconducting transition temperature $T_\mathrm{SC} = 23$ K, and deduced that the electric quadrupole $O_{x'^2-y'^2}$ associated with the degenerate $y'z$ and $zx'$ orbitals of the Fe$^{2+}$ ion coupled to the elastic strain $\varepsilon_{xy}$ of the transverse ultrasonic wave plays a role in the appearance of the superconductivity [@Goto]. Here, the $x$ and $y$ ($x'$ and $y'$) coordinates for neighboring Ba-Ba (Fe-Fe) directions are adopted. Yoshizawa *et al*. systematically investigated the elastic constant $C_{66}$ for compounds with various Co concentrations and clarified the quantum criticality of the softening of $C_{66}$ around the QCP of $x_\mathrm{QCP} = 0.061$ [@Yoshizawa].
Angle-resolved photoemission spectroscopy experiments on Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ [@ARPES_Ding], muon resonance experiments on Ba$_{1-x}$K$_x$Fe$_2$As$_2$ [@Muon], microwave penetration depth measurements of PrFeAsO$_{1-y}$ [@Microwave], and NMR relaxation rate measurements of LnFe$_{1-y}$M$_y$AsO$_{1-x}$F$_x$ [@M.; @Sato] commonly show mostly isotropic and nodeless superconducting energy gaps with an s-like shape. The robustness of the superconducting transition temperature $T_\mathrm{SC}$ for nonmagnetic impurity doping to the system is favorable for the sign-conserving s-wave state of s$_{++}$ [@M.; @Sato; @S.; @C.; @Lee; @Kawamata]. Taking the quadrupole fluctuations associated with the elastic softening of $C_{66}$ into account, the s$_{++}$ state for the superconducting energy gap has been theoretically deduced [@Yanagi_1; @Kontani; @Onari]. On the other hand, neutron scattering experiments on Ba(Fe$_{0.925}$Co$_{0.075}$)$_2$As$_2$ have shown a magnetic excitation peak around $\boldsymbol{q} = (1/2, 1/2, 1)$, indicating a role of the antiferro-magnetic fluctuations in the superconductivity [@Inosov]. Accordingly, the sign-reversing s$_\pm$-wave state has been presented while emphasizing the spin fluctuation effects due to the antiferro-magnetic interaction [@STM; @Mazin1; @Kuroki]. The clarification of the superconducting mechanism of the iron pnictides is still an important issue in solid-state physics.
Since ultrasonic waves with frequencies as high as 100 MHz easily penetrate into metals, attenuation measurements are useful for examining order parameter dynamics around phase transitions in metals. There are a few reports on ultrasonic attenuation measurements around the structural and superconducting transitions of the iron pnictides [@Simayi; @Saint-Paul]. However, the attenuation of ultrasonic waves relating the elastic constant $C_{66}$ has not been reported so far to the best of our knowledge. In the present paper, we show the critical dynamics around the structural and superconducting transitions in the iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ by means of ultrasonic measurements.
This paper is organized as follows. In Sect. 2, the experimental procedures of sample preparation and ultrasonic measurements are shown. In Sect. 3, we present experimental results of the critical slowing down of order parameters around the superconducting transition for $x = 0.071$ and the structural transition for $x = 0.036$. The critical temperatures characterizing the phase diagram of the system are shown. In Sect. 4, we demonstrate the theory treating the quantum degrees of freedom of the electric quadrupoles $O_{x'^2-y'^2}$ and $O_{x'y'}$ and the angular momentum $l_z$ associated with the degenerate $y'z$ and $zx'$ orbitals and also their interactions with the transverse acoustic phonons accompanying rotation and strain. In addition, we show plausible models of the quadrupole ordering for the structural transition and the hexadecapole ordering for the superconductivity in the iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. We show that the superconducting Hamiltonian due to the quadrupole interaction gives the superconducting ground state bearing the hexadecapole. In Sect. 5, we present conclusions.
Experiments
===========
Single crystals of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with Co concentration $x$ were grown by the Bridgman method. Energy-dispersive X-ray spectrometry showed that nominal concentrations of $x_\mathrm{nominal} = 0.03, 0.07$, and $0.10$ corresponded to actual concentrations of $x = 0.017, 0.036$, and $0.071$, respectively. A Laue X-ray camera was used to determine the crystallographic orientation of the crystals. Samples with sizes of $2.30 \times 1.05 \times 0.44$ mm$^3$ for the end material $x = 0$, $3.11 \times 5.35 \times 0.77$ mm$^3$ for $x = 0.017$, $2.13 \times 2.06 \times 0.26$ mm$^3$ for $x = 0.036$, and $1.52 \times 2.61 \times 0.83$ mm$^3$ for $x = 0.071$ were used. The elastic constant $C = \rho_\mathrm{M}(x)v^2$ was calculated from the ultrasonic velocity $v$. Mass densities of $\rho_\mathrm{M}(x) = 6.50$ g/cm$^3$ for $x =$ 0, 0.017, and 0.036 and $\rho_\mathrm{M}(x = 0.071) = 6.51$ g/cm$^3$ were estimated using the Vegard’s law for the lattice constants in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ [@Sefat]. Piezoelectric ultrasonic transducers with a 36$^\circ$ Y-cut LiNbO$_3$ plate for the generation and detection of longitudinal waves and an X-cut plate for transverse waves were glued on plane-parallel surfaces of the samples. LiNbO$_3$ plates with thicknesses of 200 and 100 $\mu$m were used to generate ultrasonic waves with frequencies as high as 260 MHz. Because the present crystal shows a large amount of softening with a considerably small elastic constant $C_{66}$, even a small external stress $\sigma_{xy}$ might easily deform the sample with the tetragonal lattice to an orthorhombic deformed lattice with finite $\varepsilon_{xy}$. This unwanted lattice deformation might easily lift the degenerate $y'z$ and $zx'$ orbital states. This lifting due to careless sample treatment would prevent accurate measurements of the softening of $C_{66}$ and the critical divergence of the ultrasonic attenuation across the structural and superconducting transitions. In order to avoid applying undesired stress $\sigma_{xy}$ to the samples, we carefully kept each sample in a brass disk during the ultrasonic experiments. A vector-type detector based on the ultrasonic pulse-echo method was used for simultaneous measurements of the attenuation coefficient $\alpha$ and velocity $v$. A $^3$He cryostat (Oxford Heliox TL) was employed for the ultrasonic measurements down to 250 mK.
Results
=======
Attenuation around superconducting transition for $x = 0.071$
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In order to investigate the dynamical features of the superconducting transition while clearly distinguishing them from the structural and antiferro-magnetic orderings, we focused on the over-doped compound $x = 0.071$, which exhibits the optimized superconducting transition temperature $T_\mathrm{SC} = 23$ K but neither a structural transition nor antiferro-magnetic ordering. We measured the attenuation coefficient $\alpha_{66}$ using the transverse ultrasonic waves with the propagation vector $\boldsymbol{q} // \left[ 100 \right]$ and polarization vector $\boldsymbol{\xi} // \left[ 010 \right]$ with frequencies of 119, 167, and 215 MHz. The attenuation coefficient $\alpha_{66}$ for $x = 0.071$ in Fig. \[Fig1\](a) increases with decreasing temperature below 80 K in the normal phase and reveals critical divergence with approaching the superconducting transition point of $T_\mathrm{SC} = 23$ K. With further lowering the temperature below $T_\mathrm{SC} = 23$ K, the attenuation coefficient $\alpha_{66}$ rapidly decreases. In both the normal and superconducting phases, the frequency dependence of the attenuation coefficient $\alpha_{66}$ obeys the $\omega^2$ law, consistent with the low-frequency regime of the Debye formula.
![ (Color online) (a) Temperature dependence of the ultrasonic attenuation coefficient $\alpha_{66}$ measured using transverse ultrasonic waves with frequencies of 119, 167, and 215 MHz, and the elastic constant $C_{66}$ in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with $x = 0.071$ having the superconducting transition temperature $T_\mathrm{SC} = 23$ K. The solid line is the softening of $C_{66}$ in the normal phase fit by $C_{66} = C_{66}^0 \bigl( T - T_\mathrm{s}^0 \bigr) / \bigl( T - \mathit{\Theta}_\mathrm{Q} \bigr)$ with $T_\mathrm{s}^0 = -26.5$ K and $\mathit{\Theta}_\mathrm{Q} = -47$ K. (b) Temperature dependence of the relaxation time $\tau$ obtained by the attenuation coefficient $\alpha_{66}$ for frequencies 119, 167, and 215 MHz. Solid lines show fits by $\tau = \tau_0 \varepsilon^{- z \nu}$ for the reduced temperature $\varepsilon = \left| T - T_\mathrm{c}^0 \right| / T_\mathrm{c}^0$ with $T_\mathrm{c}^0 = 23$ K and $z \nu = 1$. The inset in (b) shows the temperature dependence of the relaxation rate $\tau^{-1}$. Solid lines in both the normal and superconducting phases show fits by $\tau^{-1} = \tau_0^{-1} \varepsilon^{z \nu}$. []{data-label="Fig1"}](68060Fig1.pdf){width="50.00000%"}
The elastic constant $C_{66}$ in the normal phase for $x = 0.071$ exhibits the softening, which is again shown in Fig. \[Fig1\](a) [@Goto]. This softening is caused by the interaction of the quadrupole $O_{x'^2-y'^2}$ to the strain $\varepsilon_{xy}$ expressed as $$\begin{aligned}
\label{HQS exp.}
H_\mathrm{QS} = -g_{x'^2-y'^2}O_{x'^2-y'^2} \varepsilon_{xy}.\end{aligned}$$ Here, $g_{x'^2-y'^2}$ is the quadrupole-strain coupling constant. We expect the Curie-type behavior of the quadrupole susceptibility proportional to the reciprocal temperature as $\chi_\mathrm{Q} = | \langle \psi_l | O_{x'^2-y'^2} | \psi_l \rangle |^2 / T$ for suffixes $l = y'z$ and $zx'$ of the degenerate orbitals. The softening of $C_{66}$ in the normal phase is written by [@Kataoka; @and; @Kanamori] $$\begin{aligned}
\label{Temp. dep. C66}
C_{66}
= C_{66}^0 - \frac{ n_\mathrm{Q} g_{x'^2-y'^2}^2\chi_\mathrm{Q} } {1 - g'_{x'^2-y'^2}\chi_\mathrm{Q}}
= C_{66}^0\left(\frac{T - T_\mathrm{s}^0} {T - \mathit{\Theta}_\mathrm{Q} }\right).\end{aligned}$$ Here, $C_{66}^0$ is the background elastic constant when softening is absent and $g'_{x'^2-y'^2}$ is the mutual interaction coefficient between the quadrupoles $O_{x'^2-y'^2}$ at different sites. The solid line in Fig. \[Fig1\](a) is the softening of $C_{66}$ fit by Eq. (\[Temp. dep. C66\]) in the normal phase. We obtain negative values of the quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q} = - 47.0$ K and the critical temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q} = -26.5$ K. This fitting gives the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q} = 20.5$ K, which represents the coupling energy between the quadrupoles at different sites mediated by the strain $\varepsilon_{xy}$.
The negative quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q} = - 47.0$ K indicates the antiferro-type quadrupole interaction for $x = 0.071$. The negative critical temperature of $T_\mathrm{s}^0 = -26.5$ K corresponds to the fictitious critical temperature of the structural instability for $C_{66} \rightarrow 0$. Consequently, the superconducting transition temperature $T_\mathrm{SC} = 23$ K is definitely distinguished from the fictitious critical temperature $T_\mathrm{s}^0 = -26.5$ K. We conclude that the marked increase in the attenuation coefficient $\alpha_{66}$ for $x = 0.071$ in Fig. \[Fig1\](a) is caused by the critical slowing down of the order parameter around the superconducting transition, which is strictly distinguished from the quadrupole $O_{x'^2-y'^2}$ interacting to the strain $\varepsilon_{xy}$ in Eq. (\[HQS exp.\]).
Because two electrons are accommodated in the degenerate $y'z$ and $zx'$ orbitals of an Fe$^{2+}$ ion, we tentatively adopt $n_\mathrm{Q} = 2N_\mathrm{Fe} = 3.92 \times 10^{22}$ cm$^{-3}$ in Eq. (\[Temp. dep. C66\]) as the maximum number of electrons, that play a role in the softening of $C_{66}$. Here, $N_\mathrm{Fe}$ is the number of Fe$^{2+}$ ions per unit volume. The solid line in Fig. \[Fig1\](a) for the softening of $C_{66}$ fit by Eq. (\[Temp. dep. C66\]) gives the quadrupole-strain coupling constant $g_{x'^2-y'^2} = 1045$ K per electron. This coupling constant is comparable with the results for manganese compounds of $g = 1167$ K for La$_{0.88}$Sr$_{0.12}$MnO$_3$ and $g = 1020$ K for Pr$_{0.65}$Ca$_{0.35}$MnO$_3$ with considerably extended 3$d$ orbitals [@Hazama; @La; @Hazama; @Pr], but is considerably larger than $g \sim100$ K for rare-earth compounds of the form RB$_6$ with well-screened 4$f$ orbitals in the inner shell [@Nakamura].
In our attempt to clarify the order parameter dynamics around the superconducting transition, we analyzed the frequency dependence of the attenuation coefficient $\alpha_{66}$ in Fig. \[Fig1\](a) in terms of the Debye formula expressed as $$\begin{aligned}
\label{Debye-type}
\alpha_{66}
= \frac{C_{66} \left( \infty \right) - C_{66} \left( 0 \right)} {2\rho_\mathrm{M} {v_{66} \left( \infty \right) }^3} \frac{\omega^2\tau} {1 + \omega^2\tau^2}.\end{aligned}$$ Here, $C_{66} \left( \infty \right)$ and $C_{66} \left( 0 \right)$ stand for the high- and low-frequency limits of the elastic constant $C_{66}$, respectively. We regard the elastic constant experimentally observed in Fig. \[Fig1\](a) as the low-frequency limit $C_{66} \left( 0 \right)$ in Eq. (3). The background $C_{66}^0$ used in the analysis based on the quadrupole susceptibility of Eq. (\[Temp. dep. C66\]) corresponds to the high-frequency limit of the elastic constant $C_{66} \left( \infty \right)$ for the sound velocity $v_{66} \left( \infty \right) = \sqrt{C_{66}\left( \infty \right) / \rho_\mathrm{M} }$ in Eq. (\[Debye-type\]) [@Goto]. $\omega$ is the angular frequency of the employed ultrasonic waves, and $\tau$ is the relaxation time of the order parameter fluctuation of the system. We show the temperature dependence of the relaxation time $\tau$ obtained by Eq. (\[Debye-type\]) in Fig. \[Fig1\](b) and that of the relaxation rate $\tau^{-1}$ in the inset of Fig. \[Fig1\](b). The rapid increase in the relaxation time $\tau$ around $T_\mathrm{SC} = 23$ K indeed indicates the critical slowing down of the order parameter accompanying the superconducting transition in the system.
The correlation length $\zeta$ of the order parameter $Q \left( \boldsymbol{r} \right)$ for an appropriate phase transition is widely used to describe the correlation function as $G \left( \boldsymbol{r} \right)
= \left\langle Q \left( \boldsymbol{r} \right) Q \left( 0 \right) \right\rangle
\propto r^{- \left( d - 1 \right)} e^{- r / \zeta}$. Here, $| \boldsymbol{r} | = r$ stands for the spatial distance between the order parameters and $d$ denotes the dimension of the system. In the vicinity of the critical temperature $T_\mathrm{c}^0$ of the phase transition, the correlation length $\zeta$ becomes infinite due to the increase in the local order as $\zeta = \zeta_0 \varepsilon ^{- \nu}$ for a reduced temperature of $\varepsilon = \bigl| T - T_\mathrm{c}^0 \bigr| / T_\mathrm{c}^0 $. The critical index $\nu =1/2$ is expected from mean field theory [@Nishimori].
When an ultrasonic pulse wave enters a crystal, the equilibrium state of the system is instantaneously perturbed to a nonequilibrium state. After the relaxation time $\tau$, the system returns to the equilibrium state. In the vicinity of the critical point, however, the relaxation time $\tau$ becomes infinite due to the divergence of the correlation length as $\tau \propto \zeta^z$ for dynamical critical index $z$. Presumably, the critical slowing down of the relaxation time $\tau$ is explained by the critical index $z\nu$ as [@Suzuki; @Mori; @Ma; @Halperin; @and; @Hohenberg] $$\begin{aligned}
\label{Relaxation time}
\tau
= \tau_0 \left| \frac{T - T_\mathrm{c}^0} {T_\mathrm{c}^0} \right| ^{-z\nu}
= \tau_0\varepsilon^{-z\nu}.\end{aligned}$$ In mean field theory, the dynamical critical index $z = 2$ is expected. Actually, the temperature dependence of the relaxation time $\tau$ in the normal phase of Fig. \[Fig1\](b) is well reproduced by the solid line for the critical index $z\nu = 1$, the critical temperature $T_\mathrm{c}^{0+} = 23.0$ K, and the attempt time $\tau_0^+ = 6.0 \times 10^{-11}$ s. In the superconducting phase, however, we obtain $z\nu = 1/3$, $T_\mathrm{c}^{0-} = 23.5$ K, and $\tau_0^- = 5.8 \times 10^{-11}$ s. The distinct deviation of the critical index of $z\nu = 1/3$ from $z\nu = 1$ of mean field theory may be caused by the inherent property that the hexadecapole ordering appears in accordance with the superconductivity in the present iron pnictide. The analysis of the experimental results in Fig. \[Fig1\](b) gives a ratio of the attempt times of $ \tau_0^+ / \tau_0^- = 1.03$, which is distinguished from the ratio of $ \tau_0^+ / \tau_0^- = 2$ expected from mean field theory. Because the ultrasonic echo signal almost disappears in the vicinity of the superconducting transition point due to the critical slowing down, the absolute values of the attenuation coefficients inevitably include experimental errors.
Attenuation around structural transition for $x = 0.036$
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Next, we examine the critical slowing down around the structural transition from the tetragonal to orthorhombic phases of the under-doped compound $x = 0.036$. This is worth comparing with the critical slowing down around the superconducting transition of the over-doped compound $x = 0.071$ presented in Sect. 3.1.
![ (Color online) Temperature dependence of elastic constants of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with $x = 0.036$. The considerable softening of $C_{66}$ in (e) and $C_{ \mathrm{L}[110]} = \left( C_{11} + C_{12} + 2C_{66} \right) / 2$ in (d) are found to be a precursor of the structural transition at $T_\mathrm{s} = 65$ K. Anomalies associated with the antiferro-magnetic transition at $T_\mathrm{N} = 39$ K and the superconducting transition at $T_\mathrm{SC} = 16.4$ K are observed. The softening of $C_{66}$ in (e) is reproduced by a solid line fit by $C_{66} = C_{66}^0 \bigl( T - T_\mathrm{s}^0 \bigr) / \bigl( T - \mathit{\Theta}_\mathrm{Q} \bigr)$ with $T_\mathrm{s}^0 = 63$ K and $\mathit{\Theta}_\mathrm{Q} = 47$ K. []{data-label="Fig2"}](68060Fig2.pdf){width="50.00000%"}
The temperature dependence of the elastic constants for $x = 0.036$ is shown in Fig. \[Fig2\]. We denote the propagation vector as $\boldsymbol{q}$ and the polarization vector as $\boldsymbol{\xi}$ for the ultrasonic waves measured in Fig. \[Fig2\]. The elastic constant $C_{66}$ in Fig. \[Fig2\](e) measured by the transverse ultrasonic wave reveals considerable softening of 85$\%$ with decreasing temperature from 300 K down to the structural transition temperature of $T_\mathrm{s} = 65$ K. As indicated by arrows in Fig. \[Fig2\](e), the antiferro-magnetic ordering at $T_\mathrm{N} = 39$ K and the superconducting transition at $T_\mathrm{SC} = 16.4$ K are also observed as anomalies in $C_{66}$. The softening of $C_{66}$ in the tetragonal phase is reproduced by the solid line in Fig. \[Fig2\](e), which is obtained by Eq. (\[Temp. dep. C66\]) for the quadrupole interaction energy of $\mathit{\Theta}_\mathrm{Q} = 47$ K, the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q} = 16$ K, and the critical temperature of $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q} = 63$ K. The background of $C_{66}^0 = 2.87 \times 10^{10}$ J/m$^3$ is shown by the dashed line in Fig. 2(e). The critical temperature $T_\mathrm{s}^0 = 63$ K is in agreement with the experimentally observed structural transition temperature $T_\mathrm{s} = 65$ K. We deduce the quadrupole-strain coupling constant to be $g_{x'^2 - y'^2} = 920$ K per electron by adopting the electron number $n_\mathrm{Q} = 2N_\mathrm{Fe}$ in Eq. (\[Temp. dep. C66\]). The positive values of the quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q} = 47$ K and the critical temperature $T_\mathrm{s}^0 = 63$ K are consistent with the ferro-quadrupole ordering $O_{x'^2 - y'^2}$ accompanying the structural transition from the tetragonal to orthorhombic phase.
The elastic constant $C_{\mathrm{L} \left[110 \right]} = \left( C_{11} + C_{12} + 2C_{66} \right)/2$ in Fig. \[Fig2\](d) measured by a longitudinal ultrasonic wave with the propagation direction $\boldsymbol{q} // \left[110 \right]$ parallel to the polarization direction $\boldsymbol{\xi}$ also shows marked softening with decreasing temperature down to the structural transition at $T_\mathrm{s} = 65$ K. With further decreasing temperature in the distorted orthorhombic phase, the graph of the elastic constant $C_{\mathrm{L} \left[110 \right]}$ slightly bend at the antiferro-magnetic transition temperature $T_\mathrm{N} = 39$ K and distinctly decreases at the superconducting transition temperature $T_\mathrm{SC} = 16.4$ K. The longitudinal ultrasonic wave for $C_{\mathrm{L} \left[110 \right]}$ induces elastic strain of $\varepsilon_{ \mathrm{L} \left[110 \right] } = \varepsilon_\mathrm{B}/3 - \varepsilon_u/ \bigl( 2\sqrt{3} \bigr) + \varepsilon_{xy}$. The coupling of the strain $\varepsilon_{xy}$ to the quadrupole $O_{x'^2 - y'^2}$ induces the softening in $C_{\mathrm{L} \left[110 \right]}$, partly consisting of $C_{66}$. The volume strain $\varepsilon_\mathrm{B} = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz}$ and the tetragonal strain $\varepsilon_u = \bigl( 2\varepsilon_{zz} - \varepsilon_{xx} - \varepsilon_{yy} \bigr) / \sqrt{3}$ scarcely affect the softening.
We initially attempted to measure the attenuation coefficient $\alpha_{66}$ by using the pure transverse ultrasonic wave. However, the attenuation of $\alpha_{66}$ near the structural transition was too large to measure. In the present experiments, therefore, we used the longitudinal ultrasonic wave, which shows relatively moderate damping, to measure the attenuation coefficient $\alpha_{\mathrm{L} \left[110 \right]}$. In Fig. \[Fig3\](a), we show the attenuation coefficient $\alpha_{\mathrm{L} \left[110 \right]}$ for $x = 0.036$ acquired using longitudinal ultrasonic waves with frequencies of 112, 186, and 260 MHz together with the elastic constant $C_{\mathrm{L} \left[110 \right]}$. The attenuation coefficient $\alpha_{\mathrm{L} \left[110 \right]}$ increases considerably with approaching the structural transition point of $T_\mathrm{s} = 65$ K from both sides. With decreasing temperature in the orthorhombic phase, the attenuation $\alpha_{\mathrm{L} \left[110 \right]}$ exhibits a distinct peak at the superconducting transition point of $T_\mathrm{SC} = 16.4$ K. The missing ultrasonic echo signal due to considerable damping of the longitudinal ultrasonic wave prevents us from acquiring $\alpha_{\mathrm{L} \left[110 \right]}$ approaching the structural transition. Nevertheless, the distinct tendency of the divergence of $\alpha_{\mathrm{L} \left[110 \right]}$ around the structural transition at $T_\mathrm{s} = 65$ K indicates the critical slowing down of the relaxation time $\tau$ for the order parameter fluctuation.
The frequency dependence for $\alpha_{\mathrm{L} \left[110 \right]}$ for the under-doped compound $x = 0.036$ in Fig. \[Fig3\](a) is analyzed in terms of the Debye formula in Eq. (\[Debye-type\]), where we read the attenuation as $\alpha_{\mathrm{L} \left[110 \right]}$ and the elastic constant as $C_{\mathrm{L} \left[110 \right]}$. We depict the temperature dependence of the relaxation time $\tau$ in Fig. \[Fig3\](b) and the relaxation rate $\tau^{-1}$ in the inset of Fig. \[Fig3\](b). The increase in the relaxation time $\tau$ around the structural transition at $T_\mathrm{s} = 65$ K is caused by the critical slowing down of the quadrupole $O_{x'^2 - y'^2}$, which is the order parameter of the structural transition. The ultrasonic frequencies of $\omega / 2\pi$ up to 260 MHz used in the present experiments are much higher than the relaxation rate $\tau^{-1} < 100$ MHz in the vicinity of the structural transition point $T_\mathrm{s}$ for the narrow reduced temperature region of $\varepsilon = | T - T_\mathrm{c}^0|/T_\mathrm{c}^0 < 0.038$. As a result, the ultrasonic echo signal disappears on both sides around the structural transition point of $T_\mathrm{s} = 65$ K as depicted in Fig. 3(a). The solid lines in Fig. \[Fig3\](b) are fits for the relaxation time $\tau$ in terms of Eq. (\[Relaxation time\]). Supposing that the critical temperature $T_\mathrm{c}^0 = 65$ K coincides with the structural transition point and the critical indices for both phases are consistent with $z\nu = 1$ of mean field theory, we obtain the fit shown by the solid line in Fig. \[Fig3\](b) using Eq. (\[Relaxation time\]). This gives the attempt time $\tau_0^+ = 0.380 \times 10^{-9}$ s for the tetragonal phase and $\tau_0^- = 1.15 \times 10^{-9}$ s for the orthorhombic phase.
The quadrupole-strain interaction of Eq. (\[HQS exp.\]) induces the ferro-quadrupole ordering accompanying the structural transition. It is expected that the long-range Coulomb force between electrons bearing the quadrupole exhibits critical phenomena described by mean field theory. Actually, the adoption of the critical index $z\nu = 1$ for both tetragonal and orthorhombic phases consistent with mean field theory reproduces the experimental results in Fig. \[Fig3\]. The small-echo signal due to the considerable damping of the longitudinal ultrasonic waves gives inevitable errors in the absolute-value of the attenuation coefficient $\alpha_{\mathrm{L}[110]}$ in Fig. \[Fig3\]. The dispersive scattering of the ultrasonic wave by the domain wall due to the orthorhombic distortion is also included in the attenuation. These ambiguity might have resulted in the deviation of the experimentally determined ratio of $\tau_0^+ / \tau_0^- = 0.33$ from $\tau_0^+ / \tau_0^- = 2$ expected from mean field theory. The attempt time $\tau_0^+ = 0.55 \times 10^{-9}$ s of the structural transition for $x = 0.036$ in Fig. \[Fig3\](b) is one order of magnitude larger than the attempt time $\tau_0^+ = 6.0 \times 10^{-11}$ s of the superconducting transition for $x = 0.071$ in Fig. \[Fig1\](b). This notable result suggests that the order parameter showing the critical slowing down around the superconducting transition for $x = 0.071$ is distinguished from the ferro-quadrupole ordering accompanying the structural transition for $x = 0.036$.
Critical temperatures of the system
-----------------------------------
The ultrasonic measurements of the critical divergence in the attenuation coefficients and the softening in the elastic constants provided us with the critical temperatures, which characterize the structural and superconducting transitions of the present iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. In Fig. \[Fig4\], we plot the critical temperatures obtained by the experimental results for $x = 0.036$ and 0.071 together with those for $x = 0.017$ and the end material $x = 0$ [@Kurihara; @Dr].
![ (Color online) Critical temperatures relating to structural (ST) and superconducting (SC) transitions in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ determined by the present ultrasonic measurements. The quadrupole interaction (Q intr.) energy $\mathit{\Theta}_\mathrm{Q}$, quadrupole-strain interaction (Q-S intr.) energy $\mathit{\Delta}_\mathrm{Q}$, and structural transition temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q}$ associated with the ferro-quadrupole ordering are determined by the elastic constant $C_{66} \rightarrow 0$. The critical temperature $T_\mathrm{c}^0$ is determined by the critical slowing down of the relaxation time $\tau \rightarrow \infty$ measured by the ultrasonic attenuation coefficient $\alpha$. Structural transition temperatures $T_\mathrm{s}$ associated with ferro-quadrupole (FQ) ordering are indicated by black open circles. Superconducting transition temperatures $T_\mathrm{SC}$ associated with ferro-hexadecapole (FH) ordering are indicated by black open squares. Superconducting transition temperatures indicated by black open rhombuses are results of Ni $et$ $al$. [@Ni2]. \[Fig4\] ](68060Fig4.pdf){width="46.00000%"}
The softening of the elastic constant $C_{66}$ analyzed by the quadrupole susceptibility $\chi_\mathrm{Q}$ in Eq. (\[Temp. dep. C66\]) for $O_{x'^2 - y'^2}$ gives the critical temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q}$ corresponding to the structural instability point as $C_{66} \rightarrow 0$. As shown by blue closed circles and the blue solid line in Fig. \[Fig4\], the critical temperature of $T_\mathrm{s}^0 = 135$ K of the end material $x = 0$ decreases to $T_\mathrm{s}^0 = 100$ K for $x = 0.017$ and $T_\mathrm{s}^0 = 63$ K for $x = 0.036$. Beyond the QCP of $x_\mathrm{QCP} = 0.061$, the critical temperature changes its sign to $T_\mathrm{s}^0 = -26.5$ K for $x = 0.071$. The quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q}$ shown by red closed circles and the red solid line decreases from $\mathit{\Theta}_\mathrm{Q} = 115$ K for the end material $x = 0$ to $\mathit{\Theta}_\mathrm{Q} = 80$ K for $x = 0.017$, $\mathit{\Theta}_\mathrm{Q} = 47$ K for $x = 0.036$, and changes to a negative value of $\mathit{\Theta}_\mathrm{Q} = -47$ K for $x = 0.071$. On the other hand, the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q}$ shown by green closed circles and the green solid line remains at the positive value $\mathit{\Delta}_\mathrm{Q} \sim 20$ K. The positive critical temperatures $T_\mathrm{s}^0$ of the under-doped compounds are comparable with the ferro-quadrupole ordering $O_{x'^2 - y'^2}$ associated with the structural transition from the tetragonal to orthorhombic phase, while the negative critical temperature $T_\mathrm{s}^0 = -26.5$ K for the over-doped compound $x = 0.071$, suggesting absence of the ferro-quadrupole ordering, implies the fictitious lattice instability point.
The infinite divergence of the relaxation time $\tau$ due to the critical slowing down provides the critical temperature $T_\mathrm{c}^0$ at which the long-range ordering appears due to freezing of the order parameter fluctuation [@Halperin; @and; @Hohenberg; @Ma]. In the case of the structural transition for $x = 0.036$, the divergence of the relaxation time $\tau$ gives the critical temperature of $T_\mathrm{c}^0 = 65$ K, which coincides with the structural transition temperature $T_\mathrm{s} = 65$ K experimentally observed, and is in agreement with the critical temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q} = 63$ K obtained by fitting the elastic softening of $C_{66}$ by Eq. (\[Temp. dep. C66\]). This result confirms that the critical divergence of the ultrasonic attenuation coefficient $\alpha_{\mathrm{L} \left[110 \right]}$ for $x = 0.036$ is caused by the critical slowing down of the ferro-quadrupole order parameter $O_{x'^2 - y'^2}$. The critical temperature for $T_\mathrm{c}^0 = 65$ K for $x = 0.036$ is shown in Fig. \[Fig4\] by a purple closed asterisk together with the critical temperature of $T_\mathrm{s}^0 = 63$ K. Note that the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q} \sim 20$ K, which is almost independent of the Co concentration, also promotes the appearance of the structural transition at the critical temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q}$. The quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q}$ is shown by green closed circles in Fig. \[Fig4\].
In the case of the superconducting transition for $x = 0.071$, the critical temperature $T_\mathrm{c}^0 = 23$ K determined by the critical slowing down of the relaxation time $\tau$ is in good agreement with the superconducting transition temperature $T_\mathrm{SC} = 23$ K. In Fig. \[Fig4\], we show the former critical temperature $T_\mathrm{c}^0 = 23$ K for the relaxation time $\tau$ by a purple closed asterisk together with the latter superconducting transition temperature $T_\mathrm{SC}$ = 23 K by a black open square. The critical temperature $T_\mathrm{s}^0 = -26.5$ K indicated by a blue filled circle corresponds to the fictitious lattice instability point as $C_{66} \rightarrow 0$. The fictitious instability point of $T_\mathrm{s}^0 = -26.5$ K is definitely distinguished from the critical temperature $T_\mathrm{c}^0 = 23$ K of the relaxation time $\tau$ coinciding with the superconducting transition temperature $T_\mathrm{SC}$. This result evidences that the order parameter exhibiting the critical slowing down around the superconducting transition does not correspond to the electric quadrupole $O_{x'^2-y'^2}$ but to other quantum degrees of freedom, which interact with the transverse ultrasonic waves used in the experiments. It is of great importance to identify the order parameter, which brings about the critical slowing down of the relaxation time $\tau$ around the superconducting transition. In Sect. 4.6, we give the electric hexadecapole carried by a two-electron state as a plausible order parameter associated with the superconducting transition.
Theory
======
Transverse acoustic wave
------------------------
The relaxation time $\tau$ determined by the ultrasonic attenuation reveals the critical slowing down around the superconducting transition for the compound $x = 0.071$ as well as the structural transition for $x = 0.036$. In our attempt to properly explain the ultrasonic experiments, we treat the couplings between the ultrasonic waves and electrons accommodated in the degenerate $y'z$ and $zx'$ orbitals of Fe ion. We will show plausible order parameters associated with the superconducting phase and the structurally distorted phase in the system.
The transverse ultrasonic wave with the propagation vector $\boldsymbol{q} = \bigl( q_x, 0, 0 \bigr)$ and the polarization vector $\boldsymbol{\xi} = \bigl( 0, \xi_y, 0 \bigr)$ employed in the present experiments is expressed in terms of the plane wave as $$\begin{aligned}
\label{Deformation_y}
\xi_{y} = \xi_{y}^0 \exp \left[i \left( q_{x} x - \omega t \right) \right]
.\end{aligned}$$ Here, $\xi_y^0$ is the amplitude of the ultrasonic wave, $q_x$ is the wavevector component, and $\omega$ is the angular frequency. The elastic constant is determined by the sound velocity $v_x = \omega / q_x = v_{66}$ as $C_{66} = \rho v_x^2$. The transverse ultrasonic wave induces the deformation tensor $$\begin{aligned}
\label{Dif. Deformation_y}
\frac{ \partial \xi_y } { \partial x } = iq_x \xi_y^0 \exp \left[ i \left(q_x x - \omega t \right) \right] = iq_x \xi_y
.\end{aligned}$$ The deformation tensor of Eq. (\[Dif. Deformation\_y\]) associated with the transverse ultrasonic wave consists of the strain $\varepsilon_{xy}$ expressed as a symmetric tensor and the rotation $\omega_{xy}$ as an antisymmetric tensor $$\begin{aligned}
\label{Strain and rotation}
\left\{
\begin{array}{c}
\varepsilon_{xy} \\
\omega_{xy} \\
\end{array}
\right\}
= \frac{1}{2} \left( \frac{ \partial \xi_{y} } { \partial x } \pm \frac{ \partial \xi_{x} } { \partial y } \right)
.\end{aligned}$$ In experiments using the transverse ultrasonic wave with $\boldsymbol{q} = (q_x, 0, 0)$ and $\boldsymbol{\xi} = (0, \xi_y, 0)$, we tune the perturbation parameter of $\delta = q_x \xi_y^0 = (\omega / v_x) \xi_y^0 = (2 \pi / \lambda_x ) \xi_y^0$ to an infinitesimal value by controlling the amplitude of the generated ultrasonic wave $\xi_y^0$ and the frequency $\omega$ for a given phase velocity $v_x = v_{66}$. The rotation $\omega_{xy} = q_x \xi_y^0 = \sin \theta$ and the strain $\varepsilon_{xy} = q_x \xi_y^0 = \sin \theta$ are caused by slight twisting of the $x$- and $y$-axes by an angle of $\pi/2 \pm 2\theta$. This will later be illustrated in Figs. \[Fig7\](a) and \[Fig7\](b). The transverse ultrasonic wave simultaneously induces the rotation $\omega_{xy}$ and strain $\varepsilon_{xy}$, which are strictly distinguished from each other from a symmetrical viewpoint. Concerning the space group $D_{4h}^{17}$ of the tetragonal lattice, the rotation $\omega_{xy}$ is compatible with $A_2$ symmetry, while the strain $\varepsilon_{xy}$ is compatible with $B_2$ [@Lax] symmetry. Thus, it is expected that the rotation $\omega_{xy}$ and strain $\varepsilon_{xy}$ interact with the electronic states of the system in different manners. When the electron-phonon interaction is absent, both the rotation $\omega_{xy}$ and strain $\varepsilon_{xy}$ propagate in the lattice with the same velocity $v_{66}$.
In our attempt to explain the critical divergence of the ultrasonic attenuation coefficients and the marked elastic softening, we suppose that the interactions of thermally excited transverse acoustic phonons with electronic states play a role in the appearance of the structural and superconducting transitions. In the interactions of the transverse acoustic phonons with the electronic states of the iron pnictide, we take the Hamiltonian for the harmonic oscillators with polarization vectors in the $xy$ plane as $$\begin{aligned}
\label{Phonon energy}
H_\mathrm{ph}
&= \sum_{\boldsymbol{q} }
\left[
\hbar \omega_{x} \left( \boldsymbol{q} \right) \left( a_{x,\boldsymbol{q}}^\dagger a_{x,\boldsymbol{q}} + \frac{1}{2} \right)
\right.
\nonumber \\
&\left. \qquad \qquad
+ \hbar \omega_{y} \left( \boldsymbol{q} \right) \left( a_{y,\boldsymbol{q}}^\dagger a_{y,\boldsymbol{q}} + \frac{1}{2} \right)
\right]
.\end{aligned}$$ Here, $a_{i, \boldsymbol{q} }$ and $a_{i,\boldsymbol{q} }^\dagger$ respectively denote annihilation and creation operators of the transverse acoustic phonon with the polarization direction $\xi_i$ for $i = x$ and $y$ and momentum $\hbar \boldsymbol{q}$. $\hbar \omega_i \left( \boldsymbol{q} \right)$ stands for the phonon energy. The strain $\varepsilon_{xy} \left( \boldsymbol{r}, t \right)$ and rotation $\omega_{xy} \left( \boldsymbol{r}, t \right)$ at position $\boldsymbol{r}$ and time $t$ are expressed in terms of the phonon operators $a_{i, \boldsymbol{q} }$ and $a_{i,\boldsymbol{q} }^\dagger$ as [@QTS] \[Strain and Rotation 2nd quant.\] $$\begin{aligned}
\left\{
\begin{array}{c}
\varepsilon_{xy} \left( \boldsymbol{r}, t \right) \\
\omega_{xy} \left( \boldsymbol{r}, t \right) \\
\end{array}
\right\}
&= \frac{1}{2} \left[ \frac{ \partial \xi_{y}\left( \boldsymbol{r}, t \right) } { \partial x }
\pm \frac{ \partial \xi_{x}\left( \boldsymbol{r}, t \right) } { \partial y } \right]
\nonumber \\
&= \frac{i}{2} \sum_{\boldsymbol{q}} \sqrt{\mathstrut \frac{\hbar} {2V\rho_\mathrm{M} \omega_{y} \left( \boldsymbol{q} \right)} } q_{x}
\nonumber \\
&\quad \times
\left[ a_{y,\boldsymbol{q}} e^{i\boldsymbol{q}\cdot \boldsymbol{r}}e^{-i\omega_{y}\left( \boldsymbol{q} \right) t}
- a_{y,\boldsymbol{q}}^{\dagger} e^{-i\boldsymbol{q}\cdot \boldsymbol{r}}e^{i\omega_{y}\left( \boldsymbol{q} \right)t} \right]
\nonumber \\
&\pm
\frac{i}{2} \sum_{\boldsymbol{q} } \sqrt{\mathstrut \frac{\hbar} {2V\rho_\mathrm{M} \omega_{x} \left( \boldsymbol{q} \right)} } q_{y}
\nonumber \\
&\quad \times
\left[ a_{x,\boldsymbol{q}} e^{i\boldsymbol{q}\cdot \boldsymbol{r}}e^{-i\omega_{x}\left( \boldsymbol{q} \right)t}
- a_{x,\boldsymbol{q}}^{\dagger} e^{-i\boldsymbol{q}\cdot \boldsymbol{r} }e^{i\omega_{x}\left( \boldsymbol{q} \right) t} \right]
.\end{aligned}$$ Here, $\rho_M$ is the mass density and $V$ is the volume of the system.
Degenerate $\psi_{y'z}$ and $\psi_{zx'}$ orbitals
-------------------------------------------------
Many reports of band-structure calculations on the iron pnictide compounds have shown inherent 3$d$ electronic structures favorable for the superconductivity as well as the structural transition [@Nekrasov; @Mazin2; @Yanagi2; @Yanagi; @Dr.; @Miyake]. Three sheets of hole surfaces with a cylinder-type structure exist around the $\Gamma$-point of the zone center and two electron pockets exist around the X-points of the zone boundary. We focus on the degenerate $y'z$ and $zx'$ bands occupied up to half filling, which are lifted by the transverse ultrasonic waves and acoustic phonons for $C_{66}$ with a small-wavenumber limit of $|\boldsymbol{q}| \rightarrow 0$.
We show the electronic states of the degenerate $y'z$ and $zx'$ orbitals in the crystalline electric field (CEF) Hamiltonian on an Fe$^{2+}$ ion with $D_{2d}$ site symmetry, which consists of an electric monopole, quadrupole, and hexadecapole expressed in terms of spherical harmonics as $$\begin{aligned}
\label{HCEF}
H_\mathrm{CEF}
&= A_0^0
\nonumber \\
&\quad
+ A_2^0 \left( \frac{3z^2-r^2} {2r^2} \right)
\nonumber \\
&\qquad
+ A_4^0 \left( \frac{35z^4-30z^2r^2+3r^4} {8r^4} \right)
\nonumber \\
&\quad \qquad
+ A_4^4 \left( \frac{\sqrt{35} }{8} \frac{x'^4-6x'^2y'^2+y'^4} {r^4} \right)
.\end{aligned}$$ Here, $A_l^m$ denotes CEF parameters for the electric multipole potentials. The 3$d$ orbitals of the Fe$^{2+}$ ion in the CEF Hamiltonian of Eq. (\[HCEF\]) splits into five orbital states with energy $E_\mathrm{CEF}^l$ as $$\begin{aligned}
\label{ECEFl}
E_\mathrm{CEF}^l
= \int d\boldsymbol{r} \psi_{l}^{\ast} \left( \boldsymbol{r} \right) H_\mathrm{CEF} \psi_{l} \left( \boldsymbol{r} \right)
.\end{aligned}$$ Here, $l$ denotes the 3$d$ orbital suffix. The point charge model of the CEF Hamiltonian gives the low-lying singlet states of $\psi_{3z^2-r^2}$ with $A_1$ symmetry and $\psi_{x'^2 - y'^2}$ with $B_1$ symmetry and the mid-lying doublet state of $\psi_{y'z}$ and $\psi_{zx'}$ with $E$ symmetry and the excited singlet of $\psi_{x'y'}$ with $B_2$ symmetry [@Kurihara; @Dr]. Actually, the band calculations show that the three hole sheets around the $\Gamma$-point consist of the doublet state of $\psi_{y'z}$ and $\psi_{zx'}$ and the singlet state $\psi_{x'^2-y'^2}$.
In the present investigation, we suppose that the structural and superconducting transitions of the iron pnictide manifest themselves as a result of the spontaneous symmetry breaking associated with the degenerate $y'z$ and $zx'$ bands, which are mapped on the special unitary group SU(2). In order to explore the quadrupole-strain interaction in the iron pnictide, we specially treat the degenerate $\psi_{y'z}$ and $\psi_{zx'}$ orbitals of the $E$ symmetry with half filling as [@Miyake; @Inui; @Group] $$\begin{aligned}
\label{wave function y'z}
\psi_{y'z} \left( \boldsymbol{r} \right)
&= \frac{i}{\sqrt{2} } f_d \left( r \right) \left[ Y_2^1 \left( \theta, \varphi \right) + Y_2^{-1} \left( \theta, \varphi \right) \right]
\nonumber \\
&= \sqrt{\frac{15}{4\pi} } f_d \left( r \right) \frac{y'z} {r^2}
, \\
\label{wave function zx'}
\psi_{zx'} \left( \boldsymbol{r} \right)
&= \frac{-1}{\sqrt{2} } f_d \left( r \right) \left[ Y_2^1 \left( \theta, \varphi \right) - Y_2^{-1} \left( \theta, \varphi \right) \right]
\nonumber \\
&= \sqrt{\frac{15}{4\pi} } f_d \left( r \right) \frac{zx'} {r^2}
.\end{aligned}$$ Here, the polar coordinate $\left( r, \theta, \varphi \right)$ represents the position vector $\boldsymbol{r} = \left( x', y', z \right)$ of an electron. $f_d \left( r \right)$ is the radius function of a 3$d$ electron, and $Y_2^{\pm1}\left( \theta, \varphi \right)$ is the spherical harmonics with angular momentum $l = 2$ and azimuthal quantum number $m = \pm1$.
Since the direct product of the $E$ doublet is reduced as $E \otimes E = A_1 \oplus A_2 \oplus B_1 \oplus B_2$, we deduce that the degenerate $y'z$ and $zx'$ orbitals carry the electric quadrupoles $O_{3z^2 - r^2} = \bigl( 3z^2 - r^2 \bigr) / \bigl( 2r^2 \bigr)$ with $A_1$ symmetry, $O_{x'y'} = \sqrt{3} x'y'/r^2$ with $B_1$ symmetry, and $O_{x'^2-y'^2} = \sqrt{3} \bigl(x'^2 - y'^2 \bigr)/ \bigl( 2r^2 \bigr)$ with $B_2$ symmetry and the angular momentum $l_z = -i \bigl(x' \partial / \partial y' - y' \partial / \partial x' \bigr) = -i \partial / \partial \varphi$ with $A_2$ symmetry. The quadrupole $O_{x'^2-y'^2}$ couples to the strain $\varepsilon_{xy}$ of the transverse ultrasonic wave of $C_{66}$, while $O_{x'y'}$ couples to the strain $\varepsilon_{x^2 - y^2}$ of the transverse ultrasonic wave of $\left( C_{11} - C_{12} \right) /2$. The CEF Hamiltonian of Eq. (\[HCEF\]) includes $O_{3z^2 - r^2}$ with full symmetry.
Employing the identity matrix $\tau_0$ and the Pauli matrices $\tau_x$, $\tau_y$, and $\tau_z$ corresponding to the generator elements of the special unitary group SU(2), we present the quadrupoles $O_{3z^2-r^2}$, $O_{x'^2-y'^2}$, and $O_{x'y'}$ and the angular momentum $l_z$ in terms of the matrices for the orbital state $\psi_{y'z}$ of Eq. (\[wave function y’z\]) and $\psi_{zx'}$ of Eq. (\[wave function zx’\]) as $$\begin{aligned}
\label{Matrix Ou}
O_{3z^2-r^2}
&= \frac{1}{7}
\bordermatrix{
& \psi_{y'z} & \psi_{zx'} \cr
& 1 & 0 \cr
& 0 & 1 \cr
}
= \frac{1}{7} \tau_0
,
\\
%Matrix Ox'y'
\label{Matrix Ox'y'}
O_{x'y'}
&= \frac{\sqrt{3} }{7}
\left(
\begin{array}{cc}
0&1 \\
1&0 \\
\end{array}
\right)
= \frac{\sqrt{3} }{7} \tau_x
,
\\
%Matrix lz
\label{Matrix lz}
l_z
&= - \left(
\begin{array}{cc}
0&-i \\
i&0 \\
\end{array}
\right)
= - \tau_y
,
\\
%Matrix Ox'2-y'2
\label{Matrix Ox'2-y'2}
O_{x'^2-y'^2}
&= - \frac{\sqrt{3} }{7}
\left(
\begin{array}{cc}
1&0 \\
0&-1 \\
\end{array}
\right)
= - \frac{\sqrt{3} }{7} \tau_z
.\end{aligned}$$ The Pauli matrices $\tau_x$, $\tau_y$, and $\tau_z$ obey the commutation relation $$\begin{aligned}
\label{Pauli matrix Commutation relation}
\left[ \tau_i, \tau_j \right]
= 2i \varepsilon_{ijk} \tau_k \ \ \ \left( i, j, k = x, y ,z\right)
.\end{aligned}$$ Here, $\varepsilon_{ijk}$ is the Levi$-$Civita symbol. As will be shown in Sects. 4.6 and 4.10, the commutation relation among the quadrupoles $O_{x'^2 - y'^2}$ and $O_{x'y'}$ and the angular momentum $l_z$ of Eq. (\[Pauli matrix Commutation relation\]) brings about quantum fluctuations, which play a significant role in the manifestation of the specific superconductivity accompanying the hexadecapole ordering in the vicinity of the QCP.
![ (Color online) Schematic view of interactions between electric quadrupole $O_{x'y'}$ and strain $\varepsilon_{x^2 - y^2}$ with $B_1$ symmetry in (a), $O_{x'^2 - y'^2}$ and $\varepsilon_{xy}$ with $B_2$ symmetry in (b), and electric hexadecapole $H_z^\alpha \propto \partial l_z / \partial t$ and rotation $\omega_{xy}$ with $A_2$ symmetry in (c) of the space group $D_{4h}^{17}
$ in the present iron pnictide. The coordinates of $x$ and $y$ ($x'$ and $y'$) are adopted for the neighboring Ba-Ba (Fe-Fe) directions. []{data-label="Fig5"}](68060Fig5.pdf){width="46.00000%"}
The elastic strains $\varepsilon_{\mathrm{\Gamma}_\gamma}$ induced by the thermally excited transverse acoustic phonons or the experimentally generated transverse ultrasonic waves perturb the CEF Hamiltonian as $$\begin{aligned}
\label{perturbation of CEF}
H_\mathrm{CEF} \left( \boldsymbol{r}, \varepsilon_{\mathrm{\Gamma}_\gamma} \right)
= H_\mathrm{CEF} \left( \boldsymbol{r} \right)
+ \sum_{\mathrm{\Gamma}_{\gamma}}
\frac{\partial H_\mathrm{CEF} \left( \boldsymbol{r} \right) }
{\partial \varepsilon_{\mathrm{\Gamma}_\gamma} } \varepsilon_{{\mathrm{\Gamma}}_\gamma}
.\end{aligned}$$ Because electrons accommodated in the degenerate $y'z$ and $zx'$ orbitals bear the electric quadrupoles, the second term of Eq. (\[perturbation of CEF\]) proportional to the strain $\varepsilon_{\mathrm{\Gamma}_\gamma}$ is expressed in terms of the quadrupole-strain interaction as [@Dohm; @Thalmeier; @Luthi; @Phys.; @Ac.] $$\begin{aligned}
\label{HQS}
H_\mathrm{QS} = -g_{x'^2-y'^2}O_{x'^2-y'^2}\varepsilon_{xy} -g_{x'y'}O_{x'y'}\varepsilon_{x^2-y^2}.\end{aligned}$$ As the present ultrasonic experiments give a large coupling constant of $g_{x'^2 - y'^2} \sim 1 \times 10^3$ K per electron, the first term $-g_{x'^2 - y'^2} O_{x'^2 - y'^2} \varepsilon_{xy}$ in Eq. (\[HQS\]) is important for the critical divergence of the attenuation coefficient $\alpha_{66}$ and the softening of the elastic constant $C_{66}$ around the structural transition. The little softening of $\left( C_{11} - C_{12} \right) /2$ in the present experiments implies that the second term $-g_{x'y'} O_{x'y'} \varepsilon_{x^2-y^2}$ in Eq. (\[HQS\]) scarcely affects the system. We schematically show the quadrupole $O_{x'y'}$ conjugate with the strain $\varepsilon_{x^2 - y^2}$ in Fig. \[Fig5\](a) and the quadrupole $O_{x'^2 - y'^2}$ conjugate with $\varepsilon_{xy}$ in Fig. \[Fig5\](b).
Next, we consider the interaction of the rotation $\omega_{xy}$ associated with the transverse ultrasonic wave with one-electron states having the angular momentum $l_z$ in the CEF Hamiltonian. The rotation operator of $\exp \bigl[ -i l_z \omega_{xy} \bigr]$ twists the $\psi_{y'z}$ and $\psi_{zx'}$ orbital states in the CEF Hamiltonian given by Eq. (\[HCEF\]) as [@HoVO4] $$\begin{aligned}
\label{HCEF rot.}
\left\langle l \left| H_\mathrm{CEF} \bigl( \omega_{xy} \bigr) \right| l' \right\rangle
= \int d\boldsymbol{r} \psi_{l}^{\ast} \left( \boldsymbol{r} \right) e^{il_z\omega_{xy}}
H_\mathrm{CEF} e^{ -il_z\omega_{xy}} \psi_{l'} \left( \boldsymbol{r} \right)
.\end{aligned}$$ Here, ket $ | l \rangle = \psi_{l} \left( \boldsymbol{r} \right)$ stands for the orbital states $l = y'z$ of Eq. (\[wave function y’z\]) and $zx'$ of Eq. (\[wave function zx’\]). Supposing infinitesimal rotation of $\omega_{xy} \ll 1$ associated with the ultrasound in experiments, the rotation-operated Hamiltonian of Eq. (\[HCEF rot.\]) is expanded in terms of power series up to the second order of $\omega_{xy}$ as $$\begin{aligned}
\label{Rotation of HCEF}
e^{ i l_z \omega_{xy} } H_\mathrm{CEF} e^{ - i l_z \omega_{xy} }
\approx &H_\mathrm{CEF}
+ i \left[ l_z, H_\mathrm{CEF} \right] \omega_{xy}
\nonumber \\
&- \frac{1}{2} \left[ l_z, \bigl[ l_z, H_\mathrm{CEF} \bigr] \right] \bigl( \omega_{xy} \bigr)^2
.\end{aligned}$$ Here, $\left[ A, B \right] = AB - BA$ denotes the Poisson bracket. Applying the differential operation of $l_z = -i \left( x' \partial / \partial y' - y' \partial / \partial x' \right) = -i \partial / \partial \varphi$ to the CEF Hamiltonian of Eq. (\[HCEF\]), we obtain the perturbation Hamiltonian as $$\begin{aligned}
\label{Hrot from HCEF}
H_\mathrm{rot}^\mathrm{CEF} \bigl( \omega_{xy} \bigr)
= 4A_4^4 H_z^{\alpha} \omega_{xy} - 8A_4^4 H_4^4 \bigl( \omega_{xy} \bigr)^2
.\end{aligned}$$ The first term of Eq. (\[Hrot from HCEF\]) represents the linear coupling of the rotation $\omega_{xy}$ to the electric hexadecapole $H_z^\alpha = \sqrt{35}x'y' \bigl( x'^2 - y'^2 \bigr) / \bigl( 2r^4 \bigr)$ with $A_2$ symmetry of the $D_{2d}$ point group [@Kuramoto]. In Fig. \[Fig5\](c), we schematically show the hexadecapole $H_z^\alpha$ carried by the one-electron state and the rotation $\omega_{xy}$ in the interaction Hamiltonian $H_\mathrm{rot}^\mathrm{CEF} \bigl( \omega_{xy} \bigr)$ of Eq. (\[Hrot from HCEF\]). The second term in Eq. (\[Hrot from HCEF\]) is the coupling of the square of the rotation $\bigl( \omega_{xy} \bigr)^2$ to the electric hexadecapole $H_4^4 = \sqrt{35} \bigl( x'^4 - 6x'^2y'^2 + y'^4 \bigr)/ \bigl( 8r^4 \bigr)$ with full symmetry. The hexadecapole $H_4^4$ has already appeared in the CEF Hamiltonian of Eq. (\[HCEF\]).
The coefficient proportional to the rotation $\omega_{xy}$ in Eq. (\[Rotation of HCEF\]) is identified with the Heisenberg equation of the angular momentum $l_z$ for the CEF Hamiltonian of Eq. (\[HCEF\]). This gives the time derivative of the angular momentum $l_z$, which is identical to the torque $\tau_{xy}$ for a quantum system, as $$\begin{aligned}
\
\label{torque of HCEF}
i \hbar \frac{ \partial l_z} {\partial t} = \left[ l_z, H_\mathrm{CEF} \right] = -i g_\mathrm{H} H_z^{\alpha} = i \tau_{xy}
.\end{aligned}$$ Note that the coupling parameter of $g_\mathrm{H} = 4A_4^4$ in Eq. (\[torque of HCEF\]) has already been given in the CEF Hamiltonian of Eq. (\[HCEF\]). Since the angular momentum $l_z$ is constant for the motion of the one-electron states in the CEF Hamiltonian of Eq. (\[HCEF\]), $l_z$ commutes with $H_\mathrm{CEF}$ as $\left[ l_z, H_\mathrm{CEF} \right] = 0$. Consequently, the hexadecapole $H_z^{\alpha}$ proportional to the time derivative of the angular momentum $\partial l_z / \partial t$ of Eq. (\[torque of HCEF\]) is expected to vanish. This is well known as the rotational invariance for the electronic states in a central force potential such as the CEF potential. The rotation invariance can be naturally understood by the fact that the hexadecapole interaction of Eq. (\[Rotation of HCEF\]) does not change the energy of an electron moving along the contour line of the CEF potential.
The rotation $\omega_{xy}$ due to a transverse ultrasonic wave should also affect the quadrupole-strain interaction of Eq. (\[HQS\]) within the bilinear coupling between the strain and rotation as $$\begin{aligned}
\label{Hrot from HQS}
H_\mathrm{rot}^\mathrm{QS} \bigl( \omega_{xy} \bigr)
&= i \left[ l_z, H_\mathrm{QS} \right] \omega_{xy}
\nonumber \\
&= 2g_{x'^2-y'^2}O_{x'y'}\varepsilon_{xy}\omega_{xy}
\nonumber \\
&\qquad
-2g_{x'y'}O_{x'^2-y'^2}\varepsilon_{x^2-y^2}\omega_{xy}
.\end{aligned}$$ Here, the commutation relation among the Pauli matrices of Eq. (\[Pauli matrix Commutation relation\]) is employed. The perturbation energies for the orbital states $l = y'z$ and $zx'$ with energy $E_l^0$ due to the perturbation Hamiltonian of Eq. (\[Hrot from HQS\]) are written as $$\begin{aligned}
\label{Perturbation for l}
E_l \bigl( \omega_{xy} \bigr)
&= E_l^0
+ \left\langle l \left| H_\mathrm{rot}^\mathrm{QS} \bigl( \omega_{xy} \bigr) \right| l \right\rangle
\nonumber \\
&= E_l^0
+ 2g_{x'^2-y'^2} \left\langle l \left| O_{x'y'} \right| l \right\rangle \varepsilon_{xy}\omega_{xy}
\nonumber \\
&\qquad \qquad
-2g_{x'y'} \left\langle l \left| O_{x'^2-y'^2} \right| l \right\rangle \varepsilon_{x^2-y^2}\omega_{xy}
.\end{aligned}$$ The null of the diagonal elements of the Pauli matrix of $\boldsymbol{O}_{x'y'}$ of Eq. (\[Matrix Ox’y’\]) diminishes the second term $2g_{x'^2-y'^2} \left\langle l \left| O_{x'y'} \right| l \right\rangle$ in the second line in Eq. (\[Perturbation for l\]). Although the matrix of $\boldsymbol{O}_{x'^2-y'^2}$ of Eq. (\[Matrix Ox’2-y’2\]) possesses diagonal elements, we may ignore the third term $- 2g_{x'y'} \left\langle l \left| O_{x'^2-y'^2} \right| l \right\rangle$ in the second line in Eq. (\[Perturbation for l\]) because the little softening of $(C_{11}-C_{12})/2$ indicates a small quadrupole-strain coupling constant of $g_{x'y'} << 1$. Presumably, the strain-rotation bilinear coupling terms of Eq. (\[Hrot from HQS\]) have little effect on the present system with degenerate $y'z$ and $zx'$ orbitals.
The applied magnetic field, however, breaks the rotational invariance for the one-electron states in the CEF potential. This effect of rotation on the elastic properties in applied magnetic fields has been theoretically investigated [@Dohm; @Thalmeier] and experimentally verified by ultrasonic measurements on the antiferro-magnetic compound MnF$_2$ [@Melcher] and the rare-earth compounds TmSb [@Wang; @and; @Luthi], CeAl$_2$ [@Luthi; @and; @Ligner], CeB$_6$ [@Luthi; @CeB6], Ce$_{0.5}$La$_{0.5}$B$_6$ [@Goto; @Ce0.5La0.5B6], and HoVO$_4$ [@HoVO4]. Furthermore, the effect of rotation on the quantum oscillation of elastic constants of transverse ultrasonic waves for Fermi surfaces with an ellipsoidal shape has been discussed [@Kataoka; @and; @Goto]. In Sect. 4.6, we show the coupling of the rotation associated with the transverse acoustic phonons to the hexadecapole carried by two-electron states.
Quadrupole-strain interaction
-----------------------------
As shown in Eq. (20), the quadrupoles carried by electrons in the CEF Hamiltonian interact to the strains of transverse ultrasonic waves. This quadrupole-strain interaction brings about the sizable softening of $C_{66}$ and the divergence of the attenuation coefficients $\alpha_{66}$ and $\alpha_{\mathrm{L}[110]}$ around the structural transition. In order to properly describe the structural transition of the iron pnictide, we will consider the interaction of the quadrupole $O_{x'^2-y'^2} \left( \boldsymbol{r}_i \right)$ of band electrons to the transverse acoustic phonons carrying the strain $\varepsilon_{xy} \left( \boldsymbol{r}_i \right)$ for the position vector $\boldsymbol{r}_i$ per electron as $$\begin{aligned}
\label{HQS by field operator}
H_\mathrm{QS}
= - g_{x'^2-y'^2}
\sum_{l_1, l_2 = y'z, zx'} \sum_{\sigma = \uparrow, \downarrow}
& \int d\boldsymbol{r}_i
\mathit{\Psi}_{l_1, \sigma}^\ast \left( \boldsymbol{r}_i \right)
O_{x'^2-y'^2} \left( \boldsymbol{r}_i \right)
\nonumber \\
& \times \mathit{\Psi}_{l_2, \sigma} \left( \boldsymbol{r}_i \right)
\varepsilon_{x'^2-y'^2} \left( \boldsymbol{r}_i \right)
.\end{aligned}$$ The electron field operators of $\mathit{\Psi}_{y'z, \sigma} \left( \boldsymbol{r}_i \right)$ and $\mathit{\Psi}_{zx', \sigma} \left( \boldsymbol{r}_i \right)$ at position $\boldsymbol{r}_i$ acting on the degenerate $y'z$ and $zx'$ bands with spin orientation $\sigma$ in Eq. (\[HQS by field operator\]) are written as [@K.; @Yoshida] $$\begin{aligned}
\label{Filed operator 1.1}
\mathit{\Psi}_{l, \sigma} \left( \boldsymbol{r}_i \right)
= d_{i, l, \sigma} \psi_l \left( \boldsymbol{r}_i \right) v_\sigma \left( \boldsymbol{r}_i \right)
, \\
\label{Filed operator 1.2}
\mathit{\Psi}_{l, \sigma}^\ast \left( \boldsymbol{r}_i \right)
= d_{i, l, \sigma}^\dagger \psi_l^\ast \left( \boldsymbol{r}_i \right) v_\sigma^\ast \left( \boldsymbol{r}_i \right)
.\end{aligned}$$ Here, $l$ denotes the band suffix of $y'z$ and $zx'$, and $v_\sigma \left( \boldsymbol{r}_i \right)$ is the spin function of $\alpha \left( \boldsymbol{r}_i \right)$ for up-spin $\sigma = \uparrow$ or $\beta \left( \boldsymbol{r}_i \right)$ for down-spin $\sigma = \downarrow$. The annihilation operator $d_{i, l, \sigma}$ and creation operator $d_{i, l, \sigma}^\dagger$ are expressed by Fourier transforms as below $$\begin{aligned}
\label{Fourier transformation of d 1}
d_{i, l, \sigma}
&= \sum_{ \boldsymbol{k}}
d_{\boldsymbol{k}, l, \sigma} e^{i\boldsymbol{k} \cdot \boldsymbol{r}_i}
, \\
\label{Fourier transformation of d 2}
d_{i, l, \sigma}^{\dagger}
&= \sum_{\boldsymbol{k}}
d_{\boldsymbol{k}, l, \sigma}^{\dagger}e^{-i\boldsymbol{k} \cdot \boldsymbol{r}_i }
.\end{aligned}$$ Here, $\boldsymbol{k}$ is the wavevector of an electron. We express the Hamiltonian $H_\mathrm{K}$ for the electrons accommodated in the degenerate $y'z$ and $zx'$ bands, which play a significant role in the appearance of the quadrupole ordering in the system as $$\begin{aligned}
\label{HK}
H_\mathrm{K}
&= \sum_{\boldsymbol{k}} \sum_{\sigma}
\left[
\varepsilon_{y'z} \left( \boldsymbol{k} \right)
d_{\boldsymbol{k}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma} \right.
\nonumber \\
&\left. \qquad \qquad \qquad
+\varepsilon_{zx'} \left( \boldsymbol{k} \right)
d_{\boldsymbol{k}, zx', \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
\right]
.\end{aligned}$$ Here, $\varepsilon_{l} \left( \boldsymbol{k} \right)$ for suffix $l = y'z$ or $zx'$ is the energy from the Fermi level.
The quadrupole-strain interaction of Eq. (\[HQS by field operator\]) is rewritten in terms of Fourier transforms as $$\begin{aligned}
\label{HQS 2nd quant.}
H_\mathrm{QS}
= - G_{x'^2-y'^2}
\sum_{\boldsymbol{k}, \boldsymbol{q} }
O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q}} \varepsilon_{xy} \left( \boldsymbol{q} \right)
.\end{aligned}$$ Here, we use the coupling constant $G_{x'^2-y'^2} = \sqrt{3} g_{x'^2-y'^2} /7$. The interaction Hamiltonian of Eq. (\[HQS 2nd quant.\]) means that the electrons with wavevector $\boldsymbol{k}$ bearing the quadrupole $O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} }$ are scattered by the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonons with wavevector $\boldsymbol{q}$. The quadrupole $O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} }$ in Eq. (\[HQS 2nd quant.\]) is described in terms of the annihilation and creation operators of Eqs. (\[Fourier transformation of d 1\]) and (\[Fourier transformation of d 2\]) as $$\begin{aligned}
\label{Ox'2-y'2 2nd quant.}
O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} }
= \sum_{\sigma}
\left( - d_{\boldsymbol{k} + \boldsymbol{q}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma}
+ d_{\boldsymbol{k} + \boldsymbol{q}, zx', \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
\right)
.\end{aligned}$$ Note that the quadrupole $O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} }$ in Eq. (\[Ox’2-y’2 2nd quant.\]) is simply described by the difference between the occupation numbers for the degenerate $y'z$ and $zx'$ bands at the long-wavelength limit of $ \left| \boldsymbol{q} \right| = 2 \pi / \lambda \rightarrow 0$ of the transverse acoustic phonons carrying the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ as $$\begin{aligned}
\label{Ox'2-y'2 as number operator}
O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} = 0}
&= \sum_{\sigma}
\left( - d_{\boldsymbol{k}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma}
+ d_{\boldsymbol{k}, zx', \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
\right)
\nonumber \\
&= \sum_{\sigma}
\left( -n_{\boldsymbol{k}, y'z, \sigma} + n_{\boldsymbol{k}, zx', \sigma} \right)
.\end{aligned}$$ In Sect. 4.6, we will use the quadrupole $O_{x'y', \boldsymbol{k}, \boldsymbol{q} }$ expressed as $$\begin{aligned}
\label{Ox'y' 2nd quant.}
O_{x'y', \boldsymbol{k}, \boldsymbol{q} }
= \sum_{\sigma}
\left (
d_{\boldsymbol{k} + \boldsymbol{q}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
+ d_{\boldsymbol{k} + \boldsymbol{q}, zx', \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma}
\right).\end{aligned}$$
The strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ in Eq. (\[HQS 2nd quant.\]) is expressed by the annihilation and creation operators of phonons defined in Eq. (\[Phonon energy\]) as $$\begin{aligned}
\label{Strain 2nd quant.}
\varepsilon_{xy} \left( \boldsymbol{q} \right)
=& \frac{i}{2} \sqrt{\mathstrut \frac{\hbar} {2V\rho_\mathrm{M} \omega_{y} \left( \boldsymbol{q} \right)}} q_{x}
\left( a_{y,\boldsymbol{q}} - a_{y, -\boldsymbol{q}}^{\dagger} \right)
\nonumber \\
&+ \frac{i}{2} \sqrt{\mathstrut \frac{\hbar} {2V\rho_\mathrm{M} \omega_{x} \left( \boldsymbol{q} \right)}} q_{y}
\left( a_{x, \boldsymbol{q}} - a_{x, -\boldsymbol{q}}^{\dagger} \right)
.\end{aligned}$$ The strain $\varepsilon_{xy}$ induced by a transverse ultrasonic wave with frequency as low as 100 MHz is identified with $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ of Eq. (\[Strain 2nd quant.\]) in the long-wavelength limit of $ \left| \boldsymbol{q} \right| = 2 \pi / \lambda \rightarrow 0$ as $\varepsilon_{xy} = \varepsilon_{xy} \left( \boldsymbol{q} = 0 \right)$. In the case of the quadrupole-strain interaction of Eq. (\[HQS 2nd quant.\]) mediated by thermally excited transverse acoustic phonons, the emission of phonons carrying the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ scatters the electron state of momentum $\hbar \boldsymbol{k}$ to the state of $\hbar \boldsymbol{k} + \hbar \boldsymbol{q}$ and the absorption of phonons scatters the electron state of $\hbar \boldsymbol{k}$ to the state of $\hbar \boldsymbol{k} - \hbar \boldsymbol{q}$.
The canonical transformation for the electron-phonon scattering processes by the quadrupole-strain interaction $H_\mathrm{QS}$ of Eq. (\[HQS 2nd quant.\]) provides us with the electron interaction mediated by the transverse acoustic phonons as [@QTS] $$\begin{aligned}
\label{HindQQ}
H_\mathrm{ind}^\mathrm{QQ}
= & -\sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q}} \sum_{\sigma, \sigma'}
D_{y'z}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
d_{\boldsymbol{k} - \boldsymbol{q}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma}
d_{\boldsymbol{k}' + \boldsymbol{q}, y'z, \sigma'}^{\dagger} d_{\boldsymbol{k}', y'z, \sigma'}
\nonumber \\
& +\sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q}} \sum_{\sigma, \sigma'}
D_{y'z}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
d_{\boldsymbol{k} - \boldsymbol{q}, y'z, \sigma}^{\dagger} d_{\boldsymbol{k}, y'z, \sigma}
d_{\boldsymbol{k}' + \boldsymbol{q}, zx', \sigma'}^{\dagger} d_{\boldsymbol{k}', zx', \sigma'}
\nonumber \\
& +\sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q}} \sum_{\sigma, \sigma'}
D_{zx'}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
d_{\boldsymbol{k} - \boldsymbol{q} , zx', \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
d_{\boldsymbol{k}' + \boldsymbol{q}, y'z, \sigma'}^{\dagger} d_{\boldsymbol{k}', y'z, \sigma'}
\nonumber \\
&- \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q}} \sum_{\sigma, \sigma'}
D_{zx'}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
d_{\boldsymbol{k} - \boldsymbol{q}, zx', \sigma}^{\dagger} d_{\boldsymbol{k}, zx', \sigma}
d_{\boldsymbol{k}' + \boldsymbol{q}, zx', \sigma'}^{\dagger} d_{\boldsymbol{k}', zx', \sigma'}
.\end{aligned}$$ Here, the coupling coefficient $D_{l}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ in Eq. (\[HindQQ\]) for $l = y'z$ and $zx'$ is given by $$\begin{aligned}
\label{DQQ}
&
D_l^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
= - \frac{1}{2} G_{x'^2-y'^2}^2
\nonumber \\
& \times
\left\{
\frac{\hbar} {2V\rho_\mathrm{M} \omega_{y} \left( \boldsymbol{q} \right)} q_{x}^2
\frac{\hbar \omega_y \left( \boldsymbol{q} \right)}
{
\left[
\varepsilon_l \left( \boldsymbol{k} \right) - \varepsilon_l \left( \boldsymbol{k} - \boldsymbol{q} \right)
\right]^2
- \hbar^2 \omega_y \left( \boldsymbol{q} \right) ^2
} \right.
\nonumber \\
&\left.
+ \frac{\hbar} {2V\rho_\mathrm{M} \omega_{x} \left( \boldsymbol{q} \right)} q_{y}^2
\times \frac{\hbar \omega_x \left( \boldsymbol{q} \right)}
{\left[ \varepsilon_l \left( \boldsymbol{k} \right) - \varepsilon_l \left( \boldsymbol{k} - \boldsymbol{q} \right) \right]^2
- \hbar^2 \omega_x \left( \boldsymbol{q} \right)^2}
\right\}
.\end{aligned}$$ The four independent scattering processes involving virtually excited one-phonon states due to the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ in Eq. (\[HQS 2nd quant.\]) are schematically pictured in Fig. \[Fig6\]. The processes in Fig. \[Fig6\](a) indicate the scattering of the electrons accommodated in the same band $y'z$ and those in Fig. \[Fig6\](d) indicate the scattering in the same band $zx'$. The processes in Figs. 6(b) and 6(c) indicate the scattering between two electrons in the different bands $y'z$ and $zx'$.
![ The indirect quadrupole interaction $H_\mathrm{ind}^\mathrm{QQ}$ of Eq. (\[HindQQ\]) gives four scattering processes between the one-electron states through the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ of a transverse acoustic phonon, while indirect hexadecapole interaction $H_\mathrm{ind}^\mathrm{HH}$ of Eq. (\[HindHH1\]) gives four scattering processes between the two-electron states through the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ of a transverse acoustic phonon. []{data-label="Fig6"}](68060Fig6.pdf){width="48.00000%"}
Supposing the identity of the energy $\varepsilon_{y'z} \left( \boldsymbol{k} \right) = \varepsilon_{zx'} \left( \boldsymbol{k} \right)$ for electrons with a small wavevector $\boldsymbol{k}$ located near the Fermi level, we take the indirect quadrupole interaction coefficient $D_{y'z}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right) = D_{zx'}^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right) = D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ in the tetragonal phase. Using the quadrupole $O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q}}$ expressed in Eq. (\[Ox’2-y’2 2nd quant.\]), we reduce the quadrupole interaction Hamiltonian of Eq. (\[HindQQ\]) as $$\begin{aligned}
\label{HindQQ_2}
H_\mathrm{ind}^\mathrm{QQ}
= - \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q}}
D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
O_{x'^2-y'^2, \boldsymbol{k}, -\boldsymbol{q} } O_{x'^2-y'^2, \boldsymbol{k}', \boldsymbol{q} }
.\end{aligned}$$ In order to understand the frequency dependence of the ultrasonic attenuation of $\alpha_{\mathrm{L} [110]}$ in Fig. \[Fig3\](a), we will examine the indirect quadrupole interaction coefficient $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ in Eq. (\[HindQQ\_2\]), which strongly depends on the magnitude of wavevectors $\boldsymbol{k}$ and $\boldsymbol{k}'$ for the interacting electrons and $\boldsymbol{q}$ for the transverse acoustic phonons participating in the scattering. Taking the electron energies of $\varepsilon_{y'z} \left( \boldsymbol{k} \right)
= \varepsilon_{zx'} \left( \boldsymbol{k} \right)
= \varepsilon \left( \boldsymbol{k} \right)
= \hbar^2 \left| \boldsymbol{k} \right|^2 / 2m_e$ with effective electron mass $m_e$ and transverse acoustic phonon energies of $\hbar \omega_x \left( \boldsymbol{q} \right)
= \hbar \omega_y \left( \boldsymbol{q} \right)
=\hbar \omega \left( \boldsymbol{q} \right)
= \hbar v_{66} |\boldsymbol{q}|$ with ultrasonic velocity $v_{66}$, we obtain the indirect quadrupole interaction coefficient $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of Eq. (\[HindQQ\_2\]) as $$\begin{aligned}
\label{DQQ2}
D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)
&= - G_{x'^2-y'^2}^2
\frac{\hbar} {2V\rho_\mathrm{M} \omega \left( \boldsymbol{q} \right)} q^2
\nonumber \\
&\qquad \quad \times
\frac{\hbar \omega \left( \boldsymbol{q} \right)}
{
\left[
\varepsilon \left( \boldsymbol{k} \right) - \varepsilon \left( \boldsymbol{k} - \boldsymbol{q} \right)
\right]^2
- \hbar^2 \omega \left( \boldsymbol{q} \right) ^2
}
.\end{aligned}$$
In the small-wavenumber limit of $| \boldsymbol{q} | \rightarrow 0$ for the transverse acoustic phonons, the indirect quadrupole interaction coefficient of Eq. (\[DQQ2\]) is reduced to the simple formula $$\begin{aligned}
\label{DQQ(k,0)}
D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
= - \frac{G_{x'^2-y'^2}^2}{2 V \rho_\mathrm{M} }
\frac{1} {\left( \displaystyle \frac{\hbar} {m_e}\right)^2 \left( k^2 - \displaystyle \frac{m_e^2v_{66}^2} {\hbar^2} \right)}
.\end{aligned}$$ The indirect quadrupole interaction coefficient of Eq. (\[DQQ(k,0)\]) for electrons with a small wavenumber of $-\big| \boldsymbol{k}_\mathrm{b}^\mathrm{Q} \bigr| < k < \big| \boldsymbol{k}_\mathrm{b}^\mathrm{Q} \bigr| = m_e v_{66}/ \hbar$ has a positive sign, $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right) > 0$, showing the ferro-type quadrupole interaction, while the interaction coefficient for a relatively large wavenumber $k > \big| \boldsymbol{k}_\mathrm{b}^\mathrm{Q} \bigr| = m_e v_{66}/ \hbar$ possesses a negative sign, $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right) < 0$, indicating the antiferro-type quadrupole interaction. The ferro-type interaction of $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right) > 0$ brings about the ferro-quadrupole ordering accompanying the structural transition, as actually observed in the under-doped compound $x = 0.036$.
It is worth estimating the boundary wavenumber $k_\mathrm{b}^\mathrm{Q} = \left| \boldsymbol{k}_\mathrm{b}^\mathrm{Q} \right|$ and the corresponding energy for the present iron pnictide. In the compound with $x = 0.036$, the ultrasonic velocity $v_{66} = 772$ m/s at 65 K in the vicinity of the structural transition (1780 m/s at 100 K far above the transition) gives the boundary wavenumber $k_\mathrm{b}^\mathrm{Q} = 6.7 \times 10^6 $ m$^{-1}$ ($15 \times 10^6$ m$^{-1}$), the boundary wavelength $\lambda_\mathrm{b}^\mathrm{Q} = 0.94 \times 10^{-6}$ m ($0.41 \times 10^{-6}$ m), and the corresponding frequency $f_\mathrm{b}^\mathrm{Q} = \varepsilon \left( k_\mathrm{b}^\mathrm{Q} \right)/h = 0.41$ GHz (2.2 GHz). Here, we take the electron rest mass as $m_e$. In the low-energy regime below 2 GHz $\approx$ 100 mK, therefore, the electrons bearing the quadrupoles $O_{x'^2-y'^2, \boldsymbol{k}, - \boldsymbol{q} }$ and $O_{x'^2-y'^2, \boldsymbol{k}', \boldsymbol{q} }$ in Eq. (\[HindQQ\_2\]) interact with each other through the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ in the manner of the ferro-type quadrupole interaction. Actually, the relaxation rate $\tau^{-1}$ observed around the structural transition for $x = 0.036$ in Fig. \[Fig3\](b) is always smaller than the estimated crossover frequency of about 2 GHz. This implies that the transverse acoustic waves with frequencies below 2 GHz exhibit the full extent of elastic softening and considerable damping in the vicinity of the structural transition point, while the acoustic waves with frequencies higher than 2 GHz exhibit less softening and damping. As shown in Fig. \[Fig2\](e), the ultrasonic measurements with frequencies less than 260 MHz actually exhibited the full amount of softening in $C_{66}$, while the Raman scattering measurements of the transverse acoustic phonons with frequencies as high as 20 GHz in the end material $x = 0$ exhibited only 25% of the full amount of softening [@Gallais].
Quadrupole interaction
----------------------
The electrons accommodated in the degenerate $y'z$ and $zx'$ bands carry the quadrupoles. In addition to the indirect quadrupole interaction $H_\mathrm{ind}^\mathrm{QQ}$ mediated by the strain $\varepsilon_{xy}$ presented in Sect. 4.3, we expect direct quadrupole interactions through the long-range Coulomb potential between electrons at positions $\boldsymbol{r}_i$ and $\boldsymbol{r}_j$, expressed as
$$\begin{aligned}
\label{H_Coulomb}
&\left\langle l_1 \ \sigma, l_2 \ \sigma' \left| H_\mathrm{Coulomb} \right| l_3 \ \sigma', l_4 \ \sigma \right\rangle
\nonumber \\
& \ \ \ \ \ \ \ \ \ \
= \sum_{i \leq j} \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{l_1}^{\ast} \bigl( \boldsymbol{r}_i \bigr) v_\sigma^\ast \bigl( \boldsymbol{r}_i \bigr)
\psi_{l_2}^{\ast} \bigl( \boldsymbol{r}_j \bigr) v_{\sigma'}^\ast \bigl( \boldsymbol{r}_j \bigr)
\nonumber \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times \frac{e^2} {\left| \boldsymbol{r}_i - \boldsymbol{r}_j \right|}
\psi_{l_3} \bigl( \boldsymbol{r}_j \bigr) v_{\sigma'} \bigl( \boldsymbol{r}_j \bigr)
\psi_{l_4} \bigl( \boldsymbol{r}_i \bigr)v_\sigma \bigl( \boldsymbol{r}_i \bigr)
.\end{aligned}$$
Here, $l_n$ ($n = 1, 2, 3, 4$) denote suffixes of $y'z$ and $zx'$ orbital bands and $v_\sigma$ is the spin function in Eqs. (\[Filed operator 1.1\]) and (\[Filed operator 1.2\]). The electric multipole expansion for the Coulomb interaction in Eq. (\[H\_Coulomb\]) for an isotropic space with $r_i < r_j$ provides terms consisting of the monopole, dipole, and quadrupole as [@Kuramoto; @Arfken] $$\begin{aligned}
\label{Multipole expansion}
\frac{e^2} {\left| \boldsymbol{r}_i - \boldsymbol{r}_j \right| }
&= \frac{e^2} {r_j}
\sum_{k=0}^{\infty}
\frac{4\pi} {2k+1} \left( \frac{r_i} {r_j} \right)^k
\sum_{m=-k}^{k}
Y_k^m \bigl( \theta_i, \varphi_i \bigr) Y_k^{m\ast} \bigl( \theta_j, \varphi_j \bigr)
\nonumber \\
&= \frac{e^2} {r_j}
+ e^2 \frac{r_i} {r_j^2}
\left[
\frac{x_i}{r_i} \frac{x_j}{r_j} + \frac{y_i}{r_i} \frac{y_j}{r_j} + \frac{z_i}{r_i} \frac{z_j}{r_j}
\right]
\nonumber \\
&\quad
+ e^2 \frac{r_i^2} {r_j^3}
\left[
O_{3z^2-r^2} \bigl( \boldsymbol{r}_i \bigr) O_{3z^2-r^2} \bigl( \boldsymbol{r}_j \bigr)
+ O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
\right.
\nonumber \\
& \qquad \qquad
+ O_{y'z} \bigl( \boldsymbol{r}_i \bigr) O_{y'z} \bigl( \boldsymbol{r}_j \bigr)
+ O_{zx'} \bigl( \boldsymbol{r}_i \bigr) O_{zx'} \bigl( \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad \qquad \quad
\left.+ O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
\right]
.\end{aligned}$$ According to the multipole expansion of Eq. (\[Multipole expansion\]), we identify the electric quadrupole with the order parameter of the structural transition in the present iron pnictide. In our attempt to clarify the spontaneous symmetry breaking of the degenerate $y'z$ and $zx'$ orbital bands in the tetragonal lattice, we express the direct quadrupole interaction $H_\mathrm{C}^\mathrm{QQ}$ in terms of the quadrupole $O_{x'^2-y'^2}$ with the $B_2$ symmetry and $O_{x'y'}$ with $B_1$ symmetry as $$\begin{aligned}
\label{HCQQ}
H_\mathrm{C}^\mathrm{QQ}
= & - \sum_{i \leq j}
J_\mathrm{C} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
\nonumber \\
&\times
\left[
O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
+ O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
\right]
.\end{aligned}$$ Here, we adopt the Coulomb interaction coefficient $J_\mathrm{C} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)$, which depends on the relative position $\boldsymbol{r}_i - \boldsymbol{r}_j$. The quadrupole interaction of $O_{3z^2 - r^2}$ with $A_1$ symmetry is excluded because symmetry breaking is not expected. The case of quadrupoles $O_{y'z}$ and $O_{zx'}$ being absent in the $y'z$ and $zx'$ bands is beyond the scope of this study. Note that the monopole interaction among the electrons in Eq. (\[Multipole expansion\]) is effectively included in the band model concerned and that the dipole interaction is irrelevant for the present lattice bearing the inversion symmetry in the $xy$ plane.
Using the expressions $O_{x'y'} = \sqrt{3}\tau_x / 7$ in Eq. (\[Matrix Ox’y’\]) and $O_{x'^2 - y'^2} = \sqrt{3}\tau_z / 7$ in Eq. (\[Matrix Ox’2-y’2\]), we map the quadrupole interaction of Eq. (\[HCQQ\]) on the ideal $xz$ model. In the present real system, however, the deviation of the anisotropic quadrupole interaction from the ideal $xz$ model is expected. From the viewpoint of the symmetry of the tetragonal lattice, the mutual quadrupole interactions of $O_{x'^2-y'^2}$ with $B_2$ symmetry and $O_{x'y'}$ with $B_1$ symmetry may be different. Actually, it has been reported that the mixing between the 4$p$ states of As ions and the 3$d$ states of Fe ions brings about an anisotropic quadrupole interaction [@Yamada]. The quadrupoles $O_{x'^2-y'^2}$ at positions $\boldsymbol{r}_i$ and $\boldsymbol{r}_j$ interact with each other through the strain $\varepsilon_{xy}$ of the transverse acoustic phonons for $C_{66}$, which exhibits the large amount of softening. The quadrupoles $O_{x'y'}$ may also interact with each other mediated by the strain $\varepsilon_{x^2 - y^2}$ of the transverse acoustic phonons for $(C_{11} - C_{12})/2$, which monotonically increases. The former quadrupole interaction of $O_{x'^2-y'^2}$ plays a significant role in the structural transition, while the latter quadrupole interaction of $O_{x'y'}$ has a minor effect on the phase transition. Consequently, the indirect quadrupole interaction mediated by the transverse acoustic phonons also indicates the anisotropic nature of the quadrupole interaction.
Combining the direct quadrupole interaction $H_\mathrm{C}^\mathrm{QQ}$ of Eq. (\[HCQQ\]) followed by the Coulomb potential and the indirect quadrupole interaction $H_\mathrm{ind}^\mathrm{QQ}$ of Eq. (\[HindQQ\_2\]) mediated by the strain $\varepsilon_{xy}$ of the transverse acoustic phonons, we obtain the anisotropic quadrupole interaction $H_\mathrm{QQ}$ specified by parameter $\gamma$ as $$\begin{aligned}
\label{Anisotropic HQQ}
&H_\mathrm{QQ} \bigl( \gamma \bigr)
\nonumber \\
& \quad
= - \sum_{i \leq j} J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad
\times \left[
O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
+ \gamma O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
\right]
.\end{aligned}$$ The quadrupole interaction coefficient $J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)$ in Eq. (\[Anisotropic HQQ\]) is written as the sum of the Coulomb interaction coefficient $J_\mathrm{C} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)$ in the direct interaction of Eq. (\[HCQQ\]) and the indirect quadrupole interaction coefficient $D^\mathrm{QQ}\bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)$ mediated by the strain $\varepsilon_{xy}$ of the transverse acoustic phonons in Eq. (\[DQQ2\]). $$\begin{aligned}
\label{JQ = JC + DQQ}
J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
= J_\mathrm{C} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr) + D^\mathrm{QQ} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
.\end{aligned}$$ The quadrupole interaction coefficient $J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)$ and the anisotropic parameter $\gamma$ in Eq. (\[Anisotropic HQQ\]) play a substantial role in the appearance of the ferro-quadrupole ordering associated with the structural transition and the hexadecapole ordering associated with the superconducting transition.
Quadrupole ordering
-------------------
In order to explain the elastic softening of $C_{66}$ and critical slowing down of the relaxation time $\tau$ around the structural transition in the iron pnictide, we consider the quadrupole interaction Hamiltonian $H_\mathrm{QQ} (\gamma)$ of Eq. (\[Anisotropic HQQ\]) for the case of $\gamma = 0$ mapped on the Ising model. A transverse ultrasonic wave with a small-wavenumber limit of $| \boldsymbol{q} | \rightarrow 0$ induces the strain $\varepsilon_{xy} = \varepsilon_{xy} \left( \boldsymbol{q} = 0 \right)$, which is treated as a classical quantity. According to the quadrupole-strain interaction of Eq. (\[HQS 2nd quant.\]), the strain $\varepsilon_{xy}$ generated by the ultrasonic wave lifts the degenerate $y'z$ and $zx'$ bands, as schematically shown in Fig. \[Fig7\](a). This ultrasonic perturbation gives a quadrupole susceptibility proportional to the reciprocal temperature as $$\begin{aligned}
\label{chiQ}
\chi_\mathrm{Q}
=\frac{N_\mathrm{Q} G_{x'^2-y'^2}^2 / C_{66}^0} {T} = \frac{\mathit{\Delta}_\mathrm{Q} } {T}
.\end{aligned}$$ Here, $N_\mathrm{Q}$ is the number of electrons carrying the ferro-type quadrupole interaction in the small-wavenumber regime of $-k_\mathrm{b}^\mathrm{Q} < k < k_\mathrm{b}^\mathrm{Q}$ for the boundary wavenumber $k_\mathrm{b}^\mathrm{Q}$ as discussed for Eq. (\[DQQ(k,0)\]). The quadrupole-strain interaction energy of $\mathit{\Delta}_\mathrm{Q} = N_\mathrm{Q} G_{x'^2-y'^2}^2 / C_{66}^0 $ in Eq. (\[chiQ\]) is determined in terms of $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of Eq. (\[DQQ(k,0)\]) for the small-wavenumber regime of $|\boldsymbol{q}| \rightarrow 0$ as $$\begin{aligned}
\label{DQQ and deltaQ}
\frac{\mathit{\Delta}_\mathrm{Q} } {2VN_\mathrm{Q} } \frac{ C_{66}^0 } { \rho_\mathrm{M} v_{66}^2 }
&= \frac{G_{x'^2-y'^2}^2 } {2V \rho_\mathrm{M} v_{66}^2 }
\nonumber \\
&= \frac{1}{N_\mathrm{Q} } \sum_{ |\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q} }
D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\nonumber \\
&= \widetilde{D}^\mathrm{QQ}
.\end{aligned}$$ Here, $\sum_{ |\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q} }$ means the sum over the electron states with wavenumber $|\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q}$, that participate in the ferro-type quadrupole interaction. $\widetilde{D}^\mathrm{QQ}$ in Eq. (\[DQQ and deltaQ\]) stands for the effective indirect quadrupole interaction of Eq. (\[DQQ2\]). As shown in Fig. \[Fig4\], the ultrasonic experiments give a quadrupole-strain interaction energy of $\mathit{\Delta}_\mathrm{Q} \sim 20$ K that is almost independent of the Co concentration $x$. The indirect quadrupole interaction energy $D^\mathrm{QQ}$ in Eq. (\[DQQ and deltaQ\]) is enhanced by the softening of $\rho_\mathrm{M} v_{66}^2 = C_{66}$ due to the quadrupole-strain interaction of Eq. (\[HQS 2nd quant.\]).
![ (Color online) (a) Energy splitting of the degenerate $y'z$ and $zx'$ bands by the quadrupole-strain interaction $H_\mathrm{QS}$ of Eq. (\[HQS 2nd quant.\]). (b) Energy splitting of the two-electron states $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+-\]) by the hexadecapole-rotation interaction $H_\mathrm{rot} \bigl( \omega_{xy} \bigr)$ of Eq. (\[Hrot by Bij\]). The transverse ultrasonic wave with amplitude $\xi_y^0$ and frequency $\omega$ induces infinitesimal strain $\varepsilon_{xy} = \sin\theta$ and rotation $\omega_{xy} = \sin\theta$ with $\sin\theta = q_x \xi_y^0 = \left( \omega / v_x \right) \xi_y^0 = \left(2\pi / \lambda_x \right) \xi_y^0$ for a given phase velocity $v_x = v_{66}$. []{data-label="Fig7"}](68060Fig7.pdf){width="48.00000%"}
The softening of $C_{66}$ as a precursor of the structural transition is expressed as $$\begin{aligned}
\label{Elastic with chiQ tilde}
C_{66}
=C_{66}^0 \left( 1 - \widetilde{\chi}_\mathrm{Q} \right)
&= C_{66}^0\left(\frac{T - T_\mathrm{s}^0} {T - \mathit{\Theta}_\mathrm{Q} } \right)
\nonumber \\
& = C_{66}^0\left( 1- \frac{\mathit{\Delta}_\mathrm{Q} } {T - \mathit{\Theta}_\mathrm{Q} } \right)
.\end{aligned}$$ Here, we adopt the renormalized susceptibility $\widetilde{\chi}_\mathrm{Q}$ expressed by the quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q}$ and the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q}$ as $$\begin{aligned}
\label{chiQ tilde}
\widetilde{\chi}_\mathrm{Q}
= \frac{\mathit{\Delta}_\mathrm{Q} } {T-\mathit{\Theta}_\mathrm{Q} }
.\end{aligned}$$ The elastic instability point due to the full softening $C_{66} \rightarrow 0$ gives the structural transition temperature as $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q}$. This has already been used in Eq. (\[Temp. dep. C66\]) for the analysis of the experimentally observed softening of $C_{66}$.
By using the mean field approximation, we obtain the following self-consistent equation giving the temperature dependence of the quadrupole order parameter $O_{x'^2 - y'^2}$: $$\begin{aligned}
\label{Self-consistent of Ox'2-y'2}
\left\langle O_{x'^2 - y'^2} \right\rangle
= \tanh \left[
\widetilde{J}_\mathrm{Q} \left\langle O_{x'^2 - y'^2} \right\rangle /k_\mathrm{B}T \right]
.\end{aligned}$$ Here, $\langle A \rangle$ stands for the thermal average. We use the effective coupling constant $\widetilde{J}_\mathrm{Q}$, which is the average of the Fourier transformed quadrupole interaction of $
J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
= J_\mathrm{C} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr) + D^\mathrm{QQ} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
$ in Eq. (\[JQ = JC + DQQ\]) over the electron states with wavenumber $|\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q}$. $\widetilde{J}_\mathrm{Q}$ is expressed as $$\begin{aligned}
\label{JQ'}
\widetilde{J}_\mathrm{Q}
&= \widetilde{J}_\mathrm{C} + \widetilde{D}^\mathrm{QQ}
\nonumber \\
&= \frac{1}{N_\mathrm{Q} } \sum_{ |\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q} } J_\mathrm{C} \bigl( \boldsymbol{k}, \boldsymbol{q} = 0 \bigr)
+ \frac{1}{N_\mathrm{Q} } \sum_{ |\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q} } D^\mathrm{QQ} \bigl( \boldsymbol{k}, \boldsymbol{q} = 0 \bigr)
.\end{aligned}$$ Here, $\widetilde{J}_\mathrm{C}$ is the effective energy of the direct Coulomb interaction of Eq. (\[HCQQ\]). $\widetilde{D}^\mathrm{QQ}$ is the effective indirect quadrupole interaction coefficient in Eq. (\[DQQ and deltaQ\]). We suppose that the ferro-quadrupole ordering is caused by the quadrupole interaction with the positive coupling constant $\widetilde{J}_\mathrm{Q} > 0$. We expect there to be a non vanishing solution $\langle O_{x'^2 - y'^2} \rangle \neq 0$ at low temperatures below the ferro-quadrupole ordering point of $T_\mathrm{c} = \widetilde{J}_\mathrm{Q} / k_\mathrm{B}$.
The elastic instability due to the full softening $C_{66} \rightarrow 0$ brings about spontaneous lattice distortion accompanying the ferro-quadrupole ordering as $$\begin{aligned}
\label{Mean strain}
\left\langle \varepsilon_{xy} \right\rangle
= \frac{ N_\mathrm{Q} G_{x'^2 - y'^2} \left\langle O_{x'^2 - y'^2} \right\rangle } { C_{66} }
= \sqrt{ \frac{ N_\mathrm{Q} \mathit{\Delta}_\mathrm{Q} } { C_{66} } } \left\langle O_{x'^2 - y'^2} \right\rangle
.\end{aligned}$$ Here, we use the relation between the coupling constant $G_{x'^2 - y'^2}$ and the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q}$ of $G_{x'^2 - y'^2} = \sqrt{ \mathit{\Delta}_\mathrm{Q} C_{66}^0 / N_\mathrm{Q} }$. The ferro-quadrupole ordering of $\bigl\langle O_{x'^2 - y'^2} \bigr\rangle \neq 0$ associated with the spontaneous strain of $\bigl\langle \varepsilon_{xy} \bigr\rangle \neq 0$ in Eq. (\[Mean strain\]) is caused by the structural transition from the tetragonal phase with the high-symmetry space group $D_{4h}^{17}$ to the orthorhombic phase with the low-symmetry space group $D_{2h}^{23}$. The generators of the symmorphic space group $D_{4h}^{17}$ consist of the rotation, reflection, and inversion operations but do not involve the screw and glide operations. The quenching of the $B_{2g}$ symmetry in the mother phase of $D_{4h}^{17}$ associated with the ferro-quadrupole ordering of $\langle O_{x'^2-y'^2} \rangle \neq 0$ and the spontaneous strain $\langle \varepsilon_{xy} \rangle \neq 0$ across the structural transition loses the symmetry operations, which consist of rotation through $\pm \pi / 2$ about the vertical axis of $C_4$, rotations through $\pi$ about the horizontal $x$- and $y$-axes of $2C_2'$, rotation through $\pm \pi /2$ about the vertical axis followed by inversion of $2IC_4$, and mirror reflection in the vertical plane of $2\sigma_\mathrm{v}$ [@Inui; @Group; @International; @Table]. Note that the ferro-type quadrupole ordering denoted in the present paper has been frequently referred to as cooperative Jahn-Teller effects [@Gehring; @and; @Gehring], orbital orderings, or electronic nematic orders [@Murakami; @Takata; @Yi; @Zhang; @Nakayama; @Shimojima; @Kruger; @Lv; @Kontani_2; @Yamakawa; @Kasahara].
The neutron scattering experiments on the end material BaFe$_2$As$_2$ have shown a spontaneous strain $\bigl\langle \varepsilon_{xy} \bigr\rangle$ of $5.08 \times 10^{-3}$ at 5 K in the distorted orthorhombic phase [@Huang], where the order parameter is fully polarized as $\bigl\langle O_{x'^2 - y'^2} \bigr\rangle = 1$. Adopting the quadrupole-strain interaction energy $\mathit{\Delta}_\mathrm{Q} = 21$ K and the elastic constant $C_{66}^0 = 3.19 \times 10^{10}$ J/m$^3$ of BaFe$_2$As$_2$ [@Kurihara; @Dr], we estimate the number of 3$d$ electrons $N_\mathrm{Q}$ in Eq. (\[Mean strain\]) to be $2.84 \times 10^{21}$ cm$^{-3}$, which is approximately one-fourteenth of the total number of electrons of $2N_\mathrm{Fe} = 3.92 \times 10^{22}$ cm$^{-3}$. Furthermore, we deduce the coupling energy $G_{x'^2 - y'^2}$ to be $\sim 4 \times 10^3$ K per electron, which is four times larger than the tentatively estimated $G_{x'^2 - y'^2}$ of $\sim 1 \times 10^3$ K for the localized electron picture in Sect. 3.1. The large quadrupole-strain coupling energy of $G_{x'^2 - y'^2} \sim 4 \times 10^3$ K is caused by the enhanced quadrupole due to the extended orbital radius compatible with the itinerant feature of the 3$d$ electron bands. Taking the number of 3$d$ electrons $N_\mathrm{Q} = 2.84 \times 10^{21}$ cm$^{-3}$ into account, we may roughly estimate the indirect quadrupole interaction energy in BaFe$_2$As$_2$ as $D^\mathrm{QQ} \sim 220$ K at $T = T_\mathrm{s} = 135$ K for $\mathit{\Delta}_\mathrm{Q} = 21$ K, $C_{66}^0 = 3.19 \times 10^{10}$ J/m$^3$, and $C_{66} = \rho_\mathrm{M} v_{66}^2 = 0.281 \times 10^{10}$ J/m$^3$. This value is comparable with the structural transition temperature $T_\mathrm{s}$ of the under-doped compounds. It is worth referring to the extremely enhanced quadrupole-strain interaction energy of $2.8 \times 10^5$ K for the vacancy orbital of a silicon wafer with a large orbital radius, which was verified by means of bulk ultrasonic waves and surface acoustic waves [@Okabe; @Mitsumoto].
The divergence of the relaxation time $\tau$ observed via the ultrasonic attenuation coefficient $\alpha_{66}$ for $x = 0.036$ in Fig. \[Fig3\](b) is expressed in terms of the critical slowing down due to freezing of the ferro-quadrupole order parameter $O_{x'^2 - y'^2}$ at the structural transition temperature $T_\mathrm{s}^0 \cong T_\mathrm{c}^0 = 65$ K. The relaxation time $\tau_\mathrm{Q}$ is written as $$\begin{aligned}
\label{tauQ}
\tau_\mathrm{Q}
= \tau_0 \left| \frac{ T - T_\mathrm{c}^0 } {T_\mathrm{c}^0} \right|^{-1}
\propto
\frac{1} { 1 - \widetilde{\chi}_\mathrm{Q} }.\end{aligned}$$ This expression for $\tau_\mathrm{Q}$ based on the degenerate $y'z$ and $zx'$ bands well reproduces the experimental results analyzed in terms of Eq. (\[Relaxation time\]) for the critical index $z\nu =$ 1 for both tetragonal and orthorhombic phases. The critical slowing down of the quadrupole $O_{x'^2 - y'^2}$ in the present iron pnictide agrees with the critical dynamics of the kinetic Ising model [@Suzuki]. According to the dissipation fluctuation theorem, the critical slowing down of the relaxation time $\tau_\mathrm{Q}$ is expressed by the divergence of the susceptibility for the order parameter of the quadrupole $O_{x'^2-y'^2}$ due to the infinite increase in the correlation length $\zeta$ in the vicinity of the structural transition. By analogy with the ferro-magnetic spin system [@Mori; @Halperin; @and; @Hohenberg], the development of the correlation function of $\langle O_{x'^2-y'^2}(\boldsymbol{k}, t) O_{x'^2-y'^2}(-\boldsymbol{k}, 0) \rangle$ obeying the time decay factor of $\exp[-t/\tau_\mathrm{Q}(\boldsymbol{k})]$ should cause the relaxation time $\tau_\mathrm{Q}(\boldsymbol{k})$ to diverge at the ferro-type ordering point. Knowledge of the diffusion coefficient is necessary for the numerical estimation of $\tau_0$ in Eq. (\[tauQ\]).
Hexadecapole-rotation interaction of two-electron states
--------------------------------------------------------
In the over-doped compound $x = 0.071$ exhibiting superconductivity, the antiferro-type quadrupole interaction with the negative quadrupole interaction energy $\mathit{\Theta}_\mathrm{Q} = -47$ K is verified by analysis of the softening of $C_{66}$ in the normal phase. The softening of $C_{66}$ tends to zero at the fictitious structural instability point at the negative temperature $T_\mathrm{s}^0 = -26.5$ K. This means that the critical slowing down of the relaxation time $\tau$ around the superconducting transition at $T_\mathrm{c}^0 = T_\mathrm{SC} = 23$ K in Fig. \[Fig1\](b) is caused by an appropriate order parameter, which is strictly distinguished from the quadrupole. In the present section, we introduce two-electron states bound by the quadrupole interaction and consider the coupling of the hexadecapole carried by two-electron states to the rotation $\omega_{xy}$ of transverse ultrasonic waves.
Taking the Pauli exclusion principle into account, we express the energy of the anisotropic quadrupole interaction of Eq. (\[Anisotropic HQQ\]) by using the two-electron state $\psi_{\Gamma_{\gamma} }^{S, S_z} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of the Slater determinant with the irreducible representation $\Gamma_\gamma$ of the orbital part, and the spin state denoted by the total spin $S$ and $z$ component $S_z$ as $$\begin{aligned}
\label{Matrix elements of HQQ}
E_\mathrm{QQ}^{\Gamma_\gamma, S, S_z}
&= \bigl\langle \Gamma_\gamma \ S \ S_z | H_\mathrm{QQ}\bigl(\gamma \bigr) | \Gamma_\gamma \ S \ S_z \bigr\rangle
\nonumber \\
&= \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^{S, S_z \ast} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
H_\mathrm{QQ}\bigl( \gamma \bigr)
\psi_{\Gamma_\gamma }^{S, S_z} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
.\end{aligned}$$ From the viewpoint of symmetry for the orbital state, the direct product of the $E$-doublet in the point group symmetry $D_{2d}$ reduces as $E \otimes E = A_1 \oplus A_2 \oplus B_1 \oplus B_2$. Thus, the symmetric $A_1$, $B_1$, and $B_2$ orbital states exist upon exchanging $\boldsymbol{r}_i$ for $\boldsymbol{r}_j$ as well as the antisymmetric $A_2$ state. Furthermore, there are two kinds of spin states consisting of a spin singlet of total spin $S = 0$ that is symmetric upon exchanging $\boldsymbol{r}_i$ for $\boldsymbol{r}_j$ and the antisymmetric spin triplet with $S = 1$. Consequently, we deduce the two-electron state of $\psi_{\Gamma_{\gamma} }^{S, S_z} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ in Eq. (\[Matrix elements of HQQ\]), which is denoted by the orbital state of the symmetry $\Gamma_\gamma = A_1$, $B_2$, and $B_1$ with the spin singlet of $S = 0$ and the orbital state of the $A_2$ symmetry with the spin triplet of $S = 1$ for $S_z = 1$, $0$, and $-1$. As a result, we obtain the Slater determinant in terms of the one-electron orbital states of $\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr)$ and $\psi_{zx'} \bigl( \boldsymbol{r}_i \bigr)$ and the spin states of $\alpha (\boldsymbol{r}_i) $ and $\beta (\boldsymbol{r}_i)$ as $$\begin{aligned}
\label{Two-electron states A1}
%A1
\psi_{\Gamma_\gamma = A_1}^{S = 0, S_z = 0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
+ \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \quad \times \frac{1} {\sqrt{2} }
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
- \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
, \\
%B2
\label{Two-electron states B2}
\psi_{B_2}^{0, 0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
- \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \quad \times
\frac{1} {\sqrt{2} }
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
- \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
, \\
%B1
\label{Two-electron states B1}
\psi_{B_1}^{0, 0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
+ \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \quad \times
\frac{1} {\sqrt{2} }
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
- \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
, \\
%spin triplet
\label{Two-electron states S = 1}
\psi_{A_2}^{1, 1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
- \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
&\quad \times
\alpha \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
, \\
\label{Two-electron states S = 0}
\psi_{A_2}^{1, 0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
- \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \quad \times
\frac{1} {\sqrt{2} }
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
+ \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
, \\
\label{Two-electron states S = -1}
\psi_{A_2}^{1, -1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
- \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) \psi_{y'z} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \quad \times
\beta \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
.\end{aligned}$$
In order to properly describe the energy of the anisotropic quadrupole interaction of Eq. (\[Matrix elements of HQQ\]), we calculate the quadrupole interaction energy of the Hamiltonian $H_\mathrm{QQ} \bigl(\gamma \bigr)$ of Eq. (\[Anisotropic HQQ\]) in terms of the matrix representation for the two-electron states $\psi_{\Gamma_{\gamma} }^{S, S_z} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eqs. (\[Two-electron states A1\])-(\[Two-electron states S = -1\]). $$\begin{aligned}
\label{Matrix of HQQ}
&\boldsymbol{H}_\mathrm{QQ} \bigl(\gamma \bigr)
= - \sum_{i \leq j} J_\mathrm{Q}^{ij}
\nonumber \\
\times &\bordermatrix{
& \psi_{A_1}^{0, 0} & \psi_{B_2}^{0, 0} & \psi_{B_1}^{0, 0} & \psi_{A_2}^{1, 1} & \psi_{A_2}^{1, 0} & \psi_{A_2}^{1, -1} \cr
& 1 + \gamma & 0& 0 & 0 & 0 & 0 \cr
& 0 & 1 - \gamma & 0 & 0 & 0 & 0 \cr
& 0 & 0 & - 1 + \gamma & 0 & 0 & 0 \cr
& 0 & 0 & 0 & - 1 - \gamma & 0 & 0 \cr
& 0 & 0 & 0 & 0 & - 1 - \gamma & 0 \cr
& 0 & 0 & 0 & 0 & 0 & - 1 - \gamma
}
.\end{aligned}$$ Here, the quadrupole interaction energy of $J_\mathrm{Q}^{ij}$ between electrons at positions $\boldsymbol{r}_i$ and $\boldsymbol{r}_j$ is calculated in terms of the radius function $f_d \bigl( r \bigr)$ for the 3$d$ electron in Eqs. (\[wave function y’z\]) and (\[wave function zx’\]) as $$\begin{aligned}
\label{JQij}
J_\mathrm{Q}^{ij}
&= \left( \frac{ \sqrt{3} }{7} \right)^2 \iint dr_i dr_j
J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
r_i^2 f_d \bigl( r_i \bigr)^2
r_j^2 f_d \bigl( r_j \bigr)^2
\nonumber \\
&= \frac{3}{49} \iint dr dr'
J_\mathrm{Q} \bigl( \boldsymbol{r} - \boldsymbol{R}_i - \boldsymbol{r}' + \boldsymbol{R}_j \bigr)
|\boldsymbol{r} - \boldsymbol{R}_i|^2 f_d \bigl( |\boldsymbol{r} - \boldsymbol{R}_i| \bigr)^2
\nonumber \\
& \qquad\qquad\qquad \times |\boldsymbol{r}' - \boldsymbol{R}_j|^2 f_d \bigl( |\boldsymbol{r}' - \boldsymbol{R}_j| \bigr)^2
.\end{aligned}$$ In the integral in the lower line in Eq. (\[JQij\]), we take coordinates $\boldsymbol{r}$ and $\boldsymbol{r}'$ instead of $\boldsymbol{r}_i = \boldsymbol{r} - \boldsymbol{R}_i$ and $\boldsymbol{r}_j = \boldsymbol{r}' - \boldsymbol{R}_j$ for the positions $\boldsymbol{R}_i$ and $\boldsymbol{R}_j$ of Fe$^{2+}$ ion, respectively. The intra-atomic quadrupole interaction of $J_\mathrm{Q}^{ij}$ for the electrons accommodated in the orbital at the same Fe$^{2+}$ ion, $i$. $e$., $\boldsymbol{R}_i = \boldsymbol{R}_j$, is much stronger than the inter-atomic quadrupole interaction for electrons located at different Fe$^{2+}$ ions, $i$. $e$., $\boldsymbol{R}_i \neq \boldsymbol{R}_j$. It is expected that the intra-atomic quadrupole interaction $J_\mathrm{Q}^{i = j}$ for $\boldsymbol{R}_i = \boldsymbol{R}_j$, which is mostly independent of the lattice structure, will show isotropic azimuth angle dependence. This almost isotropic feature of the quadrupole interaction of $J_\mathrm{Q}^{ij}$ dominated by the intra-atomic coupling favors the s-like superconducting energy gap of the present iron pnictide. This will be discussed in Sect. 4.9.
Ultrasonic measurements under pulsed magnetic fields of up to 62 T reveal increases of 7.4% for the softened $C_{66}$ at 84 K in the tetragonal phase of $x = 0.036$ and 3.3% for the softened $C_{66}$ at 30 K in the normal phase of $x = 0.071$ [@M.; @Akatsu_Pulse]. Taking into account the robustness of the softening of $C_{66}$ against magnetic fields in the normal phase above the structural and superconducting transition temperatures, we adopt the two-electron states of the Slater determinant with the spin-singlet states of Eqs. (\[Two-electron states A1\])-(\[Two-electron states B1\]), but disregard those with the spin-triplet states of Eqs. (\[Two-electron states S = 1\])-(\[Two-electron states S = -1\]). We abbreviate the two-electron state with the spin singlet of $\psi_{\Gamma_{\gamma} }^{0, 0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ in Eqs. (\[Two-electron states A1\])-(\[Two-electron states B1\]) to $\psi_{\Gamma_{\gamma} } \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ in the following discussion.
In the case of the highly anisotropic quadrupole interaction with $\gamma = 0$, the quadrupole interaction energies of the $\psi_{A_1}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_{B_2}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ states in Eq. (\[Matrix of HQQ\]) causes the degeneration of each other and the energy of the $\psi_{B_1}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ state deviates from them. This highly anisotropic case of $\gamma \approx 0$ in Eq. (\[Anisotropic HQQ\]) leads to the ferro-quadrupole ordering of $O_{x'^2-y'^2}$ accompanying the structural transition. With increasing the Co concentration $x$ to the QCP of $x_\mathrm{QCP} = 0.061$, the quadrupole interaction of Eq. (\[Anisotropic HQQ\]) develops an almost isotropic feature with $\gamma \stackrel{<}{_\sim} 1$, where the energies of the $\psi_{B_2}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_{B_1}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ states get closer to each other and the energy of the $\psi_{A_1}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ state is different. Since the quadrupoles $O_{x'^2-y'^2}$ of Eq. (\[Matrix Ox’2-y’2\]) and $O_{x'y'}$ of Eq. (\[Matrix Ox’y’\]) and the angular momentum $l_z$ of Eq. (\[Matrix lz\]) obey the commutation relation of Eq. (\[Pauli matrix Commutation relation\]) among the Pauli matrices, we expect quantum fluctuation between the quadrupoles $O_{x'^2-y'^2}$ and $O_{x'y'}$ in the vicinity of the QCP, which is particularly important for explaining the superconductivity accompanying the hexadecapole ordering. The special case of $\gamma = 1$ mapped on the ideal $xz$ model leads to the fully degenerate $\psi_{B_2}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_{B_1}\bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ states, which causes the disappearance of the hexadecapole. This is inconsistent with the experimental observation of the critical slowing down around the superconducting transition temperature.
When the rotation $\omega_{xy}$ is induced by either the thermally excited transverse acoustic phonons or the experimentally generated transverse ultrasonic waves, the phases of the two-electron state of $\psi_{{\Gamma}_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ at positions $\boldsymbol{r}_i$ and $\boldsymbol{r}_j$ in Eq. (\[Matrix elements of HQQ\]) bound by the quadrupole interaction of Eq. (\[Anisotropic HQQ\]) are simultaneously changed by the rotation operator as follows: $$\begin{aligned}
\label{Rotation effect for two-electron state}
\psi_{\Gamma_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
& \rightarrow
\exp \left[ -i l_z \bigl( \boldsymbol{r}_i \bigr) \omega_{xy} \right]
\exp \left[ -i l_z \bigl( \boldsymbol{r}_j \bigr) \omega_{xy} \right]
\nonumber \\
& \qquad \qquad \qquad \qquad \qquad \times
\psi_{\Gamma_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
.\end{aligned}$$ Thus, an infinitesimal amount of rotation $\omega_{xy}$ perturbs the quadrupole interaction Hamiltonian of Eq. (\[Anisotropic HQQ\]) as $$\begin{aligned}
\label{Rotation of Anisotropic HQQ}
&
\left\langle \Gamma_\gamma \left| H_\mathrm{QQ} \bigl( \omega_{xy} \bigr) \right| \Gamma_{\gamma '} \right\rangle
\nonumber \\
&
= \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\exp \left\{ i \left[ l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr) \right] \omega_{xy} \right\}
\nonumber \\
& \qquad \qquad
\times H_\mathrm{QQ} \bigl( \gamma \bigr)
\exp \left\{ -i \left[ l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr) \right] \omega_{xy} \right\}
\psi_{\Gamma_{\gamma'}} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\nonumber \\
&
\approx \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
H_\mathrm{QQ} \bigl( \gamma \bigr)
\psi_{\Gamma_{\gamma'} } \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\nonumber \\
&
+ \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
i \left[
l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr), H_\mathrm{QQ} \bigl( \gamma \bigr)
\right]
\nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times
\psi_{\Gamma_{\gamma'}} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) \omega_{xy}
\nonumber \\
&
+ \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad \qquad \times \left( - \frac{1}{2} \right)
\left[
l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr),
\left[
l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr), H_\mathrm{QQ} \bigl( \gamma \bigr)
\right] \right]
\nonumber \\
& \qquad \qquad \qquad \qquad \times
\psi_{\Gamma_{\gamma'}} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) \bigl( \omega_{xy} \bigr)^2
.\end{aligned}$$ Using the commutation relation of Eq. (\[Pauli matrix Commutation relation\]) for the Pauli matrices, we reduce the perturbation Hamiltonian $H_\mathrm{rot} \bigl( \omega_{xy} \bigr)$ which depends on the rotation $\omega_{xy}$ as follows: $$\begin{aligned}
\label{Rotation Hamiltonian 1}
&H_\mathrm{rot} \bigl( \omega_{xy} \bigr)
\nonumber \\
&= 2\bigl( 1 - \gamma \bigr) \sum_{i \leq j} J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad \times
\left[
O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
+ O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
\right] \omega_{xy}
\nonumber \\
&+ 4\bigl( 1 - \gamma \bigr) \sum_{i \leq j} J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad \times
\left[
O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
- O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
\right] \bigl( \omega_{xy} \bigr)^2
.\end{aligned}$$ From the first term in Eq. (\[Rotation Hamiltonian 1\]), which linearly depends on the rotation $\omega_{xy}$, we identify the electric hexadecapole as $$\begin{aligned}
\label{Hexadecapole}
H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
= O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2-y'^2} \bigl( \boldsymbol{r}_j \bigr)
+ O_{x'^2-y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)
.\end{aligned}$$ The hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[Hexadecapole\]) is invariant upon the exchange of positions $\boldsymbol{r}_i$ and $\boldsymbol{r}_j$ but is antisymmetric upon the exchange of coordinates $x'$ and $y'$. In Fig. \[Fig5\](c), we schematically show the interaction between the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[Hexadecapole\]) carried by the two-electron states and the rotation $\omega_{xy}$ of the transverse acoustic phonons. The hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ operates on the two-electron state that is bound by the quadrupole interaction of Eq. (\[Anisotropic HQQ\]), while the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}' \bigr)$ in Eq. (\[Hrot from HCEF\]) acts on the one-electron state that is trapped in the CEF Hamiltonian of the central force. Note that the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and rotation $\omega_{xy}$ commonly belong to the $A_2$ symmetry of the point group symmetry $D_{2d}$. The coefficient of the second term proportional to $\bigl( \omega_{xy} \bigr)^2$ in Eq. (\[Rotation Hamiltonian 1\]) represents the energy modulation of the anisotropic quadrupole interaction.
Comparing the term proportional to the rotation $\omega_{xy}$ in the expressions of Eq. (\[Rotation of Anisotropic HQQ\]) with that in Eq. (\[Rotation Hamiltonian 1\]), we write the Heisenberg equation of the time derivative for the total angular momentum $l_z \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) = l_z \bigl( \boldsymbol{r}_i \bigr) + l_z \bigl( \boldsymbol{r}_j \bigr)$ which is proportional to the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ as follows: $$\begin{aligned}
\label{Torque for two-electron state}
i\hbar \frac{\partial} {\partial t} l_z \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \left[ l_z \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr), H_\mathrm{QQ} \bigl( \gamma \bigr) \right]
\nonumber \\
&= -2i \bigl( 1 - \gamma \bigr) J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr)
H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
.\end{aligned}$$ Furthermore, the time derivative of the angular momentum in Eq. (\[Torque for two-electron state\]) is identified with the torque $\tau_{xy} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ for the two-electron state bound by the anisotropic quadrupole interaction of Eq. (\[Anisotropic HQQ\]) as $$\begin{aligned}
\label{Define torque}
\hbar \frac{\partial} {\partial t} l_z \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
= \tau_{xy} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
.\end{aligned}$$ Note that the torque $\tau_{xy} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ in Eq. (\[Define torque\]) vanishes for the special case of the ideal isotropic quadrupole interaction with $\gamma = 1$ of Eq. (\[Anisotropic HQQ\]) identified with the ideal $xz$ model. The hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ proportional to the time derivative of the angular momentum in Eq. (\[Torque for two-electron state\]) conserves the time-reversal symmetry, while the angular momentum $l_z \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ itself breaks the time-reversal symmetry.
The hexadecapole-rotation interaction linearly coupled to the rotation $\omega_{xy}$ in Eq. (\[Rotation Hamiltonian 1\]) has the following matrix elements for the two-electron states of $\psi_{\Gamma_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$: $$\begin{aligned}
\label{Matrix elements of Hrot}
&\left\langle \Gamma_\gamma \left| H_\mathrm{rot}^0 \bigl( \omega_{xy} \bigr) \right| \Gamma_{\gamma '} \right\rangle
\nonumber \\
&= 2(1-\gamma) \sum_{i \leq j} \iint d\boldsymbol{r}_i d\boldsymbol{r}_j
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{r}_j \bigr) H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\nonumber \\
& \qquad \qquad \times \psi_{\Gamma_{\gamma'} } \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\omega_{xy}
\nonumber \\
&= 2(1-\gamma) \sum_{i \leq j} \iint d\boldsymbol{r} d\boldsymbol{r}'
\psi_{\Gamma_\gamma}^\ast \bigl( \boldsymbol{r} - \boldsymbol{R}_i, \boldsymbol{r}' - \boldsymbol{R}_j \bigr)
\nonumber \\
&\qquad \qquad \times J_\mathrm{Q} \bigl( \boldsymbol{r}_i - \boldsymbol{R}_i - \boldsymbol{r}_j + \boldsymbol{R}_j \bigr)
H_z^\alpha \bigl( \boldsymbol{r} - \boldsymbol{R}_i, \boldsymbol{r}' - \boldsymbol{R}_j \bigr)
\nonumber \\
&\qquad \qquad \qquad \qquad \times
\psi_{\Gamma_{\gamma'} } \bigl( \boldsymbol{r} - \boldsymbol{R}_i, \boldsymbol{r}' - \boldsymbol{R}_j \bigr)
\omega_{xy}
.\end{aligned}$$ In the integral of Eq. (\[Matrix elements of Hrot\]), we respectively take coordinates $\boldsymbol{r}$ and $\boldsymbol{r}'$ instead of $\boldsymbol{r}_i = \boldsymbol{r} - \boldsymbol{R}_i$ and $\boldsymbol{r}_j = \boldsymbol{r}' - \boldsymbol{R}_j$ for the positions $\boldsymbol{R}_i$ and $\boldsymbol{R}_j$ of Fe$^{2+}$ ions. As mentioned for Eq. (\[JQij\]), the intra-atomic quadrupole interaction between electrons accommodated in the same Fe$^{2+}$ ion with $\boldsymbol{R}_i = \boldsymbol{R}_j$ is dominant over the inter-atomic quadrupole interaction with $\boldsymbol{R}_i \neq \boldsymbol{R}_j$. The hexadecapole formed by the intra-atomic quadrupole interaction plays an important role in the appearance of the hexadecapole ordering in the superconducting state.
Among the two-electron states listed in Eqs. (\[Two-electron states A1\])-(\[Two-electron states S = -1\]), we take the spin-singlet states consisting of the two-electron states $\psi_{\Gamma_\gamma}^{0,0} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) = \psi_{\Gamma_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ for the orbital symmetries of $\Gamma_\gamma = A_1$, $B_2$, and $B_1$ compatible with the magnetic robustness of the present system, while we disregard the spin-triplet state of $\psi_{A_2}^{S = 1, \ S_z} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with $S_z = 1$, $0$, and $-1$. We deduce the hexadecapole-rotation interaction of Eq. (71) for the spin-singlet states as $$\begin{aligned}
\label{Matrix of Hrot off-diagonal}
\boldsymbol{H}_\mathrm{rot}^0 \bigl( \omega_{xy} \bigr)
= - 4 \bigl(1 - \gamma \bigr) \sum_{i \leq j} J_\mathrm{Q}^{ij}
\bordermatrix{
& \psi_{A_1} & \psi_{B_2} & \psi_{B_1} \cr
& 0 & 0 & 0 \cr
& 0 & 0 & 1 \cr
& 0 & 1 & 0
}
\omega_{xy}
.\end{aligned}$$ Here, the matrix elements $\bigl\langle \Gamma_\gamma | H_\mathrm{rot}^0 \bigl( \omega_{xy} \bigr) | \Gamma_{\gamma '} \bigr\rangle$ in Eq. (\[Matrix of Hrot off-diagonal\]) for the two-electron state $\psi_{\Gamma_\gamma} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the symmetries of $\Gamma_\gamma = A_1$, $B_2$, and $B_1$ and the spin singlet of S = 0 of Eqs. (\[Two-electron states A1\])-(\[Two-electron states B1\]) are calculated. The hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the $A_2$ symmetry possesses the off-diagonal elements between the two-electron states $\psi_{B_2} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_{B_1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$, but these matrix elements vanish for the state $\psi_{A_1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$. This is confirmed by the symmetry property of $B_2 \otimes B_1 = A_2$.
We have particular interest in the interplay of the rotation $\omega_{xy}$ with the appearance of the hexadecapole ordering and the superconductivity. Employing the unitary transformation for Eq. (\[Matrix of Hrot off-diagonal\]), we obtain the diagonal representation of the hexadecapole-rotation interaction as $$\begin{aligned}
\label{Matrix of Hrot diagonal}
\boldsymbol{H}_\mathrm{rot}\bigl( \omega_{xy} \bigr)
= - 4 \left(1 - \gamma \right) \sum_{i \leq j} J_\mathrm{Q}^{ij}
\bordermatrix{
& \psi_+ & \psi_- \cr
& 1 & 0 \cr
& 0 & -1
}
\omega_{xy}
.\end{aligned}$$ Here, we adopt the wave function $\psi_h \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with $h= \pm$ for the two-electron states as $$\begin{aligned}
\label{psi+- by y'z and zx'}
\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
\psi_{B_2} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) \pm \psi_{B_1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
&= \frac{1} {\sqrt{2} }
\left[
\frac{ \psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \pm \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) } { \sqrt{2} }
\psi_{y'z} \bigl( \boldsymbol{r}_j \bigr) \right.
\nonumber \\
& \left. \qquad \qquad \pm
\frac{ \psi_{y'z} \bigl( \boldsymbol{r}_i \bigr) \mp \psi_{zx'} \bigl( \boldsymbol{r}_i \bigr) } { \sqrt{2} }
\psi_{zx'} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
&\qquad \times
\frac{1}{\sqrt{2} }
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
- \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
.\end{aligned}$$ By analogy with the Zeeman term for the magnetic dipole moments in a magnetic field, we identify the wave function $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the eigenstate for the hexadecapole corresponding to the right-hand rotation of $h = +$ and the wave function $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with eigenstate for the hexadecapole corresponding to the left-hand rotation of $h = -$. The hexadecapole-rotation interaction of Eq. (\[Matrix of Hrot diagonal\]) indicates that the rotation $\omega_{xy}$ acts as a symmetry-breaking field on the hexadecapole carried by the two-electron states $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+- by y’z and zx’\]).
In order to properly describe the hexadecapole ordering, it is convenient to use the one-electron states $\lambda_\pm \left( \boldsymbol{r}_i \right)$ described in terms of the spherical harmonics of $Y_l^m \left( \theta_i, \varphi_i \right)$ with the orbital quantum number $l = 2$ and the azimuthal quantum number $m = \pm1$ and the radial part of $f_d \left( r_i \right)$ as $$\begin{aligned}
\label{wave function lambda}
\lambda_\pm \left( \boldsymbol{r}_i \right)
&= f_d \left( r_i \right) Y_2^{\pm1} \left( \theta_i, \varphi_i \right)
\nonumber \\
&= -\frac{1}{\sqrt{2} }
\left[
i \psi_{y'z} \left( \boldsymbol{r}_i \right) \pm \psi_{zx'} \left( \boldsymbol{r}_i \right)
\right]
.\end{aligned}$$ We rewrite the two-electron states $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+- by y’z and zx’\]) in terms of the one-electron wave function $\lambda_\pm \left( \boldsymbol{r}_i \right)$ of Eq. (\[wave function lambda\]) as $$\begin{aligned}
\label{psi+-}
\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
&= \frac{1} {\sqrt{2} }
\left[
e^{\mp i \frac{3\pi}{4} } \lambda_{+1} \bigl( \boldsymbol{r}_i \bigr) \lambda_{+1} \bigl( \boldsymbol{r}_j \bigr)
+ e^{\pm i \frac{3\pi}{4} } \lambda_{-1} \bigl( \boldsymbol{r}_i \bigr) \lambda_{-1} \bigl( \boldsymbol{r}_j \bigr)
\right]
\nonumber \\
& \qquad \qquad \times
\frac{1} {\sqrt{2}}
\left[
\alpha \bigl( \boldsymbol{r}_i \bigr) \beta \bigl( \boldsymbol{r}_j \bigr)
- \beta \bigl( \boldsymbol{r}_i \bigr) \alpha \bigl( \boldsymbol{r}_j \bigr)
\right]
.\end{aligned}$$ Note that the eigenstates of $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+-\]) consist of the two-electron states of $\lambda_{+1} \bigl( \boldsymbol{r}_i \bigr) \lambda_{+1} \bigl( \boldsymbol{r}_j \bigr)$ and $\lambda_{-1} \bigl( \boldsymbol{r}_i \bigr) \lambda_{-1} \bigl( \boldsymbol{r}_j \bigr)$, which are superposed on each other while maintaining an orthogonal relation with a phase difference of $\pm 3\pi / 2$.
Taking the spin-singlet state in Eq. (\[psi+-\]) into account, we introduce annihilation operators of $B_{ i, j, \pm, \sigma, \overline{\sigma} }$ and creation operators of $B_{ i, j, \pm, \sigma, \overline{\sigma} }^\dagger$ for the two-electron eigenstates $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of the hexadecapole as $$\begin{aligned}
\label{Operator B by l 1}
B_{i, j, \pm, \sigma, \overline{\sigma} }
= \frac{1}{\sqrt{2}}
\left(
e^{\mp i \frac{3\pi}{4} } l_{ j, +1, \overline{\sigma} } l_{ i, +1, \sigma}
+ e^{\pm i \frac{3\pi}{4} } l_{ j, -1, \overline{\sigma} } l_{ i, -1, \sigma}
\right)
, \\
\label{Operator B by l 2}
B_{i, j, \pm, \sigma, \overline{\sigma} }^\dagger
= \frac{1}{\sqrt{2}}
\left(
e^{\pm i \frac{3\pi}{4} } l_{ i, +1, \sigma}^\dagger l_{ j, +1, \overline{\sigma} }^\dagger
+ e^{\mp i \frac{3\pi}{4} } l_{ i, -1, \sigma}^\dagger l_{ j, -1, \overline{\sigma} }^\dagger
\right)
.\end{aligned}$$ Here, we use annihilation operators $l_{ i, \pm1, \sigma }$ and creation operators $l_{ i, \pm1, \sigma }^\dagger$ for the one-electron eigenstates $\lambda_{\pm1} \left( \boldsymbol{r}_i \right) v_\sigma \left( \boldsymbol{r}_i \right)$ of Eq. (\[wave function lambda\]), which obey the following anticommutation relation for fermions: $$\begin{aligned}
\label{Commutation relation of l}
\left\{
l_{ i, m, \sigma }, l_{ j, m', \sigma'}^\dagger
\right\}
= \delta_{i, j} \delta_{m,m'} \delta_{\sigma, \sigma'}
\ \ \left( m, m' = \pm 1, \ \sigma, \sigma' = \uparrow \mathrm{or} \downarrow \right)
.\end{aligned}$$ These electron operators are written in terms of $d_{ i, l, \sigma }$ and $d_{ i, l, \sigma }^\dagger$ of Eqs. (\[Fourier transformation of d 1\]) and (\[Fourier transformation of d 2\]) as $$\begin{aligned}
\label{Operator l}
l_{ i, \pm1, \sigma }
= - \frac{1}{\sqrt{2}}
\left( i d_{ i, y'z, \sigma } \pm d_{ i, zx', \sigma } \right)
.\end{aligned}$$
By using the two-electron operators of $B_{i, j, \pm, \sigma, \overline{\sigma} }$ of Eq. (\[Operator B by l 1\]) and $B_{i, j, \pm, \sigma, \overline{\sigma} }^\dagger$ of Eq. (\[Operator B by l 2\]), we rewrite the hexadecapole-rotation interaction $H_\mathrm{rot} \bigl( \omega_{xy} \bigr)$ of Eq. (\[Matrix of Hrot diagonal\]) as $$\begin{aligned}
\label{Hrot by Bij}
H_\mathrm{rot} \bigl( \omega_{xy} \bigr)
= -4 \left( 1 - \gamma \right) \sum_{i \leq j} J_\mathrm{Q}^{ij} H_{z, i, j}^\alpha \omega_{xy}
.\end{aligned}$$ Here, we describe the hexadecapole operator $H_{z, i, j}^\alpha$ in terms of the difference between the occupation numbers $N_{i, j, +, \sigma, \overline{\sigma} } = B_{i, j, +, \sigma, \overline{\sigma} }^\dagger B_{i, j, +, \sigma, \overline{\sigma} }$ of the two-electron state $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $N_{i, j, -, \sigma, \overline{\sigma} } = B_{i, j, -, \sigma, \overline{\sigma} }^\dagger B_{i, j, -, \sigma, \overline{\sigma} }$ of $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ as $$\begin{aligned}
\label{Hexadecapole by Bij}
H_{z, i, j}^\alpha
&= \frac{1}{2} \sum_{ \sigma \neq \overline{\sigma} }
\left(
B_{i, j, +, \sigma, \overline{\sigma} }^\dagger B_{i, j, +, \sigma, \overline{\sigma} }
-B_{i, j, -, \sigma, \overline{\sigma} }^\dagger B_{i, j, -, \sigma, \overline{\sigma} }
\right)
\nonumber \\
&= \frac{1}{2} \sum_{ \sigma \neq \overline{\sigma} }
\left( N_{i, j, +, \sigma, \overline{\sigma} } - N_{i, j, -, \sigma, \overline{\sigma} } \right)
.\end{aligned}$$ The two-electron wave function $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+- by y’z and zx’\]), consisting of a linear combination of $\psi_{B_2} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the $B_2$ symmetry and $\psi_{B_1} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the $B_1$ symmetry, possesses the hexadecapole with the $A_2$ symmetry of Eq. (\[Hexadecapole by Bij\]) in the diagonal elements of Eq. (\[Hrot by Bij\]). This is confirmed by the fact that the direct product of $B_1 \oplus B_2$ for the two-electron wave functions $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+- by y’z and zx’\]) is reduced as $(B_1 \oplus B_2) \otimes (B_1 \oplus B_2) = 2A_1 \oplus 2A_2$ for the point group of the $D_{2d}$ symmetry of the Fe$^{2+}$ ion site.
Fourier transforms of the two-electron operators of $B_{i, j, \pm, \sigma, \overline{\sigma} }$ of Eq. (\[Operator B by l 1\]) and $ B_{i, j, \pm, \sigma, \overline{\sigma} }^\dagger$ of Eq. (\[Operator B by l 2\]) give their momentum representations. For example, the Fourier transform of $B_{i, j, +, \sigma, \overline{\sigma} }$ is $$\begin{aligned}
\label{Fourier transformation of B+}
&
B_{i, j, +, \sigma, \overline{\sigma} }
= \frac{1} {\sqrt{2} } \sum_{\boldsymbol{k}_\mathrm{G}, \boldsymbol{k}_\mathrm{R} }
\left(
e^{- i \frac{3\pi}{4} } l_{\frac{1}{2} \boldsymbol{k}_\mathrm{G} - \boldsymbol{k}_\mathrm{R}, +1, \overline{\sigma} }
l_{\frac{1}{2} \boldsymbol{k}_\mathrm{G} + \boldsymbol{k}_\mathrm{R}, +1, \sigma }
\right.
\nonumber \\
&\left. \qquad \qquad \qquad \qquad
+
e^{i \frac{3\pi}{4} } l_{\frac{1}{2} \boldsymbol{k}_\mathrm{G} - \boldsymbol{k}_\mathrm{R}, -1, \overline{\sigma} }
l_{\frac{1}{2} \boldsymbol{k}_\mathrm{G} + \boldsymbol{k}_\mathrm{R}, -1, \sigma }
\right)
\nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \times
e^{ i \boldsymbol{k}_\mathrm{G} \cdot \boldsymbol{r}_\mathrm{G} }
e^{ i \boldsymbol{k}_\mathrm{R} \cdot \boldsymbol{r}_\mathrm{R} }
.\end{aligned}$$ Here, we use the gravity position $\boldsymbol{r}_\mathrm{G} = \bigl( \boldsymbol{r}_i + \boldsymbol{r}_j \bigr)/2$ and gravity momentum $\hbar \boldsymbol{k}_\mathrm{G} = \hbar \boldsymbol{k}_i + \hbar \boldsymbol{k}_j$. It is supposed that the two-electron states are described in terms of the relative coordinate $\boldsymbol{r}_\mathrm{R} = \boldsymbol{r}_i - \boldsymbol{r}_j$ and relative momentum $\hbar \boldsymbol{k}_\mathrm{R} = \bigl( \hbar \boldsymbol{k}_i - \hbar \boldsymbol{k}_j \bigr)/2$ under the constraint that the gravity momentum vanishes as $\hbar \boldsymbol{k}_\mathrm{G} = \hbar \boldsymbol{k}_i + \hbar \boldsymbol{k}_j = 0$. Under this constraint, the motion of the two-electron state is explained only in terms of bound states with the relative momentum $\hbar \boldsymbol{k}_\mathrm{R} = \hbar \boldsymbol{k}$. Consequently, the annihilation operators of $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }$ for the eigenstates of the hexadecapole with the right-hand rotation of $h = +$ and the left-hand rotation of $h = -$ and the creation operators of $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger$ for the conjugate eigenstates are written as $$\begin{aligned}
%B+-
\label{Fourier transformation of B 1}
B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }
= \frac{1} {\sqrt{2} }
\left(
e^{\mp i \frac{3\pi}{4} } l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
+ e^{\pm i \frac{3\pi}{4} } l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma}
\right)
, \\
%B+-*
\label{Fourier transformation of B 2}
B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger
= \frac{1} {\sqrt{2} }
\left(
e^{\pm i \frac{3\pi}{4} } l_{\boldsymbol{k}, +1, \sigma}^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger
+ e^{\mp i \frac{3\pi}{4} } l_{\boldsymbol{k}, -1, \sigma}^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\right)
.\end{aligned}$$ The two-electron operators of $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }$ and $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger$ given by Eqs. (\[Fourier transformation of B 1\]) and (\[Fourier transformation of B 2\]) satisfy the mixed commutation relations [@QTS] $$\begin{aligned}
\label{Commutation relation of B}
%B+- and B+- dagger
&\left[
B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }, B_{\boldsymbol{k}', \pm, \sigma, \overline{\sigma} }^\dagger
\right]
\nonumber \\
&= \left( 1 -
\frac{
n_{\boldsymbol{k}, +1, \sigma} + n_{-\boldsymbol{k}, +1, \overline{\sigma} }
+ n_{\boldsymbol{k}, -1, \sigma} + n_{-\boldsymbol{k}, -1, \overline{\sigma} }
}{2}
\right)
\delta_{\boldsymbol{k}, \boldsymbol{k}' }
, \\
%Commutation relation of B 2
\label{Commutation relation of B 2}
&\left[
B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }, B_{\boldsymbol{k}', \mp, \sigma, \overline{\sigma} }^\dagger
\right]
\nonumber \\
&= \mp i
\frac{
n_{\boldsymbol{k}, +1, \sigma} + n_{-\boldsymbol{k}, +1, \overline{\sigma} }
- n_{\boldsymbol{k}, -1, \sigma} - n_{-\boldsymbol{k}, -1, \overline{\sigma} }
}{2}
\delta_{\boldsymbol{k}, \boldsymbol{k}' }
.\end{aligned}$$ Here, we denote the one-electron number operator as $n_{\boldsymbol{k}, \pm1, \sigma} = l_{\boldsymbol{k}, \pm1, \sigma}^\dagger l_{\boldsymbol{k}, \pm1, \sigma}$. The mixed commutation relations of Eqs. (\[Commutation relation of B\]) and (\[Commutation relation of B 2\]) are required to calculate the hexadecapole interaction in the normal phase and describe the hexadecapole ordering associated with the superconducting transition at $T_\mathrm{SC} = 23$ K for $x = 0.071$.
Hexadecapole interaction
------------------------
In actual crystals of the iron pnictide, the thermally excited transverse acoustic phonon with wavevector $\boldsymbol{q}$ induces the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ as $$\begin{aligned}
\label{Rotation 2nd quant.}
\omega_{xy} \left( \boldsymbol{q} \right)
&= \frac{i}{2} \sqrt{
\mathstrut \frac{\hbar} {2V \rho_\mathrm{M} \omega_{y} \left( \boldsymbol{q} \right) }
} q_{x}
\left( a_{y, \boldsymbol{q}} - a_{y, -\boldsymbol{q}}^{\dagger} \right)
\nonumber \\
& \qquad - \frac{i}{2} \sqrt{
\mathstrut \frac{\hbar} {2V\rho_\mathrm{M} \omega_{x} \left( \boldsymbol{q} \right)}
} q_{y}
\left( a_{x, \boldsymbol{q}} - a_{x, -\boldsymbol{q}}^{\dagger} \right)
.\end{aligned}$$ The two-electron states $\psi_\pm \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ of Eq. (\[psi+-\]) bearing the hexadecapole are scattered by the transverse acoustic phonons carrying the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$. These scattering are expressed in terms of the hexadecapole-rotation interaction in momentum space as $$\begin{aligned}
\label{Hrot by Bk}
&H_\mathrm{rot} \bigl( \omega_{xy} \bigr)
\nonumber \\
& \qquad
= -2 \bigl( 1 - \gamma \bigr)
\sum_{\boldsymbol{k}, \boldsymbol{q}} \sum_{\sigma, \neq \overline{\sigma}}
J_\mathrm{Q} \bigl( \boldsymbol{k}, \boldsymbol{q} \bigr)
\nonumber \\
&\qquad \qquad
\times \left(
B_{\boldsymbol{k} + \boldsymbol{q}, +, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }
- B_{\boldsymbol{k} + \boldsymbol{q}, -, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }
\right) \omega_{xy} \bigl( \boldsymbol{q} \bigr)
\nonumber \\
&\qquad
= -4 \bigl( 1 - \gamma \bigr)
\sum_{\boldsymbol{k}, \boldsymbol{q}}
J_\mathrm{Q} \bigl( \boldsymbol{k}, \boldsymbol{q} \bigr)
H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha \omega_{xy} \bigl( \boldsymbol{q} \bigr)
.\end{aligned}$$ The hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha$ in Eq. (\[Hrot by Bk\]) is written in terms of the two-electron state with wavevector $\boldsymbol{k}$ involving virtually excited one-phonon states with wavevector $\boldsymbol{q}$ as $$\begin{aligned}
\label{Hexadecapole by Bk}
H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha
&= \frac{1}{2} \sum_{\sigma \neq \overline{\sigma} } H_{z, \boldsymbol{k}, \boldsymbol{q}, \sigma, \overline{\sigma} }^\alpha
\nonumber \\
&= \frac{1}{2} \sum_{\sigma \neq \overline{\sigma} }
\left(
B_{\boldsymbol{k} + \boldsymbol{q}, +, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }
- B_{\boldsymbol{k} + \boldsymbol{q}, -, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }
\right)
.\end{aligned}$$ We identify the hexadecapole of Eq. (\[Hexadecapole by Bk\]) with the difference between the occupation numbers $N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} } = B_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }$ and $N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} } = B_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }$ for the two-electron states with the small-wavenumber limit of $| \boldsymbol{q}| \rightarrow 0$ as follows: $$\begin{aligned}
\label{Hexadecapole as number operator}
H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha
&= \frac{1}{2} \sum_{\sigma \neq \overline{\sigma} } H_{z, \boldsymbol{k}, \boldsymbol{q} = 0, \sigma, \overline{\sigma} }^\alpha
\nonumber \\
&= \frac{1}{2} \sum_{\sigma \neq \overline{\sigma} }
\left( N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} } - N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} } \right)
.\end{aligned}$$
The canonical transformation involving the virtual one-phonon processes gives the indirect interactions between the two-electron states carrying the hexadecapole as [@QTS] $$\begin{aligned}
\label{HindHH1}
H_\mathrm{ind}^\mathrm{HH}
= & - \frac{1}{4} \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} } \sum_{\sigma_1 \neq \overline{\sigma}_1} \sum_{\sigma_2 \neq \overline{\sigma}_2}
D_+^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \qquad \times
B_{\boldsymbol{k} - \boldsymbol{q}, +, \sigma_1, \overline{\sigma}_1}^\dagger B_{\boldsymbol{k}, +, \sigma_1, \overline{\sigma}_1}
B_{\boldsymbol{k}' + \boldsymbol{q}, +, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', +, \sigma_2, \overline{\sigma}_2}
\nonumber \\
&+ \frac{1}{4} \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} } \sum_{\sigma_1 \neq \overline{\sigma}_1} \sum_{\sigma_2 \neq \overline{\sigma}_2}
D_+^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \qquad \times
B_{\boldsymbol{k} - \boldsymbol{q}, +, \sigma_1, \overline{\sigma}_1}^\dagger B_{\boldsymbol{k}, +, \sigma_1, \overline{\sigma}_1}
B_{\boldsymbol{k}' + \boldsymbol{q}, -, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', -, \sigma_2, \overline{\sigma}_2}
\nonumber \\
&+ \frac{1}{4} \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} } \sum_{\sigma_1 \neq \overline{\sigma}_1} \sum_{\sigma_2 \neq \overline{\sigma}_2}
D_-^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \qquad \times
B_{\boldsymbol{k} - \boldsymbol{q}, -, \sigma_1, \overline{\sigma}_1} B_{\boldsymbol{k}, -, \sigma_1, \overline{\sigma}_1}
B_{\boldsymbol{k}' + \boldsymbol{q}, +, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', +, \sigma_2, \overline{\sigma}_2}
\nonumber \\
&- \frac{1}{4} \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} } \sum_{\sigma_1 \neq \overline{\sigma}_1} \sum_{\sigma_2 \neq \overline{\sigma}_2}
D_-^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \qquad \times
B_{\boldsymbol{k} - \boldsymbol{q}, -, \sigma_1, \overline{\sigma}_1} B_{\boldsymbol{k}, -, \sigma_1, \overline{\sigma}_1}
B_{\boldsymbol{k}' + \boldsymbol{q}, -, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', -, \sigma_2, \overline{\sigma}_2}
.\end{aligned}$$ The indirect hexadecapole interaction Hamiltonian of Eq. (\[HindHH1\]) mediated by the rotation $\omega_{xy} \bigl( \boldsymbol{q} \bigr)$ of the transverse acoustic phonon consists of the four scattering processes shown in Fig. \[Fig6\]. The scattering process of the two-electron states between the same rotation direction of $h = +$ is shown in Fig. \[Fig6\](a) and the scattering between the same rotation direction of $h = -$ is in Fig. \[Fig6\](d). The scattering between opposite rotation directions of $h = +$ and $-$ are shown in Figs. \[Fig6\](b) and \[Fig6\](c).
The coupling coefficient $D_h^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of the indirect hexadecapole interaction in Eq. (\[HindHH1\]) for the right-hand rotation direction of $h = +$ and the left-hand rotation of $h = -$ is expressed as $$\begin{aligned}
\label{DHHh}
&D_h^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
&= - \frac{1}{2}
\left[
-4 \left( 1 - \gamma \right) J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\right]^2
\nonumber \\
& \times \left\{
\frac{\hbar} { 2V\rho_\mathrm{M}\omega_{y} \left( \boldsymbol{q} \right) } q_{x}^2
\frac{ 4\hbar \omega_y \left( \boldsymbol{q} \right) }
{
\left[
\varepsilon_h \left( \boldsymbol{k} \right) - \varepsilon_h \left( \boldsymbol{k} - \boldsymbol{q} \right)
\right]^2
- \hbar^2 \omega_y \left( \boldsymbol{q} \right) ^2
} \right.
\nonumber \\
&\left. \qquad
+\frac{\hbar} { 2V\rho_\mathrm{M} \omega_{x} \left( \boldsymbol{q} \right) } q_{y}^2
\frac{ 4\hbar \omega_x \left( \boldsymbol{q} \right) }
{
\left[ \varepsilon_h \left( \boldsymbol{k} \right) - \varepsilon_h \left( \boldsymbol{k} - \boldsymbol{q} \right) \right]^2
- \hbar^2 \omega_x \left( \boldsymbol{q} \right)^2
} \right\}
.\end{aligned}$$ Here, $\varepsilon_h \left( \boldsymbol{k} \right) = \hbar^2 \left| \boldsymbol{k} \right| ^2/2m^\ast$ with effective mass $m^\ast$ for wavevector $\boldsymbol{k}$ is the excitation energy of the two-electron states with the rotation direction of $h = \pm$ of Eq. (\[psi+-\]). The transverse acoustic phonon energy of $\hbar \omega_i \left( \boldsymbol{q} \right) = \hbar v_{66} q_i$ for $i = x$ or $y$ is given by the ultrasonic velocity $v_{66}$.
In the normal phase without long-range ordering, the energies of the two-electron states $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ cause the degeneration of each other as $\varepsilon_+ \left( \boldsymbol{k} \right) = \varepsilon_- \left( \boldsymbol{k} \right) = \varepsilon \left( \boldsymbol{k} \right)$. By adopting the equality $D_+^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right) = D_-^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right) = D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$, we deduce the indirect hexadecapole interaction between the two-electron states of Eq. (\[HindHH1\]) as $$\begin{aligned}
\label{HindHH}
H_\mathrm{ind}^\mathrm{HH}
=& - \frac{1}{4} \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} } \sum_{\sigma_1 \neq \overline{\sigma}_1} \sum_{\sigma_2 \neq \overline{\sigma}_2}
D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \times
\left(
B_{\boldsymbol{k} - \boldsymbol{q}, +, \sigma_1, \overline{\sigma}_1}^\dagger B_{\boldsymbol{k}, +, \sigma_1, \overline{\sigma}_1}
- B_{\boldsymbol{k} - \boldsymbol{q}, -, \sigma_1, \overline{\sigma}_1}^\dagger B_{\boldsymbol{k}, -, \sigma_1, \overline{\sigma}_1}
\right)
\nonumber \\
& \qquad \times \left(
B_{\boldsymbol{k}' + \boldsymbol{q}, +, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', +, \sigma_2, \overline{\sigma}_2}
- B_{\boldsymbol{k}' + \boldsymbol{q}, -, \sigma_2, \overline{\sigma}_2}^\dagger B_{\boldsymbol{k}', -, \sigma_2, \overline{\sigma}_2}
\right)
\nonumber \\
=& - \sum_{\boldsymbol{k}, \boldsymbol{k}', \boldsymbol{q} }
D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
H_{z, \boldsymbol{k}, -\boldsymbol{q} }^\alpha H_{z, \boldsymbol{k}', \boldsymbol{q} }^\alpha
.\end{aligned}$$ The indirect hexadecapole interaction coefficient $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ in Eq. (\[HindHH\]) is written as $$\begin{aligned}
\label{DHH}
D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)
&= - 8 \left( 1 - \gamma \right)^2 J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)^2
\nonumber \\
& \times \frac{\hbar} { 2V\rho_\mathrm{M}\omega \left( \boldsymbol{q} \right) } q^2
\frac{ 8\hbar \omega \left( \boldsymbol{q} \right) }
{
\left[
\varepsilon \left( \boldsymbol{k} \right) - \varepsilon \left( \boldsymbol{k} - \boldsymbol{q} \right)
\right]^2
- \hbar^2 \omega \left( \boldsymbol{q} \right) ^2
} .\end{aligned}$$ Here, we take the equivalence of the acoustic phonon energy $\hbar \omega_x \left( \boldsymbol{q} \right) = \hbar \omega_y \left( \boldsymbol{q} \right) = \hbar \omega \left( \boldsymbol{q} \right)$ for the tetragonal lattice. The sign of the interaction coefficient $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of Eq. (\[DHH\]) dominates the character of the hexadecapole interaction. The positive-sign regime of $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right) > 0$ corresponds to the ferro-type hexadecapole interaction, while the negative-sign regime of $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right) < 0$ corresponds to the antiferro-type hexadecapole interaction. Note that the coefficient $ \left( 1 - \gamma \right)^2 J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)^2$ in the indirect hexadecapole interaction of Eqs. (\[DHHh\]) and (\[DHH\]) is always positive for both the antiferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) < 0$ and the ferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) > 0$. Consequently, the sign of $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ is determined by the sign of the denominator $\left[ \varepsilon \left( \boldsymbol{k} \right) - \varepsilon \left( \boldsymbol{k} - \boldsymbol{q} \right) \right]^2 - \hbar^2 \omega \left( \boldsymbol{q} \right) ^2$ in Eq. (\[DHH\]). Since the two-electron state of Eq. (\[psi+- by y’z and zx’\]) with the spin-singlet state interacts with the rotation of the transverse acoustic phonons, the hexadecapole-rotation interaction of Eq. (\[Hrot by Bk\]) and the hexadecapole interaction of Eq. (\[HindHH\]) are independent of whether the spin orientation is $\sigma = \uparrow$ or $\downarrow$.
In the small-wavenumber regime of $\left| \boldsymbol{q} \right| \rightarrow 0$ for the transverse acoustic phonons, we rewrite the indirect hexadecapole interaction coefficient $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ in Eq. (\[DHH\]) as follows by disregarding high powers of $O \bigl( \left| \boldsymbol{q} \right| ^4 \bigr)$: $$\begin{aligned}
\label{DHH (q = 0)}
&D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\nonumber \\
& \qquad = - \frac{ 32 \left( 1 - \gamma \right)^2 J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right)^2} {V \rho_\mathrm{M} }
\frac{1}
{
\left( \displaystyle \frac{\hbar} {m^\ast} \right)^2 \left( k^2 - \displaystyle \frac{m^{\ast 2} v_{66}^2} {\hbar^2} \right)
}.\end{aligned}$$ This gives the boundary wavenumber $k_\mathrm{b}^\mathrm{H} = \left| \boldsymbol{k}_\mathrm{b}^\mathrm{H} \right| = m^\ast v_{66} / \hbar$, where the sign of $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q}= 0 \right)$ changes from positive to negative with increasing $\left| \boldsymbol{k} \right|$. The experimental results in Fig. \[Fig1\](a) for $x = 0.071$ give the ultrasonic velocity $v_{66} = 1750$ m/s in the vicinity of the superconducting transition point $T_\mathrm{SC} = 23$ K ($v_{66} = 1970$ m/s at $T = 80$ K in the normal phase). From these results, we estimate the excitation energy of $\varepsilon \bigl( k_\mathrm{b}^\mathrm{H} \bigr) = \hbar^2 k_\mathrm{b}^{\mathrm{H}2} / 2m^\ast$ to be $0.201$ K (0.256 K) for the boundary wavenumber $k_\mathrm{b}^\mathrm{H} = 34.0\times 10^6$ m$^{-1}$ ($30.2 \times 10^6$ m$^{-1}$) and boundary wavelength $\lambda_\mathrm{b}^\mathrm{H} = 0.185 \times 10^{-6}$ m ($0.208 \times 10^{-6}$ m). Here, we suppose that the two-electron states have an energy of $\varepsilon \left( \boldsymbol{k} \right) = \hbar^2 \left| \boldsymbol{k} \right| ^2/2m^\ast$ with the effective mass being twice the rest electron mass $m^\ast = 2m_e$.
The low-lying two-electron state with the excitation energy $\varepsilon_h \left( \boldsymbol{k} \right) < \varepsilon \bigl( k_\mathrm{b}^\mathrm{H} \bigr) \sim 0.25$ K (5 GHz) contributes to the ferro-type hexadecapole interaction with the positive $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right) > 0$. This brings about the ferro-hexadecapole ordering, which is actually confirmed by the critical slowing down around the superconducting transition temperature for $x = 0.071$ in Fig. \[Fig1\](b). The hexadecapole interaction of $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)$ proportional to $(1 - \gamma)^2J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)^2$ in Eqs. (\[DHH\]) and (\[DHH (q = 0)\]) determines the ferro-type hexadecapole transition temperature, as will be shown by Eq. (\[DHH written by DeltaH\]) of Sect. 4.8.
Hexadecapole ordering
---------------------
We use the hexadecapole susceptibility to describe the critical slowing down associated with the ferro-type hexadecapole ordering. According to the matrix representation of the hexadecapole-rotation interaction of Eq. (\[Matrix of Hrot diagonal\]), the right-hand rotation of $\omega_{xy} > 0$ splits the two-electron states of Eq. (\[psi+-\]) into a lower level with $E_+ = -4 \left( 1 - \gamma \right)J_\mathrm{Q}\omega_{xy}$ for $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the right-hand rotation direction $h = +$ and an upper level with $E_- = 4 \left( 1 - \gamma \right)J_\mathrm{Q}\omega_{xy}$ for $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ with the left-hand rotation direction $h = -$. The splitting of the two-electron states of $\psi_+ \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ and $\psi_- \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ by the rotation $\omega_{xy}$ with the symmetry-breaking character illustrated in Fig. \[Fig7\](b) gives the hexadecapole susceptibility of $\chi_\mathrm{H}$ for the two-electron state obeying the Curie law as $$\begin{aligned}
\label{chiH}
\chi_\mathrm{H}
= N_\mathrm{H}
\frac{
16 \left( 1 - \gamma \right)^2
J_\mathrm{Q}^2 / C_{66}^0
} {T}
=\frac{ \mathit{\Delta}_\mathrm{H} }{T}
.\end{aligned}$$ Here, $N_\mathrm{H}$ is the number of two-electron states participating in the ferro-type hexadecapole interaction. The indirect hexadecapole-rotation interaction energy $\mathit{\Delta}_\mathrm{H}$ in Eq. (\[chiH\]) is given by the hexadecapole interaction coefficient $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)$ of Eq. (\[DHH (q = 0)\]) as $$\begin{aligned}
\label{DHH written by DeltaH}
\frac{2 \mathit{\Delta}_\mathrm{H} } {VN_\mathrm{H} } \frac{ C_{66}^0 }{ \rho_\mathrm{M} v_{66}^2 }
&= \frac{32 \left( 1 - \gamma \right)^2 J_\mathrm{Q}^2 } {V \rho_\mathrm{M} v_{66}^2 }
\nonumber \\
&= \frac{1}{N_\mathrm{H} } \sum_{ |\boldsymbol{k}| < k_\mathrm{b}^\mathrm{H} }
D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\nonumber \\
&=\widetilde{D}^\mathrm{HH}
.\end{aligned}$$ The positive sign in the indirect hexadecapole interaction coefficient $\widetilde{D}^\mathrm{HH}$ due to $\mathit{\Delta}_\mathrm{H} > 0$ leads to the ferro-type hexadecapole ordering. Note that the softening of $\rho_\mathrm{M} v_{66}^2 = C_{66}$ due to the quadrupole-strain interaction of Eq. (\[HQS 2nd quant.\]) enhances the indirect hexadecapole interaction coefficient $\widetilde{D}^\mathrm{HH}$ in Eq. (\[DHH written by DeltaH\]).
The relaxation time $\tau$ around the superconducting transition for $x = 0.071$ diverges at the critical temperature $T_\mathrm{c}^0$ as shown in Fig. \[Fig4\]. Taking this experimental finding into account, we introduce the renormalized hexadecapole susceptibility $\widetilde{\chi}_\mathrm{H}$ as $$\begin{aligned}
\label{chiH tilde}
\widetilde{\chi}_\mathrm{H}
= \frac{ \mathit{\Delta}_\mathrm{H} } { T-\mathit{\Theta}_\mathrm{C} }
.\end{aligned}$$ Here, we introduce the critical temperature $\mathit{\Theta}_\mathrm{C}$ corresponding to the experimentally observed critical temperature $T_\mathrm{c}^0$ for $\tau_\mathrm{H}$. In the following Sects. 4.9 and 4.10, we present plausible model, where the hexadecapole ordering appears accompanying the superconductivity. Consequently, the critical temperature $\mathit{\Theta}_\mathrm{C}$ of Eq. (\[chiH tilde\]) consists of the indirect hexadecapole interaction energy $\widetilde{D}^\mathrm{HH}$ of Eq. (\[DHH written by DeltaH\]) and the superconducting transition temperature $T_\mathrm{SC}$ as $$\begin{aligned}
\label{ThetaC}
\mathit{\Theta}_\mathrm{C}
= \widetilde{D}^\mathrm{HH} + T_\mathrm{SC}
.\end{aligned}$$ Note that the superconducting transition temperature $T_\mathrm{SC}$ will be given later by the self-consistent equation for the superconducting energy gap of Eq. (\[Gap eq\]). The internal energy based on the phenomenological theory given by Eq. (\[UH\]) in Sect. 4.10 also indicates the ground state for the simultaneous ordering of the hexadecapole and superconductivity in the present iron pnictide.
The critical slowing down of the relaxation time $\tau_H$ is caused by the divergence of the correlation length associated with the ferro-type ordering of the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$. This is expressed in terms of the renormalized hexadecapole susceptibility of Eq. (\[chiH tilde\]) as $$\begin{aligned}
\label{tauH}
\tau_\mathrm{H}
= \tau_0 \left| \frac{ T - \mathit{\Theta}_\mathrm{C} } { \mathit{\Theta}_\mathrm{C} } \right| ^{-z\nu}
= \tau_0 \left| \frac{ T - \mathit{\Theta}_\mathrm{C} } { \mathit{\Theta}_\mathrm{C} } \right| ^{-1}
\propto \widetilde{\chi}_\mathrm{H}
.\end{aligned}$$ The critical index $z\nu = 1$ of Eq. (\[tauH\]) based on mean field theory well reproduces the experimental results of the relaxation time $\tau$ above the superconducting transition temperature $T_\mathrm{SC}$ in Fig. \[Fig1\](b). Because the indirect hexadecapole interaction of Eq. (\[HindHH1\]) is mediated by the rotation of the transverse acoustic phonons with a long wavelength, the critical phenomena above the transition point is well described by mean field theory. The experimentally observed $z\nu = 1/3$ in the superconducting phase below $T_\mathrm{SC}$ in Fig. \[Fig1\](b), however, distinctly deviates from $z\nu = 1$ of Eq. (\[tauH\]) obtained from mean field theory. This discrepancy is accounted for by the fact that the hexadecapole correlation due to the two-electron states develops in both the normal and superconducting phases near the transition temperature $T_\mathrm{SC}$, while the hexadecapole correlation due to the Cooper pairs develops only in the superconducting phase as will be shown in Sect. 4.10. The diffusion processes of the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ in the vicinity of the critical temperature $\mathit{\Theta}_\mathrm{C}$ determine the attempt relaxation time $\tau_0$ in Eq. (\[tauH\]) [@Mori; @Halperin; @and; @Hohenberg]. The calculation based on renormalization group theory for the inherent system exhibiting the hexadecapole ordering associated with the superconductivity may explains the experimental result of the critical index $z\nu = 1/3$ and the relative ratio $\tau_{0+} / \tau_{0-} = 1.03$ for $x = 0.071$.
![ (Color online) Divergence of the relaxation time $\tau_\mathrm{H}$ (green solid line) fit by Eq. (\[tauH\]) and the softening of the elastic constant $C_{66}$ (blue dashed line) fit by Eq. (\[Elastic with chiQ tilde\]) in the normal phase above the superconducting transition at $T_\mathrm{SC} = 23$ K for $x = 0.071$. The temperature $T_\mathrm{c}^0 = 23$ K for the divergence of $\tau_\mathrm{H}$ indicates the hexadecapole ordering point of $\mathit{\Theta}_\mathrm{C}$, while the critical temperature $T_\mathrm{s}^0 = \mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q} = -26.5$ K of $C_{66}$ (red open circles) and $\tau_\mathrm{Q}$ (orange dashed and dotted line) give the fictitious lattice instability point for $C_{66} \rightarrow 0$. []{data-label="Fig8"}](68060Fig8.pdf){width="48.00000%"}
It is worth presenting the hexadecapole susceptibility responsible for the attenuation coefficient $\alpha_{66}$ by comparing it with the quadrupole susceptibility responsible for the elastic constant $C_{66}$ in Fig. \[Fig8\]. The critical slowing down due to the ferro-type hexadecapole ordering brings about the divergence of the relaxation time $\tau_\mathrm{H}$, which is expressed in terms of the renormalized hexadecapole susceptibility $\widetilde{\chi}_\mathrm{H}$ of Eq. (\[chiH tilde\]). As shown by the solid green line in Fig. \[Fig8\], the divergence of the relaxation time $\tau_\mathrm{H}$ approaching the superconducting transition ensures the hexadecapole ordering at $\mathit{\Theta}_\mathrm{C} = T_\mathrm{c}^0 = T_\mathrm{SC} = 23$ K, consisting with the superconducting transition temperature. On the other hand, the elastic constant $C_{66}$ shown by the red solid line in Fig. \[Fig8\] exhibits softening obeying the renormalized quadrupole susceptibility $\widetilde{\chi}_\mathrm{Q}$ in Eq. (\[chiQ tilde\]). The critical temperature $T_\mathrm{s}^0$ is $\mathit{\Theta}_\mathrm{Q} + \mathit{\Delta}_\mathrm{Q} = -26.5$ K, implying that the fictitious lattice instability temperature where $C_{66} \rightarrow 0$ is strictly distinguished from the critical temperature $T_\mathrm{c}^0 = T_\mathrm{SC} = 23$ K for the divergence of the relaxation time $\tau_\mathrm{H}$. The critical slowing down due to the fictitious lattice instability may be possible, as indicated by the orange dashed and dotted line in Fig. \[Fig8\]. The ultrasonic attenuation due to the relaxation time $\tau_\mathrm{Q}$ at temperatures far above the fictitious instability point, however, is too small to detect.
The softening of $C_{66}$ in Fig. \[Fig8\] is expressed in terms of the renormalized quadrupole susceptibility $\widetilde{\chi}_\mathrm{Q}$ in Eq. (\[Elastic with chiQ tilde\]), which is caused by the quadrupole-strain interaction of Eq. (\[HQS 2nd quant.\]). The hexadecapole-rotation interaction of Eq. (\[Matrix of Hrot diagonal\]) might affect the softening of $C_{66}^\mathrm{H}$ as $$\begin{aligned}
\label{C66H}
C_{66}^\mathrm{H}
= C_{66} \left( 1- \widetilde{\chi}_\mathrm{H} \right)
= C_{66} \left( 1- \frac{ \mathit{\Delta}_\mathrm{H} } { T-\mathit{\Theta}_\mathrm{C} } \right)
.\end{aligned}$$ Here, $C_{66}$ is given by Eq. (\[Temp. dep. C66\]) or Eq. (\[Elastic with chiQ tilde\]) and represents the softening due to the quadrupole-strain interaction. Because two electrons are accommodated in the degenerate $y'z$ and $zx'$ bands, we take the number of two-electron states as $N_\mathrm{H} \sim \left( 2N_\mathrm{Fe} \right) / 2 = 2\times10^{22}$ cm$^{-3}$ as an upper limit. Adopting the quadrupole interaction energy $J_\mathrm{Q} = \mathit{\Theta}_\mathrm{Q} = -47$ K for $x = 0.071$ and the anisotropic parameter $\gamma = 0.9$ as a tentative value, we deduce that the hexadecapole-rotation interaction energy in Eq. (\[chiH\]) is as small as $\mathit{\Delta}_\mathrm{H}
= 16N_\mathrm{H} \left( 1 - \gamma \right)^2 J_\mathrm{Q}^2 / C_{66}^0 \sim 10^{-3}$ K. This is too small to sizably affect the softening of the elastic constant $C_{66}$. This small $\mathit{\Delta}_\mathrm{H}$ of $\sim 10^{-3}$ K is in strongly contrast to the considerable quadrupole-strain interaction energy of $\mathit{\Delta}_\mathrm{Q} \sim 20$ K, which brings about appreciable softening of $C_{66}$ with decreasing temperature.
Superconductivity due to quadrupole interaction
-----------------------------------------------
In out attempt to show the superconducting state compatible with the hexadecapole ordering, we will solve the superconducting Hamiltonian for a pair of electrons coupled to each other through the quadrupole interaction $H_\mathrm{QQ} \left( \gamma \right)$ of Eq. (\[Anisotropic HQQ\]). To this end, we treat band electrons of the orbital state $\lambda_m \left( \boldsymbol{r} \right)$ in Eq. (\[wave function lambda\]) with the angular momentum $l = 2$ and the azimuthal quantum number $m = \pm1$. The corresponding bare electron Hamiltonian is expressed in terms of the electron operators of $l_{\boldsymbol{k}, m, \sigma }$ and $l_{\boldsymbol{k}, m, \sigma }^\dagger$ for $m = \pm1$ as $$\begin{aligned}
\label{HK'}
H_\mathrm{K}'
& = \sum_{\boldsymbol{k} } \sum_{\sigma}
\left[
\varepsilon_{+1, \sigma} \left( \boldsymbol{k} \right)
l_{\boldsymbol{k}, +1, \sigma}^\dagger l_{\boldsymbol{k}, +1, \sigma} \right.
\nonumber \\
&\left. \qquad \qquad \qquad
+ \varepsilon_{-1, \sigma} \left( \boldsymbol{k} \right)
l_{\boldsymbol{k}, -1, \sigma}^\dagger l_{\boldsymbol{k}, -1, \sigma}
\right]
.\end{aligned}$$ Here, the electron energy $\varepsilon_{m, \sigma} \left( \boldsymbol{k} \right)$ is measured from the Fermi energy.
The quadrupoles expressed by the electron operators of $d_{i, l, \sigma}$ and $d_{i, l, \sigma}^\dagger$ $( l = y'z $ and $zx')$ in Eqs. (\[Ox’2-y’2 2nd quant.\]) and (\[Ox’y’ 2nd quant.\]) are rewritten in terms of the electron operators of $l_{\boldsymbol{k}, m, \sigma }$ and $l_{\boldsymbol{k}, m, \sigma }^\dagger$ with $m = \pm1$ as $$\begin{aligned}
\label{quadrupole Ov' by l}
O_{x'^2-y'^2, \boldsymbol{k}, \boldsymbol{q} }
&= - \sum_{\sigma}
\left(
l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{\boldsymbol{k}, -1, \sigma}
+ l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{\boldsymbol{k}, +1, \sigma}
\right)
, \\
\label{quadrupole Ox'y' by l}
O_{x'y', \boldsymbol{k}, \boldsymbol{q} }
&= i \sum_{\sigma}
\left(
l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{\boldsymbol{k}, -1, \sigma}
- l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{\boldsymbol{k}, +1, \sigma}
\right)
.\end{aligned}$$ Thus, we obtain an alternative expression for the quadrupole interaction Hamiltonian $H_\mathrm{QQ} \left( \gamma \right)$ with the anisotropic parameter $\gamma$ of Eq. (\[Anisotropic HQQ\]) as $$\begin{aligned}
\label{HQQ by l}
H_\mathrm{QQ} \left( \gamma \right)
=& -\frac{1}{2} \sum_{ \boldsymbol{k}, \boldsymbol{q} } \sum_{\sigma \neq \overline{\sigma} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \quad \times
\left(
l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma } \right.
\nonumber \\
& \qquad + l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma }^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma}
\nonumber \\
&\quad \qquad +l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\nonumber \\
&\qquad \qquad \left.
+l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\right)
\nonumber \\
& -\frac{1}{2} \gamma \sum_{ \boldsymbol{k}, \boldsymbol{q} } \sum_{\sigma \neq \overline{\sigma} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
& \quad \times
\left(
- l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma} \right.
\nonumber \\
&\qquad + l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, -1,\sigma}
\nonumber \\
&\quad \qquad +l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\nonumber \\
&\qquad \qquad \left.
- l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\right)
.\end{aligned}$$ The quadrupole interaction $H_\mathrm{QQ} \left( \gamma \right)$ of Eq. (\[HQQ by l\]) gives four independent scattering processes between electrons with azimuthal quantum numbers of $m = +1$ and $-1$, wavevectors $\boldsymbol{k}$ of an electron bearing the electric quadrupoles and $\boldsymbol{q}$ of a transverse acoustic phonon, and spin orientations of $\sigma$ and $\overline{\sigma}$. In order to account for the critical slowing down around the superconducting transition point, we notice that the two scattering processes consisting of $l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma} $ and $l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}$ in Eq. (\[HQQ by l\]) give the superconducting ground state bearing the hexadecapole. The former term of $l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma} $ annihilates two electrons with the same azimuthal quantum number of $m = -1$ and creates two electrons with the same number of $m = +1$, while the latter term of $l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}$ causes the reverse process. On the other hand, we disregard the scattering of $l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1,\sigma}$ involved in exchange process from the electron state with $m = +1$ to the opposite state with $m = -1$, and that of $l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1,\sigma}$ for the exchange from $m = -1$ to $m = +1$. These latter scattering lead to a superconducting ground state that does not carry the hexadecapole, which is incompatible with the critical slowing down due to the hexadecapole ordering. Consequently, we adopt the restricted quadrupole interaction Hamiltonian consisting of the former process to properly describe the superconductivity accompanying the hexadecapole ordering: $$\begin{aligned}
\label{Restricted HQQ}
&H_\mathrm{QQ}' \left( \gamma \right)
\nonumber \\
&\quad
= -\frac{1}{2} \left( 1 - \gamma \right)
\sum_{ \boldsymbol{k}, \boldsymbol{q} } \sum_{\sigma \neq \overline{\sigma} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\nonumber \\
&\qquad \quad \times
\left(
l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, +1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma } \right.
\nonumber \\
&\left. \qquad \qquad \quad
+l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, -1, \overline{\sigma} }^\dagger l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\right)
.\end{aligned}$$ Here, the gravity momentum for two-electron states bound by the restricted quadrupole interaction of Eq. (\[Restricted HQQ\]) is constrained to vanish as $\hbar \boldsymbol{k}_\mathrm{G} = 0$. Note that the restricted Hamiltonian $H_\mathrm{QQ}' \left( \gamma \right)$ of Eq. (\[Restricted HQQ\]) satisfies the criterion of the $A_1$ symmetry of point group $D_{2d}$.
We solve the superconducting Hamiltonian $H_\mathrm{SC}$ consisting of the bare electron Hamiltonian $H_\mathrm{K}'$ of Eq. (\[HK’\]) and the restricted quadrupole interaction Hamiltonian $H_\mathrm{QQ}' \left( \gamma \right)$ of Eq. (\[Restricted HQQ\]): $$\begin{aligned}
\label{HSC}
&H_\mathrm{SC}
\nonumber \\
&= H_\mathrm{K}' + H_\mathrm{QQ}' \left(\gamma \right)
\nonumber \\
&= \sum_{\boldsymbol{k} } \sum_{\sigma \neq \overline{\sigma} }
\left(
\begin{array}{c}
l_{\boldsymbol{k}, +1, \sigma}^\dagger \\
l_{-\boldsymbol{k}, +1, \overline{\sigma} } \\
l_{\boldsymbol{k}, -1, \sigma}^\dagger \\
l_{-\boldsymbol{k}, -1, \overline{\sigma} }
\end{array}
\right)^\mathrm{T} \times
\nonumber \\
&\left(
\begin{array}{cccc}
\varepsilon \left( \boldsymbol{k} \right) & - \mathit{\Delta}_{-1, -1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right) & 0 & 0 \\
- \mathit{\Delta}_{-1, -1}^{\sigma, \overline{\sigma} \ast} \left( \boldsymbol{k} \right) & - \varepsilon \left( \boldsymbol{k} \right) & 0 & 0 \\
0 & 0 & \varepsilon \left( \boldsymbol{k} \right) & - \mathit{\Delta}_{+1, +1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right) \\
0 & 0 & - \mathit{\Delta}_{+1, +1}^{\sigma, \overline{\sigma} \ast} \left( \boldsymbol{k} \right) & - \varepsilon \left( \boldsymbol{k} \right) \\
\end{array}
\right)
\nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad \times \left(
\begin{array}{c}
l_{\boldsymbol{k}, +1, \sigma} \\
l_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger \\
l_{\boldsymbol{k}, -1, \sigma} \\
l_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\end{array}
\right)
.\end{aligned}$$ Here, the electron state vector of $\boldsymbol{l}_{\boldsymbol{k} } \equiv \left(
l_{\boldsymbol{k}, +1, \sigma},
l_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger,
l_{\boldsymbol{k}, -1, \sigma},
l_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\right)^\mathrm{T}$ is employed. We use the equivalence of the energy $\varepsilon_{+1, \sigma} \left( \boldsymbol{k} \right)
= \varepsilon_{-1, \sigma} \left( \boldsymbol{k} \right)
= \varepsilon \left( \boldsymbol{k} \right)$ for the electrons with the azimuthal quantum numbers of $m = +1$ and $-1$. There are two different superconducting energy gaps of $\mathit{\Delta}_{+1, +1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right)$ denoted by the Cooper pairs with the right-hand azimuthal quantum number $m = +1$ and $\mathit{\Delta}_{-1, -1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right)$ for the left-hand azimuthal quantum number of $m = -1$ in Eq. (\[HSC\]) as $$\begin{aligned}
\label{Energy gap 1}
\mathit{\Delta}_{\pm 1, \pm 1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right)
= \frac{1}{2} \left( 1-\gamma \right)
\sum_{\boldsymbol{q} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\bigl\langle
l_{-\boldsymbol{k} - \boldsymbol{q}, \pm1, \overline{\sigma} } l_{\boldsymbol{k} + \boldsymbol{q}, \pm1, \sigma }
\bigr\rangle
, \\
\label{Energy gap 2}
\mathit{\Delta}_{\pm 1, \pm 1}^{\sigma, \overline{\sigma} \ast} \left( \boldsymbol{k} \right)
= \frac{1}{2} \left( 1-\gamma \right)
\sum_{\boldsymbol{q} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\bigl\langle
l_{\boldsymbol{k} + \boldsymbol{q}, \pm1, \sigma }^\dagger l_{-\boldsymbol{k} - \boldsymbol{q}, \pm1, \overline{\sigma} }^\dagger
\bigr\rangle
.\end{aligned}$$ Here, $\bigl\langle l_{-\boldsymbol{k}, \pm1, \overline{\sigma} } l_{\boldsymbol{k}, \pm1, \sigma } \bigr\rangle$ and $\bigl\langle l_{\boldsymbol{k}, \pm1, \sigma }^\dagger l_{-\boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger \bigr\rangle$ stand for the mean-field values of the Cooper pair indicating the off-diagonal long-range order parameter of the superconducting phase. The quadrupole interaction coefficient $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ and the anisotropic parameter $\gamma$ in Eq. (\[Restricted HQQ\]) dominate the two energy gaps of $\mathit{\Delta}_{\pm1, \pm1}^{\sigma, \overline{\sigma} } \left( \boldsymbol{k} \right)$ in Eq. (\[Energy gap 1\]) and $\mathit{\Delta}_{\pm1, \pm1}^{\sigma, \overline{\sigma} \ast} \left( \boldsymbol{k} \right)$ in Eq. (\[Energy gap 2\]), which characterize the inherent superconductivity of the system. The appearance of the energy gap $\mathit{\Delta}_{\pm 1, \pm 1}^{\sigma, \overline{\sigma} } \neq 0$ indicates the symmetry breaking of the U(1) gauge, where the electron number is not conserved across the superconducting transition [@P.; @W.; @Anderson].
Supposing a Cooper pair consisting of two electrons with opposite spin orientations, we set the energy gap to $\mathit{\Delta}_{\pm1, \pm1}^{\uparrow, \downarrow} \left( \boldsymbol{k} \right)
= \mathit{\Delta}_{\pm1, \pm1}^{\downarrow, \uparrow} \left( \boldsymbol{k} \right)
= \mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right) $ while omitting the spin orientations. The superconducting Hamiltonian $H_\mathrm{SC}$ of Eq. (\[HSC\]) is expressed by Bogoliubov quasiparticles of $\boldsymbol{L}_{\boldsymbol{k} } \equiv \left(
L_{\boldsymbol{k}, +1, \sigma},
L_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger,
L_{\boldsymbol{k}, -1, \sigma},
L_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\right)^\mathrm{T}$ as $$\begin{aligned}
\label{HSC diagonal}
&H_\mathrm{SC}
= \sum_{\boldsymbol{k} } \sum_{\sigma \neq \overline{\sigma} }
\left(
\begin{array}{c}
L_{\boldsymbol{k}, +1, \sigma}^\dagger \\
L_{-\boldsymbol{k}, +1, \overline{\sigma} } \\
L_{\boldsymbol{k}, -1, \sigma}^\dagger \\
L_{-\boldsymbol{k}, -1, \overline{\sigma} }
\end{array}
\right)^\mathrm{T}
\nonumber \\
&\times \left(
\begin{array}{cccc}
E_{+1} \left( \boldsymbol{k} \right) & 0 & 0 & 0 \\
0 & - E_{+1} \left( \boldsymbol{k} \right) & 0 & 0 \\
0 & 0 & E_{-1} \left( \boldsymbol{k} \right) & 0 \\
0 & 0 & 0 & -E_{-1} \left( \boldsymbol{k} \right) \\
\end{array}
\right)
\nonumber \\
&\qquad \qquad \qquad \times
\left(
\begin{array}{c}
L_{\boldsymbol{k}, +1, \sigma}\\
L_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger \\
L_{\boldsymbol{k}, -1, \sigma} \\
L_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\end{array}
\right).\end{aligned}$$ Here, we obtain the excitation energies $\pm E_{+1} \left( \boldsymbol{k} \right)$ and $\pm E_{-1} \left( \boldsymbol{k} \right)$ of the Bogoliubov quasiparticles as $$\begin{aligned}
\label{E_Bogo}
E_{\pm1} \left( \boldsymbol{k} \right)
= \sqrt{
\varepsilon \left( \boldsymbol{k} \right)^2
+ \left| \mathit{\Delta}_{\mp1, \mp1} \left( \boldsymbol{k} \right) \right|^2
}
.\end{aligned}$$ Here, we adopt double-sign correspondence. The Bogoliubov transformation from the electron state vector of $\boldsymbol{l}_{\boldsymbol{k} } = \bigl(
l_{\boldsymbol{k}, +1, \sigma},
l_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger,
l_{\boldsymbol{k}, -1, \sigma},
l_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\bigr)^T$ to the Bogoliubov quasiparticles of $\boldsymbol{L}_{\boldsymbol{k} } = \bigl(
L_{\boldsymbol{k}, +1, \sigma},
L_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger,
L_{\boldsymbol{k}, -1, \sigma},
L_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\bigr)^T$ is expressed in terms of an unitary matrix as $$\begin{aligned}
\label{Vector L}
&\left(
\begin{array}{c}
L_{\boldsymbol{k}, +1, \sigma}\\
L_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger \\
L_{\boldsymbol{k}, -1, \sigma} \\
L_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\end{array}
\right)
\nonumber \\
&= \left(
\begin{array}{cccc}
u_{+1} \left( \boldsymbol{k} \right) & -v_{+1}^\ast \left( \boldsymbol{k} \right) & 0 & 0 \\
v_{+1} \left( \boldsymbol{k} \right) & u_{+1} \left( \boldsymbol{k} \right) & 0 & 0 \\
0 & 0 & u_{-1} \left( \boldsymbol{k} \right) & -v_{-1}^\ast \left( \boldsymbol{k} \right) \\
0 & 0 & v_{-1} \left( \boldsymbol{k} \right) & u_{-1} \left( \boldsymbol{k} \right) \\
\end{array}
\right)
\left(
\begin{array}{c}
l_{\boldsymbol{k}, +1, \sigma} \\
l_{-\boldsymbol{k}, +1, \overline{\sigma} }^\dagger \\
l_{\boldsymbol{k}, -1, \sigma} \\
l_{-\boldsymbol{k}, -1, \overline{\sigma} }^\dagger
\end{array}
\right)
\nonumber \\
&= \left(
\begin{array}{c}
u_{+1} \left( \boldsymbol{k} \right) l_{\boldsymbol{k}, +, \sigma}
- v_{+1}^\ast \left( \boldsymbol{k} \right) l_{-\boldsymbol{k}, +, \overline{\sigma} }^\dagger \\
v_{-1} \left( \boldsymbol{k} \right) l_{\boldsymbol{k}, +, \sigma}
+ u_{+1} \left( \boldsymbol{k} \right) l_{-\boldsymbol{k}, +, \overline{\sigma} }^\dagger \\
u_{-1} \left( \boldsymbol{k} \right) l_{\boldsymbol{k}, -, \sigma}
- v_{-1}^\ast \left( \boldsymbol{k} \right) l_{-\boldsymbol{k}, -, \overline{\sigma} }^\dagger \\
v_{-1} \left( \boldsymbol{k} \right) l_{\boldsymbol{k}, -, \sigma}
+ u_{-1} \left( \boldsymbol{k} \right) l_{-\boldsymbol{k}, -, \overline{\sigma} }^\dagger \\
\end{array}
\right)
.\end{aligned}$$ Taking the constraint of $\left| u_{\pm1} \left( \boldsymbol{k} \right) \right|^2 + \left| v_{\pm1} \left( \boldsymbol{k} \right) \right|^2 = 1$ for fermion quasiparticles into account, we set the elements of the Bogoliubov transformation in Eq. (\[Vector L\]) as $$\begin{aligned}
%u
\label{u}
u_{\pm1} \left( \boldsymbol{k} \right)
&= \frac{ E_{\pm1} \left( \boldsymbol{k} \right) + \varepsilon \left( \boldsymbol{k} \right) }
{
\sqrt{
\left[ E_{\pm1} \left( \boldsymbol{k} \right) + \varepsilon \left( \boldsymbol{k} \right) \right] + \left| \mathit{\Delta}_{\mp1, \mp1} \left( \boldsymbol{k} \right) \right|^2
}
}
,\\
%v
\label{v}
v_{\pm1}^\ast \left( \boldsymbol{k} \right)
&= \frac{ \mathit{\Delta}_{\mp1, \mp1}^\ast \left( \boldsymbol{k} \right) }
{
\sqrt{
\left[ E_{\pm1} \left( \boldsymbol{k} \right) + \varepsilon \left( \boldsymbol{k} \right) \right] + \left| \mathit{\Delta}_{\mp1, \mp1} \left( \boldsymbol{k} \right) \right|^2
}
}
.\end{aligned}$$ Here, $u_{\pm1} \left( \boldsymbol{k} \right)$ is a real number and $v_{\pm1} \left( \boldsymbol{k} \right)$ is a complex number. The Bogoliubov quasiparticles of $\boldsymbol{L}_{\boldsymbol{k} }$ in Eq. (\[HSC diagonal\]) obey the fermion commutation relations $$\begin{aligned}
\label{Commutation relation of L 1}
\left\{ L_{\boldsymbol{k}, m, \sigma }, L_{\boldsymbol{k}', m', \sigma' }^\dagger \right\}
&= & \delta_{\boldsymbol{k}, \boldsymbol{k}' } \delta_{m, m'} \delta_{\sigma, \sigma' }
, \\
\label{Commutation relation of L 2}
\left\{ L_{\boldsymbol{k}, m, \sigma }, L_{\boldsymbol{k}', m', \sigma' } \right\}
&= &\left\{ L_{\boldsymbol{k}, m, \sigma }^\dagger, L_{\boldsymbol{k}', m', \sigma' }^\dagger \right\} = 0
.\end{aligned}$$
In the superconducting phase, the energy gaps of Eqs. (\[Energy gap 1\]) and (\[Energy gap 2\]) show finite values of $\mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right) \neq 0$ and $\mathit{\Delta}_{\pm1, \pm1}^\ast \left( \boldsymbol{k} \right) \neq 0$ for both the ferro-type quadrupole interaction $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) > 0$ and the antiferro-type quadrupole interaction $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) < 0$. The slight deviation of the anisotropic feature of $\gamma \stackrel{<}{_\sim} 1$ from the ideal $xz$ model is necessary for the manifestation of the superconductivity in the vicinity of the QCP. The superconducting ground state $\mathit{\Phi}_0$ given by the restricted quadrupole interaction Hamiltonian of Eq. (\[Restricted HQQ\]) is described in terms of the annihilation operators $L_{- \boldsymbol{k}, \pm1, \overline{\sigma} }L_{\boldsymbol{k}, \pm1, \sigma}$ of the Bogoliubov quasiparticles acting on the vacuum state $\mathit{\Phi}_\mathrm{vac}$. An alternative expression for the superconducting ground state is obtained in terms of the electron creation operators $l_{\boldsymbol{k}, \pm1, \sigma}^\dagger l_{- \boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger$ of the Cooper pair acting on $\mathit{\Phi}_\mathrm{vac}$. Consequently, the grand state $\mathit{\Phi}_0$ is expressed as $$\begin{aligned}
\label{Superconducting ground state}
\left| \mathit{\Phi}_0 \right\rangle
&= C \prod_{\boldsymbol{k} } \prod_{m = \pm1} \prod_{\sigma \neq \overline{\sigma} }
L_{-\boldsymbol{k}, m, \overline{\sigma} } L_{\boldsymbol{k}, m, \sigma}
\left| \mathit{\Phi}_\mathrm{vac} \right\rangle
\nonumber \\
&= C \prod_{\boldsymbol{k} } \prod_{m = \pm1} \prod_{\sigma \neq \overline{\sigma} }
\left[ - v_m^\ast \left( \boldsymbol{k} \right) \right]
\nonumber \\
&\qquad \qquad \times
\left[
u_m \left( \boldsymbol{k} \right)
+ v_m^\ast \left( \boldsymbol{k} \right)
l_{\boldsymbol{k}, m, \sigma}^\dagger l_{-\boldsymbol{k}, m, \overline{\sigma} }^\dagger
\right]
\left| \mathit{\Phi}_\mathrm{vac} \right\rangle
.\end{aligned}$$ Here, we take the available operations over the azimuthal quantum numbers of $m = +1$ and $-1$, spin orientations of $\sigma$ and $\overline{\sigma}$, and wavevector $\boldsymbol{k}$. The coefficient of $C^{-2} = \prod_{\boldsymbol{k} } \prod_{m = \pm1} \prod_{\sigma \neq \overline{\sigma} }\left| v_m \left( \boldsymbol{k} \right)\right|^2$ stands for the normalized factor of $\mathit{\Phi}_0$. The superconducting ground state $\mathit{\Phi}_0$ of Eq. (\[Superconducting ground state\]) consists of the creation operators $l_{\boldsymbol{k}, \pm1, \sigma}^\dagger l_{-\boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger$ of the Cooper pairs with probability weight density $\left| v_m \left( \boldsymbol{k} \right)\right|^2$ of Eq. (\[v\]). This is the coherent state treated in the standard BCS theory [@BCS; @QTS; @Tinkham; @Schrieffer].
The restricted quadrupole interaction Hamiltonian of Eq. (\[Restricted HQQ\]) brings about the energy gaps $\mathit{\Delta}_{\pm1, \pm1} \neq 0$ of Eq. (\[Energy gap 1\]) and their complex conjugate energy gaps $\mathit{\Delta}_{\pm1, \pm1}^\ast \neq 0$ of Eq. (\[Energy gap 2\]). The corresponding mean fields of the off-diagonal long-range order parameters of $\bigl\langle l_{-\boldsymbol{k}, \pm1, \overline{\sigma} } l_{\boldsymbol{k}, \pm1, \sigma } \bigr\rangle$ and $\bigl\langle l_{\boldsymbol{k}, \pm1, \sigma }^\dagger l_{-\boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger \bigr\rangle$ are written in terms of the Bogoliubov quasiparticles of Eq. (\[Vector L\]) as $$\begin{aligned}
\label{For gap eq 1}
\bigl\langle l_{-\boldsymbol{k}, \pm1, \overline{\sigma}} l_{\boldsymbol{k}, \pm1, \sigma} \bigr\rangle
=u_{\pm1}^\ast \left( \boldsymbol{k} \right) v_{\pm1}^\ast \left( \boldsymbol{k} \right)
\left(
1
- \bigl\langle L_{\boldsymbol{k}, \pm1, \sigma}^\dagger L_{\boldsymbol{k}, \pm1, \sigma} \bigr\rangle \right.
\nonumber \\
\quad \left.
- \bigl\langle L_{-\boldsymbol{k}, \pm1, \sigma}^\dagger L_{-\boldsymbol{k}, \pm1, \sigma} \bigr\rangle
\right),
\\
\label{For gap eq 2}
\bigl\langle l_{\boldsymbol{k}, \pm1, \sigma}^\dagger l_{-\boldsymbol{k}, \pm1, \overline{\sigma}}^\dagger \bigr\rangle
=u_{\pm1} \left( \boldsymbol{k} \right) v_{\pm1} \left( \boldsymbol{k} \right)
\left(
1
- \bigl\langle L_{\boldsymbol{k}, \pm1, \sigma}^\dagger L_{\boldsymbol{k}, \pm1, \sigma} \bigr\rangle \right.
\nonumber \\
\quad \left.
- \bigl\langle L_{-\boldsymbol{k}, \pm1, \sigma}^\dagger L_{-\boldsymbol{k}, \pm1, \sigma} \bigr\rangle
\right)
.\end{aligned}$$ Here, we used the fact that the mean-field values for the off-diagonal excitation of $\bigl\langle L_{\boldsymbol{k}, \pm1, \sigma}^\dagger L_{-\boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger \bigr\rangle$ and $\bigl\langle L_{-\boldsymbol{k}, \pm1, \overline{\sigma} } L_{\boldsymbol{k}, \pm1, \sigma} \bigr\rangle$ for the Bogoliubov quasiparticles vanish. The Bogoliubov quasiparticle numbers of $ \bigl\langle L_{\boldsymbol{k}, \pm1, \sigma}^\dagger L_{\boldsymbol{k}, \pm1, \sigma} \bigr\rangle$ and $\bigl\langle L_{-\boldsymbol{k}, \pm1, \overline{\sigma} }^\dagger L_{-\boldsymbol{k}, \pm1, \overline{\sigma} } \bigr\rangle$ in Eqs. (\[For gap eq 1\]) and (\[For gap eq 2\]) are calculated in terms of the Fermi distribution function $f \left( E_{\pm1} \left( \boldsymbol{k} \right) \right) = \left\{ \exp \left [ E_{\pm1} \left( \boldsymbol{k} \right) / k_\mathrm{B}T \right] +1 \right\}^{-1}$ for the excitation energy $E_{\pm 1} \left( \boldsymbol{k} \right)$ in Eq. (\[E\_Bogo\]). By using the alternative expressions for the Bogoliubov-transform elements $\left| u_{\pm1} \left( \boldsymbol{k} \right) \right|^2 - \left| v_{\pm1} \left( \boldsymbol{k} \right) \right|^2
= \varepsilon \left( \boldsymbol{k} \right) / E_{\pm1} \left( \boldsymbol{k} \right)$ and $2u_{\pm1} \left( \boldsymbol{k} \right) v_{\pm1} \left( \boldsymbol{k} \right)
= \mathit{\Delta}_{\mp1, \mp1}^\ast \left( \boldsymbol{k} \right) / E_{\pm1} \left( \boldsymbol{k} \right)$, we obtain the self-consistent equation for the energy gap of Eq. (\[Energy gap 2\]) as $$\begin{aligned}
\label{Gap eq}
&\mathit{\Delta}_{\pm 1, \pm 1}^\ast \left( \boldsymbol{k} \right)
\nonumber \\
&= \frac{1}{2} \left( 1-\gamma \right) \sum_{\boldsymbol{q} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\left\langle
l_{\boldsymbol{k} + \boldsymbol{q}, \pm1, \sigma } l_{-\boldsymbol{k} - \boldsymbol{q}, \pm1, \overline{\sigma} }
\right\rangle
\nonumber \\
&= \frac{1}{4} \left( 1-\gamma \right) \sum_{\boldsymbol{q} } J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\frac{ \mathit{\Delta}_{\mp 1, \mp 1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) }
{ \sqrt{ \varepsilon \left( \boldsymbol{k} + \boldsymbol{q} \right)^2 + \left| \mathit{\Delta}_{\mp1, \mp1} \left( \boldsymbol{k} + \boldsymbol{q} \right) \right|^2 } }
\nonumber \\
&\qquad \quad \times
\tanh \left[
\frac{ \sqrt{ \varepsilon \left( \boldsymbol{k} + \boldsymbol{q} \right)^2 + \left| \mathit{\Delta}_{\mp1, \mp1} \left( \boldsymbol{k} + \boldsymbol{q} \right) \right|^2 } }
{ 2k_\mathrm{B}T }
\right]
.\end{aligned}$$ The quadrupole interaction $J_\mathrm{Q} \left( \boldsymbol{k},\boldsymbol{q} \right)$ and the anisotropic parameter $\gamma$ used in the restricted quadrupole interaction Hamiltonian $H_\mathrm{QQ}'(\gamma)$ of Eq. (\[Restricted HQQ\]) dominate the self-consistent equation for the superconducting energy gaps of $\mathit{\Delta}_{\pm1, \pm1}^\ast \left( \boldsymbol{k} \right)$ in Eq. (\[Gap eq\]). The microscopic mechanism for the Cooper pair formation in the present treatment is in good agreement with the previous theoretical studies based on the quadrupole interaction [@Yanagi_1; @Kontani; @Onari] but are rather different from the theory based on the spin fluctuation [@Mazin1; @Kuroki; @Fernandes]. The superconducting transition temperature $T_\mathrm{SC}$ determined by Eq. (\[Gap eq\]) in the present model and the hexadecapole interaction $\widetilde{D}^\mathrm{HH}$ of Eq. (\[DHH written by DeltaH\]) both contribute to the critical temperature $\mathit{\Theta}_\mathrm{C}$ of the renormalized hexadecapole susceptibility of Eq. (\[chiH tilde\]) as $\mathit{\Theta}_\mathrm{C} = \widetilde{D}^\mathrm{HH} + T_\mathrm{SC}$ as shown in Eq. (\[ThetaC\]).
In order to characterize the superconducting state of the present iron pnictide, we calculate the eigenenergy of the restricted quadrupole interaction Hamiltonian $H_\mathrm{QQ}' \left( \gamma \right)$ of Eq. (\[Restricted HQQ\]) for the superconducting ground state $\mathit{\Phi}_0$ of Eq. (\[Superconducting ground state\]) as $$\begin{aligned}
\label{Expectation value of HQQ}
&\left\langle \mathit{\Phi}_0 \left| H_\mathrm{QQ}' \left( \gamma \right) \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
& \quad
= -\frac{1}{8} \sum_{\boldsymbol{k}, \boldsymbol{q} } \sum_{\sigma \neq \overline{\sigma} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\left[
\frac{ \mathit{\Delta}_{-1, -1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) }{ E_{+1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) }{ E_{-1} \left( \boldsymbol{k} \right) } \right.
\nonumber \\
&\qquad \qquad \qquad \qquad \left.
+
\frac{ \mathit{\Delta}_{+1, +1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) }{ E_{-1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) }{ E_{+1} \left( \boldsymbol{k} \right) }
\right]
.\end{aligned}$$ The antiferro- and ferro-quadrupole interactions of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ for various wavevectors $\boldsymbol{q}$ of the transverse acoustic phonons over the Brillouin zone participate in the Cooper pair formation. However, a small-wavenumber limit of $|\boldsymbol{q}| \rightarrow 0$ corresponding to the measured ultrasonic wave particularly plays a significant role to cause the the critical slowing down due to the ferro-hexadecapole ordering. Therefore, we calculate the quadrupole interaction energy of $\left\langle \mathit{\Phi}_0 \left| H_\mathrm{QQ}' \left( \gamma \right) \right| \mathit{\Phi}_0 \right\rangle$ of Eq. (\[Expectation value of HQQ\]) for the Cooper pairs bound by electrons with wavevectors $\boldsymbol{k}$ and $\boldsymbol{k} + \boldsymbol{q}$ for the small-wavenumber limit of $|\boldsymbol{q}| \rightarrow 0$.
In out attempt to examine the interference between the phase $\varphi_{+1,+1} \left( \boldsymbol{k} \right)$ of the energy gap $\mathit{\Delta}_{+1,+1} \left( \boldsymbol{k} \right)$ due to the off-diagonal long-range order parameter $\left\langle l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma } \right\rangle$ and the phase $\varphi_{-1,-1} \left( \boldsymbol{k} \right)$ of $\mathit{\Delta}_{-1,-1} \left( \boldsymbol{k} \right)$ due to $\left\langle l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma } \right\rangle$, we take polar form of the energy gaps. $$\begin{aligned}
\label{Polar form of gap}
\mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right)
= \left| \mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right) \right|
\exp \left[ i \varphi_{\pm1, \pm1} \left( \boldsymbol{k} \right) \right]
.\end{aligned}$$ Thus, we deduce the energy of the restricted quadrupole interaction of Eq. (\[Restricted HQQ\]) for the superconducting ground state $\mathit{\Phi}_0$ in the small-wavenumber limit of $\left| \boldsymbol{q} \right| \rightarrow 0$ as $$\begin{aligned}
\label{HQQ by cos}
&
\left\langle \mathit{\Phi}_0 \left| H_\mathrm{QQ}' \left( \gamma \right) \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
& \qquad
= -\frac{1}{2} \sum_{\boldsymbol{k} }
J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right)
\frac{ \left| \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) \right| }{ E_{+1} \left( \boldsymbol{k} \right) }
\frac{ \left| \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) \right| }{ E_{-1} \left( \boldsymbol{k} \right) }
\nonumber \\
&\qquad \qquad \qquad \times
\cos \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right]
.\end{aligned}$$ The restricted quadrupole interaction energy of Eq. (\[HQQ by cos\]) depends on $\cos \left[ \varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right) \right]$ for the phase difference $\varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right)$ between the two energy gaps in Eq. (\[Polar form of gap\]). [@Leggett]. The Cooper pairs dominated by the ferro-type quadrupole interaction with $J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right) > 0$ for the small-wavenumber regime of $|\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q} $ bring about the state for $\cos \left[ \varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right) \right] = 1$ with the phase difference corresponding to the stationary point of $\varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right) = s \pi$ for an even integer $s = 2n$. In the opposite case, however, the antiferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right) < 0$, which is relevant for relatively high wavenumbers of $|\boldsymbol{k}| > k_\mathrm{b}^\mathrm{Q}$, has $\cos \left[ \varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right) \right] = -1$ with the stationary point of $\varphi_{+1,+1} \left( \boldsymbol{k} \right) - \varphi_{-1,-1} \left( \boldsymbol{k} \right) = s\pi$ for an odd integer $s = 2n + 1$.
The indirect quadrupole interaction coefficient $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of Eq. (\[DQQ2\]) in the Hamiltonian of Eq. (\[HindQQ\_2\]) may possess positive or negative sign depending on the sign of its denominator. Since the coefficient $D^\mathrm{QQ} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ dominates the quadrupole interaction coefficient $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)$, there are expected two cases of the ferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) > 0$ and antiferro-type of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) < 0$ in the restricted quadrupole interaction energy of Eq. (\[HQQ by cos\]). These both cases of $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) > 0$ and $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right) < 0$ are commonly available for the superconducting Cooper pair formation in the system. The restricted quadrupole interaction $H_\mathrm{QQ}' \left( \gamma \right)$ of Eq. (\[Restricted HQQ\]) is ruled by the quadrupole interaction coefficient $J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ and the anisotropic coefficient $\gamma \stackrel{<}{_\sim} 1$ indicating the slight deviation from the ideal $xz$ model. Consequently, the superconductivity energy gap caused by the restricted quadrupole interaction $H_\mathrm{QQ}' \left( \gamma \right)$ is favorable for the s-like shape superconducting energy gap reflecting the almost isotropic feature in the $x'y'$ plane.
Superconducting ground state and hexadecapole ordering
------------------------------------------------------
In order to examine the hexadecapole ordering in the superconducting phase, we investigate whether or not the superconducting ground state $\mathit{\varPhi}_0$ of Eq. (\[Superconducting ground state\]) due to the restricted quadrupole interaction Hamiltonian $H_\mathrm{QQ}' \left( \gamma \right)$ of Eq. (\[Restricted HQQ\]) bears the hexadecapole. The annihilation operators $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }$ of Eq. (\[Fourier transformation of B 1\]) and the creation operators $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger$ of Eq. (\[Fourier transformation of B 2\]) describing the hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha$ of Eq. (\[Hexadecapole by Bk\]) possess the following expectation values for the superconducting ground state $\mathit{\Phi}_0$ of Eq. (\[Superconducting ground state\]): $$\begin{aligned}
%B
\label{Expectation value of B 1}
&
\left\langle \mathit{\Phi}_0 \left| B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} } \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
& \quad \qquad
= \frac{1}{2\sqrt{2} }
\left[
e^{\mp i \frac{3\pi}{4} }
\frac{ \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) } { E_{+1} \left( \boldsymbol{k} \right) }
+ e^{\pm i \frac{3\pi}{4} }
\frac{ \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) } { E_{-1} \left( \boldsymbol{k} \right) }
\right]
, \\
%B*
\label{Expectation value of B 2}
&
\left\langle \mathit{\Phi}_0 \left| B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
& \quad \qquad
= \frac{1}{2\sqrt{2} }
\left[
e^{\pm i \frac{3\pi}{4} }
\frac{ \mathit{\Delta}_{-1, -1}^\ast \left( \boldsymbol{k} \right) } { E_{+1} \left( \boldsymbol{k} \right) }
+ e^{\mp i \frac{3\pi}{4} }
\frac{ \mathit{\Delta}_{+1, +1}^\ast \left( \boldsymbol{k} \right) } { E_{-1} \left( \boldsymbol{k} \right) }
\right]
.\end{aligned}$$ The finite values of $\bigl\langle \mathit{\Phi}_0 | B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} } | \mathit{\Phi}_0 \bigr\rangle \neq 0$ and $\bigl\langle \mathit{\Phi}_0 | B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger | \mathit{\Phi}_0 \bigr\rangle \neq 0$ are expected with the appearance of the energy gaps $\mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right) \neq 0$ of Eq. (\[Energy gap 1\]) and $\mathit{\Delta}_{\pm1, \pm1}^\ast \left( \boldsymbol{k} \right) \neq 0$ of Eq. (\[Energy gap 2\]) below the superconducting transition temperature. This result shows that the superconducting ground state $\mathit{\Phi}_0$ of Eq. (\[Superconducting ground state\]) actually bears the off-diagonal long-range order parameters $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }$ and $B_{\boldsymbol{k}, \pm, \sigma, \overline{\sigma} }^\dagger$ for the Cooper pairs.
The hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q}}^\alpha$ of Eq. (\[Hexadecapole by Bk\]) is expressed in terms of the difference in the numbers of Cooper pairs as $$\begin{aligned}
\label{Hza by l}
H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha
&=\frac{1}{2} \sum_{\sigma \neq \overline{\sigma} }
\left(
B_{\boldsymbol{k} + \boldsymbol{q}, +, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }
- B_{\boldsymbol{k} + \boldsymbol{q}, -, \sigma, \overline{\sigma} }^\dagger B_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }
\right)
\nonumber \\
&= - \frac{i}{2} \sum_{\sigma \neq \overline{\sigma} }
\left(
l_{\boldsymbol{k} + \boldsymbol{q}, +1, \sigma}^\dagger l_{-\boldsymbol{k} -\boldsymbol{q}, +1, \overline{\sigma} }^\dagger
l_{-\boldsymbol{k}, -1, \overline{\sigma} } l_{\boldsymbol{k}, -1, \sigma} \right.
\nonumber \\
&\left. \qquad \qquad
- l_{\boldsymbol{k} + \boldsymbol{q}, -1, \sigma}^\dagger l_{-\boldsymbol{k} -\boldsymbol{q}, -1, \overline{\sigma} }^\dagger
l_{-\boldsymbol{k}, +1, \overline{\sigma} } l_{\boldsymbol{k}, +1, \sigma}
\right)
.\end{aligned}$$ Thus, we obtain the expectation value for the hexadecapole for the superconducting ground state $\mathit{\Phi}_0$ as $$\begin{aligned}
\label{Expectation value of Hza}
\left\langle \mathit{\Phi}_0 \left| H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha \right| \mathit{\Phi}_0 \right\rangle
&= -\frac{i}{4}
\left[
\frac{ \mathit{\Delta}_{-1, -1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) } { E_{+1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) } { E_{-1} \left( \boldsymbol{k} \right) } \right.
\nonumber \\
&\qquad \quad \left.
- \frac{ \mathit{\Delta}_{+1, +1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) } { E_{-1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) } { E_{+1} \left( \boldsymbol{k} \right) }
\right]
.\end{aligned}$$ The appearance of the energy gaps $\mathit{\Delta}_{\pm1, \pm1} \left( \boldsymbol{k} \right) \neq 0$ in the superconducting phase simultaneously brings about a finite eigenvalue of the hexadecapole $\bigl\langle \mathit{\Phi}_0 | H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha | \mathit{\Phi}_0 \bigr\rangle \neq 0$.
The critical slowing down of the relaxation time $\tau_\mathrm{H}$ around the superconducting transition point in Fig. \[Fig1\](b) is observed via the transverse ultrasonic waves with wavelengths as long as $\lambda \sim 10$ $\mu$m. Therefore, we treat the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonons in Eqs. (\[Hza by l\]) and (\[Expectation value of Hza\]) for the small-wavenumber limit of $\left| \boldsymbol{q} \right| = 2 \pi / \lambda \rightarrow 0 $. Furthermore, we assume equivalence in the amplitudes of $\left| \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) \right|
= \left| \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) \right|
= \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|$ for the two energy gaps of Eq. (\[Energy gap 1\]) and in the excitation energies of $E_{+1} \left( \boldsymbol{k} \right) = E_{-1} \left( \boldsymbol{k} \right) = E \left( \boldsymbol{k} \right)$ for the Bogoliubov quasiparticles of Eq. (\[Vector L\]). The hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha $ of Eq. (\[Hza by l\]) corresponding to the difference in the numbers of Cooper pairs $N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }$ and $N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }$ has an expectation value for the superconducting ground state $\mathit{\Phi}_0$ of $$\begin{aligned}
\label{Hza by sin}
\left\langle \mathit{\Phi}_0 \left| H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha \right| \mathit{\Phi}_0 \right\rangle
&= \left\langle \mathit{\Phi}_0 \left|
\frac{1}{2} \sum_{\sigma \neq \overline{\sigma} }
\left( N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} } - N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} } \right)
\right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
&= \frac{1}{2}
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 } { E \left( \boldsymbol{k} \right)^2 }
\sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right]
.\end{aligned}$$ Thus, the hexadecapole ordering corresponding to the finite difference in the numbers of Cooper pairs $N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} } - N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }$ leads to $\sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] \neq 0$ for the finite deviation of the phase difference of $\varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right)$ from the stationary point of $s\pi$ with an even integer $s = 2n$ for $J_\mathrm{Q} \bigl( \boldsymbol{k}, 0 \bigr) > 0$ or with an odd integer $s = 2n+1$ for $J_\mathrm{Q} \bigl( \boldsymbol{k}, 0 \bigr) < 0$. On the other hand, when the number of Cooper pairs $N_{\boldsymbol{k}, +, \sigma, \overline{\sigma} }$ is equivalent to the counter number of $N_{\boldsymbol{k}, -, \sigma, \overline{\sigma} }$, we expect the equilibrium state of $\sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] = 0$ in accordance with the phase difference located at the stationary point of $\varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) = 2n\pi$ or $(2n+1)\pi$.
The interaction of the hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha$ with the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonons of Eq. (\[Hrot by Bk\]) generates a perturbation in the quadrupole interaction energy of Eq. (\[HQQ by cos\]) in the superconducting phase. The expectation value of the hexadecapole-rotation interaction Hamiltonian $H_\mathrm{rot} \bigl( \omega_{xy} \bigr)$ of Eq. (\[Hrot by Bk\]) for the superconducting ground state $\mathit{\Phi}_0$ is calculated as $$\begin{aligned}
\label{Expectation value of Hrot}
&
\left\langle \mathit{\Phi}_0 \left| H_\mathrm{rot} \bigl( \omega_{xy} \bigr) \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
&\quad
= i \left( 1 - \gamma \right) \sum_{\boldsymbol{k}, \boldsymbol{q} } J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} \right)
\left[
\frac{ \mathit{\Delta}_{-1, -1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) }{ E_{+1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) }{ E_{-1} \left( \boldsymbol{k} \right) } \right.
\nonumber \\
&\qquad \qquad \qquad \left. -
\frac{ \mathit{\Delta}_{+1, +1}^\ast \left( \boldsymbol{k} + \boldsymbol{q} \right) }{ E_{-1} \left( \boldsymbol{k} + \boldsymbol{q} \right) }
\frac{ \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) }{ E_{+1} \left( \boldsymbol{k} \right) }
\right] \omega_{xy} \left( \boldsymbol{q} \right)
.\end{aligned}$$ Since the rotation $\omega_{xy} \left( 0 \right)$ of the transverse acoustic phonons in the small-wavenumber regime of $|\boldsymbol{q}| \rightarrow 0$ participates in the ferro-type hexadecapole interaction, the hexadecapole-rotation interaction energy $\langle \mathit{\Phi}_0 | H_\mathrm{rot} \bigl( \omega_{xy} \bigr) | \mathit{\Phi}_0 \rangle$ in the superconducting state of Eq. (\[Expectation value of Hrot\]) is reduced as $$\begin{aligned}
\label{Hrot by sin}
&
\left\langle \mathit{\Phi}_0 \left| H_\mathrm{rot} \bigl( \omega_{xy} \bigr) \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
&\quad
= - 2\left( 1 - \gamma \right) \sum_{\boldsymbol{k} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 }{ E \left( \boldsymbol{k} \right)^2 }
\nonumber \\
&\qquad \qquad \qquad \qquad \times
\sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right]
\omega_{xy} \left( 0 \right)
\nonumber \\
&\quad
= -4 \left( 1 - \gamma \right) \sum_{\boldsymbol{k} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\left\langle \mathit{\Phi}_0 \left| H_{z, \boldsymbol{k}, \boldsymbol{q} = 0}^\alpha \right| \mathit{\Phi}_0 \right\rangle
\omega_{xy} \left( 0 \right)
.\end{aligned}$$ Here, we used the expectation value of the hexadecapole of Eq. (\[Hza by sin\]) for the superconducting ground state $\mathit{\Phi}_0$ of Eq. (\[Superconducting ground state\]). As discussed for Eqs. (\[Torque for two-electron state\]) and (\[Define torque\]), the torque $\tau_{xy} \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)$ is proportional to the hexadecapole. It is meaningful to show the expectation value of the torque $\tau_{xy}$ for the superconducting ground state $\mathit{\Phi}_0$ as follows: $$\begin{aligned}
\label{Expectation value of torque}
&
\left\langle \mathit{\Phi}_0 \left| \tau_{xy} \right| \mathit{\Phi}_0 \right\rangle
\nonumber \\
& \quad
= -4 \left( 1 - \gamma \right) \sum_{\boldsymbol{k} }
J_\mathrm{Q} \left( \boldsymbol{k}, \boldsymbol{q} = 0 \right)
\left\langle \mathit{\Phi}_0 \left| H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha \right| \mathit{\Phi}_0 \right\rangle.\end{aligned}$$ Thus, we have an elementary expression for the hexadecapole-rotation interaction energy in terms of the torque $\tau_{xy}$ and rotation $\omega_{xy} \left( \boldsymbol{q} = 0 \right)$ as $$\begin{aligned}
\label{Hrot by torque}
\left\langle \mathit{\Phi}_0 \left| H_\mathrm{rot} \bigl( \omega_{xy} \bigr) \right| \mathit{\Phi}_0 \right\rangle
= \left\langle \mathit{\Phi}_0 \left| \tau_{xy} \right| \mathit{\Phi}_0 \right\rangle
\omega_{xy} \left( 0 \right)
.\end{aligned}$$ Note that the expectation value of the torque $\bigl\langle \mathit{\Phi}_0 | \tau_{xy} | \mathit{\Phi}_0 \bigr\rangle $ defined in Eq. (\[Expectation value of torque\]) spontaneously becomes finite with the appearance of the hexadecapole ordering $\bigl\langle \mathit{\Phi}_0 | H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha | \mathit{\Phi}_0 \bigr\rangle \neq 0$ due to the superconducting energy gaps of $| \mathit{\Delta}_{+1, +1} \left( \boldsymbol{k} \right) |
= | \mathit{\Delta}_{-1, -1} \left( \boldsymbol{k} \right) |
= | \mathit{\Delta} \left( \boldsymbol{k} \right) | \neq 0$ in the superconducting phase.
In order to specify the hexadecapole ordering in the superconducting phase far below the transition temperature, we adopt a phenomenological treatment using the microscopic theory described above. The internal energy $U$ includes both the restricted quadrupole interaction energy of Eq. (\[HQQ by cos\]) and the hexadecapole-rotation interaction energy for the rotation $\omega_{xy} \left( 0 \right)$ in the small-wavenumber limit of $\left| \boldsymbol{q} \right| \rightarrow 0$ in Eq. (\[Expectation value of Hrot\]) as $$\begin{aligned}
\label{HQQ + Hrot in SC}
U &= \sum_{ \boldsymbol{k} }U \left( \boldsymbol{k} \right)
\nonumber \\
&
= - \frac{1}{2} \left( 1 - \gamma \right) \sum_{\boldsymbol{k} }
J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right)
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 }{ E \left( \boldsymbol{k} \right)^2 }
\nonumber \\
&\qquad \times
\left\{
\cos \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] \right.
\nonumber \\
&\left. \qquad \qquad
+ 4 \sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] \omega_{xy} \left( 0 \right)
\right\}
\nonumber \\
& \quad
+ \frac{1}{2} C_{66}^0 \omega_{xy} \left( 0 \right)^2 + \frac{1}{4} \beta \omega_{xy} \left( 0 \right)^4
.\end{aligned}$$ Here, we add the harmonic rotation energy of $C_{66}^0 \omega_{xy} \left( 0 \right)^2/2$ and the higher-order term of $\beta \omega_{xy} \left( 0 \right)^4/4$ with $\beta > 0$ to avoid instability of the system. Since the present system has the spin-singlet property of $S = 0$, we take the sum of the spin orientation as $\sum_{\sigma \neq \overline{\sigma} } = 2$ in the internal energy $U$ of Eq. (\[HQQ + Hrot in SC\]). We seek the minimum of the internal energy $U \left( \boldsymbol{k} \right)$ of Eq. (\[HQQ + Hrot in SC\]) in finite variations of the phase difference of $\delta \varphi \left( \boldsymbol{k} \right)
= \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) - s\pi \neq 0$ from the stationary point of $s\pi = 2n\pi$ for the ferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0\right) > 0$ or $s\pi = (2n+1)\pi$ for the antiferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0\right) < 0$ and the finite rotation $\omega_{xy} \left( 0 \right) \neq 0$.
The internal energy $U \left( \boldsymbol{k} \right)$ characterized by wavevector $\boldsymbol{k}$ in Eq. (\[HQQ + Hrot in SC\]) is expanded up to the second order of $\delta \varphi \left( \boldsymbol{k} \right)$ as $$\begin{aligned}
\label{HQQ + Hrot in SC with k}
U \left( \boldsymbol{k} \right)
&= - \frac{1}{2} \left( 1 - \gamma \right) J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right) \cos \left( s\pi \right)
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 }{ E \left( \boldsymbol{k} \right)^2 }
\nonumber \\
&\qquad \quad \times
\left[ 1 - \frac{1}{2} \delta \varphi \left( \boldsymbol{k} \right)^2 + 4 \delta \varphi \left( \boldsymbol{k} \right) \omega_{xy} \left( 0 \right) \right]
\nonumber \\
&+ \frac{1}{2} \frac{ C_{66}^0 }{ N_\mathrm{C} } \omega_{xy} \left( 0 \right)^2
+ \frac{1}{4} \frac{ \beta }{ N_\mathrm{C} } \omega_{xy} \left( 0 \right)^4
.\end{aligned}$$ Here, $N_\mathrm{C} = \sum_{\boldsymbol{k} }$ is the number of Cooper pairs participating in the superconductivity. For simplicity, we use the abbreviated notation $S_\mathrm{Q} \left( \boldsymbol{k} \right)$ proportional to the square of the energy gap $\left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2$ as $$\begin{aligned}
\label{SQ}
S_\mathrm{Q} \left( \boldsymbol{k} \right)
&
= \frac{1}{2} \left( 1 - \gamma \right) J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right) \cos \left( s\pi \right)
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 } {E \left( \boldsymbol{k} \right)^2 }
\nonumber \\
&
= \frac{1}{2} \left( 1 - \gamma \right) \left| J_\mathrm{Q} \left( \boldsymbol{k}, 0 \right) \right|
\frac{ \left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right|^2 } {E \left( \boldsymbol{k} \right)^2 }
.\end{aligned}$$ Since the quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0\right)$ is dominated by the indirect quadrupole interaction of $D^\mathrm{QQ} \left( \boldsymbol{k}, 0\right)$ of Eq. (\[DQQ(k,0)\]), the ferro-type quadrupole interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0\right) > 0$ for the small-wavenumber regime of $|\boldsymbol{k}| < k_\mathrm{b}^\mathrm{Q}$ causes the energy gaps with the phase difference around the stationary point of $s\pi = 2n\pi$ for $\cos \left( 2n\pi \right) = 1$, while the antiferro-type interaction of $J_\mathrm{Q} \left( \boldsymbol{k}, 0\right) < 0$ with the relatively large-wavenumber regime of $|\boldsymbol{k}| > k_\mathrm{b}^\mathrm{Q}$ brings about the energy gaps with the phase difference around the stationary point of $s\pi = \left(2n+1 \right) \pi$ for $\cos \left[ \left( 2n + 1 \right)\pi \right] = -1$. The experimentally determined phase diagram as shown in Fig. \[Fig4\] indicates that the ferro-type quadrupole interaction for the Co concentration $x$ below the QCP of $x_\mathrm{QCP} = 0.061$ changes to the antiferro-type quadrupole interaction with increasing $x$ across $x_\mathrm{QCP}$. Therefore, the inherent property, that both the antiferro- and ferro-quadrupole interactions participate in the Cooper pair formation in Eq. (\[SQ\]), is available for the appearance of the superconductivity in the vicinity of the QCP.
Thus, the internal energy $U \left( \boldsymbol{k} \right)$ of Eq. (\[HQQ + Hrot in SC with k\]) is deduced as $$\begin{aligned}
\label{HQQ + Hrot in SC with k 2}
U \left( \boldsymbol{k} \right)
&
= \frac{1}{2} S_\mathrm{Q} \left( \boldsymbol{k} \right)
\left[ \delta \varphi \left( \boldsymbol{k} \right) - 4 \omega_{xy} \left( 0 \right) \right]^2
\nonumber \\
&\quad
+ \frac{1}{4} \frac{ \beta }{N_\mathrm{C} }
\left\{
\omega_{xy} \left( 0 \right)^2 - 16 \frac{N_\mathrm{C} }{ \beta }
\left[ S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{1}{16} \frac{ C_{66}^0}{ N_\mathrm{C} }
\right]
\right\}^2
\nonumber \\
&\qquad
- S_\mathrm{Q} \left( \boldsymbol{k} \right)
- 64 \frac{N_\mathrm{C} }{ \beta }
\left[ S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{1}{16} \frac{ C_{66}^0}{ N_\mathrm{C} }
\right]^2
.\end{aligned}$$ The appearance of the energy gap $\left| \mathit{\Delta} \left( \boldsymbol{k} \right) \right| \neq 0$ in the superconducting phase leads to a finite value of $S_\mathrm{Q} \left( \boldsymbol{k} \right) \neq 0$ in Eq. (\[SQ\]). The minimum of the internal energy $U \left( \boldsymbol{k} \right)$ of Eq. (\[HQQ + Hrot in SC with k 2\]) is satisfied by the following constraint for the phase deviation of $\delta \varphi \left( \boldsymbol{k} \right)$: $$\begin{aligned}
\label{Value of omega1}
\delta \varphi \left( \boldsymbol{k} \right)
= 4 \omega_{xy} \left( 0 \right)
.\end{aligned}$$ This result means that the finite phase deviation of $\delta \varphi \left( \boldsymbol{k} \right) = \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) - s\pi \neq 0$ from the stationary point of $s\pi = 2n\pi$ or $s\pi = (2n+1) \pi$ associated with the hexadecapole ordering brings about the spontaneous rotation of $\omega_{xy} \left( 0 \right) \neq 0$ in the superconducting state.
The energy $U \left( \boldsymbol{k} \right)$ of Eq. (\[HQQ + Hrot in SC with k 2\]) is optimized by the spontaneous rotation $\omega_{xy} \left( 0 \right)$ as $$\begin{aligned}
\label{Value of omega 2}
\omega_{xy} \left( 0 \right)
= \pm 4 \sqrt{
\frac{N_\mathrm{C} }{ \beta }
\left[
S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{1}{16} \frac{ C_{66}^0}{ N_\mathrm{C} }
\right]
}
.\end{aligned}$$ Here, we need the criterion $$\begin{aligned}
\label{Condition of SQ}
S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{1}{16} \frac{ C_{66}^0}{ N_\mathrm{C} } > 0
.\end{aligned}$$ When the appropriate magnitude of $S_\mathrm{Q} \left( \boldsymbol{k} \right)$ due to the superconducting energy gap $\left| \mathit{\Delta}\left( \boldsymbol{k} \right) \right|$ in Eq. (\[SQ\]) satisfies the criterion of Eq. (\[Condition of SQ\]), the spontaneous rotation $\omega_{xy} \left( 0 \right) \neq 0$ of Eq. (\[Value of omega 2\]) of the macroscopic superconducting state occurs with respect to the host lattice. The occurrence of the spontaneous rotation $\omega_{xy} \left( 0 \right) \neq 0$ gives the optimized internal energy associated with the superconductivity bearing the hexadecapole as $$\begin{aligned}
\label{UH}
U_\mathrm{H} \left( \boldsymbol{k} \right)
&= - S_\mathrm{Q} \left( \boldsymbol{k} \right)
- 64 \frac{N_\mathrm{C} }{ \beta }
\left[ S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{1}{16} \frac{ C_{66}^0}{ N_\mathrm{C} }
\right]^2
\nonumber \\
&= - S_\mathrm{Q} \left( \boldsymbol{k} \right) - \frac{\beta}{4 N_\mathrm{C} } \omega_{xy} (0)^4
.\end{aligned}$$ The first term $- S_\mathrm{Q} \left( \boldsymbol{k} \right)$ in Eq. (\[UH\]) is the restricted quadrupole interaction energy with the null phase difference of $\delta \varphi \left( \boldsymbol{k} \right) = \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) - s\pi = 0$ corresponding to $\cos \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] = 1$ for the stationary point of $s\pi = 2n \pi$ or $\cos \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right] = -1$ for the stationary point of $s\pi = (2n+1) \pi$ in Eq. (\[HQQ by cos\]).
The hexadecapole ordering of $\langle H_z^\alpha \rangle \neq 0$ due to the spontaneous deviation $\delta \varphi \left( \boldsymbol{k} \right) \neq 0$ from the stationary point of $s\pi = 2n \pi$ or $s\pi = (2n+1) \pi$ leads to the spontaneous rotation $\omega_{xy} \left( 0 \right) \neq 0$ with respect to the container of the host lattice. According to the second term in Eq. (\[UH\]), the spontaneous rotation $\omega_{xy} \left( 0 \right) \neq 0$ further lowers the internal energy $U\left( \boldsymbol{k} \right)$ from the stationary energy $- S_\mathrm{Q} \left( \boldsymbol{k} \right)$ of the restricted quadrupole interaction for the superconducting ground state of Eq. (\[Superconducting ground state\]). In the framework of the phenomenological theory for the second-order transition [@Landau; @and; @Lifshitz], the quenching of the pair-electron state with the $A_2$ symmetry of the higher-symmetry space group $D_{4h}^{17}$ associated with the spontaneous rotation $\omega_{xy} (0) \neq 0$ brings about a twisted phase with the lower-symmetry space group $C_{4h}^5$ [@Inui; @Group; @International; @Table]. The spontaneous breaking of the $A_2$ symmetry associated with $\langle H_z^\alpha \rangle \neq 0$ and $\omega_{xy} \left( 0 \right) \neq 0$ loses the symmetry operations consisting of $2C_2'$, $2C_2''$, $2\sigma_\mathrm{v}$, and $2\sigma_\mathrm{d}$ in the space group $D_{4h}^{17}$ of the mother phase. Consequently, the macroscopic superconducting state spontaneously twists by $\omega_{xy}(0) \neq 0$ with respect to the host lattice. The hexadecapole of Eq. (\[Hexadecapole\]) is carried by the two-electron states of Eq. (\[psi+- by y’z and zx’\]), which consist of two electrons accommodated in the degenerate $y'z$ and $zx'$ band orbitals mapped on the special unitary group SU(2). Therefore, the appearance of the hexadecapole ordering $\langle H_z^\alpha \rangle \neq 0$ of Eq. (\[Hza by sin\]) is explained by the symmetry breaking in the direct product of the special unitary group SU(2) $\otimes$ SU(2).
The critical slowing down in the relaxation time $\tau_\mathrm{H}$ observed via the ultrasonic attenuation coefficient $\alpha_{66}$ of the transverse ultrasonic wave around the superconducting transition is explained in terms of the hexadecapole susceptibility for the ferro-type ordering of the hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha$. The finite expectation value of the hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha$ of Eq. (\[Hza by sin\]) means that the superconducting ground state of Eq. (\[Superconducting ground state\]) actually bears the hexadecapole. This result convincingly accounts for why the critical slowing down due to the hexadecapole ordering is observed around the superconducting transition. The direct detection of the spontaneous rotated state $\omega_{xy} \left( 0 \right) \neq 0$ due to the hexadecapole ordering in the superconducting phase is strongly required in future.
As is presented in Fig. \[Fig1\](b), the onset of the critical slowing down in the relaxation time $\tau_\mathrm{H}$ appears in the normal phase far above the superconducting transition temperature of $T_\mathrm{SC} = 23$ K, where fermion quasiparticles are relevant but the off-diagonal long-range ordering of the Cooper pairs does not exist. This means that the hexadecapole $H_{z, \boldsymbol{k}, \boldsymbol{q} }^\alpha$ of Eq. (\[Hexadecapole by Bk\]) in the normal phase is actually carried by the fermion quasiparticles but not by the Cooper pairs. The indirect hexadecapole interaction $D^\mathrm{HH} \left( \boldsymbol{k}, \boldsymbol{q} \right)$ of Eq. (\[DHH\]) between two-electron states mediated by the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonons with the small-wavenumber limit of $|\boldsymbol{q}| \rightarrow 0$ favors the ferro-type hexadecapole ordering. This ferro-type hexadecapole interaction reduces the energy of the two-electron states bearing the hexadecapole in both normal and superconducting phases. This plausibly explains why the system specially favors the Cooper pairs bearing the hexadecapole among the various types of Cooper pairs due to the quadrupole interaction of Eq. (\[HQQ by l\]).
The indirect hexadecapole interaction Hamiltonian of Eq. (\[HindHH\]) responsible for the hexadecapole ordering of $H_{z, \boldsymbol{k}, \boldsymbol{q} = 0 }^\alpha$ is mapped to the Ising model. The critical index of $z\nu = 1$ determined by the relaxation time $\tau_\mathrm{H}$ in the normal phase above the superconducting transition temperature for $x = 0.071$ is in good agreement with mean field theory [@Suzuki] and reasonable consistent with the three-dimensional Ising model with $z\nu = 1.2$ for $\nu = 0.63$ and $z = 2.04$ [@G.; @S.; @Pawley; @S.; @Wansleben]. The critical index of $z\nu = 1/3$ in the superconducting phase below $T_\mathrm{SC}$, however, considerably deviates from the standard scaling theory with the same critical indices in both the normal and ordered phases [@Halperin; @and; @Hohenberg]. This inconsistency of $z\nu = 1/3$ below $T_\mathrm{SC}$ and $z\nu = 1$ above $T_\mathrm{SC}$ is accounted for by the inherent properties of the present system where the superconductivity and the hexadecapole ordering simultaneously appear.
The quenching of the $A_2$ symmetry of the hexadecapole ordering $\langle H_z^\alpha \rangle \neq 0$ accompanied the superconducting transition for $x = 0.071$ changes the symmetry of the mother tetragonal phase with the space group $D_{4h}^{17}$ to the ordered tetragonal phase with the space group $C_{4h}^5$, while the quenching of the $B_2$ symmetry of the quadrupole ordering $\langle O_{x'^2-y'^2} \rangle \neq 0$ accompanying the structural transition for $x = 0.036$ gives the orthorhombic phase with the space group $D_{2h}^{23}$. Note that the space groups $C_{4h}^5$ for the hexadecapole ordered phase and $D_{2h}^{23}$ for the quadrupole ordered phase are subfamilies of the space group $D_{4h}^{17}$ for the mother tetragonal phase. The orthorhombic phase of the space group $D_{2h}^{23}$ due to the ferro-quadrupole ordering changes to the superconducting phase below $T_\mathrm{SC} =$ 16.4 K for $x = 0.036$. The orthorhombicity of $\delta = (a - b)/(a + b) \sim 10^{-3}$ for the lattice parameters $a$ and $b$ in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ with $x = 0.047$-$0.063$ reveals the tendency of $\delta \rightarrow 0$ with decreasing temperature in the superconducting phase [@Nandi]. This reentrant property favoring the tetragonal lattice instead of the distorted orthorhombic lattice in the superconducting phase is consistent with the tetragonal structure $C_{4h}^5$ proposed as the hexadecapole ordering phase in the present paper. In short, we conclude that the unconventional superconductivity accompanying the hexadecapole ordering in the present system is a common feature across the QCP for $x = 0.061$.
Conclusion
==========
In the present work, we investigated the order parameter dynamics around the superconducting and structural transitions in the iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ by means of ultrasonic measurements. The critical slowing down of the relaxation time $\tau$ due to freezing of the order parameter fluctuations associated with the superconducting and structural transitions was verified by the divergence of the ultrasonic attenuation coefficients. In the analysis of the experiments, we employed the significant fact that the transverse ultrasonic waves used to measure the elastic constant $C_{66}$ simultaneously induce the rotation $\omega_{xy}$ of the antisymmetric tensor and the strain $\varepsilon_{xy}$ of the symmetric tensor. Taking the band calculations on the iron pnictide into account, we suppose that the degenerate $y'z$ and $zx'$ bands carrying the electric quadrupoles $O_{x'^2 - y'^2}$ and $O_{x'y'}$ and the angular momentum $l_z$ play an essential role in the appearance of the superconducting and structural transitions in the system. We attempted to clarify the order parameters caused by the symmetry breaking of the space group $D_{4h}^{17}$ of the tetragonal mother phase and the special unitary group SU(2) of the degenerate $y'z$ and $zx'$ bands.
The structural transition in the compound $x = 0.036$ is caused by the ferro-quadrupole ordering of $O_{x'^2 - y'^2}$, which was identified as one of the generator elements of SU(2). The interaction of the quadrupole $O_{x'^2 - y'^2}$ to the strain $\varepsilon_{xy}$ leads to a structural transition due to the ferro-quadrupole ordering. The critical slowing down of the relaxation time $\tau_\mathrm{Q}$ by the ultrasonic attenuation coefficient $\alpha_{66}$ and the large softening of the elastic constant $C_{66}$ are well described in terms of the divergence of the quadrupole susceptibility. The quenching of the quadrupole $O_{x'^2 - y'^2}$ belonging to the $B_2$ irreducible representation of the tetragonal mother phase of the high-symmetry space group $D_{4h}^{17}$ brings about a distorted orthorhombic phase with the low-symmetry space group $D_{2h}^{23}$.
The rotation $\omega_{xy}$ of the transverse ultrasonic waves or transverse acoustic phonons twists the electronic states with angular momentum $l_z$. The rotation operator of $\exp \bigl[ - i l_z \omega_{xy} \bigr]$ acting on the Hamiltonian $H$ leads to the perturbation of $H_\mathrm{rot} \bigl( \omega_{xy} \bigr) = i \left[ l_z, H \right] \omega_{xy} =\tau_{xy}\omega_{xy}$, where the torque $\tau_{xy}$ is described by the Heisenberg equation $i\hbar \partial l_z/ \partial t = \left[ l_z, H \right] = i \tau_{xy}$. For the CEF Hamiltonian of $H = H_\mathrm{CEF}$, the torque $\tau_{xy}$ corresponding to the hexadecapole of one-electron states, however, vanishes because of the rotational invariance for states in the central force in the absence of an external magnetic field.
The rotation operation on the anisotropic quadrupole interaction Hamiltonian consisting of $O_{x'^2 - y'^2}$ and $O_{x'y'}$ gives the interaction of the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr)
= O_{x'y'} \bigl( \boldsymbol{r}_i \bigr) O_{x'^2 - y'^2} \bigl( \boldsymbol{r}_j \bigr)
+ O_{x'^2 - y'^2} \bigl( \boldsymbol{r}_i \bigr) O_{x'y'} \bigl( \boldsymbol{r}_j \bigr)$ to the rotation $\omega_{xy}$ of the transverse ultrasonic waves. The critical slowing down of the relaxation time $\tau_\mathrm{H}$ around the superconducting transition is well described in terms of the divergence of the hexadecapole susceptibility. This result indicates that the ferro-type hexadecapole ordering is caused by the quenching of the two-electron state belonging to the $A_2$ irreducible representation of the tetragonal phase with space group $D_{4h}^{17}$.
Supposing that the quadrupole interaction of $O_{x'^2 - y'^2}$ mediated by the strain $\varepsilon_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonons with a small wavevector favors the attractive force for pairs of electrons, we solve the superconducting Hamiltonian based on the anisotropic quadrupole interaction. The superconducting ground state consisting of two different energy gaps for the Cooper pairs possesses the energy dependence of $\cos \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right]$ for the phase difference of $ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right)$ between the two energy gaps. The Cooper pair bearing the hexadecapole proportional to $\sin \left[ \varphi_{+1, +1} \left( \boldsymbol{k} \right) - \varphi_{-1, -1} \left( \boldsymbol{k} \right) \right]$ couples to the rotation of $\omega_{xy} \left( \boldsymbol{q} \right)$ of the transverse acoustic phonon. The hexadecapole interaction mediated by the rotation $\omega_{xy} \left( \boldsymbol{q} \right)$ with the small-wavenumber limit of $\left| \boldsymbol{q} \right| \rightarrow 0$ causes the ferro-type hexadecapole ordering in the superconducting phase, which brings about the spontaneous rotation $\omega_{xy} \left( 0 \right) \neq 0$ of the macroscopic superconducting state with respect to the host tetragonal lattice. This is in good agreement with the ground scenario that the critical slowing down around the superconducting transition is caused by the hexadecapole ordering. The simultaneous symmetry breaking of the U(1) gauge for the off-diagonal long-range ordering of the Cooper pairs and the freezing of the $A_2$ symmetry of space group $D_{4h}^{17}$ for the ferro-type hexadecapole ordering specify the unconventional superconductivity of the present iron pnictide.
The present experiments on the iron pnictide Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ showed that ultrasonic measurements by transverse ultrasonic waves carrying rotation as well as strain are a powerful means of detecting the hexadecapole and quadrupole effects of quantum systems with orbital degrees of freedom. We hope that the measurements of the hexadecapole as well as the quadrupole by the transverse ultrasonic waves will lead to the discovery of various exotic phenomena associated with magnetic and quadrupole orderings and unconventional superconductivity in strongly correlated electron physics.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors specially thank Prof. Hidetoshi Fukuyama, Prof. Yoshiaki Ōno, and Prof. Haruhiko Suzuki for stimulating and helpful discussions in the present work. We are grateful for financial support by Grants-in-Aid for Specially Promoted Research entitled “Strongly correlated quantum phase associated with charge fluctuation” (JSPS KAKENHI Grant Number JP18002008), Young Scientists (B) entitled “Elucidation of electric quadrupole effects in multiband superconductors by ultrasonic measurement” (JP26800184), and Scientific Research (C) entitled “Elucidation of superconductivity in multiband superconductors by ultrasonic measurements under hydrostatic pressures” (JP17K05538) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. This work was also supported in part by Grants-in-Aid for Scientific Research (A) (JP26247059), Scientific Research (B) (JP21340097), Young Scientists (B) (JP15K17704), Challenging Exploratory Research (JP23654116 and JP15K13518), and the Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation from the Japan Society for the Promotion of Science.
Most of this work is based on a thesis submitted to partially fulfill the requirement for the Ph.D degree in Science of Ryosuke Kurihara of the Graduate School of Science and Technology, Niigata University, Niigata (2016).
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[^1]: E-mail : r.kurihara@phys.sc.niigata-u.ac.jp
[^2]: Corresponding author, Emeritus Prof. of Niigata Univ., E-mail : goto@phys.sc.niigata-u.ac.jp
[^3]: Emeritus Prof. of Nagoya Univ., E-mail : msatojun7@gmail.com
|
---
abstract: |
Taking precautions before or during the start of a virus outbreak can heavily reduce the number of infected. The question which individuals should be immunized in order to mitigate the impact of the virus on the rest of population has received quite some attention in the literature. The dynamics of the of a virus spread through a population is often represented as information spread over a complex network. The strategies commonly proposed to determine which nodes are to be selected for immunization often involve only one centrality measure at a time, while often the topology of the network seems to suggest that a single metric is insufficient to capture the influence of a node entirely.
In this work we present a generic method based on a genetic algorithm (GA) which does not rely explicitly on any centrality measures during its search but only exploits this type of information to narrow the search space. The fitness of an individual is defined as the estimated expected number of infections of a virus following SIR dynamics. The proposed method is evaluated on two contact networks: the Goodreau’s Faux Mesa high school and the US air transportation network. The GA method manages to outperform the most common strategies based on a single metric for the air transportation network and its performance is comparable with the best performing strategy for the high school network. **Keywords:** Complex networks; Genetic algorithm; Node centrality; SIR model; Damage control
author:
- 'V.V. Kashirin[^1]'
- 'L.J. Dijkstra'
nocite:
- '[@Hethcote2000]'
- '[@Brandes2001]'
title: A heuristic optimization method for mitigating the impact of a virus attack
---
Introduction {#sec:introduction}
============
Many viruses spread through human population by means of personal contact between infectious individuals (those who carry the virus) and susceptibles (those who are not ill at the moment but can catch the disease) [@Hethcote2000; @Newman2002]. A concept that proved to be very valuable in order to gain a better understanding of the virus spread process is the complex network [@Newman2010]. Nodes within this graph structure represent individuals and are associated with a certain state (e.g., susceptible or infectious). Edges between these nodes account for social interactions. The percolation of the virus through the population (network) is then predicated by a fixed set of rules [@Newman2002; @Boccaletti2006; @Pastor-Satorras2001; @Tao2006].
This approach has not only been employed in order to better understand the dynamics of such a disease spread, but also gives rise to a field of research with a more proactive attitude: which individuals in the population should be immunized to limit the spread of the virus most effectively? Or, to frame it a bit differently, which nodes in the network should be protected in order to limit the damage done as much as possible?
The concept commonly used in order to find those nodes that are in need of protection (or removed in some cases) is *node centrality*[^2] [@Holme2002; @Memon2006; @Kitsak2010; @Bright2011; @Hou2012; @Chen2012]. Node centralities express to what extent the node facilitates the spread of information over the network (note that information spread is, to some extent, alike to a virus spread [@Daley1964]). In this article we consider three common traditional variants[^3] used in the literature, each of which formalizes the concept of centrality in a (slightly) different manner: 1) *degree centrality* which is equal to the node’s degree, 2) *betweenness centrality* which focusses rather on to what extent the node could influence the communication between other nodes in the network, and 3) *eigenvector centrality* which expresses the relative importance of a node in terms of importance of its direct neighbors (a having a few connections with important nodes is more important than the one having the same number of connections with less important ones). The definitions of these centrality measures are given in section \[sec:centrality\]. The usual approach to locate those nodes that influence the spread of information the most is to determine for all nodes in the network their score on one centrality measure only. A fraction of nodes that scored the highest are then proposed to be immunized. This approach has its drawbacks. Take, for example, the network in Fig. \[fig:example\]. It consists of two clusters connected by one single node. If one would only use a ranking based on degree centralities (as it rather common in the literature), one would start by immunizing several nodes in the clusters. The node in the center of the graph is left untouched, while immunizing it at an early phase, would prohibit the virus to spread from one cluster to the other. The importance of this node would be noted only when one would take betweenness centralities into account as well. Of course, this example is rather simple and one would be able to come up with such an analysis by simply examining the graph visually. When the network gets large, however, (e.g., 500 nodes as in one of the networks examined later – see section \[sec:networks\]) it is rather hard to make these kinds of assertions and a more structural approach is required.
![A rather simple hypothetical network. The values in each node represent the node’s degree and betweenness centrality, respectively.[]{data-label="fig:example"}](Figure1.pdf){width=".6\textwidth"}
In this paper we propose a genetic algorithm [@SivanandamS.N.andDeepa2007] (GA) for finding those individuals that need to be protected in order to limit the spread of the virus through the rest of the network as much as possible. This heuristic search approach differs from the methods proposed in the literature, since the GA is allowed to search through a wide range of subsets of nodes and does not rely explicitly on any kind of centrality metrics during its search; centrality measures are only used to narrow the search space. Nodes that score low on all measures of centrality most likely do not play an important role in the spread of the virus and can, thus, be neglected.
In order to emulate the spread of the virus over the network we define an adapted version of one of the most common models in epidemiology: the *SIR model* [@Hethcote2000; @Newman2002; @Newman2010; @Daley1964]. The model was first introduced in the twenties by Lowell Reed and Wade Hampton Frost. They proposed to divide the entire population into the following three distinct classes:
- The *susceptibles*, $S$: the group of individuals who have not been infected but can catch the disease.
- The *infectives*, $I$: the group consisting of all individuals that are currently infected by the disease and could infect others from the susceptible class.
- The *removed*, $R$ (sometimes referred to in the literature as recovered): those who had the virus but either recovered and gained immunity or died.
Note that an individual in the model can go through three strictly sequential phases: susceptible individuals can get infected by the virus and either recover or die; removed individuals never loose their gained immunity[^4]. In section \[sec:sir\] we discuss our adaptation of this model for simulating the spread on networks.
The structure of the paper is organized as follows. In the following section we discuss the methods used. First we define the optimization task at hand formally in section \[sec:formal\]. Section \[sec:networks\] introduces the two networks that are used for validation of the methods. We proceed with defining the three different centrality measures discussed earlier in the text. In section \[sec:sir\] we formally define an adaption of the traditional SIR compartmental model for network structures. Section \[sec:ga\] contains a detailed description of the genetic algorithm for finding the optimal set of nodes that need to be protected using the various centrality rankings. The results are presented in section \[sec:results\]. We finish this paper with our conclusions, some discussion and a few pointers for future research.
Methods {#sec:methods}
=======
The optimization task {#sec:formal}
---------------------
The optimization task presented in the introduction can be expressed a bit more formally in the following manner. Suppose we are dealing with a network $G = \left<V, E\right>$ where $V$ denotes the set of vertices in the network and $E$ is the set of (bidirectional) edges between pairs of nodes in $V$. The size of the network is given by $N = |V|$. The number of nodes that can be protected is limited due to time and resource constraints to a total of $k$ nodes. We are, thus, interested in finding a $k$-subset of nodes $I \subseteq V$ that, when protected, will most effectively limit the spread of the virus through the network $G$. This task is by no means easy. The number of possible $k$-subsets of nodes is $\binom{N}{k}$ which, for most real-world networks, is simply too large to brute-force. In addition, one is generally unaware where the virus starts which needs to be accounted for when assessing the successfulness of a solution.
In the following sections we describe a method to find a (near-)optimal $k$-subset of nodes.
The data sets used for validation {#sec:networks}
---------------------------------
In this paper we consider two contact networks to validate our approach, one social and one transportation network:
1. The Goodreau’s Faux Mesa high school network which is the result of a simulation of a in-school friendship network based on data from a high school in the rural western United States [@Resnick1997]. The network consists of $147$ nodes and $202$ undirected edges (representing mutual friendship). See Fig. \[fig:topo\]a for a visual representation of the graph.
2. The US air transportation network [@Colizza2007], which consists of $500$ nodes (US airports with the highest amount of traffic) and $2980$ undirected edges (representing air travel connections). See Fig. \[fig:topo\]b for a visual representation of the graph.
Both data sets are freely available and can be found, respectively, at the website of the CASOS group[^5] and the Cx-Nets website[^6].
Please, note that the models, methods and measures discussed in the following sections can be applied (or extended) to any network structure. The networks chosen here are only used as examples.
Node centrality {#sec:centrality}
---------------
Measures for centrality try to formalize the (relative) importance of a node in the network. In this paper we consider the following usual measures: 1) degree, 2) betweenness and 3) eigenvector centrality.
Degree centrality, $C_D$, formalizes the importance of a node by setting it equal to the number of neighbors the node has; the more connections, the more important the node: $$C_D (v) = \mathrm{deg}(v).
\label{eq:d}$$ Betweenness centrality, $C_B$, tries to capture a different kind of ‘importance’. The idea behind this formalism is that the importance of a node depends on to what extent it can influence the communication between the other nodes in the network [@Newman2010]. The measure is computed by determining the fraction of the shortest paths between each pair of nodes in the network that contains the node $v$, i.e.: $$C_B (v) = \displaystyle\sum_{s\neq v\neq t \in V} \frac{\sigma_{s,t} (v)}{\sigma_{s,t}},
\label{eq:b}$$ where $\sigma_{s,t}$ is the number of the shortest paths between the nodes $s$ and $t$ and $\sigma_{s,t}(v)$ returns the number of shortest paths that passes through the vertex of interest, $v$. As one can imagine, this measure for centrality is computationally rather hard to determine. In 2001 Ulrik Brandes presented a fast version of the algorithm; computation time is upper bounded by $O(|V||E|)$. We will use this algorithm for determining the betweenness centrality throughout this paper.
Eigenvector centrality, $C_E$, distinguishes itself from the other measures discussed here, since it takes the importance of its neighbors explicitly into account: $$C_E (v) = \frac{1}{\lambda} \displaystyle\sum_{w \in V} a_{v,w} C_E (w),
\label{eq:e1}$$ where $a_{v,w}$ is the $(v,w)$-entry of the adjacency matrix of the network in question. Rewriting this expression to matrix form yields $$\mathbf{A}\mathbf{x} = \lambda \mathbf{x},
\label{eq:e2}$$ where the $i$-th entry of $\mathbf{x}$ is equal to $C_E(i)$. In other words, the eigenvector centrality of a node $v$ is the $v$-th entry of the right eigenvector associated with the first eigenvalue $\lambda$ of the adjacency matrix $\mathbf{A}$ of the network.
An adaption of the SIR model for networks {#sec:sir}
-----------------------------------------
The SIR model is one of the most often used models for simulating the spread of a virus through a population. The original model in terms of a set of ordinary differential equations makes the assumptions that 1) the population size, $N$, is large and 2) the population is ‘fully mixed’, meaning that every individual has the same number of (randomly picked) connections and have (approximately) the same number of contacts at the same time [@Hethcote2000; @Newman2002]. Of course, these assumptions are rather unsatisfactory and can be overcome by extending the SIR model to networks [@Newman2002; @Newman2010].
Suppose we have the following network $G = \left<V,E\right>$ where the set $V$ denotes the vertices of the network and $E$ is the set of bidirectional edges between pairs of vertices in $V$. Each node $v$ in the set of vertices $V$ is associated with either the state $S$ (susceptible), $I$ (infectious) or $R$ (removed) at each moment in time, i.e., $\mathrm{St}\left( v, t\right) \in \left\{S, I, R\right\}$. The initial state of the network is given by $\{S_0 , I_0 , R_0 \}$, i.e., the sets of nodes who’s state at the start of the simulation, $t = 0$, are either $S, I$ or $R$. The dynamics of the virus are emulated by updating the state of all the nodes in the network simultaneously while stepping forward in time. When the state of node $v \in V$ at moment $t$ is susceptible, i.e., $\mathrm{St}(v, t) = S$, the state of this node at the next time step is given by $$\mathrm{St}\left(v, t + 1 \right) := \begin{cases}
I & \text{with probability }1 - (1 - \beta)^{| \mathcal{N}_v(I) |}, \\
S & \text{otherwise,}
\end{cases}
\label{eq:s}$$ where $\beta$ is the chance of the susceptible node to be infected by a single infectious neighbor. The function $\mathcal{N}_v(I)$ returns the set of infectious neighbors in the direct vicinity of node $v$. The operator $|.|$ returns the cardinality of a set. Note that the chance of *not* becoming infected decreases exponentially with the number of infectious neighbors.
When node $v$ is at time $t$ infective, its state becomes $$\mathrm{St}\left(v, t + 1\right) := \begin{cases}
R & \text{with probability }\gamma, \\
I & \text{otherwise,}
\end{cases}
\label{eq:i}$$ where $\gamma$, thus, denotes the chance of recovering (or dying) from the virus during one time step. Since we assumed that once a node reached the removed state $R$, it is either indefinitely immune for the virus or dead, the state of that node will not change till the end of the simulation. The simulation ends when there are no more infective nodes in the networks; the spread of the virus grinded to a halt and the number of susceptible and removed nodes in the network will not change. The total number of casualties is equal to the number of nodes in state $R$.
Genetic algorithm {#sec:ga}
-----------------
In order to find a (near-optimal) set of nodes that need to be protected from the virus we employ a genetic algorithm [@SivanandamS.N.andDeepa2007]. First, we compute the degree, betweenness and eigenvector centrality for each node in the network (see section \[sec:centrality\]). By ranking the nodes according to their centrality scores, we obtain the following three ascending rankings: $$R_D = \{v_{(1)}^D, v_{(2)}^D, \dots, v_{(N)}^D\}, \quad R_B = \{v_{(1)}^B, v_{(2)}^B, \dots, v_{(N)}^B\} \quad \text{and} \quad R_E = \{v_{(1)}^E, v_{(2)}^E, \dots, v_{(N)}^E\},
\label{eq:rankings}$$ where $R_D$, $R_B$ and $R_E$ are the rankings based on, respectively, the degree (\[eq:d\]), betweenness (\[eq:b\]) and eigenvector (\[eq:e1\]) centrality. We limit the search space of the GA by taking into account that nodes with low centrality scores on all three centrality measure do most likely not play an important role in spreading the virus over the network. We, thus, keep only the first $l < N$ nodes in each ranking, i.e., $$R'_D = \{v_{(1)}^D, v_{(2)}^D, \dots, v_{(l)}^D\}, \quad R'_B = \{v_{(1)}^B, v_{(2)}^B, \dots, v_{(l)}^B\} \quad \text{and} \quad R'_E = \{v_{(1)}^E, v_{(2)}^E, \dots, v_{(l)}^E\}.
\label{eq:r}$$ The genetic algorithm will only search for an optimal $k$-subset of nodes in the union of these reduced rankings: $$R' = R'_D \cup R'_B \cup R'_E.
\label{eq:reduced}$$ Each individual in the GA is represented by a chromosome with $k$ genes: $$I = \{v_1, v_2, \ldots, v_k\} \qquad \text{where }v_i \in R'.
\label{eq:individual}$$ The genes, thus, represent the nodes that need to be protected from the virus where we only take into account those nodes that score relatively high on at least one of the centrality measures.
In order to determine the fitness of an individual, firstly we take the original network $G = \left<V, E\right>$ and remove the nodes present in the chromosome $I$. (Removing nodes corresponds here to immunizing the nodes from the virus). Secondly, in order to account for the fact that one is normally unaware where the virus starts, we select one node at random in the new graph to be infectious while keeping the other nodes susceptible. Thirdly, we apply the SIR model as described in section \[sec:sir\] and store the number of removed individuals at the end of the simulation which we denote with $N_{\text{casualties}}$. (Recall that the simulation ends when there are no more infectious nodes in the network). Since the selection of infectious individuals is a stochastic process we repeat the last two steps $m$ times. The fitness of an individual in the GA population is then defined as the expected number of casualties: $$\text{fitness}(I) = \mathrm{E}\left[N_{\text{casualties}} \right] \approx \frac{1}{m} \displaystyle\sum_{i = 1}^{m} N_{\text{casualties}}^{(i)}
\label{eq:fitness}$$ where $N_\text{casualties}^{(i)}$ is the number of recovered nodes at the end of the $i$-th simulation of the SIR model applied to the updated graph $G$.
\[tab:parameters\]
In summary, the objective of the GA is find that subset $I^\ast$ that minimizes the expected number of casualties[^7]: $$I^\ast = \operatorname*{arg\,min}_{I \subseteq R', |I| = k} \text{fitness}(I).$$
The genetic algorithm simulates $100$ generations each with a total of $100$ individuals. Selection for mating is performed by applying tournament selection where the tournament size is set to $4$. As crossover operator we use an adaptation of uniform crossover. Suppose we selected two parents to mate, e.g., $P_1 = \{1,2,3,4\}$ and $P_2 = \{3,4,5,6\}$. We then concatenate the chromosomes of these parents and sort the resulting array. In our example, we find $\{1,2,3,3,4,4,5,6\}$. The chromosome of the first child, $C_1$, consists of the odd entries of the array; the chromosome of the second child, $C_2$, consists of the even entries, e.g., $C_1 = \{1,3,4,5\}$ and $C_2 = \{2,3,4,6\}$. This approach guarantees the absence of duplicates in the new chromosomes. The mutation rate per gene is set to $1/k$, i.e., each chromosome undergoes on average one mutation in one gene every generation. The mutation of a gene entails that its value is randomly set to a node in the set $R'$ that is not in the individual’s chromosome already.
The initial population is randomly generated except for three individuals: their chromosomes are set to the first $k$ nodes in, respectively, the degree, betweenness and eigenvector centrality rankings as given in eq. (\[eq:r\]). In addition, we keep for every generation the $10$ best performing individuals from the previous generation in order not to loose good solutions that were found earlier.
Results {#sec:results}
=======
Table \[tab:parameters\] provides an overview of the parameters used in the SIR model and the GA introduced earlier. In addition, it presents the parameter settings that were used for producing the results presented in this section.
Fig. \[fig:corr\] depicts the correlations between the degree, betweenness and eigenvector centralities for the nodes in both the networks. The reported $R^{2}$’s are determined by applying linear regression. Note that in both cases degree centrality seems to correlate with its betweenness and eigenvector counterparts while betweenness and eigenvector centralities do not seem to correspond. It is interesting to see that degree and eigenvector centralities seem to correlate more for the air transportation network (e) than for the high school network (b). This might be due to the higher number of edges in the air transportation network (2980 against 500 nodes) in contrast to the high school network (202 edges against 147 nodes). The main point of these figures is to show that there are discrepancies between the various centralities measures, which the genetic algorithm from section \[sec:ga\] is set out to exploit in order to find a more optimal set of nodes.
So, to what extent does combining centralities measures help in finding a more optimal set of nodes that need to be protected from the virus? In order to answer this question, we compare four different strategies:
1. Provide protection for the nodes with the highest degree centrality, see eq. (\[eq:d\]);
2. Provide protection for the nodes with the highest betweenness centrality, see eq. (\[eq:b\]);
3. Provide protection for the nodes with the highest eigenvector centrality, see eq. (\[eq:e1\]);
4. Provide protection for the nodes found be the genetic algorithm, see section \[sec:ga\].
In order to keep the comparison fair, each strategy is allowed to protect exactly $k$ nodes. The parameters $k$ and $l$ are set to $10$ and $100$, respectively, for the high school network and to $50$ and $200$ for the air transportation network (see Table \[tab:parameters\]).
![The topology of the Goodreau’s Faux Mesa high school network (a) and the US air transportation network (b). The nodes depicted with are the nodes that were suggested by both the genetic algorithm and the strategy based on degree centrality. The nodes denoted with $\blacksquare$ were only selected by the GA and not the degree strategy. Symbol $\square$ denotes the nodes that were only selected by the degree strategy. []{data-label="fig:topo"}](Figure4.pdf){width="\textwidth"}
![The degree distributions of the high school and air transportation networks. The bars marked black denote the number of nodes with that particular degree that were targeted by the genetic algorithm.[]{data-label="fig:degreeDistr"}](Figure5.pdf){width=".9\textwidth"}
After applying a strategy on a network, we run $500$ SIR simulations and count the number of nodes that got infected by the virus. Before each SIR run we randomly select a single node to be infectious (the others are susceptible) in order to account for the fact that one is normally unaware where the virus starts. The results are presented in Fig. \[fig:boxplot\]. The ‘no protection’ case represents the situation when no preparations were undertaken. For both networks this would have disastrous consequences. In the air transportation network, almost all nodes will definitely get infected. The genetic algorithm and the degree centrality approach seem to be preferred for both networks. The GA seems to outperform the degree strategy, but only slightly. The methods based on betweenness and eigenvector centralities do improve the situation, but are clearly not optimal. Note that the performance of these two approaches switch places between the graphs. The reason why eigenvector centralities might outperform the betweenness approach when applied to the air transportation network might lie in the fact that degree and eigenvector centrality expose a higher correlation in this graph than in the high school network, see Fig. \[fig:corr\]b and \[fig:corr\]e.
The genetic algorithm outperforms the strategy based on degree centrality only in the case of the US air transportation network. For the high school network their performances are comparable. Fig. \[fig:topo\] presents the topology of both networks and the differences between the sets of nodes proposed by the GA and the degree centrality strategy. The nodes depicted with are the ones that were suggested by both strategies. The nodes depicted with $\blacksquare$ were only selected by the GA and not the degree strategy. Symbol $\square$ denotes the nodes that were only selected by the degree strategy. Note that the GA manages to locate nodes that when protected will help to ‘separate’ clusters, i.e., it becomes harder for the virus to move from one cluster to the next.
Fig. \[fig:degreeDistr\] depicts the degree distributions of both the high school and the air transportation network. The bars marked black denote the number of nodes with that particular degree that were targeted by the genetic algorithm. Note that the nodes selected by the GA are not all in the tail but also appear at the beginning and in the middle of the degree distribution.
Conclusions & Discussion {#sec:conclusions}
========================
The work presented in this paper was undertaken to design and evaluate the effectiveness of a heuristic optimization method for virus spread inhibition, which, in contrast to earlier research [@Holme2002; @Memon2006; @Bright2011; @Hou2012; @Chen2012; @Kitsak2010], does not rely explicitly on any centrality measures during its search.
We found that the genetic algorithm proposed in this paper seems to outperform the standard approaches in the literature. This is a remarkable feat since the genetic algorithm searches through the immense set of possible subsets of nodes and does not base its decision directly on various centrality rankings. Fig. \[fig:topo\] shows that the genetic algorithm is able to find an adequate subset of nodes that need to be protected: it is able to construct a set of nodes that not only scores high on degree, but also on betweenness centrality. The virus is, thus, restricted in its movements within a cluster (degree) but also from one cluster to the other (betweenness).
The degree distributions in Fig. \[fig:degreeDistr\] show the interesting result that the nodes with the biggest influence on the spread of the virus over the network are not necessarily to be found in the tail of the degree distribution; among them there are the nodes with no high degree at all, still, by being immunized, they seem to reduce the number of infected significantly. Future research is required to identify why exactly these nodes are of importance, and, most importantly, whether there are ways to simplify the identification of these individuals. Perhaps (non-topological) node attributes can provide important clues for the selection of individuals to be immunized? Being able to locate these individuals effectively without the need for the full topology of the network will aid tremendously in mitigating the impact of the virus on a population.
The developed method could be applied rather easily to a wide variety of situations that require urgent computing since the method is not limited to a specific type of network: (near-)optimal solutions can be found for both regular and heterogenous networks. In addition, one does not have to restrict to one particular centrality measure. The search space of the genetic algorithm can be seamlessly extended for various fields of applications, regardless of the type of metrics used.
The presented method is recommended for topologically heterogenous networks that are characterized by the absence of strong correlations between the various node centralities and the presence of modularity. The authors expect that for homogenous networks the traditional degree centrality approach (i.e., immunizing the nodes with the highest degree) is to be preferred. In addition, the GA requires more computation time. Especially when time is pressing, one is, therefore, recommended to resort to the traditional and fast-to-compute strategies.
Further research might be done to explore different sets of metrics and heuristics in order to widen or narrow the search space. In addition, it might be interesting to take edge centralities into account as well, although the authors expect that this would not make a significant difference, since edge and node centralities are often highly correlated. The parameters for the genetic algorithm were set rather intuitively. Better results might be obtained when different parameter settings are applied. In this paper we only explored one particular compartmental model, the SIR model (see section \[sec:sir\]). The method can be extended to different kind of models as well [@Hethcote2000; @Newman2010]. It would be interesting to see how the method proposed here would perform when different virus dynamics are taking into account. The applicability and performance of other optimization methods such as simulated annealing might be a subject of future research as well.
This paper has shown the promise of applying a genetic algorithm in order to mitigate the impact of a virus attack on a population. The authors would like to express their hope that continuing research along the lines set out in this paper will assist in gaining a better understanding of virus inhibition, and will ultimately provide better ways to avert future epidemics.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Prof. dr. A.V. Boukhanovsky from NRU ITMO for his insightful comments and support. This work is supported by the *Leading Scientist Program* of the Russian Federation, contract 11.G34.31.0019.
[^1]: Corresponding author. Tel.: +7-965-073-0861. E-mail address: kashirin.victor@gmail.com.
[^2]: Measures for edge centrality exist as well, see [@Newman2010].
[^3]: More complicated centrality measures have been proposed as well, see, for example, [@Kitsak2010; @Hou2012; @Chen2012].
[^4]: Several extensions of the models exist, e.g., the SIRS model where recovered individuals can return to the susceptible state. See the paper by H.W. Hethcote (2000) for a splendid overview of these so-called compartmental models.
[^5]: http://www.casos.cs.cmu.edu/computational\_tools/datasets/external/Goodreau/index11.php
[^6]: http://sites.google.com/site/cxnets/usairtransportationnetwork
[^7]: Although it is more common to maximize a fitness function rather than minimizing it, we felt minimization would be more appropriate, since we are dealing with the expected number of casualties.
|
---
abstract: 'The European Solar Telescope (EST) is a project of a new-generation solar telescope. It has a large aperture of 4 m, which is necessary for achieving high spatial and temporal resolution. The high polarimetric sensitivity of the EST will allow to measure the magnetic field in the solar atmosphere with unprecedented precision. Here, we summarise the recent advancements in the realisation of the EST project regarding the hardware development and the refinement of the science requirements.'
address:
- 'Astronomical Institute of the Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic'
- |
Instituto de Astrofísica de Canarias (IAC), Vía Lactéa, 38200 La Laguna (Tenerife), Spain\
Departamento de Astrofísica, Universidad de La Laguna, 38205 La Laguna (Tenerife), Spain
- 'Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University Centre, SE-106 91 Stockholm, Sweden'
- 'Max-Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany'
- 'Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany'
author:
- Jan Jurčák
- Manuel Collados
- Jorrit Leenaarts
- Michiel van Noort
- Rolf Schlichenmaier
bibliography:
- 'manuscript.bib'
title: Recent advancements in the EST project
---
instrumentation ,the Sun ,EST
=0.5 cm
Overview of the EST project development
=======================================
The development of EST is organized by the European Association for Solar Telescopes (EAST). The EAST was formed in June 2006 and is currently a consortium of 23 solar research centres in 17 European countries. To ensure access of European solar physicists to world-class observing facilities, the primary goal of the EAST consortium is to develop, construct and operate the next generation 4-metre class solar telescope on the Canary Islands.
The conceptual design study of EST was realised in the frame of the FP7 project “EST: The large aperture European Solar Telescope”[^1] supported by the European Commission. This project was realised between February 2008 and July 2011 and involved 14 research institutions and 15 industrial partners. The conceptual design phase was summarised in the “EST: Conceptual Design Study Report” containing the proposed solutions for the telescope itself and all its subsystems along with management plans, socio-economic impact, and financial feasibility. A detailed “Report on technical, financial, and socio-economic aspects”[^2] was also one of the conceptual phase project deliverables. The conceptual design of the EST has been positively evaluated in 2011 by a panel formed by prestigious external reviewers.
The follow up project of the European solar physics community to prepare for the EST research infrastructure (RI) was the SOLARNET (High-Resolution Solar Physics Network), a network aimed at bringing together and integrating the major European research infrastructures in the field of high-resolution solar physics in order to promote their coordinated use and to secure the future development of the next-generation RI in the form of EST. The project was carried out between April 2013 and March 2017 and funded from the EC Integrated Infrastructures Initiative. It involved 32 partners from 16 countries: 24 EU research institutions, 6 EU private companies, and 2 USA research institutions.
GREST (Getting Ready for EST) is an on-going project, funded by the EC H2020 program from June 2015 till June 2018. GREST is taking the EST to the next level of development by undertaking crucial activities to improve the performance of current state-of-the-art instrumentation. Also legal, industrial, and socio-economic issues are addressed in the frame of this project. GREST involves 13 partners from 6 EU countries, 3 of them are private companies.
EST formally entered the active project list of the ESFRI (the European Strategy Forum on Research Infrastructures) roadmap in March 2016 as the flagship project for the European Solar Physics community. Acceptance of EST project to the list of ESFRI roadmap is an important milestone in the implementation of the EST as the primary task of ESFRI is to support and help the projects on the roadmap move towards realisation.
On 1st April 2017, the PRE-EST project (Preparatory phase for EST) was commenced. It aims to provide both the EST international consortium and the funding agencies with a detailed plan regarding the implementation of EST. Moreover, PRE-EST will lead the detailed design of the EST key elements to the required level of definition and validation for their final implementation. This project is partly funded under the H2020 framework till March 2021. One of the major decisions in this ongoing project is the plan to establish an ERIC (European Research Infrastructure Consortium) as the legal entity of the EST RI. The plan is to establish the ERIC with Spain as the host state.
Currently, there is a proposal for continuation of the SOLARNET project applying for the call H2020-INFRAIA-2018-2020. The goals of this project are comparable to the original SOLARNET project and aim at integrating the major European infrastructures in the field of high-resolution solar physics.
The most recent information about the project development and status can be found on the project web-site[^3], followed on various social media like Facebook[^4], Twitter[^5], Youtube[^6], and Linkedin[^7].
Overview of the EST science goals
=================================
The EST’s main research goal is to observe the Sun. The Sun is the only star that can be studied in high resolution. We can study fundamental interactions between plasma, magnetic field, and radiation in the solar atmosphere. Although we are not able to spatially resolve the intrinsic scales of these magneto-hydrodynamic processes yet, the magnetic Sun forms the basis of our knowledge of the cosmic magnetic field. With the EST we will improve the resolution and enhance our understanding of fundamental science.
A variety of structures and physical processes can be observed on the Sun. These phenomena are connected by a variable magnetic field, which becomes most prominent during the maximum of the Sun’s magnetic activity: Energetic events are the result of interaction of the magnetic field with ionised plasma and its radiation field. The EST will focus on observing this interaction. Magnetic fields are detected and characterised by analysing their imprint on the polarisation, therefore EST is designed to achieve the polarimetric accuracy of $10^{-4}$.
The optical design and instruments of the EST are optimised for observing the physical coupling of the atmospheric layers from the deepest photosphere to the upper chromosphere. Suitable techniques have been developed to determine the thermal, dynamic, and magnetic properties of the plasma over many scale heights with imaging, spectroscopy, and spectro-polarimetry instrumentation. The EST will be equipped with a suite of instruments to simultaneously observe in various wavelengths from 380 nm to 2 $\mu$m, so that the solar photon flux can be exploited more efficiently than at other current ground-based or space telescopes.
Hardware development {#hardware}
====================
The design of EST and its sub-systems was summarised in the conceptual design study. However, the detailed design of the sub-systems is now in progress and feasibility of the individual sub-systems is studied. There are studies on the purely technical sub-systems like the heat rejecter at the primary focus of the EST [@Berrilli:2010; @Berrilli:2017]. Studies on the sub-systems that directly influence the observations, e.g., the multi-conjugate adaptive optics that can correct for the atmospheric aberrations in the whole field of view of the telescope [@Stangalini:2010; @Soltau:2010; @Montilla:2012; @Montilla:2013; @Montilla:2014; @Rosa:2016; @Stangalini:2016].
![Schemes of the IFU units that are developed for EST. Adapted from @Calcines:2013.[]{data-label="ifu_scheme"}](ifu_scheme.eps){width="\linewidth"}
We describe here in more details the development of instruments that will allow for integral field spectropolarimetry, i.e., simultaneous observations in both bi-dimensional spatial and spectral domain. Schemes of two technical solutions of these integral field units (IFU) that are under development for the EST are shown in Fig. \[ifu\_scheme\].
Micro-lens array
----------------
One of the options is so-called micro-lensing, which is developed at MPS, primarily by M. van Noort. The principle of this technology is shown in the upper row of Fig. \[ifu\_scheme\]. Every pixel (resolution element) at the telescope focal plane is re-imaged to a smaller area, the resulting sparse image is fed to a spectrograph, and the dispersion angle of the spectrograph is then optimised to minimise the mixture of spatial and spectral information. This concept necessitates the use of a narrow-band pre-filter to ensure that the spectra of different spatial pixels do not overlap. Since the micro-lens array is a refraction-based device, it is not achromatic. The periodic design unavoidably produces Moire fringes and the instrument must be kept in stable conditions so that the fringes do not change in time. On the other hand, the optical design of micro-lens arrays is very simple and massively parallel, allowing it to be scaled to the order of $10^5$ image elements. In addition, since the image has already been sampled by the micro-lens array, the optical quality of most of the optical elements after the micro-lens array is not very critical for the image quality.
![Photos of the prototype of the micro-lens array tested at SST by M. van Noort.[]{data-label="micro-lens-array"}](MLA_1.eps "fig:"){height="4.3cm"} ![Photos of the prototype of the micro-lens array tested at SST by M. van Noort.[]{data-label="micro-lens-array"}](MLA_2.eps "fig:"){height="4.3cm"}
![Test data obtained with the micro-lens array at SST. Image of the whole CCD chip is on the left, amplified sample in the middle, and reconstructed images of the field of view on the right.[]{data-label="MLA_data"}](microlens_data.eps){width="\linewidth"}
The concept of micro-lens array was already tested by Y. Suematsu [@Suematsu:1999; @Suematsu:2011] using a single micro-lens array, who found his instrument to have very limited performance. This was primarily due to his specific optical configuration, which caused the image elements to be re-imaged onto a grating, resulting in a variable, generally poor, spectral performance. To avoid this problem, a second micro-lens array is needed and it must be aligned to the first one with formidable accuracy. The two micro-lens arrays are therefore fabricated on two sides of a single substrate to ensure accurate alignment (see Fig. \[micro-lens-array\]), and it in addition contains a straylight mask. Manufacturing of such a device has become technologically possible only recently and a prototype of this instrument was constructed by M. van Noort and tested on the Swedish Solar Telescope (SST).
Some results of the test observations are shown in Fig. \[MLA\_data\], where the image on the left corresponds to the whole sensor image, and a sample of the spectra at individual spatial pixels are shown in the middle. These data are then rearranged to create spatial maps at individual wavelengths as shown in the right part of Fig. \[MLA\_data\]. Details about the instrument will be summarised in an up-coming paper from M. van Noort.
Image slicer
------------
The other option that is considered for an IFU for EST is a so-called image slicer. Its concept is shown in the bottom row of Fig. \[ifu\_scheme\]. An image slicer uses an array of thin mirrors at the focal plane to redistribute the light from the entrance field of view along one long spectrograph slit. Since the image slicer has a mirror-based design, it is in principle achromatic. Each slit retains an image information, which is not periodic, therefore there are no Moire fringes. If single-slit version of the IFU is used, there is no overlap of the spectra. However, this advantage is lost once the thin mirrors redistribute the light to number of slits. This IFU design is very challenging optically. Each slit requires slit-specific optics and alignment. The optical quality has to be high throughout the instrument as image information is propagated. This IFU is developed by IAC and details about this instrument can be found in @Calcines:2013 [@2014SPIE.9147E..3IC].
In Fig. \[image\_slicer\_photo\] we show the prototype of the image slicer for the EST that was tested at the GREGOR telescope. In Fig. \[image\_slicer\_data\] there are snapshots from the test observations made with the prototype of image slicer IFU attached to the GREGOR telescope. On the left is the image from the "slit-jaw“ camera depicting the field of view of one IFU tile. Panels on the right are maps of $I$, $Q$, $U$, and $V$ intensities from a field of view composed of $3 \times 3$ IFU tiles. Note that the image slicer provides simultaneous spatial and spectral observations for a single tile and it took nine sequential measurements to construct the right panels in Fig. \[image\_slicer\_data\].
![Photos of the prototype of the image slicer developed at IAC and tested on GREGOR.[]{data-label="image_slicer_photo"}](image_slicer_1.eps "fig:"){height="3.3cm"} ![Photos of the prototype of the image slicer developed at IAC and tested on GREGOR.[]{data-label="image_slicer_photo"}](image_slicer_2.eps "fig:"){height="3.3cm"}
![Test data obtained with the image slicer IFU at GREGOR. ”Slit-jaw“ image on the left shows the size of one tile. Reconstructed maps of $I$, $Q$, $U$, and $V$ intensities from a FOV composed of $3 \times 3$ IFU tiles are shown on the right.[]{data-label="image_slicer_data"}](image_slicer_data_1.eps "fig:"){height="4cm"} ![Test data obtained with the image slicer IFU at GREGOR. ”Slit-jaw“ image on the left shows the size of one tile. Reconstructed maps of $I$, $Q$, $U$, and $V$ intensities from a FOV composed of $3 \times 3$ IFU tiles are shown on the right.[]{data-label="image_slicer_data"}](image_slicer_data_2.eps "fig:"){height="4cm"}
Update of the Science Requirement Document
==========================================
The design of the EST including the set-up of the light distribution system and the set of instruments intended for the first light is summarised in the “EST: Conceptual Design Study Report” and was based on the Science Requirement Document (SRD) that was also developed during the conceptual design study of EST.
The advancements in the hardware development of new instruments (like those described in Sect. \[hardware\]) along with the advancements in our understanding of processes occurring in the solar atmosphere require an update of the original SRD. This update is now in progress and will be presented to the community shortly before the EST science meeting on June 11-15, 2018. This update will result into the refinement of the light distribution system and the set of post-focal instruments required for the first-light science.
However, this update will not affect the main strengths of the EST baseline design. The entire telescope is designed to be polarimetrically compensated, i.e., the Müller matrix of the telescope will be unity and independent on the wavelength and time. Due to the alt-azimuthal mount the field of view is rotating in time. To compensate for this effect, seven mirrors of the transfer optics before the Coudé focus will be arranged on a rotating platform and thus used as an optical field-of-view de-rotator. This design allows to mount the instruments on a fixed platform, which is advantageous in terms of simplicity, instrument stability, higher number of instruments, and flexibility of the instruments upgrades. Another advantage of the EST design is the height of the tower of approximately 35 m that will reduce the ground layer effect of the local seeing conditions.
![Snapshot of the photon flux calculator developed by J. Leenaarts.[]{data-label="photonflux"}](photon_flux_calculator.eps){width="\linewidth"}
In view of the update of the SRD, J. Leenaarts developed a photon flux calculator that provides an immense help to estimate the capabilities of the 4 m class solar telescope. It is a freely available[^8] python routine (it is executed by photoncount.py; additional python packages might be necessary to run it). Snapshot of this photon flux calculator is shown in Fig. \[photonflux\].
In the photon count calculator you can adjust:
- the aperture diameter (by default set to the EST aperture of 4 m)
- the wavelength range of the observations
- polarimetric or spectroscopic mode
- spatial binning (by default the pixel size is set to critically sample the spatial resolution for a given wavelength)
- spectral resolution R ($\lambda / \Delta\lambda$)
- total transmission of the telescope at the given wavelength (see Table \[throughput\] for the throughput values of the whole system, i.e., including the transfer optics, the light distribution system, and the individual instruments. This table will be updated based on the new design of the light distribution system and the set of instruments.)
- desired signal-to-noise ratio
- evolution speed (the transversal motion of the fine structures in the solar atmosphere)
As an output, the routine provides:
- resulting spatial pixel size (function of the wavelength, aperture size, and spatial binning)
- resulting spectral pixel size (function of the wavelength and spectral resolution)
- (top-left plot) the FTS atlas spectrum [@Neckel:1999] at the given range of observed wavelengths and the spectrum smeared to the given spectral resolution R
- (lower-left plot) the integration time necessary to reach the desired signal-to-noise ratio for a given spatial and spectral pixel
- (right plots) for these plots, the diameter of the telescope, the throughput, the spectral resolution, the desired signal-to-noise ratio and the evolution speed are taken into account. Given these parameters, the routine computes the optimal integration time and pixel size from the constraints that a structure crosses exactly one spatial pixel during the integration time, which means that the image is not smeared owing to the evolution of solar structure.
Instrument $\lambda$ band \[nm\] Individual obs. Simultaneous obs.
------------ ----------------------- ----------------- -------------------
SP 380-400 0.020 0.005
400-500 0.080 0.020
500-620 0.098 0.025
620-800 0.098 0.024
NB1 380-400 0.020 0.011
400-500 0.080 0.044
BB1 380-400 0.019 0.001
400-500 0.076 0.003
BB2 380-400 0.017 0.000
400-500 0.069 0.001
BB3 500-620 0.098 0.006
620-800 0.098 0.006
NB2 500-620 0.098 0.046
NB3 620-800 0.103 0.043
IRSP 800-1100 0.068 0.014
1100-1500 0.088 0.018
1500-1800 0.090 0.018
1800-2300 0.090 0.018
IRNB1 800-1100 0.061 0.037
IRNB2 1500-1800 0.085 0.052
: Throughput values for the current EST design.
\[throughput\]
Conclusions
===========
The project of the EST is advancing quickly in the recent years thanks to the joint effort of people united in the EAST consortium and those working on the realisation of projects like SOLARNET, GREST, and PRE-EST. We are entering a crucial phase of the project when the commitment of the funding agencies of individual member states have to be secured for the preparatory phase, the construction phase, and the operating costs.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is carried out as a part of the project Preparatory phase of the EST (PRE-EST) funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 739500. EST is an ambitious project to build a 4-m class solar telescope, to be erected in the Canary Islands. The project is promoted by the European Association for Solar Telescopes, formed by 23 research institutions from Austria, Belgium, Croatia, Czech Republic, France, Germany, Great Britain, Greece, Hungary, Italy, Netherlands, Norway, Poland, Slovakia, Spain, Sweden, and Switzerland. Part of this work was supported by the Czech institutional project RVO:67985815.
References {#references .unnumbered}
==========
[^1]: http://istar.ll.iac.es/files/58cef4ec1579d9e39754c76d5.pdf
[^2]: www.est-east.eu/est/images/media/pdf/EST\_socio-economic\_web\_version.pdf
[^3]: <http://www.est-east.eu>
[^4]: <https://www.facebook.com/EuropeanSolarTelescope>
[^5]: <https://twitter.com/estsolarnet>
[^6]: <https://www.youtube.com/user/ESTtvCHANNEL>
[^7]: <https://www.linkedin.com/company/european-solar-telescope/>
[^8]: <http://dubshen.astro.su.se/~jleen/sag/>
|
---
abstract: |
The aim of this paper is to establish the range of $p$’s for which the expansion of a function $f\in L^p$ in a generalized prolate spheroidal wave function (PSWFs) basis converges to $f$ in $L^p$. Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases cover the classical PSWFs for which the corresponding results has been previously established by Barceló and Cordoba.
To establish those results, we prove a general result that allows to extend mean convergence in a given basis (e.g. Jacobi polynomials or Bessel basis) to mean convergence in a second basis (here the generalized PSWFs).
address:
- 'Mourad Boulsane Address: University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Bizerte, Tunisia.'
- 'Philippe Jaming Address: Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France. CNRS, IMB, UMR 5251, F-33400 Talence, France.'
- 'Ahmed Souabni Address: University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, Bizerte, Tunisia.'
author:
- 'Mourad Boulsane, Philippe Jaming & Ahmed Souabni'
title: Mean convergence of prolate spheroidal series and their extensions
---
Introduction
============
In their seminal work from the 70s, Landau, Pollak and Slepian [@LP1; @LP2; @Slepian1] have shown that the orthonormal basis that is best concentrated in the time-frequency plane is given by the Prolate Spheroidal Wave Functions (PSFWs). This basis therefore provides an efficient tool for signal processing. Since then, the PSFWs have proven useful in many applications ranging from random matrix theory ([*e.g.*]{} [@dCM; @Me; @Dy]) to numerical analysis ([*e.g.*]{} [@XRY; @Wang]). While taking naturally place in an $L^2$ setting, one may also consider the behavior of expansions of functions in the PSFW basis in the $L^p$-setting. This has been done by Barcelo and Cordoba for the usual PSFWs. Our aim here is to extend this work to two natural generalizations of the PSFWs, namely, the Hankel-PSFWs introduced by Slepian [@Slepian2] and the weighted PSFWs recently introduced by Wang and Zhang [@Wang2].
Let us now be more precise with the results in this paper. First let us recall that the prolate spheroidal wave functions $(\psi_{n,c})_{n\geq 0}$ are eigenvectors of an integral operator. Using the min-max theorem, they can thus be obtained as solutions of an extremal problem: for $c>0$, recall that the Paley-Wiener space $PW_c=\{f\in L^2(\R)\,: \operatorname{supp}\widehat{f}\subset [-c,c]\}$ where $\widehat{f}$ stands for the Fourier transform of $f$. Then one sets $$\psi_{n,c}=\mathrm{argmax}\left\{\frac{{{\left\|{f}\right\|}}_{L^2(I)}}{{{\left\|{f}\right\|}}_{L^2(\R)}}\,: f\in PW_c,\
f\in\operatorname{span}\{\psi_{k,c},k<n\}^\perp\right\}.$$ A fundamental fact discovered by Landau, Pollak and Slepian is that they are also eigenfunctions of a Sturm-Liouville operator, a fact tagged as a “happy miracle” by Slepian [@Slepian4]. Another key fact four our purpuses is that $(\psi_{n,c})_{n\geq 0}$ is an orthonormal basis of $PW_c$ and this basis is the best concentrated in the time domain.
In this paper, we are interested in two generalizations of the PSFWs. For both cases, the basis is constructed as a set of eigenvectors of an integral operator, the happy miracle occurs so that they are also eigenvectors of a Sturm-Liouville operator and, more important for us, they form an orthonormal basis of a Paley-Wiener type of space.
The first basis we consider was introduced by Slepian [@Slepian2]. It is an analogue of the classical PSFWs adapted to higher dimensional [*radial*]{} Fourier analysis. To introduce them, we need some further notation. First, we replace the Fourier transform by the [*Hankel transform*]{} defined for $f\in L^1(0,+\infty)$ by $$\mathcal H^{\alpha}f(x)=\int_0^{+\infty}\sqrt{xy}J_{\alpha}(xy) f(y)\d y$$ where $J_\alpha$ is the Bessel function and $\alpha>-1/2$. Like the usual Fourier transform, the Hankel transform extends into a unitary operator on $L^2(0,+\infty)$. The corresponding Paley-Wiener space is then denoted by $$HB_c^{(\alpha)} =\left\{f\in L^2(0,\infty); \operatorname{supp}\mathcal{H}^{\alpha}(f)\subseteq[0,c]\right\}.$$ Finally, the Circular (Hankel) Prolate Spheroidal Wave Functions (CPSWFs) are defined by $$\psi_{n,c}^\alpha=\mathrm{argmax}\left\{\frac{{{\left\|{f}\right\|}}_{L^2(0,1)}}{{{\left\|{f}\right\|}}_{L^2(0,+\infty)}}\,: f\in HB_c^{(\alpha)},\
f\in\operatorname{span}\{\psi_{k,c}^\alpha,k<n\}^\perp\right\}.$$ Then $(\psi_{n,c}^\alpha)_{n\geq 0}$ is an orthonormal basis of $HB_c^{(\alpha)}$. Note also that when $\alpha=0$, these are usual PSFWs, more precisely, $\psi_{n,c}^0=\psi_{2n,c}$.
The second basis we consider, the Weighted Prolate Spheroidal Wawe Functions (WPSFWs), is defined in a similar fashion. We first introduce the weighted Paley-Wiener spaces $$wPW_c^{(\alpha)}=\left\{f\in L^2(\R); \operatorname{supp}\widehat{f}\subseteq[-c,c],
\ \widehat{f}\in L^2\bigl((-c,c),(1-x^2/c^2)^{-\alpha}\d x\bigr)
\right\}.$$ The WPSFWs are defined by $$\Psi_{n,c}^\alpha=\mathrm{argmax}\left\{\frac{{{\left\|{f}\right\|}}_{L^2\bigl((-1,1),(1-x^2)^\alpha\d x\bigr)}}
{{{\left\|{\widehat{f}}\right\|}}_{L^2\bigl((-c,c),(1-x^2/c^2)^{-\alpha}\d x\bigr)}}\,: f\in wPW_c^{(\alpha)},\
f\in\operatorname{span}\{\Psi_{k,c}^\alpha,k<n\}^\perp\right\}.$$ Again, $\Psi_{n,c}^\alpha$ is an orthonormal basis of $wPW_c^{(\alpha)}$ and $\Psi_{n,c}^0=\psi_{n,c}$.
The aim of this paper is to characterize the range of $p$’s for which prolate spheroidal wave functions converge in $L^p$. The subject of the $L^p$-convergence (also called mean convergence of order $p$) of orthogonal series, is a central subject in harmonic analysis. This kind of convergence is briefly described as follows. Let $1 < p <\infty$ , $a,b \in \overline{\mathbb{R}}$, $I=(a,b)$, and $\{\phi_n\}$ an orthonormal set of the weighted Hilbert space $L^2(I, \omega)$-space, where $\omega$ is a positive weight function. We define the kernel $$K_N(x,y)=\sum_{n=0}^N\phi_n(x)\overline{\phi_n(y)}$$ so that the orthogonal projection of $f\in L^2(I,\omega)$ on the span of $\{\phi_0,\ldots,\phi_N\}$ is given by $$\mathcal{K}_N(f)(x)=\int_I K_N(x,y)f(y)\omega(y)\d y=\sum_{n=0}^N a_n(f) \phi_n(x)$$ with $$a_n(f) = \int_I f(y)\overline{\phi_n(y)} \omega(x)\d y.$$ Now, this last expression may be well defined even for $f\in L^p(I,\omega)$, $p\not=2$ and then $\mathcal{K}_N(f)$ is also well defined. This happens for instance if $\phi_n\in L^p(I,\omega)$ for every $p$ which is often the case in practice. The orthonormal set $\{\phi_n\}$ is said to have mean convergence of order $p$, or $L^p$-convergence over the Banach space $L^p(I,d\omega)$ if for every $f \in L^p(I,d\omega)$, $\mathcal{K}_N(f)$ is well defined and $$\lim_{N \to \infty}{{\left\|{f-\mathcal{K}_Nf}\right\|}}_{L^p(I,\omega)} = \lim_{N \to \infty} \bigg[ \int_{a}^{b}|f(x)- \mathcal{K}_N(f)(x) |^p \omega (x)\d x\bigg]^{1/p} =0.$$ This concept of mean convergence is also valid on a subspace $\mathcal B,$ rather than the whole Banach space $L^p(I,d\omega)$.
To the best of our knowledge, M. Riesz was the first in the late 1920’s, to investigate this problem in the special case of the trigonometric Fourier series over $L^p(\T),$ $1\leq p < +\infty.$ More precisely, in [@Riesz], it has been shown that the Hilbert transform over the torus $\tt$ is bounded on $L^p(\T)$ if and only if $ p>1$. Further, the $L^p-$boundedness of the Hilbert transform is equivalent to the mean convergence of the Fourier series on $L^p(\T)$. In the late 1940’s, H. Pollard, in a series of papers [@Pollard1; @Pollard2; @Pollard3], has studied the mean convergence of some classical orthogonal polynomials, such as Legendre and Jacobi polynomials. In particular, in the later case, he has shown that if $\alpha \geq -\frac{1}{2}$ and $\omega_{\alpha}(x)=(1-x^2)^{\alpha}$, $x\in I=[-1,1]$ is the Jacobi weight, then the mean convergence over $L^p(I,\omega_{\alpha,})$ of Jacobi series expansion holds true, whenever $$m(\alpha):= 4 \frac{\alpha+1}{2\alpha+3} < p <
M(\alpha) := 4 \frac{\alpha+1}{2\alpha+1}.$$ He has also shown that the previous conclusion fails if $p< m(\alpha)$ or $p>M(\alpha)$. In [@Milton], the authors have shown that the mean convergence of the Bessel series expansion over the space $L^p([0,1], x\,\mathrm{d}x)$ holds true whenever $4/3 < p < 4$. Later on, Newman and Rudin [@Newman] have shown that the mean convergence fails for the critical values of $p= m(\alpha), p=M(\alpha)$ in the Jacobi case and for $p=4/3,\, p=4$ for the Bessel case. More recently, in [@J.L.VARONA] Varona has extended the mean convergence of Bessel series for $ \alpha > -1/2 $ over the Hankel Paley-Wiener space of functions from $L^p([0,\infty),x^\alpha\,\mathrm{d}x )$ with compactly supported Hankel transforms.
An other important extension has been given by Barcelo and Cordoba [@BC] where they have shown that the series expansion in terms of the classical prolate spheroidal wave functions (PSWFs) has the mean convergence property over the previous Fourier Paley- Wiener space, holds true if and only if $ 4/3 < p < 4 $. This is the main source of inspiration for our work, so let us detail the ideas behind [@BC]. Barcelo and Cordoba first determine the expansion of PSFWs in a basis consisting of Bessel functions. It turns out that the kernel of the projection onto this second basis is given by a Christoffel-Darboux like formula so that it’s mean convergence properties can be deduced from estimates for weighted Hilbert transforms. The last step of the proof is a sort of transference principle which allows to show that the PSWFs have the mean convergence property of order $p$ exactly when the Bessel basis has this property.
Our first aim here is to formalize this transference principle. We consider two orthonormal bases $(\ffi_n)_{n\geq 0}$ and $(\psi_n)_{n\geq 0}$ of $L^2(\Omega,\mu)$. Then, we establish a fairly general principle giving several conditions on $(\ffi_n)_{n\geq 0}$ and $(\psi_n)_{n\geq 0}$ that will ensure the mutual mean convergence property of order $p$ associated for the two bases.
The second part of the paper then consists in applying this principle to the two extensions of PSFWs mentionned above. For the Circular PSFWs the second basis consists again of a basis built from Bessel functions for which we have to adapt the proof of Barcelo-Cordoba to establish the range of $p$’s for which mean convergence holds. The case of Weighted PSFWs is a bit simpler as the second basis consists of Jacobi polynomials for which the mean convergence property is already known. As this case is simpler, it will be treated first. We may now state our main result:
[**Theorem.**]{}
*Let $\alpha>-1/2$, $c>0$, $N\geq 0$. Let $I=(-1,1)$ and $\omega_\alpha(x)=(1-x^2)^\alpha$.*
- Let $p_0=2-\frac{1}{\alpha+3/2}$ and $p_0^{\prime}=\dst 2+\frac{1}{\alpha+1/2}$. Let $(\Psi_{n,c}^{(\alpha)})_{n\geq 0}$ be the family of weighted prolate spheroidal wave functions. For a smooth function $f$ on $I=(-1,1)$, define $$\Psi^{(\alpha)}_Nf=\sum_{n=0}^N{{\left\langle{f,\Psi_{n,c}^{(\alpha)}}\right\rangle}}_{L^2(I,\omega_\alpha)}\Psi_{n,c}^{(\alpha)}.$$ Then, for every $p\in(1,\infty)$, $\Psi^{(\alpha)}_N$ extends to a bounded operator $L^p(I,\omega_\alpha(x)\,\mathrm{d}x)\to L^p(I,\omega_\alpha(x)\,\mathrm{d}x)$. Further $$\Psi^{(\alpha)}_Nf\to f\qquad \mbox{in }L^p(I,\omega_\alpha(x)\,\mathrm{d}x)$$ for every $f\in L^p(I,\omega_\alpha(x)\,\mathrm{d}x)$ if and only if $p\in(p_0,p_0^\prime)$.
- Let $(\psi_{n,c}^{(\alpha)})_{n\geq 0}$ be the family of Hankel prolate spheroidal wave functions. For a smooth function $f$ on $I=(0,\infty)$, define $$\Psi^{(\alpha)}_Nf=\sum_{n=0}^N{{\left\langle{f,\psi_{n,c}^{(\alpha)}}\right\rangle}}_{L^2(0,\infty)}\psi_{n,c}^{(\alpha)}.$$ Then, for every $p\in(1,\infty)$, $\Psi^{(\alpha)}_N$ extends to a bounded operator $L^p(0,\infty)\to L^p(0,\infty)$. Further $$\Psi^{(\alpha)}_Nf\to f\qquad \mbox{in }L^p(0,\infty)$$ for every $f\in B_{c,p}^{\alpha}$ if and only if $p\in(4/3,4)$.
This work is organized a follows. In section 2, we study a general principle that ensure the mutual $L^p$-convergence of two series expansion with respect to two different orthonormal bases of a Hilbert space $L^2(\mu) $. In section 3, we give a list of technical lemmas that ensure or simplify the conditions given in the general principle of the previous section. In section 4, we apply the results of sections 2 and 3 and check in detail that the conditions that we have established in the case of general principle hold true for the series expansion in the weighted PSWFs. Finally in section 5, we prove that this mean convergence property holds also true for circular PSWFs series.
The General principal
=====================
The setting and the main result {#sec:mainth}
-------------------------------
As already explained, to prove the $L^p$-convergence of the expansion in a prolate basis, we will expend the prolates in a second basis for which this $L^p$-convergence is easier to study. This idea is formalized in the following setting:
We consider a measure space $(\Omega,\mu)$ and assume that, for every $1<p<\infty$, $L^p(\Omega,\mu)$ is infinite dimensional and separable. The dual index of $p$ will be denoted by $p'=\dst\frac{p}{p-1}$. We consider two orthonormal bases $(\ffi_n)_{n\geq 0}$ and $(\psi_n)_{n\geq 0}$ of $L^2(\Omega,\mu)$. For $N\geq 0$ we denote by $\Phi_N$ (resp. $\Psi_N$) both the orthogonal projection on $\operatorname{span}\{\ffi_0,\ldots,\ffi_N\}$ (resp. $\operatorname{span}\{\psi_0,\ldots,\psi_N\}$) and its kernel $$\Phi_N(x,y)=\sum_{n=0}^N\ffi_n(x)\overline{\ffi_n(y)}
\quad\mbox{resp.}\quad
\Psi_N(x,y)=\sum_{n=0}^N\psi_n(x)\overline{\psi_n(y)}.$$
Our aim in this section is to define several conditions on $\ffi_n,\psi_n$ that will ensure that, for any $1<p<\infty$, $\Phi_Nf\to f$ in some $L^p$ if and only if $\Psi_Nf\to f$ in $L^p$. The first condition is of course that this makes sense. The second one is that some relation exists between the two bases. The other conditions are technical and are those that will be the most difficult to check in practice.
- For every $1<p<\infty$, and every $n$, $\ffi_n\in L^p(\mu)$. Further, we assume that there is a $0\leq \gamma_p<1$ such that $$\label{eq:lpind}
{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}\lesssim n^{\gamma_p}.$$ Finally, we assume that $0<\alpha_p:=\gamma_p+\gamma_{p'}<1$ and that there is a $p_0$ such that if $p\in (p_0,p_0^{\prime})$, $\alpha_p=0$. In other words, for $p\in (p_0,p_0^{\prime})$, $$\label{eq:lp1}
{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_n}\right\|}}_{L^{p'}(\mu)}\lesssim 1
$$ while for $p\notin (p_0,p_0^{\prime})$, $$\label{eq:lp2}
{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_n}\right\|}}_{L^{p'}(\mu)}\lesssim n^{\alpha_p},\qquad \alpha_p<1.
$$
- Let $\alpha_k^n={{\left\langle{\psi_n,\ffi_k}\right\rangle}}_{L^2(\mu)}$ so that $\psi_n=\sum_{n=0}^\infty \alpha_k^n\ffi_k$. We assume that there exists an integer $n_0$ and $\kappa,\kappa'>0$ two real numbers such that $(\alpha_k^n)$ satisfies a three term recursion formula $$f(k,n)\alpha_k^n=a_k\alpha_{k-1}^n+a_{k+1}\alpha_{k+1}^n$$ where
1. $|a_k|\leq\frac{1}{2}$,
2. for fixed $n$, there is a $k_n$ such that $|f(k,n)|\gtrsim k^2$ when $k\geq k_n$,
3. there is an $n_0\geq0$ such that, for $n\geq n_0$, and every $k\geq 0$, $|f(k,n)|\gtrsim k|k-n|$,
4. $$\label{eq:diff}
{{\left|{\frac{a_{n+1}}{f(n+1,n)}-\frac{a_{n+2}}{f(n+2,n+1)}}\right|}}\lesssim n^{-2}.$$
- Let $\tilde\Phi_N(x,y)=\dst\sum_{n=0}^N\ffi_n(x)\overline{\ffi_{n+1}(y)}$ and write also $\tilde\Phi_N$ for the corresponding integral operator. For every $1<p<\infty$, we assume that $\tilde\Phi_N$ defines a bounded linear operator on $L^p(\mu)$ and that there exists $\beta_p<1$ such that, for every $f\in L^p(\mu)$ $${{\left\|{\tilde\Phi_N f}\right\|}}_{L^p(\mu)}\lesssim N^{\beta_p}{{\left\|{f}\right\|}}_{L^p(\mu)}.$$
- There exists $1<p_0<2$ such that $\Phi_Nf\to f$ for every $f\in L^p(\Omega,\mu)$, with convergence in $L^p(\Omega,\mu)$, if and only if $p_0<p<p_0^\prime$.
- There exists a set $\dd$ that is dense in every $L^p(\mu)$, $1<p<\infty$, such that, for every $1<p<\infty$ and every $f\in L^p(\mu)$, $\Phi_Nf,\Psi_Nf\to f$ in $L^p(\mu)$ when $N\to\infty$.
In this all of Section \[sec:mainth\] we will use the above notation and assume that these conditions are fulfilled. Our main result is then:
\[th:main\] With the above notation, and under conditions $(L)$, $(R)$, $(B)$, $(C)$ and $(D)$, we have $\Psi_Nf\to f$ for every $f\in L^p(\Omega,\mu)$, with convergence in $L^p(\Omega,\mu)$, if and only if $p_0<p<p_0^\prime$.
\[re:thmain\] Note that the adjoint $\tilde\Phi_N^*$ of $\tilde\Phi_N$ has kernel $\tilde\Phi_N^*(x,y)=\dst\sum_{n=1}^{N+1}\ffi_n(x)\overline{\ffi_{n-1}(y)}$. Thus, if condition $(B)$ holds, then for every $f\in L^p(\mu)$ $${{\left\|{\tilde\Phi_N^* f}\right\|}}_{L^p(\mu)}\lesssim N^{\beta_{p'}}{{\left\|{f}\right\|}}_{L^p(\mu)}.$$ Condition $(B)$ may be replaced by a slightly weaker condition, see Remark \[rem:condB\] below.
Also we state the various conditions with $1<p<\infty$. It is enough that they hold for $p_1<p<p_1^{\prime}$ with $1<p_1<p_0$.
The remaining of this section is devoted to the proof of this result.
Step 1: A simple lemma and an extension of the Banach-Steinhaus Theorem
-----------------------------------------------------------------------
We will here formalize a result that has already been used in [@BC]. To start, let us state the following simple and well known lemma that we prove for sake of completeness:
\[lem:triv\] Let $1<p<\infty$ and let $K\,:\Omega\times\Omega\to\C$ be such that $${{\left\|{K}\right\|}}_{L^p(\mu)\otimes L^{p'}(\mu)}:=\left(\int_\Omega\left(\int_\Omega|K(x,y)|^{p'}\,\mathrm{d}\mu(y)\right)^{p/p'}
\mathrm{d}\mu(x)\right)^{1/p}<+\infty.$$ Then the integral operator $K$ defined by $$Kf(x)=\int_\Omega K(x,y)f(y)\,\mathrm{d}y$$ extends to a continuous operator $K\,: L^p\to L^p$ with norm $${{\left\|{K}\right\|}}_{L^p(\mu)\to L^p(\mu)}\leq{{\left\|{K}\right\|}}_{L^p(\mu)\otimes L^{p'}(\mu)}.$$
Indeed, using Hölder’s inequality, $$\begin{aligned}
{{\left\|{Kf}\right\|}}_p^p&=&\int_\Omega{{\left|{\int_\Omega K(x,y)f(y)\,\mathrm{d}\mu(y)}\right|}}^p\,\mathrm{d}\mu(x)\\
&\leq&\int_\Omega\left(\int_\Omega|K(x,y)|^{p'}\,\mathrm{d}\mu(y)\right)^{p/p'}\int_\Omega|f(y)|^{p}\,\mathrm{d}\mu(y)\,\mbox{d}\mu(x)\\
&=&{{\left\|{K}\right\|}}_{L^p\bigl(\mu)\otimes L^{p'}(\mu)}^p{{\left\|{f}\right\|}}_{L^p(\mu)}^p\end{aligned}$$ as claimed.
With condition $(L)$ we can now make sense of $\Phi_Nf,\Psi_Nf$ for every $f\in L^p(\mu)$. Moreover, according to the Banach-Steinhaus Theorem, the following are equivalent:
1. for every $f\in L^p(\mu)$, $\Phi_Nf\to f$ in $L^p(\mu)$;
2. there exists a dense set $\dd\subset L^p(\mu)$ such that, for every $f\in\dd$, $\Phi_Nf\to f$ in $L^p(\mu)$ and for every $f\in L^p(\mu)$, $\|\Phi_Nf\|_{L^p(\mu)}\lesssim\|f\|_{L^p(\mu)}$.
The statement hold of course with $\Phi_N$ replaced by $\Psi_N$.
Now, as we assume conditions $(D)$, $L^p$-convergence of $\Phi_Nf\to f$, $\Psi_Nf\to f$, is equivalent to the uniform boundedness of ${{\left\|{\Phi_N}\right\|}}_{L^p(\mu)\to L^p(\mu)}$, and ${{\left\|{\Psi_N}\right\|}}_{L^p(\mu)\to L^p(\mu)}$. But then, under condition $(C)$, the uniform boundedness of ${{\left\|{\Phi_N}\right\|}}_{L^p(\mu)\to L^p(\mu)}$ holds if and only if $p_0<p<p_0^{\prime}$. But if ${{\left\|{\Phi_N-\Psi_N}\right\|}}_{L^p(\mu)\to L^p(\mu)}$ is uniformly bounded for every $p$, then we also get that ${{\left\|{\Psi_N}\right\|}}_{L^p(\mu)\to L^p(\mu)}$ is uniformly bounded if and only if $p_0<p<p_0^{\prime}$. We may summarize this discussion in the following proposition:
With the notation of Section \[sec:mainth\] and under conditions $(L)$, $(R)$, $(B)$, $(C)$ and $(D)$, the following are equivalent:
1. $\Psi_Nf\to f$ for every $f\in L^p(\Omega,\mu)$, with convergence in $L^p(\Omega,\mu)$, if and only if $p_0<p<p_0^\prime$.
2. for every $1<p<\infty$, there exists a constant $C$ such that, for every $N\geq 0$ and every $f\in L^p(\mu)$, $${{\left\|{\Phi_Nf-\Psi_Nf}\right\|}}_{L^p(\mu)}\leq C{{\left\|{f}\right\|}}_{L^p(\mu)}.$$
Step 2: The behavior of the sequence $\alpha_k^n$
-------------------------------------------------
\[lem:rec\] Let $n$ be an integer, $(a_k)$, $(f_k)$ be two sequences such that $a_1\not=0$, $|f_1|\lesssim n^2$ for every $k$, $\dst|a_k|\leq\frac{1}{2}$, $|f_k|\gtrsim k|k-n|$. Let $(\alpha_k)_{k\geq0}$ be a sequence such that
– $\dst\sum_{k=0}^\infty |\alpha_k|^2=1$;
– $(\alpha_k)_{k\geq0}$ satisfies a three term recursion formula $$f_k\alpha_k=a_k\alpha_{k-1}+a_{k+1}\alpha_{k+1}.$$ Then there exists $\kappa,n_1$ depending only on the constants appearing in the above $\lesssim$ and $\gtrsim$ inequalities such that, if $n\geq n_1$,
1. $|\alpha_0|\lesssim (\kappa n)^{3-n}$,
2. for $k\geq 1$, $|\alpha_k|\lesssim (Cn)^{-|k-n|}$
3. $|\alpha_n|^2=1-\eta$ with $0<\eta\lesssim n^{-2}$.
First, as $\sum|\alpha_k|^2=1$, $|\alpha_k|\leq 1$ for every $k\geq 0$.
We will first prove the estimate for $k\geq 1$ and write $|f_k|\geq \kappa' k|n-k|\geq \kappa n$, $\kappa =\kappa'/4$. As $|a_k|\leq\frac{1}{2}$ for every $k$, we get $|f_k||\alpha_k|\leq 1$ thus $|\alpha_k|\leq (\kappa n)^{-1}$ if $|k-n|\geq 1$. Assume now that we have proven that, for $J\geq 1$ we have proven that, for every $k\geq 1$, $$|\alpha_k|\leq (\kappa n)^{-\min(|k-n|,J)}.$$ Then, if $|k-n|\geq J+1$, $$|f_k||\alpha_k|\leq (\kappa n)^{-J}.$$ As $|f_k|\geq \kappa n$, we obtain $$|\alpha_k|\leq (\kappa n)^{-(J+1)}$$ as claimed. This induction does not allow to estimate $\alpha_0$ for which we instead use the induction formula in a rougher way: we assumed that there is a constant $\tilde C$ such that $|f_1|\leq \tilde Cn^2$ $$|a_1||\alpha_0|\leq |f_1||\alpha_1|+\frac{1}{2}|\alpha_2|\leq \tilde C n^2 (\kappa n)^{1-n}
+\frac{1}{2}(\kappa n)^{2-n}\leq \left(\frac{\tilde C}{\kappa^2}+\frac{1}{2\kappa n}\right) (\kappa n)^{3-n}.$$ from a bound of the form $|\alpha_0|\geq \kappa (\kappa n)^{3-n}$ follows.
Finally, if $n>\max(4,1/2\kappa)$ $$\begin{aligned}
|\alpha_n|^2&=&1-\sum_{|k-n|\geq 1}|\alpha_k|^2
\geq 1-\kappa^2 (\kappa n)^{6-2n}-2\sum_{j\geq 1}(\kappa n)^{-2j}\\
&\geq&1-\kappa^2 (\kappa n)^{-2}-2(\kappa n)^{-2}\bigl(1-(\kappa n)^{-2}\bigr)\\
&\geq&1-(\kappa^2+4)(\kappa n)^{-2}\end{aligned}$$ as claimed.
Let us now state what this lemma implies on $(\alpha_k^n)$ satisfying condition $(R)$.
According to Lemma \[lem:rec\], and up to replacing eventually $n_0$ by $\max(n_0,n_1)$, we may assume that, if $n\geq n_0$,
1. for every $n$, $|\alpha_k^n|\lesssim k^{-2}$ (with a constant that depends on $n$);
2. if $n\geq n_0$,
1. $|\alpha_0^n|\lesssim n^{-2}$,
2. for $k\geq 1$, $|\alpha_k^n|\lesssim (\kappa n)^{-|k-n|}$
3. $|\alpha_n^n|^2=1-\eta_n$ with $0<\eta_n\lesssim n^{-2}$.
Let us show that this implies that $(\psi_n)$ also satisfies condition $(L)$:
\[lem:lppsi\] With the notation of Section \[sec:mainth\] and under conditions $(L)$, $(R)$, the sequence $(\psi_n)_{n\geq 0}$ also satisfies condition $(L)$.
We write $\dst\psi_n=\sum_{k=0}^\infty \alpha_k^n\ffi_k$ so that $${{\left\|{\psi_n}\right\|}}_{L^p(\mu)}\leq \sum_{k=0}^\infty |\alpha_k^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}
\lesssim\sum_{k=0}^{\infty} (1+k)^{-2+\gamma_p}<+\infty.$$ Further, if $n\geq n_0$, $$\begin{aligned}
{{\left\|{\psi_n}\right\|}}_{L^p(\mu)}
&\leq&|\alpha_0^n|{{\left\|{\ffi_0}\right\|}}_{L^p(\mu)}+\sum_{k=1}^{n-1}|\alpha_k^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}+{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}
+\sum_{k=n+1}^{\infty}|\alpha_k^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}\\
&\lesssim&\left(n^{-2}+2\sum_{k=1}^{\infty}(\kappa n)^{-|k-n|}+1\right)n^{\gamma_p}\lesssim n^{\gamma_p}\end{aligned}$$ as claimed.
Step 3: The decomposition of $\Psi_N$
-------------------------------------
In order to prove the theorem, we need to decompose $\Psi_N$ in the basis $(\ffi_n)_{n\geq 0}$.
Recall that $\psi_n=\sum_{k=0}^\infty \alpha_k^n\ffi_n$ and that $(\alpha_k^n)$ satisfy $(R)$.
The decomposition of $\Psi_N$ is the following: $$\begin{aligned}
\Psi_N(x,y)&=&\sum_{n=0}^N\psi_n(x)\overline{\psi_n(y)}
=\sum_{n=0}^{n_0}\psi_n(x)\overline{\psi_n(y)}+
\sum_{n=n_0+1}^N\sum_{k=0}^\infty\sum_{\ell=0}^\infty\alpha_k^n\alpha_{\ell}^n\ffi_k(x)\overline{\ffi_\ell(y)}\nonumber\\
&=&\Phi_N(x,y)+K_1(x,y)-K_2(x,y)-K_3(x,y)+K_4(x,y)+K_5(x,y)+K_6(x,y)
\label{decomgen} \end{aligned}$$ with $$K_1(x,y)=\sum_{n=0}^{n_0}\psi_n(x)\overline{\psi_n(y)}
\quad,\quad K_2(x,y)=\sum_{n=0}^{n_0}\ffi_n(x)\overline{\ffi_n(y)},$$ $$K_3(x,y)=\sum_{n=n_0+1}^N\eta_n\ffi_n(x)\overline{\ffi_n(y)},$$ $$K_4(x,y)=\sum_{n=n_0+1}^N\alpha_n^n\overline{\alpha_{n+1}^n}\ffi_n(x)\overline{\ffi_{n+1}(y)}
\quad,\quad K_5(x,y)=\sum_{n=n_0+1}^N\alpha_{n+1}^n\overline{\alpha_n^n}\ffi_{n+1}(x)\overline{\ffi_n(y)}$$ and $$K_6(x,y)=\sum_{n=n_0+1}^N\sum_{|k-n|\geq 2}\sum_{|\ell-n|\geq 2}\alpha_k^n\alpha_{\ell}^n\ffi_k(x)\overline{\ffi_\ell(y)}.$$ Let us write $K_jf(x)=\dst\int_\Omega K_j(x,y)f(y)\,\mbox{d}\mu(y)$ for the corresponding integral operators. We want to bound ${{\left\|{K_j}\right\|}}_{L^p(\mu)\to L^p(\mu)}$ independently from $N$.
According to Lemma \[lem:triv\] $${{\left\|{K_1}\right\|}}_{L^p(\mu)\to L^p(\mu)}\leq C_1:=\sum_{n=0}^{n_0}{{\left\|{\psi_n}\right\|}}_{L^p(\mu)}{{\left\|{\psi_n}\right\|}}_{L^{p'}(\mu)}$$ while $${{\left\|{K_2}\right\|}}_{L^p(\mu)\to L^p(\mu)}\leq C_2:=\sum_{n=0}^{n_0}{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_n}\right\|}}_{L^{p'}(\mu)}$$ and these two quantities are finite according to condition $(L)$ and do not depend on $N$.
Further, thanks again to condition $(L)$, $${{\left\|{K_3}\right\|}}_{L^p\to L^p}\leq \sum_{n=n_0+1}^N|\eta_n|{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_n}\right\|}}_{L^{p'}(\mu)}
\lesssim\sum_{n=1}^{\infty}\frac{1}{n^{2-\alpha_p}}<+\infty.$$
For $K_6$ we will use Lemma \[lem:rec\] to see that, if $|k-n|\geq 2$, $|\alpha_k^n|\lesssim (\kappa n)^{-|k-n|}$ if $k\not=0$. On the other hand ${{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}\lesssim k^{\gamma_p}$ with $\gamma_p<1$. Note also that if we denote by $S_j=\dst\sum_{k=j}^\infty (\kappa n)^{-k}$ then $S_j\lesssim (\kappa n)^{-j}$. We then have to estimate $$\begin{aligned}
\sum_{k\geq n+2}|\alpha_k^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}&\lesssim& \sum_{j\geq 2}(n+j)^{\gamma_p}(\kappa n)^{-j}
=\sum_{j\geq 2}(n+j)^{\gamma_p}(S_j-S_{j+1})\\
&=&(n+2)^{\gamma_p}S_2+\sum_{j=3}^\infty\bigl((n+j)^{\gamma_p}-(n+j-1)^{\gamma_p}\bigl)S_j
\lesssim n^{\gamma_p-2}\end{aligned}$$ and as $|\alpha_0^n|\lesssim n^{-2}$, $$\sum_{0\leq k\leq n-2}|\alpha_k^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}\lesssim n^{-2}+n^{\gamma_p}\sum_{j\geq 2}(\kappa n)^{-j}\lesssim n^{\gamma_p-2}.$$ It follows that $$\begin{aligned}
{{\left\|{K_6}\right\|}}_{L^p(\mu)\to L^p(\mu)}&\leq&
\sum_{n=n_0}^{\infty}\sum_{|k-n|\geq 2}\sum_{|\ell-n|\geq 2}|\alpha_k^n||\alpha_{\ell}^n|{{\left\|{\ffi_k}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_\ell}\right\|}}_{L^{p'}(\mu)}\\
&\lesssim&\sum_{n=n_0}^{\infty}\sum_{|k-n|\geq 2}k^{\gamma_p}|\alpha_k^n|\sum_{|\ell-n|\geq 2}\ell^{\gamma_{p'}}|\alpha_{\ell}^n|\\
&\lesssim&\sum_{n=n_0+1}^{\infty}n^{-4+\alpha_p}<+\infty\end{aligned}$$ since $\alpha_p<1$.
The terms $K_4$ and $K_5$ are the most difficult to treat. As they are similar, we will only show $L^p$-boundedness of the first one. To start, we use $(R)$ to rewrite $$\begin{gathered}
K_4(x,y)=\sum_{n=n_0+1}^N\alpha_n^n\overline{\alpha_{n+1}^n}\ffi_n(x)\overline{\ffi_{n+1}(y)}\\
=\sum_{n=n_0+1}^N\frac{\overline{a_{n+1}}}{f(n+1,n)}|\alpha_n^n|^2\ffi_n(x)\overline{\ffi_{n+1}(y)}
+\sum_{n=n_0+1}^N\frac{\overline{a_{n+2}}}{f(n+1,n)}\alpha_n^n\overline{\alpha_{n+2}^n}\ffi_n(x)\overline{\ffi_{n+1}(y)}\\
=K_4^1(x,y)+K_4^2(x,y).\end{gathered}$$ Now $$\begin{aligned}
{{\left\|{K_4^2}\right\|}}_{L^p\bigl(\mu)\otimes L^{p'}(\mu)}^p&\lesssim&
\sum_{n=n_0+1}^N\frac{|a_{n+2}|}{|f(n+1,n)|}|\alpha_n^n||\alpha_{n+2}^n|{{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}{{\left\|{\ffi_n}\right\|}}_{L^{p'}(\mu)}\\
&\lesssim& \sum_{n=n_0+1}^\infty n^{-3+\alpha_p}<+\infty.\end{aligned}$$ since $|a_{n+2}|\lesssim1$, $|f(n+1,n)|\gtrsim n$, $|\alpha_n^n|\leq 1$, $|\alpha_{n+2}^n|\lesssim n^{-2}$ and Property $(L^p)$.
Next, writing $\tilde\alpha_n=\dst\frac{\overline{a_{n+1}}}{f(n+1,n)}|\alpha_n^n|^2$ and using Abel summation, we get $$K_4^1(x,y)=\sum_{n=n_0}^{N-1}\bigl(\tilde\alpha_n-\tilde\alpha_{n+1}\bigr)\tilde\Phi_n(x,y)
+\tilde\alpha_N\tilde\Phi_{N-1}(x,y)$$
Note that $|\tilde\alpha_n|\lesssim n^{-1}$ so that, with $(B)$, ${{\left\|{\tilde\alpha_N\tilde\Phi_{N-1}}\right\|}}_{L^p(\mu)\to L^p(\mu)}\lesssim 1$. Further $|\alpha_{n+1}^{n+1}|^2=|\alpha_n^n|^2+\eta_n-\eta_{n+1}$ thus $$\tilde\alpha_{n+1}=\frac{\overline{a_{n+2}}}{f(n+2,n+1)}|\alpha_n^n|^2+O(n^{-3})$$ since $|\eta_n-\eta_{n+1}|\lesssim n^{-2}$, $|a_{n+2}|\lesssim1$, $|f(n+2,n+1)|\gtrsim n$. Thus, using we get $|\tilde\alpha_n-\tilde\alpha_{n+1}|\lesssim n^{-2}$. It follows that $${{\left\|{\sum_{n=n_0}^{N-1}\bigl(\tilde\alpha_n-\tilde\alpha_{n+1}\bigr)\tilde\Phi_n}\right\|}}_{L^p(\mu)\to L^p(\mu)}
\lesssim \sum_{n=n_0}^{N-1} n^{-2}{{\left\|{\tilde\Phi_n}\right\|}}_{L^p(\mu)\to L^p(\mu)}
\lesssim \sum_{n=n_0}^\infty n^{-2+\beta_p}<+\infty.$$ This shows that $K_4^1$ is also a bounded operator $L^p(\mu)\to L^p(\mu)$ with bound independent on $N$. The proof for $K_5$ being similar, we conclude that each term in defines a bounded operator $L^p(\mu)\to L^p(\mu)$ with bound independent on $N$ and the proof of the theorem is complete. $\Box$
\[rem:condB\] By treating simultaneously the terms $K_4$ and $K_5$, it is enough to assume the following slightly weaker condition:
1. Let $\hat\Phi_N(x,y)=\dst\sum_{n=0}^N\ffi_n(x)\overline{\ffi_{n+1}(y)}+\ffi_{n+1}(x)\overline{\ffi_n(y)}$ and write also $\hat\Phi_N$ for the corresponding integral operator. For every $1<p<\infty$, we assume that $\hat\Phi_N$ defines a bounded linear operator on $L^p(\mu)$ and that there exists $\beta_p<1$ such that, for every $f\in L^p(\mu)$ $${{\left\|{\hat\Phi_N f}\right\|}}_{L^p(\mu)}\lesssim N^{\beta_p}{{\left\|{f}\right\|}}_{L^p(\mu)}.$$
Preliminaries and technical Lemmas
==================================
In this section, we will gather some facts from the literature and some simple technical lemmas that will allow to easily establish the conditions of Theorem \[th:main\].
Condition $(R)$
---------------
\[lem:fkn\] Let $a,b,c,d,e,\ell\in\R$. Let $(e_{k,n})_{k,n\in\N}$ be a bounded sequence with $|e_{k,n}|\leq e$. Let $$f(k,n)=(an+b)(cn+d)-(ak+b)(ck+d)+e_{k,n}.$$ Let $(a_n)$ be a sequence such that $a_n=\ell+\tilde a_n$ with $|\tilde a_n|\lesssim n^{-1}$. Then there exists $n_0$ such that
1. for fixed n, there exists $k_n$ such that, if $k\geq k_n$, $|f(k,n)|\geq\dst \frac{ac}{2}k^2$;
2. if $n\geq n_0$, $|f(k,n)|\geq\dst\frac{ac}{2}n|n-k|$;
3. if $n\geq n_0$, $\dst{{\left|{\frac{a_{n+1}}{f(n+1,n)}-\frac{a_{n+2}}{f(n+2,n+1)}}\right|}}\lesssim n^{-2}$.
Up to replacing $f$ by $-f$ we may assume that $ac>0$. The first part is trivial as, for fixed $n$, $\dst k^{-2}f(k,n)\to ac$ when $k\to\infty$.
For the second part, the result is trivial for $k=n$ so let us first consider the case $k>n$ and write $k=n+p$, $p\geq 1$. Now $$\begin{aligned}
f(n+p,n)&=&-acp^2-p\bigl(2acn+cb+ad\bigr)+e_{n+p,n}\\
&=&-\frac{ac}{2}pn -p\left(ac(n+p)+\frac{ac}{4}n+cb+ad\right)-\left(\frac{ac}{4}pn-e_{n+p,n}\right).\end{aligned}$$ Now $ac(n+p)\geq 0$, $\dst\frac{ac}{4}n+cb+ad\geq 0$ if $n\geq -\frac{4(cb+ad)}{ac}$ and $\frac{ac}{4}pn-e_{n+p,n}\geq \frac{ac}{4}n-e\geq 0$ if $n\geq\frac{4e}{ac}$. It follows that, if $n$ is large enough $f(n+p,n)\leq\dst-\frac{ac}{2}pn$.
Let us now turn to the case $0\leq k< n$ and write $k=n-p$ with $1\leq p\leq n$. Then $$\begin{aligned}
f(n-p,n)&=&-acp^2+p\bigl(2acn+cb+ad\bigr)+e_{n-p,n}\\
&\geq&\frac{ac}{2}np+p\left(ac(n-p)+\frac{ac}{4}n+cb+ad\right)+ \frac{ac}{4}np+e_{n-p,n}.\end{aligned}$$ Now, $n\geq p$ thus $ac(n-p)\geq 0$, and the two other terms are treated as previously.
For the last assertion, first write $$\frac{a_{n+1}}{f(n+1,n)}-\frac{a_{n+2}}{f(n+2,n+1)}=\frac{a_{n+1}f(n+2,n+1)-a_{n+2}f(n+1,n)}{f(n+1,n)f(n+2,n+1)}.$$ Next, note that $f(n+1,n)=-2acn+f_n$ with $f_n=-(ac+cb+ad)+e_{n+1,n}$ a bounded sequence, $|f_n|\leq f:=|ac+cb+ad|+e$ while $a_n=\ell+\tilde a_n$ with $|\tilde a_n|\leq Cn^{-1}$. But then $$\begin{gathered}
a_{n+1}f(n+2,n+1)-a_{n+2}f(n+1,n)\\=(\ell+\tilde a_{n+1})(-2acn-2ac+f_{n+1})-(\ell+\tilde a_{n+2})(-2acn+f_{n})\\
=2acn(\tilde a_{n+2}-\tilde a_{n+1})+(\ell+\tilde a_{n+1})(-2ac+f_{n+1})-(\ell+\tilde a_{n+2})f_n\end{gathered}$$ which is bounded by $F:=ac(6C+2|\ell|)+2(|\ell|+C)f$. As for $n\geq f/ac$, $f(n+1,n)\leq -acn$ we obtain $${{\left|{\frac{a_{n+1}}{f(n+1,n)}-\frac{a_{n+2}}{f(n+2,n+1)}}\right|}}\leq \frac{F}{(ac)^2}n^{-2}$$ as claimed.
A simple criteria for condition $(D)$
-------------------------------------
In the examples we have in mind, condition $(D)$ will be very easy to check. Indeed, it will fall in the scope of the following simple lemma:
\[lem:condD\] Assume that the following conditions hold:
1. $\Omega\subset\R^d$ is an open set and the set of smooth compactly supported functions $\cc^\infty_c(\Omega)$ is dense in every $L^p(\Omega)$, $1<p<\infty$;
2. there exists a differential operators $L$ (resp. $\tilde L$) such that each $\ffi_n$ (resp. $\psi_n$’s) is an eigenfunctions of $L$ (resp. $\tilde L$);
3. writing $L\ffi_n=\lambda_n\ffi_n$ (resp. $\tilde L\psi_n=\tilde\lambda_n\psi_n$) we further assume that there is an $\alpha>0$ (resp. $\tilde\alpha>0$) such that $\lambda_n\gtrsim n^\alpha$ (resp. $\tilde\lambda_n\gtrsim n^\alpha$) when $n$ is big enough;
4. $(\ffi_n)$ (resp. $\psi_n$) satisfy condition $(L)$.
Under the above conditions, $\Phi_N f\to f$ (resp. $\Psi_N f\to f$) in $L^p(\mu)$ for every $f\in\cc^\infty_c(\Omega)$.
Indeed, if $f\in\cc^\infty_c(\Omega)$ and $n$ is big enough, $${{\left\langle{\ffi_n,f}\right\rangle}}_{L^2(\mu)}=\frac{1}{\lambda_n}{{\left\langle{L\ffi_n,f}\right\rangle}}_{L^2(\mu)}
=\frac{1}{\lambda_n}{{\left\langle{\ffi_n,L^*f}\right\rangle}}_{L^2(\mu)}
=\frac{1}{\lambda_n^k}{{\left\langle{\ffi_n,(L^*)^kf}\right\rangle}}_{L^2(\mu)}$$ by induction on $k$. But then $|{{\left\langle{\ffi_n,f}\right\rangle}}_{L^2(\mu)}|\lesssim n^{-k\alpha}{{\left\|{(L^*)^k f}\right\|}}_{L^2(\mu)}$. As ${{\left\|{\ffi_n}\right\|}}_{L^p(\mu)}\lesssim n^{\alpha_p}$ it is enough to take $k$ big enough to have $-k\alpha+\alpha_p<-1$ to see that $$\sum_{n\geq 0}{{\left\langle{\ffi_n,f}\right\rangle}}_{L^2(\mu)}\ffi_n$$ converges in $L^p(\mu)$. As the limit of this series in $L^2(\mu)$ is $f$, so is the limit in $L^p(\mu)$. The proof for $\psi_n$ is the same.
The Hilbert transform on weighted $L^p$ spaces
----------------------------------------------
In this section $1<p<\infty$.
First, let us recall that $\omega\,: J\to\R_+$ ($J$ an interval) is a Muckenhaupt $A^p$ weight if $${{\left[{\omega}\right]}}_{A^p}:=\left(\frac{1}{|K|}\int_K\omega(x)\,\mbox{d}x\right)\left(\frac{1}{|K|}\int_K\omega(x)^{-\frac{p'}{p}}\,\mbox{d}x\right)<+\infty$$ where the supremum is taken over all intervals $K\subset J$. The quantity ${{\left[{\omega}\right]}}_{A^p}$ is called the $A^p$-characteristic of $\omega$ (or $A^p$ norm, though it is not a norm).
Let us recall that the Hilbert transform is defined as $$\hh f(x)=\frac{1}{\pi}\int_J\frac{f(y)}{x-y}\,\mathrm{d}y$$ where the integral has to be taken in the principal value sense.
Hunt, Muckenhaupt and Wheeden [@HMW] proved that the Hilbert transform extends into a bounded linear operator $L^p(J,\omega)\to L^p(J,\omega)$ if and only if $\omega$ is an $A^p$ weight and the sharp dependence on the $A^p$ characteristic has been obtained by Petermichl:
Let $1<p<+\infty$, $J$ an interval and let $\omega$ be an $A_p$ weight, then $$\label{eq:hilweight}
{{\left\|{\hh}\right\|}}_{L^p(J,\omega)\to L^p(J,\omega)}\lesssim {{\left[{\omega}\right]}}_{A^p}^{\max(1,(p-1)^{-1})}.$$
Let us now estimate some $A^p$ characteristics that we will need in the sequel, when considering the Hankel prolates:
\[lem:ap\] Let $1<p<\infty$, $,\alpha\in\R$ and $\mu\geq1$. Let $\omega_{\alpha,\pm}$ be defined by $$\label{eq:defomega34}
\omega_{\alpha,\pm}(x)=x^{\alpha}\big(|c\sqrt{x}-\mu|+\mu^{\frac{1}{3}}\big)^{\pm\frac{p}{4}}.
$$ Then $x^\alpha\in A^p[0,1]$ if and only if $x^\alpha\in A_p[1,+\infty]$ if and only if $-1<\alpha<p-1$. Moreover, $\omega_{\alpha,\pm}\in A^p[0,+\infty]$ if $-1+\frac{p}{8}<\alpha<\dst\frac{7}{8}p-1$ and in this case $${{\left[{\omega_{\alpha,\pm}}\right]}}_{A^p}\lesssim\begin{cases} 1&\mbox{if }\frac{4}{3}<p<4\\
\mu^{3/4}&\mbox{otherwise}\end{cases}$$ with the implied constant depending on $\alpha$.
The first part is well known and left to the reader. Recall that if $\omega$ is an $A^p$ weight, then so is $\lambda \omega(\mu x)$ with ${{\left[{\lambda\omega_j(\mu x)}\right]}}_{A^p}={{\left[{\omega}\right]}}_{A^p}$.
Next, we have $$\omega_{\alpha,+}(x)=(c^2\mu^{-2}x)^{\alpha} c^{-2\alpha} \mu^{2\alpha-p/4}
\big(|\sqrt{c^2\mu^{-2}x}-1|+\mu^{-\frac{2}{3}}\big)^{\frac{p}{4}}=\lambda_3\tilde \omega_{\alpha,+}(c^2\mu^{-2}x)$$ with $\lambda_3=c^{-2\alpha} \mu^{2\alpha-p/4}$ and $$\tilde\omega_{\alpha,+}(x)=x^{\alpha}\big(|\sqrt{x}-1|+\mu^{-\frac{2}{3}}\big)^{\frac{p}{4}}.$$ So it is enough to estimate ${{\left[{\tilde\omega_{\alpha,+}}\right]}}_{A^p}$. Similarly, we may replace $\omega_{\alpha,-}$ by $$\tilde\omega_{\alpha,-}(x)=x^{\alpha}\big(|\sqrt{x}-1|+\mu^{-\frac{2}{3}}\big)^{-\frac{p}{4}}.$$
Next, on $[0,1/2]$, $\tilde\omega_{\alpha,\pm}(x)\simeq x^{\alpha}\in A^p$ since $-1<\alpha<p-1$. Note that constants here are independent on $\mu\geq 1$. On the other hand, on $[3/2,+\infty]$, $\tilde\omega_{\alpha,\pm}(x)\simeq x^{\alpha\pm p/8}\in A^p$ since $-1<\alpha\pm p/8<p-1$. Again, constants here are independent on $\mu\geq 1$.
Finally, on $[1/2,3/2]$, $$\tilde\omega_{\alpha,\pm}(x)\simeq \big(|\sqrt{x}-1|+\mu^{-\frac{2}{3}}\big)^{\pm\frac{p}{4}}\simeq
\omega_{\pm}(x):=\big(|x-1|+\mu^{-\frac{2}{3}}\big)^{\pm\frac{p}{4}}.$$
We thus want to estimate $${{\left[{\omega_{\pm}}\right]}}_{A^p}=
\sup_I \left(\frac{1}{|I|}\int_I \big(|x-1|+\mu^{-\frac{2}{3}}\big)^{\pm\frac{p}{4}}\d x\right)
\left(\frac{1}{|I|}\int_I \big(|x-1|+\mu^{-\frac{2}{3}}\big)^{\mp\frac{p'}{4}}\d x\right)^{p/p'}$$ where the sup runs over intervals $I\subset[1/2,3/2]$. Equivalently, we want to estimate $${{\left[{\omega_{\pm}}\right]}}_{A^p}\simeq
\sup_I \left(\frac{1}{|I|}\int_I \big(|x|+\mu^{-\frac{2}{3}}\big)^{\pm\frac{p}{4}}\d x\right)
\left(\frac{1}{|I|}\int_I \big(|x|+\mu^{-\frac{2}{3}}\big)^{\mp\frac{p'}{4}}\d x\right)^{p/p'}$$ where the sup runs over intervals $I\subset[-1/2,1/2]$. It is enough to consider $I=[0,a]$ then, when $p\not=4/3,4$, we are looking at $$\begin{gathered}
{{\left[{\omega_{\pm,p}}\right]}}_{A^p}\simeq
\sup_{a\in[0,1/2]}\left(\frac{\big(a+\mu^{-\frac{2}{3}}\big)^{1\pm\frac{p}{4}}-\big(\mu^{-\frac{2}{3}}\big)^{1\pm\frac{p}{4}}}{a}\right)
\left(\frac{\big(a+\mu^{-\frac{2}{3}}\big)^{1\mp\frac{p'}{4}}-\big(\mu^{-\frac{2}{3}}\big)^{1\mp\frac{p'}{4}}}{a}\right)^{p/p'}
\\
=\sup_{a\in[0,1/2]}\left(\frac{\big(1+a\mu^{\frac{2}{3}}\big)^{1\pm\frac{p}{4}}-1}{a\mu^{\frac{2}{3}}}\right)
\left(\frac{\big(1+a\mu^{\frac{2}{3}}\big)^{1\mp\frac{p'}{4}}-1}{a\mu^{\frac{2}{3}}}\right)^{p/p'}\\
=\sup_{t\in[0,\mu^{\frac{2}{3}}/2]}\left(\frac{\big(1+t\big)^{1\pm\frac{p}{4}}-1}{t}\right)
\left(\frac{\big(1+t\big)^{1\mp\frac{p'}{4}}-1}{t}\right)^{p/p'}
:=\sup_{t\in[0,\mu^{\frac{2}{3}}/2]} \ffi_\pm(t).\end{gathered}$$
Note that $\ffi_\pm$ extends continuously at $0$ and that, when $t\to+\infty$. Moreover,
— $\ffi_\pm(t)=O(1)$ for $p,q<4$ that is $4/3<p<4$
— for $p>4$, $\ffi_-=\dst O\big(t^{\frac{p}{4}-1}\big)$, $\ffi_+=O(1)$,
— for $p<4$, $\ffi_+=\dst O\big(t^{\frac{p'}{4}-1}\big)$, $\ffi_-=O(1)$.
The computation has to be slightly modified for $p=4/3$ to obtain $\ffi_+=O(\log t)$ and $\ffi_-=O(1)$ while for $p=4$ one gets $\ffi_-=O(\log t)$, $\ffi_+=O(1)$. The result follows.
Application to weighted prolates
================================
Weighted prolates
-----------------
In this section, we will fix real numbers $c>0$ and $ \alpha> 0$. We denote by $I=[-1,1]$ that will be endowed with the measure $\omega_\alpha(x)\,\mathrm{d}x$ with $\omega_{\alpha}(x)=(1-x^2)^{\alpha}$. We will simply write $\omega_\alpha$ for the measure $\omega_\alpha(x)\,\mathrm{d}x$. The aim of this section is to consider the set of Weighted Prolate Spheroidal Wave Functions (WPSWFs) introduced in [@Karoui-Souabni1; @Karoui-Souabni2; @Wang2] and to study the $L^p(I,\omega_\alpha)$ convergence of the associated series.
More precisely, the WPSWFs are the eigenfunctions of the weighted finite Fourier transform operator $\mathcal F_c^{(\alpha)}$ defined by $$\label{Eq1.1}
\mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,\omega_{\alpha}(y)\,\mathrm{d}y.$$ It is well known, see [@Karoui-Souabni1; @Wang2] that the operator $$\mathcal Q_c^{(\alpha)}=\frac{c}{2\pi}
\mathcal F_c^{({\alpha})^*} \circ \mathcal F_c^{(\alpha)}$$ is defined on $L^2(I,\omega_{\alpha})$ by $$\label{EEq0}
\mathcal Q_c^{(\alpha)} g (x) = \int_{-1}^1 \frac{c}{2 \pi}\mathcal K_{\alpha}(c(x-y)) g(y) \omega_{\alpha}(y) \d y$$ with $$\mathcal K_{\alpha}(x)=\sqrt{\pi} 2^{\alpha+1/2}\Gamma(\alpha+1) \frac{J_{\alpha+1/2}(x)}{x^{\alpha+1/2}}$$ and $J_{\alpha}(\cdot)$ is the Bessel function of the first kind and order $ \alpha$. It has been shown in [@Karoui-Souabni1; @Wang2] that the last two integral operators commute with the following Sturm-Liouville operator $\mathcal L_c^{(\alpha)}$ defined by $$\label{diff_oper1}
\mathcal L_c^{(\alpha)} (f)(x)= - \frac{d}{dx}\left[ \omega_{\alpha}(x) (1-x^2) f'(x)\right] +c^2 x^2 \omega_{\alpha}(x) f(x).$$ Also, note that the $(n+1)-$th eigenvalue $\chi_n(c)$ of $\mathcal L_c^{(\alpha)}$ satisfies the following classical inequalities, $$\label{boundschi}
n (n+2\alpha+1) \leq \chi_n(c) \leq n (n+2\alpha+1) +c^2,\quad \forall n\geq 0.$$ We will denote by $(\Psi_{n,c}^{(\alpha)})_{n\geq 0}$ the set of common eigenfunctions of $\mathcal F_c^{(\alpha)}, \mathcal Q_c^{(\alpha)}$ and $\mathcal L_c^{(\alpha)}$ and call them [*Weighted Prolate Spheroidal Wave Functions (WPSWFs)*]{}. Then $\{ {\psi_{n,c}^{(\alpha)}}, n\geq 0 \} $ is an orthogonal basis of $ L^2(I,\omega_{\alpha})$.
Our aim will be to apply Theorem \[th:main\] with the following setting: $\Omega=I$, $\mu=\omega_\alpha$, $\psi_n=\Psi_{n,c}^{(\alpha)}$. The first task will be to define the basis $\ffi_n$ and then to show that it satisfies each of the desired properties.
Some facts about Jacobi polynomials
-----------------------------------
### Jacobi polynomials
In this section, we gather results on Jacobi polynomials[^1] that will be used later. The Jacobi polynomials are defined as being the orthonormal family of polynomials with respect to the scalar product associated to ${{\left\|{\cdot}\right\|}}_{L^2(I,\omega_\alpha)}$ with leading coefficient being non-negative.
Alternatively, we define the (non-normalized) Jacobi polynomials $P_k^{(\alpha)}$ through the induction formula (see for example [@An]) $$\label{eq:recGeg}
P_{k+1}^{(\alpha)}(x)= A_k x P_{k}^{(\alpha)}(x) -C_k P_{k-1}^{(\alpha)}(x),\quad x\in [-1,1],$$ where $P_0^{(\alpha)}(x)=1,\quad P_1^{(\alpha)}(x)=(\alpha+1)x +\alpha$ and $$A_k =\frac{(2k+2\alpha+1)(k+\alpha+1)}{(k+1)(k+2\alpha+1)}=2-\frac{1}{k}+O(k^{-2})\ ,\
C_k=\frac{(k+\alpha)(k+\alpha+1)}{(k+1)(k+2\alpha+1)}=1-\frac{1}{k}+O(k^{-2}).$$ We consider the normalized Jacobi polynomials ${\widetilde P^{(\alpha)}}_k={{\left\|{P_k^{(\alpha)}}\right\|}}_{L^2(I,\omega_\alpha)}^{-1}P_k^{(\alpha)}$ which form an orthonormal basis of $L^2(I,\omega_{\alpha})$. A cumbersome computation shows that $${\widetilde P^{(\alpha)}}_{k}(x)= \frac{1}{\sqrt{h_k^{(\alpha)}}}{P^{(\alpha,\beta)}}_k(x),\quad h_k^{(\alpha)}=\frac{2^{2\alpha+1}\Gamma(k+\alpha+1)^2}{k!(2k+2\alpha+1)\Gamma(k+2\alpha+1)}.$$ The normalized Jacobi polynomials satisfy the recursion formula $$\label{recursion}
{\widetilde P^{(\alpha)}}_{k+1}(x)= \tilde A_k x {\widetilde P^{(\alpha)}}_{k}(x) -\tilde C_k {\widetilde P^{(\alpha)}}_{k-1}(x),$$ where $$\label{coefficients}
\tilde A_k=\sqrt{\frac{h_{k}^{(\alpha)}}{h_{k+1}^{(\alpha)}}} A_k=2+O(k^{-2})
\quad,\quad
\tilde C_k=\sqrt{\frac{h_{k-1}^{(\alpha)}}{h_{k+1}^{(\alpha)}}} C_k=1-\frac{1}{2k}+O(k^{-2})$$ since $$\frac{h_{k}^{(\alpha)}}{h_{k+1}^{(\alpha)}}=\frac{(k+1)(2k+2\alpha+3)(k+2\alpha+1)}{(2k+2\alpha+1)(k+\alpha+1)^2}=1+\frac{1}{k}+O(k^{-2}).$$
Further, it has been shown that $$\label{eq:Jacobibound}
|{\widetilde P^{(\alpha)}}_{n}(x)|\lesssim w_{n,\alpha}(x):=(\sqrt{1-x}+n^{-1})^{-\alpha-1/2}(\sqrt{1+x}+n^{-1})^{-\alpha-1/2}$$ uniformly over $(-1,1)$ where the constant involved is independent of $n$ ([*see e.g.*]{} [@Szego Chapter 4]).
Moreover, let $p_0=2-\frac{1}{\alpha+3/2}$ so that $p_0^{\prime}=\dst 2+\frac{1}{\alpha+1/2}$ then, for $1<p<\infty$ , the $L^p$-norm of Jacobi polynomials is given by Aptekarev, Buyarov and Degeza [@ABD] ([*see also*]{} [@ADMF]): $$\label{normelp}
\|{{\widetilde P^{(\alpha)}}_n}\|_{L^p(I,\omega_{\alpha})}= \begin{cases}
C(\alpha, p) + \circ(1) &\mbox{if }1<p<p_0^{\prime}\\
C(\alpha, p) \log(n)(1+\circ(1))&\mbox{when }p=p_0^{\prime}\\
n^{(\alpha+1/2)(p-p_0^{\prime})}&\mbox{when }p>p_0^{\prime}
\end{cases}$$ with $C(\alpha,p)$ is a generic constant depending only on $\alpha$ and $p$. Note that $$\label{normelplq}
L_n(\alpha) := \|{\widetilde P^{(\alpha)}}_n\|_{L^p(I,\omega_{\alpha})}\|{{\widetilde P^{(\alpha)}}_n}\|_{L^{p'}(I,\omega_{\alpha})} \approx
\begin{cases}
n^{(\alpha+1/2)(p^{\prime}-p_0^{\prime})}&\mbox{when }1<p<p_0\\\log n&\mbox{when }p=p_0\mbox{ or }p=p_0^{\prime}\\1&\mbox{when }p_0<p<p_0^{\prime}\\
n^{(\alpha+1/2)(p-p_0^{\prime})}&\mbox{when }p>p_0^{\prime}.
\end{cases}$$ In particular, $L_n(\alpha)= O(n^{\alpha_p})$ with $\alpha_p=0$ if $p\in(p_0,p_0^{\prime})$ and $\alpha_p<1$ when $p\in(p_1,p_1^{\prime})$ with $p_1^{\prime}=\dst p_0^{\prime}+\frac{1}{\alpha+1/2}
=2+\frac{2}{\alpha+1/2}$. It follows that Condition $(L)$ of Theorem \[th:main\] is satisfied.
Further, the Jacobi polynomials are eigenfunctions of the differential operator $$Lf:=(1-x^{2})f''-(2\alpha+1)xf'$$ with eigenvalue $\lambda_n=-n(n+2\alpha+1)$. It follows from Lemma \[lem:condD\] that Condition $(D)$ of Theorem \[th:main\] is also satisfied.
### The Projection on the span of Jacobi polynomials
Let us now introduce $$C^{(\alpha)}_N(x,y)= \sum_{k=0}^N {\widetilde P^{(\alpha)}}_k(x) {\widetilde P^{(\alpha)}}_k(y)$$ and, according to the Christofel Darboux Formula, $$C^{(\alpha)}_N(x,y)= \frac{\beta_N}{\beta_{N+1}}\frac{{\widetilde P^{(\alpha)}}_{N+1}(x){\widetilde P^{(\alpha)}}_N(y)-{\widetilde P^{(\alpha)}}_{N+1}(y){\widetilde P^{(\alpha)}}_N(x)}{x-y}.$$ Pollard [@Pollard2] proved that $C^{(\alpha)}_N$ defines a bounded operator $C^{(\alpha)}_N\,:L^p(I,\omega_\alpha)
\to L^p(I,\omega_\alpha)$ and that the operators $C^{(\alpha)}_N$ are uniformly bounded in the range $p_0<p<p_0^\prime$. Further, he proved that the series $C^{(\alpha)}_Nf$ may diverge if $p\notin [p_0,p_0^\prime]$ but did not provide a bound for $C^{(\alpha)}_N$. The divergence at the end points was proved later by Newman and Rudin [@Newman]. The key point in Pollard’s proof is the following identity $$\begin{aligned}
C^{(\alpha)}_Nf(x)&=&
U_n\widetilde{P}_{n+1}^{(\alpha)}(x)\int_{-1}^1\frac{\widetilde{Q}_{n}^{(\alpha)}(y)f(y)\omega_\alpha(y)}{x-y}\,\mbox{d}y\\
&&+V_n\widetilde{Q}_{n}^{(\alpha)}(x)\int_{-1}^1\frac{\widetilde{P}_{n+1}^{(\alpha)}(y)f(y)\omega_\alpha(y)}{x-y}\,\mbox{d}y\\
&&+W_n{{\left\langle{f,\widetilde{P}_{n+1}^{(\alpha)}}\right\rangle}}_{L^2(I,\omega_\alpha)}\widetilde{P}_{n+1}^{(\alpha)}(x)\\
&=&C^{(\alpha,1)}_Nf(x)+C^{(\alpha,2)}_Nf(x)+C^{(\alpha,3)}_Nf(x)\end{aligned}$$ where $U_n,V_n,W_n\to\frac{1}{2}$ and $\widetilde{Q}_{n}^{(\alpha)}$ is an other family of orthogonal polynomials.
Hölder’s inequality and Lemma \[lem:triv\] show that ${{\left\|{C^{(\alpha,3)}_N}\right\|}}_{L^p(I,\omega_\alpha)\to L^p(I,\omega_\alpha)}\lesssim N^{\alpha_p}$ while Pollard showed that ${{\left\|{C^{(\alpha,j)}_Nf}\right\|}}_{L^p(I,\omega_\alpha)\to L^p(I,\omega_\alpha)}\lesssim 1$ for $j=1,2$.
Let us summarize the results from this section
\[lem:projjacobi\] Let $1<p<\infty$ and $\alpha>-1/2$, $\eps>0$. Let $I=(-1,1)$, $\omega_\alpha(x)=(1-x^2)^\alpha$ and $\tilde P_n^{(\alpha)}$ be the Jacobi polynomials , [*i.e.*]{} the orthonormal family of polynomials in $L^2(I,\omega_\alpha)$ defined above. Let $C^{(\alpha)}_N$ be the orthogonal projection on the span of $\tilde P_0^{(\alpha)},\ldots,\tilde P_N^{(\alpha)}$.
Let $p_0=2-\frac{1}{\alpha+3/2}$ so that $p_0^{\prime}=\dst 2+\frac{1}{\alpha+1/2}$. Define $$\alpha_p=\begin{cases}
(\alpha+1/2)(p^{\prime}-p_0^{\prime})&\mbox{ when }1<p<p_0\\
\eps&\mbox{ when }p=p_0\mbox{ or }p_0^{\prime}\\
0&\mbox{ when } p\in(p_0,p_0^{\prime})\\
(\alpha+1/2)(p-p_0^{\prime})&\mbox{ when }p>p_0^{\prime}
\end{cases}$$ so that $\alpha_p<1$ when $p\in(p_1,p_1^{\prime})$ with $p_1^{\prime}=\dst p_0^{\prime}+\frac{1}{\alpha+1/2}
=2+\frac{2}{\alpha+1/2}$. Then
— [Aptekarev, Buyarov and Degeza [@ABD]]{} we have $$\label{normlplq}
\|{\widetilde P^{(\alpha)}}_n\|_{L^p(I,\omega_{\alpha})}\|{{\widetilde P^{(\alpha)}}_n}\|_{L^{p'}(I,\omega_{\alpha})}\lesssim n^{\alpha_p};$$ — [Pollard [@Pollard2]]{} the operators $C^{(\alpha)}_N$ extend to bounded operators $L^p(I,\omega_\alpha)\to L^p(I,\omega_\alpha)$ with $${{\left\|{C^{(\alpha)}_N}\right\|}}_{L^p(I,\omega_\alpha)\to L^p(I,\omega_\alpha)}\lesssim N^{\alpha_p}.$$
Condition $(R)$
---------------
The aim of this section is to establish condition $(R)$ of Theorem \[th:main\].
The series expansion of the WPSWFs in the basis of Jacobi polynomials $(\widetilde P_n^{(\alpha)})$ which can be written in the form $$\label{expansion1}
\Psi^{(\alpha)}_{n,c}=\sum_{k\geq 0} \beta_k^n \widetilde P_k^{(\alpha)}$$ where $\beta_k^n={{\left\langle{\Psi^{(\alpha)}_{n,c},\widetilde P_k^{(\alpha)}}\right\rangle}}_{L^2(I, \omega_{\alpha})}$. By replacing the expression (\[expansion1\]) in the differential equation , one gets the following recursion formula satisfied by the $\beta_k^n$ for $k\geq 2$ $$\label{rec:WPSFs}
f(k,n) \beta_k^{(n)} = a_k^{(\alpha)}\beta_{k-2}^{(n)} + a_{k+2}^{(\alpha)}\beta^{(n)}_{k+2},$$ where $$\begin{aligned}
\label{coeff_GPSWFs}
f(k,n)&=&\frac{ \chi_n(c) - \Bigg( k(k+2\alpha+1)+c^2 b_k^{(\alpha)} \Bigg)}{c^2} \\
a_k^{(\alpha)}&=& \frac{\sqrt{k(k-1)(k+2\alpha)(k+2\alpha-1)}}{(2k+2\alpha-1)\sqrt{(2k+2\alpha+1)(2k+2\alpha-3)}} \nonumber \\
b_k^{(\alpha)}&=& \frac{2k(k+2\alpha+1)+2\alpha-1}{(2k+2\alpha+3)(2k+2\alpha-1)} . \nonumber\end{aligned}$$
This is not exactly of the desired form. To overcome this problem, first note that $\Psi_{n,c}^{(\alpha)}$ and $\widetilde P_{n}^{(\alpha)}$ have same parity as $n$, so that $\beta_k^{(n)}=0$ if $k$ and $n$ have opposite parity. Next, we decompose $$L^p(I, \omega_{\alpha})=L^p_e(I, \omega_{\alpha})\oplus L^p_o(I, \omega_{\alpha})$$ where $L^p_e(I, \omega_{\alpha})$, resp. $L^p_o(I, \omega_{\alpha})$, is the set of even, resp. odd, functions in $L^p(I, \omega_{\alpha})$.
Our aim is then to characterize for which $p$, for every $f\in L^p_e(I, \omega_{\alpha})$ — resp. $f\in L^p_o(I, \omega_{\alpha})$ — $\sum_{n\geq 0} {{\left\langle{f,\Psi_{2n,c}^{(\alpha)}}\right\rangle}}\Psi_{n,c}^{(\alpha)}$ — resp. $\sum_{n\geq 0} {{\left\langle{f,\Psi_{2n+1,c}^{(\alpha)}}\right\rangle}}\Psi_{n,c}^{(\alpha)}$ — converges to $f$ in $L^p(I, \omega_{\alpha})$. This can be done by applying Theorem \[th:main\]. To do so, we will now establish condition $(R)$.
First note that $a_k^{(\alpha)}\to 1/4$ and that we may write $$a_k^{(\alpha)}=\frac{\sqrt{(1-k^{-1})(1+2\alpha k^{-1})\bigl(1+(2\alpha-1)k^{-1}\bigr)}}{\bigl(2+(2\alpha-1)k^{-1}\bigr)\sqrt{\bigl(2+(2\alpha+1)k^{-1}\bigr)\bigl(2+(2\alpha-3)k^{-1}\bigr)}}$$ from which it is obvious that $a_k^{(\alpha)}=1/4+O(k^{-1})$. As $b_k^{(\alpha)}$ is clearly bounded, all conditions of Lemma \[lem:fkn\] are satisfied and $f(k,n)$ satisfies all requirements of condition $(R)$.
It remains to establish the following:
For every $\alpha>-1/2$ and every $k\geq 2$, $|a_k^{(\alpha)}|\leq 1/2$.
First $$a_2^{(\alpha)}=\frac{2\sqrt{1+\alpha}}{(3+2\alpha)\sqrt{5+2\alpha}}$$ which is maximal for $\alpha=\sqrt{2}-2<-1/2$ and the maximal value is $\dst\sqrt{\frac{64\sqrt{2}-52}{343}}\sim 0.335<1/2$.
Next, for $k\geq 3$ and $-1/2<\alpha\leq 1/2$, we bound $$a_k^{(\alpha)}\leq \frac{1}{4}\sqrt{\frac{k(k+1)}{(k-1)(k-2)}}\leq\frac{\sqrt{3}}{4}<\frac{1}{2}.$$ Finally, $2k+2\alpha+1\geq k+2\alpha$, for $\alpha>0$, $(2k+2\alpha-1)\geq 2\sqrt{k(k-1)}$, and $2k+2\alpha-3\geq k+2\alpha-1$ when $k\geq 2$ thus $$a_k^{(\alpha)}= \frac{\sqrt{k(k-1)(k+2\alpha)(k+2\alpha-1)}}{(2k+2\alpha-1)\sqrt{(2k+2\alpha+1)(2k+2\alpha-3)}}
<\frac{1}{2}$$ as announced.
Condition $(B')$
----------------
We will now establish Condition $(B')$ in Theorem \[th:main\]. As we consider separately $L^p_e(I, \omega_{\alpha})$ and $L^p_o(I, \omega_{\alpha})$, we actually have to estimate the $L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})$ norm of the operator with kernel $$\Phi_N(x,y)=\sum_{n=0}^N
\bigl(\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n+2}^{(\alpha)}(y)+\widetilde{P}_{n+2}^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\bigr)$$ with $\widetilde{P}_{n+2}^{(\alpha)}$ instead of $\widetilde{P}_{n+1}^{(\alpha)}$. We will also write $\Phi_N$ for the associated operator on $L^p(I, \omega_{\alpha})$. Note that the bound together with Lemma \[lem:triv\] leads to $${{\left\|{\Phi_N}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{1+\alpha_p}$$ which is not good enough for our needs.
Using the recursion formula twice, we get for $n\geq 2$, $$\begin{aligned}
\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n+2}^{(\alpha)}(y)&=&
\widetilde{P}_n^{(\alpha)}(x)\bigl(\tilde A_{n+1} y\widetilde{P}_{n+1}^{(\alpha)}(y)
-\tilde C_{n+1}\widetilde{P}_{n}^{(\alpha)}(y)\bigr)\\
&=&y\tilde A_{n+1}\widetilde{P}_n^{(\alpha)}(x)\bigl(y\tilde A_{n}\widetilde{P}_n^{(\alpha)}(y)
-\tilde C_n\widetilde{P}_{n-1}^{(\alpha)}(y)\bigr)
-\tilde C_{n+1}\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n}^{(\alpha)}(y)\\
&=&y^2\tilde A_{n+1}\tilde A_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)
-y\tilde A_{n+1}\tilde C_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-1}^{(\alpha)}(y)
-\tilde C_{n+1}\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n}^{(\alpha)}(y).\end{aligned}$$ Next note that $$y\widetilde{P}_{n-1}^{(\alpha)}(y)=\frac{1}{\tilde A_{n-1}}\widetilde{P}_n^{(\alpha)}(y)
+\frac{\tilde C_{n-1}}{\tilde A_{n-1}}\widetilde{P}_{n-2}^{(\alpha)}(y)$$ so that $$\begin{gathered}
\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n+2}^{(\alpha)}(y)
=y^2\tilde A_{n+1}\tilde A_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)
-\left(\tilde C_{n+1}+\frac{\tilde A_{n+1}\tilde C_n}{\tilde A_{n-1}}\right)\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\\
-\frac{\tilde A_{n+1}\tilde C_n\tilde C_{n-1}}{\tilde A_{n-1}}\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-2}^{(\alpha)}(y).\end{gathered}$$
Let us define $$\begin{aligned}
\kappa_n&=&\dst-\Big(\tilde C_{n+1}+\frac{\tilde A_{n+1}\tilde C_n}{\tilde A_{n-1}}\Big)=-1+\frac{1}{n}+O(n^{-2})\\
\tilde\kappa_n&=&\dst 1-\frac{\tilde A_{n+1}\tilde C_n\tilde C_{n-1}}{\tilde A_{n-1}}=\frac{1}{n}+O(n^{-2}).\end{aligned}$$ Then $$\begin{aligned}
\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n+2}^{(\alpha)}(y)+\widetilde{P}_n^{(\alpha)}(y)\widetilde{P}_{n+2}^{(\alpha)}(x)
&=&y^2\tilde A_{n+1}\tilde A_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)
+\kappa_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\\
&&+\tilde\kappa_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-2}^{(\alpha)}(y).\end{aligned}$$ Summing over $n$, we conclude that $$\begin{aligned}
\Phi_N(x,y)&=&\widetilde{P}_0^{(\alpha)}(x)\widetilde{P}_{2}^{(\alpha)}(y)
+\sum_{n=2}^N\bigl(\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n+2}^{(\alpha)}(y)+\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-2}^{(\alpha)}(y)\bigr)-\widetilde{P}_N^{(\alpha)}(x)\widetilde{P}_{N+2}^{(\alpha)}(y)\\
&=&\widetilde{P}_0^{(\alpha)}(x)\widetilde{P}_{2}^{(\alpha)}(y)-\widetilde{P}_N^{(\alpha)}(x)\widetilde{P}_{N+2}^{(\alpha)}(y)
+y^2\sum_{n=2}^N\tilde A_{n+1}\tilde A_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\\
&&+\sum_{n=2}^N\kappa_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)
+\sum_{n=2}^N\tilde\kappa_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-2}^{(\alpha)}(y).\end{aligned}$$ Further, exchanging the roles of $x$ and $y$ and summing, we obtain $2\Phi_N(x,y)=\Phi_N^1(x,y)+\cdots+\Phi_N^6(x,y)$ where $$\begin{aligned}
\Phi_N^1(x,y)&=&\widetilde{P}_0^{(\alpha)}(x)\widetilde{P}_{2}^{(\alpha)}(y)+\widetilde{P}_2^{(\alpha)}(x)\widetilde{P}_0^{(\alpha)}(y)\\
&&-(4x^2+4y^2-2)\bigl(\widetilde{P}_0^{(\alpha)}(x)\widetilde{P}_0^{(\alpha)}(y)+\widetilde{P}_1^{(\alpha)}(x)\widetilde{P}_1^{(\alpha)}(y)\bigr)\\
\Phi_N^2(x,y)&=&-\widetilde{P}_N^{(\alpha)}(x)\widetilde{P}_{N+2}^{(\alpha)}(y)-\widetilde{P}_{N+2}^{(\alpha)}(x)\widetilde{P}_N^{(\alpha)}(y)\\
\Phi_N^3(x,y)&=&(x^2+y^2)\sum_{n=2}^N\tilde A_{n+1}\tilde A_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\\
\Phi_N^4(x,y)&=&2\sum_{n=2}^N\kappa_n\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\\
\Phi_N^5(x,y)&=&\sum_{n=2}^N\tilde\kappa_n\bigl(\widetilde{P}_n^{(\alpha)}(x)\widetilde{P}_{n-2}^{(\alpha)}(y)
+\widetilde{P}_{n-2}^{(\alpha)}(x)\widetilde{P}_n^{(\alpha)}(y)\bigr).\end{aligned}$$
We also write $\Phi_N^j$ for the corresponding integral operators and will now estimate their norm as operators $L^p(I,\omega_\alpha)
\to L^p(I,\omega_\alpha)$.
Using the bound together with Lemma \[lem:triv\] we get $${{\left\|{\Phi_N^1}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim 1$$ and $${{\left\|{\Phi_N^2}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{\alpha_p}.$$
Using Abel summation, we can write $$\begin{aligned}
\Phi_N^3(x,y)&=&-\tilde A_3\tilde A_2 (x^2+y^2)C_1^{(\alpha)}(x,y)+
(x^2+y^2)\sum_{n=2}^N\tilde A_{n+1}(\tilde A_n-\tilde A_{n+2})C_n^{(\alpha)}(x,y)\\
&&+\tilde A_{N+1}\tilde A_N (x^2+y^2)C_N^{(\alpha)}(x,y)\\
&=&\Phi_N^{3,1}(x,y)+\Phi_N^{3,2}(x,y)+\Phi_N^{3,3}(x,y).\end{aligned}$$ Of course $${{\left\|{\Phi_N^{3,1}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim 1$$ while Lemma \[lem:projjacobi\] shows that $${{\left\|{\Phi_N^{3,2}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim
\sum_{n=2}^N \frac{1}{n^2}n^{\alpha_p}\lesssim N^{\alpha_p-1}$$ since $|\tilde A_{n+1}(\tilde A_n-\tilde A_{n+2})|\lesssim n^{-2}$ and $${{\left\|{\Phi_N^{3,3}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{\alpha_p}.$$
Using Abel summation again, we write $$\begin{aligned}
\Phi_N^4(x,y)&=&-2\kappa_2 C_1^{(\alpha)}(x,y)+2\sum_{n=2}^N(\kappa_n-\kappa_{n+1})C_n^{(\alpha)}(x,y)
+2\kappa_N y^2C_N^{(\alpha)}(x,y)\\
&=&\Phi_N^{4,1}(x,y)+\Phi_N^{4,2}(x,y)+\Phi_N^{4,3}(x,y).\end{aligned}$$ Again $${{\left\|{\Phi_N^{4,1}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim 1$$ while Lemma \[lem:projjacobi\] shows that $${{\left\|{\Phi_N^{4,2}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim
\sum_{n=2}^N \frac{1}{n^2}n^{\alpha_p}\lesssim N^{\alpha_p-1}$$ since $|\kappa_n-\kappa_{n+1}|\lesssim n^{-2}$ and $${{\left\|{\Phi_N^{4,3}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{\alpha_p}.$$
A last use of Abel summation leads to $$\begin{aligned}
\Phi_N^5(x,y)&=&-\tilde\kappa_2 \Phi_1^{(\alpha)}(x,y)+\sum_{n=2}^N(\tilde\kappa_n-\tilde\kappa_{n+1})\Phi_n^{(\alpha)}(x,y)
+\tilde\kappa_N \Phi_N^{(\alpha)}(x,y)\\
&=&\Phi_N^{5,1}(x,y)+\Phi_N^{5,2}(x,y)+\Phi_N^{5,3}(x,y).\end{aligned}$$ Of course $${{\left\|{\Phi_N^{5,1}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim 1.$$ For the two other terms, we will use the fact that ${{\left\|{\Phi_N}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{1+\alpha_p}$ and that $\tilde\kappa_n=n^{-1}+O(n^{-2})$, in particular $|\tilde\kappa_n-\tilde\kappa_{n+1}|\lesssim n^{-2}$. It follows that $${{\left\|{\Phi_N^{5,2}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim \sum_{n=2}^Nn^{-2}n^{1+\alpha_p}\lesssim N^{\alpha_p}$$ and $${{\left\|{\Phi_N^{5,3}}\right\|}}_{L^p(I, \omega_{\alpha})\to L^p(I, \omega_{\alpha})}\lesssim N^{-1}N^{1+\alpha_p}\lesssim N^{\alpha_p}.$$
Summing all terms, Condition $(B)$ of Theorem \[th:main\] is satisfied.
Conclusion
----------
It remains to conclude, all conditions of Theorem \[th:main\] are satisfied. Therefore, the Weighted prolate spheroidal series converges in $L^p(I,\omega_\alpha)$ if and only if the Jacobi series converge. The later ones converge in $L^p(I,\omega_\alpha)$ if and only if $p\in(p_0,p_0^\prime)$. We have thus proved the following:
Let $\alpha>-1/2$ and $c>0$, $N\geq 0$. Let $p_0=2-\frac{1}{\alpha+3/2}$ so that $p_0^{\prime}=\dst 2+\frac{1}{\alpha+1/2}$.
Let $(\psi_{n,c}^{(\alpha)})_{n\geq 0}$ be the family of weighted prolate spheroidal wave functions. For a smooth function $f$ on $I=(-1,1)$, define $$\Psi^{(\alpha)}_Nf=\sum_{n=0}^N{{\left\langle{f,\psi_{n,c}^{(\alpha)}}\right\rangle}}_{L^2(I,\omega_\alpha)}\psi_{n,c}^{(\alpha)}.$$ Then, for every $p\in(1,\infty)$, $\Psi^{(\alpha)}_N$ extends to a bounded operator $L^p(I,\omega_\alpha)\to L^p(I,\omega_\alpha)$. Further $$\Psi^{(\alpha)}_Nf\to f\qquad \mbox{in }L^p(I,\omega_\alpha)$$ for every $f\in L^p(I,\omega_\alpha)$ if and only if $p\in(p_0,p_0^\prime)$.
Application to circular prolate spheroidal wave functions
=========================================================
For two real numbers $c>0$ and $\alpha>-\frac{1}{2}$, the family of the circular prolate spheroidal wave functions (CPSWFs), introduced by D. Slepian [@Slepian3] and denoted by ${\psi^{(\alpha)}_{n,c}}$, are the eigenfunctions of the finite Hankel transform $\mathcal H_c^{\alpha}$, the operator on $L^2[0,1]$ with kernel given by $\mathcal H_c^{\alpha}(x,y)= \sqrt{cxy}J_{\alpha}(cxy)$. On other words $$\mathcal H_c^{\alpha}f(x)=\int_0^1
\sqrt{cxy}J_{\alpha}(cxy) f(y)\d y.$$ We denote by $\mu_{n,\alpha}(c)$ the family of the eigenvalues of the operator $\mathcal H_c^{\alpha}$, that is $\mathcal H_c^{\alpha}{\psi^{(\alpha)}_{n,c}}=
\mu_{n,\alpha}(c){\psi^{(\alpha)}_{n,c}}$. The functions ${\psi^{(\alpha)}_{n,c}}$ satisfy the following orthogonality relations: $$\int_{0}^{1}\psi_{n,c}^{\alpha}(x)\psi_{m,c}^{\alpha}(x)\d x
=\delta_{n,m}\quad\mbox{and}\quad
\int_{0}^{+\infty}\psi_{n,c}^{\alpha}(x)\psi_{m,c}^{\alpha}(x)\d x=
\frac{\delta_{n,m}}{c\mu_{n,\alpha}^2(c)}$$ and the ${\psi^{(\alpha)}_{n,c}}$’s constitute a complete orthonormal system in $L^2[0,1]$.
The ${\psi^{(\alpha)}_{n,c}}$’s are also related to the Hankel operator $\mathcal{H}^{\alpha}$, the integral operator on $L^2[0,+\infty[$ with kernel given by $\mathcal H^{\alpha}(x,y)=\sqrt{xy}J_{\alpha}(xy)$. More precisely, $$\mathcal H^{\alpha}(\psi_{n,c}^{\alpha})(x)=\frac{1}{c\mu_{n,\alpha}(c)}\psi_{n,c}^{\alpha}\left(\frac{x}{c}\right)\chi_{[0,c]}(x).$$ According to Plancherel’s theorem, the family $\psi_{n,c}^{(\alpha)}=\sqrt{c}|\mu_{n,\alpha}(c)|{\psi^{(\alpha)}_{n,c}}$ constitute a complete orthonormal system in $\mathcal B_c^{\alpha}$ defined by: $$\mathcal B_c^{\alpha}=\{ f\in L^2(0,\infty);\,\ \operatorname{supp}(\mathcal H^{\alpha}(f))\subset [0,c]\}.$$ Fore more details, see for example [@Slepian3; @Karoui-Boulsen]. Our first aim in this section is to prove that in the case of CPSWFs, we have mean convergence in the Hankel Paley-Wiener space $B_{c,p}^{\alpha}$ defined by $$B_{c,p}^{\alpha}=\left\{f\in L^p(0,\infty); \operatorname{supp}(\mathcal{H^{\alpha}}(f))\subseteq[0,c]\right\},$$ if and only if $ 4/3< p <4.$
Some facts about Spherical Bessel function
------------------------------------------
### Spherical Bessel function
The spherical Bessel function is defined as $$\label{eq:sphbess}
{j^{(\alpha)}_{n,c}}(x)=\sqrt{2(2n+\alpha+1)}\frac{J_{2n+\alpha+1}(cx)}{\sqrt{cx}}.$$ Here, $J_{\alpha}$ is the Bessel function of the first kind and order $\alpha.$ The spherical Bessel functions satisfy the orthogonality relation, $$\int_{0}^{+\infty}{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{m,c}}(x)\d x=\delta_{n,m}.$$ Their Hankel transforms are given by, see for example [@Slepian3] $$\label{eq:besseljacobi}
\mathcal H^{\alpha}({j^{(\alpha)}_{n,c}})(x)=\frac{\sqrt{2(2n+\alpha+1)}}{c}\left(\frac{x}{c}\right)^{\alpha+\frac{1}{2}}
P_n^{(\alpha,0)}\left(1-2\left(\frac{x}{c}\right)^2\right)
\chi_{[0,c]}(x).$$ where $P_n^{(\alpha,0)}$ is the Jacobi polynomials of degree $n$ and parameter $\alpha$, normalized so that $P_n^{(\alpha,0)}(1)=\dst\frac{\Gamma(n+\alpha+1)}{\Gamma(n+1)\Gamma(\alpha+1)}$. Introducing $$\label{eq:tk}
T_{n,\alpha}(x)=(-1)^n\sqrt{2(2n+\alpha+1)}x^{\alpha+\frac{1}{2}}P_n^{(\alpha,0)}(1-2x^2).$$ We thus get $\hh^{\alpha}({j^{(\alpha)}_{n,c}})(x)=c^{-1}\chi_{[0,1]}(x/c) T_{n,\alpha}(x/c)$. Note that the orthogonality relations of the ${j^{(\alpha)}_{n,c}}$’s and the unitary character of $\hh^{\alpha}$ imply that $(T_{n,\alpha})_{n\ge 0}$ is an orthonormal basis of $L^2[0,1]$ while the spherical Bessel functions ${j^{(\alpha)}_{n,c}}$ form a complete orthonormal system in $\mathcal B_c^{\alpha}$.
Further, using the induction property $\dst \frac{2\beta}{x}J_\beta(x)=J_{\beta-1}(x)+J_{\beta+1}(x)$, we get the following induction formula $${j^{(\alpha)}_{n+1,c}}=\frac{2\sqrt{(2n+\alpha+2)(2n+\alpha+3)}}{cx}{j^{(\alpha)}_{n+1/2,c}}-\frac{\sqrt{2n+\alpha+3}}{\sqrt{2n+\alpha+1}}{j^{(\alpha)}_{n,c}}.$$ Moreover, for $1<p<\infty$, we have $$\label{eq:lpnorm}
{{\left\|{{j^{(\alpha)}_{n,c}}}\right\|}}_{L^p(0,\infty)} \sim \begin{cases}
n^{-\frac{1}{2}+\frac{1}{p}}&\mbox{when }1<p<4\\
n^{-\frac{1}{4}}\log n&\mbox{when }p=4\\
n^{-\frac{1}{3}+\frac{1}{3p}}&\mbox{when }p>4
\end{cases}.$$ Note that, if $\frac{1}{p}+\frac{1}{q}=1$, then for $\ell\in\Z$, we have $$\label{eq:lpnormprod}
{{\left\|{{j^{(\alpha)}_{n+\ell,c}}}\right\|}}_{L^p(0,\infty)}{{\left\|{{j^{(\alpha)}_{n,c}}}\right\|}}_{L^q(0,\infty)} \sim
\begin{cases}
n^{\frac{2}{3p}-\frac{1}{2}}&\mbox{when }1<p<\frac{4}{3}\\
\log n&\mbox{when }p=\frac{4}{3}\mbox{ or }p=4\\
1&\mbox{when }\frac{4}{3}<p<4\\
n^{\frac{1}{6}-\frac{2}{3p}}&\mbox{when }p>4
\end{cases}=o(n^{\frac{1}{6}})$$ where the constants depend on $\ell$, for more details see [@BC]. We can now see that condition (L) is satisfied.
The expansion of $\psi_{n,c}^{(\alpha)}$ in the basis of the spherical Bessel functions is done as follows. First, by using , we calculate the scalar product $$\begin{aligned}
{{\left\langle{\psi_{n,c}^{(\alpha)},{j^{(\alpha)}_{n,c}}}\right\rangle}}_{L^2[0,+\infty[}
&=&\int_{0}^{+\infty}\sqrt{c}|\mu_{n,\alpha}(c)|\psi_{n,c}^{\alpha}(x){j^{(\alpha)}_{n,c}}(x)\d x\\
&=&\sqrt{c}|\mu_{n,\alpha}(c)|\int_{0}^{+\infty}\psi_{n,c}^{\alpha}(x)\sqrt{2(2n+\alpha+1)}\frac{J_{2n+\alpha+1}(cx)}{\sqrt{cx}}\d x.\end{aligned}$$ Writing $\dst \nu=\sqrt{c}\frac{|\mu_{n,\alpha}(c)|}{\mu_{n,\alpha}(c)}\sqrt{2(2n+\alpha+1)}$ and since $$\dst\psi_{n,c}^{\alpha}(x)=\frac{1}{\mu_{n,\alpha}(c)}\hh_c^\alpha\psi_{n,c}^{\alpha}(x)
=\frac{1}{\mu_{n,\alpha}(c)}\int_0^1
\sqrt{cxy}J_{\alpha}(cxy) \psi_{n,c}^{\alpha}(y)\d y,$$ then Fubini’s theorem together with , we get $$\begin{aligned}
{{\left\langle{\psi_{n,c}^{(\alpha)},{j^{(\alpha)}_{n,c}}}\right\rangle}}_{L^2[0,+\infty[}
&=&\nu\int_{0}^{1}\sqrt{y}\psi_{n,c}^{\alpha}(y)
\int_{0}^{+\infty}J_{2n+\alpha+1}(cx)J_{\alpha}(cxy)\d x\d y\\
&=&\frac{\nu}{c}\int_{0}^{1}y^{\alpha+\frac{1}{2}}P_n^{(\alpha,0)}(1-2y^2)\psi_{n,c}^{\alpha}(y)\d y.\end{aligned}$$ We thus have $${{\left\langle{\psi_{n,c}^{(\alpha)},{j^{(\alpha)}_{n,c}}}\right\rangle}}_{L^2[0,+\infty[}=
(-1)^k\frac{|\mu_{n,\alpha}(c)|}{\sqrt{c}\mu_{n,\alpha}(c)}{{\left\langle{{\psi^{(\alpha)}_{n,c}},T_{n,\alpha}}\right\rangle}}_{L^2[0,1]}$$ where $T_{n,\alpha}$ has been defined in . Writing $d_k^n={{\left\langle{{\psi^{(\alpha)}_{n,c}},T_{k,\alpha}}\right\rangle}}_{L^2[0,1]}$, we thus get the following expansion on $[0,+\infty)$: $$\psi_{n,c}^{(\alpha)}(x)=\frac{|\mu_{n,\alpha}(c)|}{\sqrt{c}\mu_{n,\alpha}(c)}\sum_{k\ge 0}{(-1)^kd_k^n{j^{(\alpha)}_{k,c}}(x)}.$$ Consider the differential operator given by $$\mathcal D_c^{\alpha}(\phi)(x)=-\frac{\d}{\d x}{{\left[{(1-x^2)\frac{\d}{\d x}}\right]}}\phi(x)-\left(\frac{\frac{1}{4}-\alpha^2}{x^2}-c^2x^2\right)\phi(x).$$ We know from [@Slepian3] that the operators $\mathcal D_c^{\alpha}$ and $\mathcal H_c^{\alpha}$ commute so that ${\psi^{(\alpha)}_{n,c}}$ are eigenvectors of both operators and we denote by $\chi_{n,\alpha}(c)$ the corresponding eigenvalue of $\dd_c^\alpha$, that is $$\label{eq:eigcircprol}
\dd_c^\alpha({\psi^{(\alpha)}_{n,c}})=\chi_{n,\alpha}(c){\psi^{(\alpha)}_{n,c}}.$$ Further more, we have the following inequality see [@Slepian3]: $$\label{eq:slepeigest}
\left(\alpha+2n+\frac{1}{2}\right)\left(\alpha+2n+\frac{3}{2}\right)\le\chi_{n,\alpha}(c)
\le \left(\alpha+2n+\frac{1}{2}\right)\left(\alpha+2n+\frac{3}{2}\right)+c^2.$$ According to [@Watson], the Spherical Bessel functions are the eigenfunctions of the differential operator given by $$\mathcal{D}_c(\omega)=\left(\frac{x}{c}\right)^2\frac{d^2}{d^2x}(\omega)+\frac{(1+c)}{\sqrt{c}}\left(\frac{x}{c}\right)
\frac{d}{dx}(\omega)+c^2x^2\omega$$ and the corresponding eigenvalues are given by $\Big((2n+\alpha+1)^2-\frac{c^2-2c}{2c^3}\Big)\sqrt{2(2n+\alpha+1)}$. It follows from Lemma \[lem:condD\] that Condition $(D)$ of Theorem \[th:main\] is also satisfied.
If we substitute the expression of ${\psi^{(\alpha)}_{n,c}}$ as a series of Jacobi polynomials into , we obtain the relations satisfied by the coefficients $d_k^n$. More precisely, from [@Slepian3], we obtain the three term recurrence relation $$f(k,n,c,\alpha)d_k^n=a_{k,\alpha}d_{k-1}^n+a_{k+1,\alpha}d_{k+1}^n,~ \forall k\ge 0$$ where $d_{-1}^{n}=0$ and $$\begin{aligned}
\label{coeff-CPSWFs}
f(k,n,c,\alpha)&=&\displaystyle\frac{\chi_{n,\alpha}(c)-(\alpha+2k+\frac{1}{2})(\alpha+2k+\frac{3}{2})-c^2b_{k,\alpha}}{c^2} \\
a_{k,\alpha}&=&\frac{k(k+\alpha)}{(\alpha+2k)\sqrt{\alpha+2k+1}\sqrt{\alpha+2k-1}} \nonumber \\
b_{k,\alpha}&=&\frac{1}{2}{{\left[{\frac{\alpha^2}{(\alpha+2k+1)(\alpha+2k)}+1}\right]}} \nonumber.\end{aligned}$$
### The projection on the span of spherical Bessel functions
Let $1<p<\infty$ and $\alpha\ge -\frac{1}{2}$. For $n\geq 0$, let $$P_n^{(\alpha)}(x,y):=\sum_{k=0}^n{j^{(\alpha)}_{k,c}}(x){j^{(\alpha)}_{k,c}}(y)$$ and $\pp_n^{(\alpha)}$ be the operator with kernel $P_n^{(\alpha)}(x,y)$. That is, $\pp_n^{(\alpha)}$ is the projection on the span of $\{{j^{(\alpha)}_{0,c}},\ldots,{j^{(\alpha)}_{n,c}}\}$.
\[prop:projbessel\] Let $1<p<\infty$, $\alpha>-1/2$. Then the following estimate holds for every $n$ and every $f\in L^p(0,\infty)$ $${{\left\|{\pp_n^{(\alpha)}(f)}\right\|}}_{L^p(0,\infty)}\lesssim \begin{cases}{{\left\|{f}\right\|}}_{L^p(0,\infty)}&\mbox{if }\dst\frac{4}{3}<p<4\\
n^{3/4}{{\left\|{f}\right\|}}_{L^p(0,\infty)}&\mbox{otherwise}
\end{cases}$$ with the implied constant independent of $f$ and $n$.
The projection on the span of spherical Bessel functions has been studied by Varona [@J.L.VARONA] with a different normalization. He considered $$j_n^{\alpha}(x)=\sqrt{2n+\alpha+1}J_{2n+\alpha+1}(\sqrt{x})x^{-\alpha/2-1/2}$$ so that $${j^{(\alpha)}_{n,c}}=\sqrt{2}j_n^{\alpha}(c^2x^2)(cx)^{\alpha+1/2}.$$
Next, if we define $$K_n(x,y)=\sum_{k=0}^nj_n^{\alpha}(x)j_n^{\alpha}(y)$$ (with Varona’s notation) then $P_n^{(\alpha)}(x,y)
=2c^{2\alpha+1}(xy)^{\alpha+1/2}K_n(c^2x^2,c^2y^2)$.
Using Varona’s computation [@J.L.VARONA page 69] we get
$$\begin{aligned}
P_n^{(\alpha)}(x,y)&=&\frac{(xy)^{1/2}}{x^2-y^2}\{xJ_{\alpha+1}(cx)J_{\alpha}(cy)-yJ_{\alpha+1}(cy)J_{\alpha}(cx)\}\nonumber\\
&+&\frac{(xy)^{1/2}}{x^2-y^2}\{xJ^\prime_{\alpha+2n+2}(cx)J_{\alpha+2n+2}(cy)-yJ_{\alpha+2n+2}(cx)J^\prime_{\alpha+2n+2}(cy)\}.
\label{eq:decomppn}\end{aligned}$$
Now, recalling that $\hh$ denotes the Hilbert transform, it follows from that $$\mathcal{P}_n^{(\alpha)}(f)(x)=\Omega_1(f)(x)-\Omega_2(f)(x)+\Omega_{3}(f)(x)-\Omega_{4}(f)(x)$$ where $$\begin{aligned}
\Omega_1(f)(x)&=&\int_0^{\infty}\frac{(xy)^{1/2}}{x^2-y^2}xJ_{\alpha+1}(cx)J_{\alpha}(cy)f(y)\d y\\
&=&\frac{x^{\frac{3}{2}}}{2}J_{\alpha+1}(cx)\mathcal{H}\big[y^{-1/4}J_{\alpha}(cy^{1/2})f(y^{1/2})\big](x^2).\\
\Omega_2(f)(x)&=&\int_0^{\infty}\frac{(xy)^{1/2}}{x^2-y^2}yJ_{\alpha}(cx)J_{\alpha+1}(cy)f(y)\d y\\
&=&\frac{x^{\frac{1}{2}}}{2}J_{\alpha}(cx)\mathcal{H}\big[y^{1/4}J_{\alpha+1}(cy^{1/2})f(y^{1/2})\big](x^2).\\
\Omega_{3}(f)(x)&=&\int_0^{\infty}\frac{(xy)^{1/2}}{x^2-y^2}xJ'_{\alpha+2n+2}(cx)J_{\alpha+2n+2}(cy)f(y)\d y\\
&=&\frac{x^{\frac{3}{2}}}{2}J'_{\alpha+2n+2}(cx)\mathcal{H}\big[y^{-1/4}J_{\alpha+2n+2}(cy^{1/2})f(y^{1/2})\big](x^2).\\
\Omega_{4}(f)(x)&=&\int_0^{\infty}\frac{(xy)^{1/2}}{x^2-y^2}yJ_{\alpha+2n+2}(cx)J'_{\alpha+2n+2}(cy)f(y)\d y\\
&=&\frac{x^{\frac{1}{2}}}{2}J_{\alpha+2n+2}(cx)\mathcal{H}\big[y^{1/4}J'_{\alpha+2n+2}(cy^{1/2})f(y^{1/2})\big](x^2).\end{aligned}$$ Note that each of these operators is of the form $$\Omega_j(f)(x)=G_j(x)\hh\big[\ffi_j\big](x^2)$$ so that $$\begin{aligned}
{{\left\|{\Omega_j(f)}\right\|}}_{L_p(0,\infty)}^p&=&\int_0^\infty |G_j(x)|^p|\hh\big[\ffi_j\big](x^2)|^p\d x\\
&=&\int_0^\infty \frac{|G_j(\sqrt{x})|^p}{2\sqrt{x}}|\hh\big[\ffi_j\big](x)|^p\d x.\end{aligned}$$ But then, if we are able to find an upper bound $\omega_j\in A^p$ (see [@J.L.VARONA]) of $\dst \frac{|G_j(\sqrt{x})|^p}{2\sqrt{x}}\lesssim \omega_j(x)$, we obtain $${{\left\|{\Omega_j(f)}\right\|}}_{L_p(0,\infty)}^p\lesssim {{\left\|{\hh\big[\ffi_j\big]}\right\|}}_{L_p\bigl((0,\infty),\omega_j(x)\d x\bigr)}^p
\lesssim {{\left[{\omega_j}\right]}}_{A^p}^{\max(p,p')} {{\left\|{\ffi_j}\right\|}}_{L_p\bigl((0,\infty),\omega_j(x)\d x\bigr)}^p.$$ It then remains to prove that ${{\left\|{\ffi_j}\right\|}}_{L_p\bigl((0,\infty),w_j(x)\d x\bigr)}^p\lesssim{{\left\|{f}\right\|}}_{L_p(0,\infty)}^p$.
For $\Omega_1$, $\ffi_1(y)=y^{-1/4}J_{\alpha}(cy^{1/2})f(x^{1/2})$. Further, we use the bound $|J_{\alpha+1}(t)|\leq C_\alpha t^{-1/2}$ which allows us to chose $\omega_1(y)=y^{\frac{p-1}{2}}\in A^p$ since $-1<\dst\frac{p-1}{2}<p-1$. Further $$\begin{aligned}
{{\left\|{\ffi_1}\right\|}}_{L^p(\R_+,\omega_1(y)\d y)}^p&=&
{{\left\|{y^{-1/4}J_{\alpha}(cy^{1/2})f(y^{1/2})}\right\|}}_{L^p(\R_+,\omega_1(y)\d y)}^p
\lesssim \int_0^\infty {{\left|{\frac{f(y^{1/2})}{(cy)^{1/2}}}\right|}}^px^{\frac{p-1}{2}}\d y\\
&\lesssim&{{\left\|{f}\right\|}}_{L^p(0,+\infty)}^p.\end{aligned}$$
We will now take care of $\Omega_2$. In this case with $\ffi_2(y)=y^{1/4}J_{\alpha+1}(cy^{1/2})f(y^{1/2})$ and the same bound on the Bessel function shows that we can chose $\omega_2(x)=x^{-\frac{1}{2}}\in A^p$. $$\begin{aligned}
{{\left\|{\ffi_2}\right\|}}_{L^p(\R_+,\omega_2(x)\d x)}^p&=&
{{\left\|{x^{1/4}J_{\alpha+1}(cx^{1/2})f(x^{1/2})}\right\|}}_{L^p(\R_+,\omega_2(x)\d x)}^p
\lesssim \int_0^\infty {{\left|{f(x^{1/2})}\right|}}^px^{-\frac{1}{2}}\d x\\
&\lesssim&{{\left\|{f}\right\|}}_{L^p(0,+\infty)}^p.\end{aligned}$$
The same reasoning would apply to $\Omega_{3},\Omega_{4}$ but with a bound that depends on $n$. We thus need a more refined estimate which follows from [@BC]: $$\begin{aligned}
|J_{\mu}(x)|&\lesssim& x^{-\frac{1}{4}}\big(|x-\mu|+\mu^{\frac{1}{3}}\big)^{-\frac{1}{4}}\\|J'_{\mu}(x)|&\lesssim& x^{-\frac{3}{4}}\big(|x-\mu|+\mu^{\frac{1}{3}}\big)^{\frac{1}{4}}.$$ Set $\mu=\alpha+2n+2$. We may then take $$\omega_3(x)=x^{\frac{3p}{8}-\frac{1}{2}}\big(|c\sqrt{x}-\mu|+\mu^{\frac{1}{3}}\big)^{\frac{p}{4}}\quad\mbox{and}\quad
\omega_4(x)=x^{\frac{p}{8}-\frac{1}{2}}\big(|c\sqrt{x}-\mu|+\mu^{\frac{1}{3}}\big)^{-\frac{p}{4}}$$ By the Lemma , $\omega_3$ and $\omega_4$ $\in A_p$ with ${{\left[{\omega_j}\right]}}_{A^p}\lesssim 1$ if $\dst\frac{4}{3}<p<4$ and ${{\left[{\omega_j}\right]}}_{A^p}\lesssim\mu^{3/4}$ otherwise.
Finally $$\ffi_3(x)=x^{-1/4}J_{\alpha+2n+2}(cx^{1/2})f(x^{1/2})
\quad\mbox{and}\quad
\ffi_4(x)=x^{1/4}J'_{\alpha+2n+2}(cx^{1/2})f(x^{1/2}).$$ Note that $$|\ffi_3(x)|\lesssim x^{-3/8}\big(|c\sqrt{x}-\mu|+\mu^{\frac{1}{3}}\big)^{-\frac{1}{4}}|f(x^{1/2})|
\quad\mbox{and}\quad
|\ffi_4(x)|\lesssim x^{-1/8}\big(|c\sqrt{x}-\mu|+\mu^{\frac{1}{3}}\big)^{\frac{1}{4}}|f(x^{1/2})|.$$ so that $${{\left\|{\ffi_j}\right\|}}_{L^p(\R_+,\omega_j(x)\d x)}^p
\lesssim \int_0^\infty x^{-\frac{1}{2}}|f(x^{1/2})|\d x\lesssim {{\left\|{f}\right\|}}_{L^p(0,+\infty)}^p.$$ It follows that ${{\left\|{\Omega_j(f)}\right\|}}_{L^p(0,\infty)}\lesssim 1$ if $\dst\frac{4}{3}<p<4$ and ${{\left\|{\Omega_j(f)}\right\|}}_{L^p(0,\infty)}\lesssim n^{3/4}{{\left\|{f}\right\|}}_{L^p(0,\infty)}$ for $1<p\leq \dst\frac{4}{3}$. Grouping all estimates, the same holds for $\pp_n^{(\alpha)}$. Finally, as $\pp_n^{(\alpha)}$ is self-adjoint, we also get the estimate ${{\left\|{\pp_n^{(\alpha)}(f)}\right\|}}_{L^p(0,\infty)}\lesssim n^{3/4}{{\left\|{f}\right\|}}_{L^p(0,\infty)}$ for $p\geq 4$.
Condition $(R)$
---------------
We will now show that conditions $(R)$ of Theorem \[re:thmain\] are satisfied, this is done in three lemmas.
For every $k\geq 1$ and $\alpha>\dst-\frac{1}{2}$, $0\leq a_k^{(\alpha)}\leq \dst\frac{1}{2}$.
For $k=1$, $a_{1,\alpha}=\frac{\sqrt{1+\alpha}}{(2+\alpha)\sqrt{\alpha+3}}$ which is clearly $\leq 1/2$ when $\alpha\geq0$. It is easy to see that $a_{1,\alpha}$ is increasing with $\alpha\in(-1/2,\alpha_0)$ and decreasing with $\alpha\in(\alpha_0,0)$ where $\alpha_0=\frac{-3+\sqrt{5}}{2}$. Finally, $a_{1,\alpha_0}\sim0.3$ so that $a_{1,\alpha}\leq 1/2$ for every $\alpha$. Write $$|a_{k,\alpha}|
=\frac{1}{4}\frac{1+\frac{2\alpha}{2k}}{\left(1+\frac{\alpha}{2k}\right)\sqrt{1+\frac{\alpha+1}{2k}}\sqrt{1+\frac{\alpha-1}{2k}}}
=\frac{1}{4}\psi(1/2k),$$ where $\dst\psi(x)=\frac{1+2\alpha x}{(1+\alpha x)\big(1+(\alpha-1)x\big)^{1/2}\big(1+(\alpha+1)x\big)^{1/2}}$. It is thus enough to show that ${{\left|{\psi(x)}\right|}}\leq 2$ for $x\in[0,1/4]$. Note that $\psi$ is non-negative for $\alpha>-1/2$ and $x\leq 1$. When $-1/2<\alpha\leq 0$, as $1+2\alpha x\leq 1+\alpha x$ and $1+(\alpha+1)x\geq 1$, $$\psi(x)\leq\frac{1}{\sqrt{1+(\alpha-1)x}}
\leq \frac{1}{\sqrt{1-3x/2}}\leq\frac{1}{\sqrt{5/8}}<2$$ when $x\leq1/4$. When $\alpha> 0$, we first bound $$\psi(x)\leq\frac{1+2\alpha x}{(1+\alpha x)\big(1+(\alpha-1)x\big)}$$ and it is enough to prove that, for $x>0$, we have $$1+2\alpha x\leq 2(1+\alpha x)\big(1+(\alpha-1)x\big)
=2+(4\alpha-2)x+2\alpha(\alpha-1)x^2.$$ or, equivalently, $1+2(\alpha-1)x+2\alpha(\alpha-1)x^2\geq 0$. When $\alpha\geq1$ this is obvious, while for $0<\alpha<1$, the roots of this equation are $$-\frac{1-\alpha+\sqrt{1-\alpha^2}}{2\alpha(1-\alpha)}<0<\frac{\sqrt{1-\alpha^2}-(1-\alpha)}{2\alpha(1-\alpha)}
=\frac{\sqrt{1+\alpha}-\sqrt{1-\alpha}}{2\alpha\sqrt{1-\alpha}}.$$ The inequality is therefore satisfied as soon as $x\leq 1/2k$ with $$k\geq k_\alpha
:=\frac{\alpha\sqrt{1-\alpha}}{\sqrt{1+\alpha}-\sqrt{1-\alpha}}=\frac{\alpha(1-\alpha+\sqrt{1-\alpha^2})}{2}.$$ As $0<\alpha<1$, it is easy to see that $k_\alpha\leq 1$.
Let us now estimate the $b_{k,\alpha}$’s:
\[lem:2\] For every $k$ and every $\alpha>-1/2$, $b_{k,\alpha}=\frac{1}{2}+\tilde\eta_{k,\alpha}$ with $|\tilde\eta_{k,\alpha}|\leq \frac{1}{2}$.
From the definition of $b_{k,\alpha}$, $\tilde\eta_{k,\alpha}=\frac{\alpha^2}{2(\alpha+2k+1)(\alpha+2k)}$. When $\alpha= 0$, $\eta=0$ and when $\alpha>0$ we directly get $\tilde\eta_{k,\alpha}\leq \frac{1}{2}$. When $-1/2<\alpha<0$, $\alpha+j>1/2>|\alpha|$ for every $j\geq 1$ thus $|\tilde\eta_{0,\alpha}|=\frac{|\alpha|}{2(\alpha+1)}\leq\frac{1}{2}$ while for $k\geq 1$ we directly get $0\leq\tilde\eta_{k,\alpha}=\frac{1}{2}\frac{|\alpha|}{\alpha+2k}\frac{|\alpha|}{\alpha+2k+1}\leq\frac{1}{2}$.
The last step consists in establishing the bounds for ${{\left|{f(k,n,c,\alpha)}\right|}}$. But from and , it is straightforward to see that $f(k,n,c,\alpha)$ satisfies the conditions of Lemma \[lem:fkn\]. In summary
\[lem:3\] For every $\alpha>-1/2$, every $c>0$,
— for fixed $n$, $f(k,n,c,\alpha)\gtrsim k^2$, when $k$ is large enough;
— for every $n\geq c^2/2$, $k\geq0$, $k\not=n$, we have $${{\left|{f(k,n,c,\alpha)}\right|}}\geq 4\frac{|k-n|k+c^2}{c^2};$$
— for every $n\geq c^2/2$, $${{\left|{\frac{a_{n+1}^{(\alpha)}}{f(n+1,n,c,\alpha)}-\frac{a_{n+2}^{(\alpha)}}{f(n+2,n+1,c,\alpha)}}\right|}}\lesssim n^{-2}.$$
Condition $(B')$
----------------
It remains to check condition $(B')$ that is, to estimate the $L^p$ norm of the operator with kernel $$Q_N^{(\alpha)}(x,y)=\sum_{n=n_0}^N\big({j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n+1,c}}(y)+{j^{(\alpha)}_{n+1,c}}(x){j^{(\alpha)}_{n,c}}(y)\big).$$
\[lem:BC\] Let $1<p<\infty$ then, for every $f\in L^p(0,\infty)$ $$\left(\int_0^{\infty}{{\left|{\int_0^{\infty}Q_N^{(\alpha)}(x,y)f(y)\d y}\right|}}^p\d x\right)^{1/p}\lesssim N^{2/3}{{\left\|{f}\right\|}}_{L^p(0,\infty)}.$$ and the implied constant is independent of $N$ and $f$.
First, using the identity (see [@Watson]) $$\label{eq:watson}
\frac{2\nu}{x}J_{\nu}(x)=J_{\nu+1}(x)+J_{\nu-1}(x)$$ twice, one gets $$\begin{aligned}
J_{2n+\alpha+3}(x)&=&\frac{4(2n+\alpha+1)(2n+\alpha+2)}{x^2}J_{2n+\alpha+1}(x)\\
&&-\frac{2(2n+\alpha+2)}{x}J_{2n+\alpha}(x)-J_{2n+\alpha+1}(x).\end{aligned}$$ so that $$\begin{aligned}
\lefteqn{J_{2n+\alpha+1}(x)J_{2n+\alpha+3}(y)+J_{2n+\alpha+1}(y)J_{2n+\alpha+3}(x)} \\
&&\qquad\qquad=4(2n+\alpha+1)(2n+\alpha+2)\left(\frac{1}{x^2}+\frac{1}{y^2}\right)J_{2n+\alpha+1}(x)J_{2n+\alpha+1}(y) \\
&&\qquad\qquad\qquad
-2(2n+\alpha+2)\left(\frac{1}{y}J_{2n+\alpha}(y)J_{2n+\alpha+1}(x)+\frac{1}{x}J_{2n+\alpha}(x)J_{2n+\alpha+1}(y)\right) \\
&&\qquad\qquad\qquad -2J_{2n+\alpha+1}(x)J_{2n+\alpha+1}(y).\end{aligned}$$ Using again for the middle term, we get $$\begin{aligned}
\lefteqn{J_{2n+\alpha+1}(x)J_{2n+\alpha+3}(y)+J_{2n+\alpha+1}(y)J_{2n+\alpha+3}(x)} \\
&&\qquad\qquad =4(2n+\alpha+1)(2n+\alpha+2)\left(\frac{1}{x^2}+\frac{1}{y^2}\right)J_{2n+\alpha+1}(x)J_{2n+\alpha+1}(y) \\
&&\qquad\qquad\ -2\frac{(2n+\alpha+2)}{2n+\alpha}\Bigl(\bigl(J_{2k+\alpha+1}(y)+J_{2k+\alpha-1}(y)\bigr)J_{2n+\alpha+1}(x)\\
&&\qquad\qquad\ +\bigl(J_{2n+\alpha+1}(x)+J_{2n+\alpha-1}(x)\bigr)J_{2n+\alpha+1}(y)\Bigr) \\
&&\qquad\qquad\ -2J_{2n+\alpha+1}(x)J_{2n+\alpha+1}(y).\end{aligned}$$
Next, since ${j^{(\alpha)}_{n,c}}(x)=\sqrt{2(2n+\alpha+1)}\frac{J_{2n+\alpha+1}(cx)}{\sqrt{cx}}$, one obtains $$\begin{aligned}
\lefteqn{{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n+1,c}}(y)+{j^{(\alpha)}_{n,c}}(y){j^{(\alpha)}_{n+1,c}}(x)}\nonumber \\
&&\qquad\qquad= \frac{4}{c^2}\sqrt{(2n+\alpha+1)(2n+\alpha+3)}(2n+\alpha+2)\left(\frac{1}{x^2}+\frac{1}{y^2}\right)
{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n,c}}(y) \nonumber\\
&&\qquad\qquad\ -2\frac{(2n+\alpha+2)}{2n+\alpha}\sqrt{\frac{2n+\alpha+3}{2n+\alpha-1}}\Bigl({j^{(\alpha)}_{n-1,c}}(y){j^{(\alpha)}_{n,c}}(x)+{j^{(\alpha)}_{n-1,c}}(x){j^{(\alpha)}_{n,c}}(y)\Bigr) \nonumber\\
&&\qquad\qquad\ -2\sqrt{\frac{2n+\alpha+3}{2n+\alpha+1}}\left(\frac{2(2n+\alpha+2)}{2n+\alpha}+1\right){j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n,c}}(y).
\label{eq:jcnjcn+1}\end{aligned}$$ Now we write $$\gamma_n=\frac{4}{c^2}\sqrt{(2n+\alpha+1)(2n+\alpha+3)}(2n+\alpha+2),$$ so that $|\gamma_n|\lesssim n^2$, $$2\frac{(2n+\alpha+2)}{2n+\alpha}\sqrt{\frac{2n+\alpha+3}{2n+\alpha-1}}=2+\kappa_n,\qquad 0\leq\kappa_n\lesssim n^{-1}$$ and $$2\sqrt{\frac{2n+\alpha+3}{2n+\alpha+1}}\left(\frac{2(2n+\alpha+2)}{2n+\alpha}+1\right)
=6+\tilde\kappa_n,\qquad 0\leq\tilde\kappa_n\lesssim n^{-1}.$$ Then, summing from $n=n_0$ to $n=N$ gives $$\begin{aligned}
Q_N^{(\alpha)}(x,y)&=&\left(\frac{1}{x^2}+\frac{1}{y^2}\right)\sum_{n=n_0}^N \gamma_n{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n,c}}(y)\\
&&-(2+\kappa_{n_0})\Bigl({j^{(\alpha)}_{n_0-1,c}}(y){j^{(\alpha)}_{n_0,c}}(x)+{j^{(\alpha)}_{n_0-1,c}}(x){j^{(\alpha)}_{n_0,c}}(y)\Bigr)\\
&&-2Q_N^{(\alpha)}(x,y)-\sum_{n=n_0}^N\kappa_n\Bigl({j^{(\alpha)}_{n,c}}(y){j^{(\alpha)}_{n+1,c}}(x)+{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n+1,c}}(y)\Bigr)\\
&&+(2+\kappa_{N})\Bigl({j^{(\alpha)}_{N,c}}(y){j^{(\alpha)}_{N+1,c}}(x)+{j^{(\alpha)}_{N,c}}(x){j^{(\alpha)}_{N+1,c}}(y)\Bigr)\\
&&-6P_N^{(\alpha)}(x,y)+6P_{n_0-1}(x,y)-\sum_{n=n_0}^N\tilde\kappa_n{j^{(\alpha)}_{n,c}}(x){j^{(\alpha)}_{n,c}}(y).\end{aligned}$$ It follows that $$Q_N^{(\alpha)}(x,y)=Q_{N,1}(x,y)+\cdots+Q_{N,7}(x,y).$$ Using and Lemma \[lem:triv\] we get that $${{\left\|{Q_{N,2}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)},{{\left\|{Q_{N,6}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim 1$$ and $${{\left\|{Q_{N,4}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim N^{1/6}$$ while $${{\left\|{Q_{N,3}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim \sum_{n=n_0}^N\frac{1}{n}n^{1/6}\lesssim N^{1/6}$$ and $${{\left\|{Q_{N,7}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim \sum_{n=n_0}^N\frac{1}{n}n^{1/6}\lesssim N^{1/6}.$$ We have seen in Proposition \[prop:projbessel\] that ${{\left\|{Q_{N,5}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim N^{3/4}$.
Concerning $Q_{N,1}$, we will use the following equality, see [@BC], $${{\left\|{x^{-2}j_n^{\alpha}}\right\|}}_{L^p(0,\infty)}{{\left\|{j_n^{\alpha}}\right\|}}_{L^q(0,\infty)}+{{\left\|{j_n^{\alpha}}\right\|}}_{L^p(0,\infty)}
{{\left\|{x^{-2}j_n^{\alpha}}\right\|}}_{L^q(0,\infty)}=O(n^{-\frac{7}{3}})$$ from which we deduce that $${{\left\|{Q_{N,1}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim \sum_{n=n_0}^Nn^2n^{-7/3}\lesssim N^{2/3}.$$ By grouping all estimates, we obtain ${{\left\|{Q_{N}^{(\alpha)}}\right\|}}_{L^p(0,\infty)\to L^p(0,\infty)}\lesssim N^{2/3}$ as claimed.
Conclusion
----------
It remains to conclude. ll conditions of Theorem \[th:main\] are satisfied. Therefore, the Hankel prolate spheroidal series converges in $L^p(0,\infty)$ if and only if the Bessel series converge. The later ones converge in $L^p(0,\infty)$ if and only if $p\in(4/3,4)$. We have thus proved the following:
Let $\alpha>-1/2$ and $c>0$, $N\geq 0$.
Let $(\psi_{n,c}^{(\alpha)})_{n\geq 0}$ be the family of circular prolate spheroidal wave functions. For a smooth function $f$ on $I=(0,\infty)$, define $$\Psi^{(\alpha)}_Nf=\sum_{n=0}^N{{\left\langle{f,\psi_{n,c}^{(\alpha)}}\right\rangle}}_{L^2(0,\infty)}\psi_{n,c}^{(\alpha)}.$$ Then, for every $p\in(1,\infty)$, $\Psi^{(\alpha)}_N$ extends to a bounded operator $L^p(0,\infty)\to L^p(0,\infty)$. Further $$\Psi^{(\alpha)}_Nf\to f\qquad \mbox{in }L^p(0,\infty)$$ for every $f\in B_{c,p}^{\alpha}$ if and only if $p\in(4/3,4)$.
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[^1]: We only use a particular subfamily of Jacobi polynomials and may as well call them ultra-spherical or Gegenbauer polynomials.
|
---
abstract: 'Narasimhan and Ramadas showed in [@NR] that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3\times SU(2)\to S^3$. Instead of a base manifold $S^3$, we consider here a base manifold of dimension $n\ge 2$ with a boundary and use *Dirichlet boundary conditions* on connections as defined by Marini in [@Ma]. A key step in the method of Narasimhan and Ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy Lie algebra with respect to an appropriate Sobolev norm. Our objective is to explore the effect of the presence of a boundary on this construction of the holonomy Lie algebra. Fixing appropriate Sobolev norms, it will be shown that the space spanned, linearly, by the curvature form at any one connection is never dense in the holonomy Lie algebra. In contrast, the linear space spanned by the curvature form and its first commutators at the flat connection is dense and, in the $C^\infty$ category, is in fact the entire holonomy Lie algebra. The former, negative, theorem is proven for a general principle bundle over $M$, while the latter, positive, theorem is proven only for a product bundle over the closure of a bounded open subset of $\mathbb{R}^n$. Our technique for proving absence of density consists in showing that the linear space spanned by the curvature form at one point is contained in the kernel of a linear map consisting of a third order differential operator, followed by a restriction operation at the boundary; this mapping is determined by the mean curvature of the boundary.'
author:
- 'William E. Gryc[^1]'
bibliography:
- 'Will-ref.bib'
title: On the Holonomy of the Coulomb Connection over Manifolds with Boundary
---
Introduction {#intro}
============
In this paper we study the space of connections $\mathcal{A}$ of a certain type of principal bundle and the set of gauge transformations $\mathcal{G}$ that act on these connections. In particular, we are interested in the quotient $\mathcal{A}\to\mathcal{A}/\mathcal{G}$. This quotient is important in classical Yang-Mills theory; the equivalence classes of $\mathcal{A}/\mathcal{G}$ are physically observable, while individual members of $\mathcal{A}$ are not. This distinction has led to complications in Yang-Mills theory, such as the Gribov ambiguity (see [@Gribov], [@NR], [@Singer1], for example).
In [@NR], Narasimhan and Ramadas considered the Coulomb connection on the quotient $\mathcal{A}^k\to\mathcal{A}^k/\mathcal{G}^{k+1}$, where $\mathcal{A}^k$ and $\mathcal{G}^{k+1}$ are certain Sobolev spaces of generic connections and gauge transformations, respectively, of the trivial $SU(2)$ principal bundle over $S^3$ (note that the Coulomb connection is a connection over a space of connections $\mathcal{A}^k$ of the bundle $S^3\times SU(2)\to S^3$). They showed that in this case the image of the curvature form of the Coulomb connection at the Maurer-Cartan connection is dense in the gauge algebra. Since the image of the curvature form is contained in the holonomy algebra, this fact implies that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of $\mathcal{G}^k$. They described the physical ramifications of this density, and call it a “maximal ambiguity” in reference to the Gribov ambiguity.
Here we are interested in the Coulomb connection when the principal bundle $P$ is over a compact manifold $M$ with *non-empty boundary*, and the structure group $K$ is a compact subgroup of $SO(m)$ or $SU(m)$. This bundle need not be trivial. In this with boundary case, we will want our connections to satisfy *Dirichlet boundary conditions* as defined by Marini in [@Ma]. In this case, we see that, unlike in the case found in [@NR], the image of the curvature form of the Coulomb connection cannot be dense in the gauge algebra at any one fixed point. Indeed, we will show\
\
[**Theorem.**]{} *Let $M$ be a compact $n$-dimensional manifold with boundary with $n\ge 2$ and let $\nabla_{A}$ be a connection of Sobolev class $k$ for an integral $k>n/2+1$ that satisfies Dirichlet boundary conditions. Define a set $\mathcal{L}_A$ as $$\mathcal{L}_A = \Span(\mathrm{Im} (\mathcal{R}_A)),$$ where $\mathcal{R}_A$ is the curvature of the Coulomb connection at $\nabla_A$. Let $\Gk$ be the gauge transformations of Sobolev class $k+1$ that satisfy Dirichlet boundary conditions. There exists a bounded nonzero operator $T_A:\Lie(\Gk)\to L^2_{k-\frac{5}{2}}(\kb|_{\partial M})$ such that $\mathcal{L}_A\subseteq \mathrm{ker}(T_A)$, and thus $\mathcal{L}_A$ cannot be dense in $\Lie(\Gk)$. This linear map is given by $$T_A(f)= d_A\Delta_A f + 2(n-1)H_{}\Delta_A f,$$ where $H_{}$ is the mean curvature of the boundary of $M$.*\
The above theorem will be restated as Lemma \[bct\] and Theorem \[bigcor\] of Section \[2.3\]. While the image of the curvature form $\mathcal{R}_A$ at a fixed connection $\nabla_A$ may not be dense, the holonomy algebra may still be dense as the span of $\mathcal{R}_A$ for a fixed connection $\nabla_A$ is not the entire holonomy algebra (indeed, this is not even an algebra). The denseness of the gauge algebra was the physically relevent result of Narasimhan and Ramadas, and this denseness will still hold in at least one specific case despite the presence of a boundary and Dirichlet boundary conditions.\
[**Theorem.**]{} *Consider the trivial principal bundle $\bar{O}\times K\to\bar{O}$, where $O\subseteq\mathbb{R}^n$ is a bounded open set with smooth boundary, $n\ge 2$, and $K$ is a compact subset of $SO(m)$ or $SU(m)$. The linear space spanned by the curvature form and its first commutators at the flat connection generates all $C^\infty$ elements of the gauge algebra. Furthermore, the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of $\Gk$.*\
The above theorem will be restated as Lemma \[mainthm\] and Theorem \[lastcor\] in Section \[2.5\].
This paper is a condensed version of the author’s doctoral thesis [@GrycThesis]. One difference is that in [@GrycThesis] only dimension $3$ is considered, while in this paper we consider any dimension $n$ greater than $2$. Certain arguments that are omitted or shortened in this paper can be found in [@GrycThesis] for the specific case of $n=3$, and where appropriate a reference to [@GrycThesis] will be given if more detail can be found there. However, please note that some of the notation has been changed between the thesis and this paper.
Background and Notation {#BN}
=======================
The Differential Geometric Setting of Yang-Mills Theory {#1.1}
-------------------------------------------------------
The differential geometric set-up of Yang-Mills theory can be found in [@BL]. Here we give a brief review as well as establish notation.
Let $M$ be a compact oriented $n$-dimensional Riemannian manifold with boundary for $n\ge 2$, and let $P\to M$ be a principal bundle with a semisimple compact connected structure group $K$. We also assume that $K$ acts faithfully on a finite dimensional real (or complex) inner product space $V$ by isometries, and thus we view $K$ as a compact matrix group and a subgroup of $SO(m)$ (or $SU(m)$, respectively). The natural action of $K$ on $V:=\mathbb{R}^m$ (or $V:=
\mathbb{C}^m$) induces a vector bundle $E:=P\times_{K} V$ (for the definition of these associated bundles, see Chapter 1.5 in [@KN]). $K$ also acts on itself and its Lie algebra $\mathfrak{k}$ via the adjoint representation, and thus we have the corresponding bundles $\Kb := P\times_{K} K$ and $\kb :=
P\times_{K} \mathfrak{k}$.
Note that $\kb$ is a vector bundle, while $\Kb$ is not. However, both $\kb$ and $\Kb$ are subbundles of the vector bundle $\Mb:=P\times_K
\mathrm{End}(V)$, where again $K$ acts by the adjoint action.
Recall the exponential map $\exp :\mathfrak{k}\to K $. Since $\Ad_ (k)\circ \exp = \exp\circ
\mathrm{Ad }(k)$, for any $k\in K$, we have an induced map $\exp :\kb\to\Kb$.
As $\mathrm{End}(V)$ acts on $V$ in an obvious way, fibers of $\Mb$ act on fibers of $E$. Viewing $\kb$ and $\Kb$ as subbundles of $\Mb$, fibers of $\kb$ and $\Kb$ also act on fibers of $E$. Similar reasoning also tells us that given two elements $\phi,
\psi$ in the same fiber of $\kb$, the Lie bracket $[\phi,\psi]$ is well-defined.
A Koszul connection $\nabla_A$ on $E$ induces a Koszul connection also called $\nabla_A^{Hom}$ on $\mathrm{Hom}(E,E)$ (see [@BL], [@Dar] for more background on $\nabla_A^{Hom}$). Often, we will write $\nabla_A$ for $\nabla_A^{Hom}$ if it is clear we are using this induced connection. The connection $\nabla_A^{Hom}$ induces a connection on $\kb$, and allows us to calculate $\nabla_A g$ for $g\in\Gamma(\Kb)$. Note that $\nabla_A g$ is not necessarily a section of $\Kb$, but a section of $\mathrm{Hom}(E,E)$.
Of special interest is the trivial bundle $P:=\bar{O}\times K\to\bar{O}$, where $O$ is a bounded open subset of $\mathbb{R}^n$ with smooth boundary. In this case, the induced bundles $E$, $\kb$, and $\Kb$ are also direct products of the appropriate sort. For example, $E=\bar{O}\times V\to \bar{O}$. In this case, the *flat connection* on $P$ is the Ehresmann connection whose kernel is the tangent bundle of $\bar{O}$ embedded in the tangent bundle of $\bar{O}\times K$. Using parallel transport, one can check that the Koszul connection $\nabla_0$ induced on the product bundle $E$ is given by $$\nabla_0 \sigma = d \sigma\mbox{, $\sigma\in\Gamma(E)$}$$ where $d \sigma$ is the standard exterior derivative. Thus we will often use $\nabla_0$ and $d$ interchangeably and call them the *flat connection* on $E$.
We are only concerned with connections $\nabla_A$ on $E$ that are induced by connections on $P$. Such connections are called *$K$-connections*, and one can show that $\nabla_A$ is a $K$-connection if and only if the local connection form is $\mathfrak{k}$-valued.
As a vector bundle, we may equip $\kb$ with a metric. Any Ad-invariant inner product on $\mathfrak{k}$ will induce a Riemannian metric on $\kb$. In particular, we can use the trace inner product $(A,B)=\tr (A^*B)$ to induce a metric on $\kb$. We now view $\kb$ as equipped with the metric induced by the trace inner product on $\mathfrak{k}$. Similarly, we equip the bundle $\Mb$ with the trace inner product.
For any vector bundle $\xi$ over $M$, we may consider the associated vector bundle $\mathrm{Hom}(\Lambda^j(TM),\xi)$ for any $j\ge 1$ and define $\mathrm{Hom}(\Lambda^0(TM),\xi):=\xi$. We call sections of $\mathrm{Hom}(\Lambda^j(TM),\xi)$ *$\xi$-valued $j$-forms*, and generally call them *vector-valued forms*. We denote these sections by $\Omega^j(\xi)$. Any connection on $\xi$ induces a connection on $\mathrm{Hom}(\Lambda^j(TM),\xi)$ that involves the Levi-Civita connection on $M$. See [@Dar] for more about these forms and the induced connections.
There are certain operations we will like to define on forms. Given any $\alpha\in\Omega^1(\kb)$ and $\phi\in\Gamma(\kb)$, we define the $1$-form $[\alpha,\phi]$ by $$[\alpha,\phi](X) = [\alpha(X),\phi],$$ for any $X\in TM$. Also, given any $\alpha,\beta\in\Omega^1(\kb)$, we define the product $[\alpha\cdot\beta]\in \Gamma(\kb)$ in the following way: Suppose locally $\alpha = \sum_i \alpha_i dx_i$, and $\beta = \sum_i \beta_i dx_i$, and the associated metric tensor is $g_{ij}$. Then, we set $$\label{dotproduct}
[\alpha\cdot\beta] = \sum_{i,j} [\alpha_i,\beta_j]g^{ij},$$ noting that $<dx_i,dx_j>=g^{ij}$ where the matrix $(g^{ij})$ is inverse to $(g_{ij})$ . One can verify that this globally defines $[\alpha\cdot\beta]$ as a section of $\kb$.
We will often be looking at the difference between two $K$-connections, and the following will be useful in looking at such differences. If $\nabla_{A_1}$ and $\nabla_{A_2}$ are two $K$-connections, using the local characterization of $K$-connections, one can show that the difference $\nabla_{A_1}-\nabla_{A_2}$ is a $\kb$-valued $1$-form. Furthermore, if we set $\alpha := \nabla_{A_1}-\nabla_{A_2}$, we have for any $\phi\in\Gamma(\kb)$ $$\label{dprop}
(\nabla_{A_1}^{Hom}-\nabla_{A_2}^{Hom})(\phi)=[\alpha,\phi].$$ Similarly, if $\beta\in\Omega^1(\kb)$, one can show that $$\label{d*prop}
((\nabla_{A_1}^{Hom})^*-(\nabla_{A_2}^{Hom})^*)(\beta) = -[\alpha\cdot\beta].$$ The previous two equations are ubiquitous in what follows. On sections we have $d_A=\nabla_A$ and on $1$-forms we have $d_A^*=(\nabla_A)^*$. We will use both notations interchangably on these respective domains. The curvature $R_A$ of a $K$-connection $\nabla_A$ is a $\kb$-valued $2$-form.
Using , one can show that a $K$-connection $\nabla_A^{Hom}$ is compatible with the metric on $\Mb$ induced by the trace inner product. Furthermore, a $K$-connection $\nabla_A$ on $E$ and the Levi-Civita connection on $M$ induce a connection on $\mathrm{Hom}(\Lambda^j(TM),\kb)$ that is compatible with the induced metric on $\mathrm{Hom}(\Lambda^j(TM),\kb)$.
Dirichlet Boundary Conditions and Sobolev Spaces {#1.2}
------------------------------------------------
We define Sobolev spaces of sections of vector bundles as Palais does in [@Palais]. Using the notation of [@Palais], the space $L^p_k(\xi)$ is the space of sections of $\xi$ with $k$ Sobolev derivatives under the $L^p$ norm, and $L^p_k(\xi)^0$ is the completion of $C^\infty_c(\xi|_{\mathrm{int}(M)})$ in the $L^p_k$ norm. As usual, we define $H^k(\xi) := L^2_k(\xi)$, where the latter notation is what [@Palais] uses. Also converting from Palais’s notation, we set $H^k_0(\xi) := L^2_k(\xi)^0$.
Marini in [@Ma] defines Dirichlet boundary conditions on Sobolev spaces, which we will denote $H^k_{Dir}(\mathrm{Hom}(\Lambda^j(TM),\xi))$ for appropriate vector bundles $\xi$. Specifically, we define the *Dirichlet Sobolev space $H^k_{Dir}(\mathrm{Hom}(\Lambda^j(TM),\xi))$* for $k\ge 1$ as $$\begin{aligned}
H^k_{Dir}(\mathrm{Hom}(\Lambda^j(TM),\xi))&:=& \{\alpha\in H^k(\mathrm{Hom}(\Lambda^j(TM),\xi)): \iota^*(\alpha)=0, \\
& &\mbox{ where $\iota:\partial M\to M$ is the inclusion}\}.\end{aligned}$$ Since $k\ge 1$, $\alpha|_{\partial M}$ is defined in the trace sense, so $\iota^*(\alpha)$ is defined almost everywhere. Note that $\iota^*(\alpha)=0$ is equivalent to saying that $\alpha$ vanishes on wedges of vectors $X_1\wedge\ldots\wedge X_j$, where all $X_i$ are tangent to $\partial M$. For a $0$-form $\sigma$ (i.e. a section $\sigma$), $\iota^*(\sigma)=0$ if and only if $\sigma|_{\partial M}=0$. Hence, we see that $$H^k_{Dir}(\xi) = H^1_0(\xi)\cap H^k(\xi)\mbox{, $k\ge 1$}\label{pB2}.$$
In what follows we use $k>n/2 +1$ where $k$ is an integer so we can use the multiplication theorem of Sobolev spaces for $H^{k-1}$ (see Corollary 9.7 in [@Palais]).
Since we will be using Dirichlet boundary conditions, we need a fixed smooth connection $\nabla_{A_0}$. Set $$\Ck := \nabla_{A_0} + H^k_{Dir}(\mathrm{Hom}(TM,\kb)).$$ Note that all the connections in $\Ck$ will be equal to $\nabla_{A_0}$ in tangential directions on the boundary. Also $\Ck$ is an affine space and is therefore a $C^\infty$-Hilbert manifold. We will call any connection $\nabla_{A}$ *$C^\infty$-smooth* (resp. *$H^k$-smooth, $L^p$-smooth*) if $\nabla_A-\nabla_{A_0}\in C^\infty(\mathrm{Hom}(TM,\kb))$ (resp. $\in H^k(\mathrm{Hom}(TM,\kb))$, $L^p(\mathrm{Hom}(TM,\kb))$. Note that if $\nabla_{A}$ is $C^\infty$-smooth, then it is a Koszul connection in the usual Riemannian geometry sense.
The sections of $\Kb$ are the *gauge transformations*. The Sobolev regularity and boundary conditions we will need is set in the following definition:
Let $\nabla_{A_0}$ be a fixed smooth $K$-connection on $E$. Let $g\in H^{k+1}(\Kb)$, with $g|_{\partial M}\equiv e$, where $e$ is the identity on $K$. Then we say that $g\in\Gk$, and call $\Gk$ the *(Dirichlet) gauge group*.
The Sobolev space $H^{k+1}(\Kb)$ is defined as in [@MV] as the completion of smooth sections of $\Kb$ in the norm $H^{k+1}(\Mb)$. This completion without the boundary conditions we will call $\mathcal{G}^{k+1}$, as it is called in [@MV].
\[blgroup\] For $k>n/2+1$, the group $\Gk$ is a Hilbert Lie group whose Lie algebra is identifiable with $H^{k+1}_{Dir}(\kb)$.
As proven in [@MV], $\mathcal{G}^{k+1}$ is Hilbert Lie group. Since $\Gk$ is a closed topological subgroup of $\mathcal{G}^{k+1}$, it carries a topology. To give $\Gk$ Hilbert coordinates in a neighborhood of the identity, we show that the exponential map takes $H^{k+1}_{Dir}(\kb)$ into $\Gk$ and is a local homeomorphism at $0$. In [@MV] it is shown that $\exp$ is a $C^\infty$ local diffeomorphism $\exp: H^{k+1}(\kb)\to\mathcal{G}^{k+1}$ without boundary conditions. Hence, we need only show that $\exp$ maps $H^{k+1}_{Dir}(\kb)$ into $\Gk$, and for a neighborhood $U$ of the identity in $\mathcal{G}^{k+1}$, $\exp^{-1}\equiv\log$ maps $U$ into $H^{k+1}_{Dir}(\kb)$.
To prove the first assertion, let $f\in H^{k+1}_{Dir}(\kb)$. Since $k+1>n/2 + 2$ we have $f\in C^2(\kb)$ with $f|_{\partial M}\equiv 0$. Then if $g:=\exp(f)$, we have that $g\in C^2(\Kb)\subseteq C^2(\Mb)$, and $g|_{\partial M}\equiv e$, where $e$ is the identity element of $K$, proving $g\in\Gk$.
To prove the second assertion, since $\exp:H^{k+1}(\kb)\to\mathcal{G}^{k+1}$ is a local diffeomorphism between the spaces without boundary conditions, we need only show that for small $\mu\in H^{k+1}(\kb)$, if $g:=\exp(\mu)\in\Gk$, then $\mu\in H^{k+1}_{Dir}(\kb)$. Note that $\sup |\mu|\le C\|\mu\|_
{H^{k+1}}$. So for small enough $\|\mu\|_{H^{k+1}}$, we can use the fact that the “pointwise” map $\exp:\mathfrak{k}\to
K$ is local diffeomorphism at $0$ to say that since $g|_{\partial M}\equiv e$, we have $\mu|_{\partial M}\equiv 0$. Since $\mu\in H^{k+1}(\kb)$, this vanishing on the boundary implies that $\mu\in H^1_0(\kb)\cap H^{k+1}(\kb)=
H^{k+1}_{Dir}(\kb)$.
To give $\Gk$ an atlas, we transport these coordinates via right translation as is done in [@MV].
$\Gk$ acts on $\Ck$ on the right in the following way. Suppose that we have a $1$-form $\eta\in H^k_{Dir}(\mathrm{Hom}(TM,\kb))$. Then the action is $$(\nabla_{A_0} + \eta)\cdot g = \nabla_{A_0} + (g^{-1}\nabla^{Hom}_{A_0}g +
\mathrm{Ad }(g^{-1})\eta).\label{gaction}$$ By the same reasoning as found in [@MV], this action is smooth. Note that for $(\nabla_{A_0} + \eta)\cdot g$ to remain in $\Ck$, we need to have $g^{-1}\nabla^{Hom}_{A_0}g $ satisfy Dirichlet boundary conditions. The following proposition shows that this is the case.
\[gaugeEZ\] Let $k>n/2+1$ and suppose $g\in\Gk$. Then we have $g^{-1}\nabla_{A_0}g\in H^k_{Dir}(\mathrm{Hom}(TM,\kb))$,
Let $g\in\Gk$. Since $g\in H^{k+1}(\Kb)$, there exist smooth $g_m\in H^{k+1}(\Kb)$ such that $g_m\to g$ in $H^{k+1}(\Mb)$. It is shown in [@MV] that inversion is continuous on $\mathcal{G}^{k+1}$. Hence, $(g_m)^{-1}\to g^{-1}$ in $H^{k+1}(\Mb)$. Since $\nabla_{A_0}$ is a smooth $K$-connection, we see that $g_m^{-1}\nabla_{A_0}^{Hom} g_m\in\Omega^1(\kb)$, and by the multiplication theorem, $$\begin{aligned}
||g^{-1}\nabla_{A_0}^{Hom} g - g_m^{-1}\nabla_{A_0}^{Hom} g_m||_{H^k} &\le& ||g^{-1}\nabla_{A_0}^{Hom} g - g_m^{-1}\nabla_{A_0}^{Hom} g||_{H^k}+\\ & &||g_m^{-1}\nabla_{A_0}^{Hom} g - g_m^{-1}\nabla_{A_0}^{Hom} g||_{H^k}\\
&\le& C(||g^{-1}-g_m^{-1}||_{H^{k+1}}||\nabla_{A_0}^{Hom}g||_{H^k} + \\
& & ||g_m^{-1}||_{H^{k+1}}||g_m - g||_{H^{k+1}})\to 0.\end{aligned}$$ Thus, $g^{-1}\nabla_{A_0}^{Hom} g\in H^k(\mathrm{Hom}(TM,\kb))$.
We now show that $g^{-1}\nabla_{A_0}^{Hom} g\in H^k_{Dir}(\mathrm{Hom}(TM,\kb))$. Locally, we have $\nabla_{A_0} = d + A_0$, where $A_0$ is a $C^\infty$-smooth $\mathfrak{k}$-valued $1$-form. Let $X$ be a tangential direction at a boundary point. Since $g\equiv e$ on $\partial M$, we have $dg(X)=0$. Also on the boundary, $[A_0(X),g]=[A_0(X),e]=0$, since $e$ commutes with everything. Hence, globally, $\iota^*(\nabla_{A_0}^{Hom} g) \equiv 0$, and thus $\iota^*(g^{-1}\nabla_{A_0}^{Hom} g) \equiv 0$. This proves that $g^{-1}\nabla_{A_0}^{Hom} g\in H^k_{Dir}(\mathrm{Hom}(TM,\kb))$.
Using Dirichlet boundary conditions gives us a Sobolev-inequality.
\[SPprop\] Let $\nabla_{A}$ be a $L^q$-smooth connection on a vector bundle $\xi\to M$ compatible with the metric on $\xi$, where $q\ge n$. Then there exists a $\kappa_p > 0$ such that for any $f\in L^p_1(\xi)^0$ with $1<p<n$, we have $$\|f\|_{L^p}\le\kappa_p\|\nabla_A f\|_{L^p}\label{SP},$$ where $\kappa_p$ does not depend on the connection, but does depend on $p$.
This is done for real-valued functions by a standard inequality argument that can be found in, for example, [@Evans] and [@He]. This shows that there exists a $\kappa_p >0$ such that $$\|g\|_{L^p}\le\kappa_p\|dg\|_{L^p},\label{HSP}$$ for any $g\in L^p_1(M\times\mathbb{R})^0$ where $M\times\mathbb{R}\to M$ is the trivial vector bundle. (The references above prove for real-valued functions $g$. But real-valued functions on $M$ and sections of $M\times\mathbb{R}$ are the same.)
Now let $f\in C^1_0(\xi|_{\mathring{M}})$, where $\mathring{M}$ is the interior of $M$. Then the function $|f|$ is globally Lipschitz, so by Lemma 2.8 in [@He] we have $|f|\in L^p_1(M\times\mathbb{R})$. Since $|f|$ is continuous and $0$ on $\partial M$, Theorem 5.5.2 in [@Evans] tells us that $|f|\in L^p_1(M
\times\mathbb{R})^0$. Hence, yields $$\begin{aligned}
\|f\|_{L^p}\le\kappa_p\|d|f|\|_{L^p}\le\kappa_p\|\nabla_A f\|_{L^p}.\end{aligned}$$ The second inequality is *Kato’s inequality*. This inequality only requires that $\nabla_A$ is compatible with the metric. For a proof of this inequality, see [@Par]. Since $C^1_0(\xi|_{\mathring{M}})$ is dense in $L^p_1(\xi)^0$, the proposition has been proven. (The condition $q\ge n$ ensures that $\nabla_A f \in L^p$).
The Sobolev-inequality immediately tells us
\[gfree\] For $k > n/2 + 1$, the action of $\Gk$ on $\Ck$ is free.
Suppose $\nabla_{A_0} + \eta\in\Ck$, $g\in\Gk$, and $(\nabla_{A_0} +\eta)\cdot g = \nabla_A + \eta$. By (\[gaction\]), this means that $$(\nabla^{Hom}_{A_0} +\eta)g = \nabla^{Hom}_{A_0} g + [\eta,g] = 0.$$ By (\[SP\]), we have $$\|g-e\|_{L^2} \le \kappa_2\|(\nabla^{Hom}_{A_0} +\eta)g\|_{L^2} = 0$$ Since $g$ is continuous, the above shows $g\equiv e$ and thus the corollary is proven.
The freeness of this action allows us to directly use $\Ck$ and $\Gk$, instead of so-called generic (or irreducible) connections and modified gauge groups as found in [@MV], [@NR], and [@Par].
Given a $K$-connection $\nabla_{A}\in\Ck$, we can define the Laplacian $$\Delta_A=\nabla_A^*\nabla_A = d^*_A d_A:H^{m+1}_{Dir}(\kb)\to H^{m-1}(\kb)\mbox{ for $1\le m\le k$.}$$ The regularity is correct by the following: Since $\nabla_{A_0}$ is a smooth connection, clearly $\Delta_{A_0}$ is a bounded map from $H^{m+1}$ into $H^{m-1}$. Suppose $h=\nabla_A-\nabla_{A_0}\in H^k_{Dir}(\kb)$. Recalling equations and and the comment following them, for $f\in H^{m+1}$ we have $$\label{localLap}
\Delta_{A} f = \Delta_{A_0}f + [d_{A_0}^*h,f] - [h\cdot[h,f]] - 2[h\cdot
d_{A_0}f].$$ So, we have (allowing $||\cdot||_i$ to denote the $H^i$ norm) $$\begin{aligned}
||\Delta_A f||_{m-1} &\le& ||\Delta_{A_0}f||_{{m-1}}+||[d_{A_0}^*h,f]||_
{{m-1}} \\
& &+ ||[h\cdot[h,f]]||_{m-1} + 2||[h\cdot d_{A_0}f]||_{m-1} \\
&\le& ||\Delta_{A_0}f||_{{m-1}}+C(||d_{A_0}^*h||_{k-1}||f||_{{m-1}} \\
& &+ ||h||_k||[h,f]||_{m-1} + 2||h||_k||d_{A_0}f]||_{m-1}) \\
&\le& ||\Delta_{A_0}f||_{{m-1}}+C(||h||_{k}||f||_{{m-1}} \\
& &+ ||h||_k^2||f||_{m-1} + 2||h||_k||f||_{m}) < \infty,\end{aligned}$$ where we used the fact that $H^{m-1}$ is a $H^{k-1}$ module, which is the case since $k-1\ge m-1$ and $k-1 > n/2$ (this is a critical point where we need $k > n/2+1$). Thus, $\Delta_A$ is bounded from $H^{m+1}_
{Dir}(\kb)$ to $H^{m-1}(\kb)$. Furthermore, we have
\[Greenexist\] Let $\nabla_{A}\in\Ck$ for an integer $k > n/2 + 1$, and suppose $1\le m\le k$. Then the mapping $\Delta_A:H^{m+1}_{Dir}(\kb)\to H^{m-1}(\kb)$ is an isomorphism. Furthermore, if $f\in H^2_{Dir}(\kb)$ and $\Delta_A f\in H^{m-1}(\kb)$, then $f\in
H^{m+1}_{Dir}(\kb)$ and $$\|f\|_{H^{m+1}}\le C(\|\Delta_A f\|_{H^{m-1}} + \|f\|_{H^0}).$$
We set $G_A:=(\Delta_A)^{-1}$ and call it a *Green operator*.
The proof of the existence of Green operators follows from variations of standard argument for weak solutions to elliptic equations that can be found in [@Evans] and [@GT]. We omit the details although they can be found in [@GrycThesis]. Nominally, the proof in [@GrycThesis] is for $n=3$. But replacing “$3$” with “$n$” in the proof gives the general result.
We emphasize here that *every* connection $\nabla_A\in\Ck$ has an associated Green operator. We need not restrict our space of connections in this with-boundary case since we are imposing Dirichlet boundary conditions.
The Quotient $\Ck\to\Ck/\Gk$ {#1.3}
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We are now in a position to consider the structure of the quotient $\Ck\to\Ck/\Gk$.
\[bigpB\] Let $k>n/2+1$ for an integer $k$. The quotient space $\Ck/\Gk$ is a $C^\infty$ Hilbert manifold, and $\pi: \Ck\to
\Ck/\Gk$ is a principal bundle with structure group $\Gk$.
The proof is a straightforward adaptation of the no-boundary case of Mitter and Viallet in [@MV]. Others have proven similar statements in more specific situations (see [@AHS], [@NR], and [@Par]). Since the techniques for proving Proposition \[bigpB\] are very little changed from those employed by the above authors, we omit the proof. However, for those interested in the proof, it is in [@GrycThesis]. Again, the proof in [@GrycThesis] is nominally for $n=3$, but an examination of the proof shows that the dimension is not mentioned and the only relevent contribution of the dimension is that we have $k>n/2+1$ as we have here. Thus, the proof works for the $n\ge 2$ as well. We again note that with our Dirichlet boundary conditions on the connections and the gauge group, we need not restrict the space of connections to generic connections nor restrict the gauge group further. This situation is unlike the no-boundary situations as found in [@AHS], [@NR], and [@Par]. Also, the presence of the Sobolev-inequality of Proposition \[SPprop\] leads to some simplifications of the arguments.
The Coulomb Connection and Its Holonomy {#Coulomb}
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Since the bundle $\Ck\to \Ck/\Gk$ is a principal bundle, we can consider holonomy groups of a fixed connection upon it. The connection we will consider is the *Coulomb connection* whose connection form at $\nabla_A$ is defined as $\displaystyle G_A d^*_A$. The corresponding horizontal subspace at $\nabla_{A}$ we will denote as $H_A$. Note that since $\Ck = \nabla_{A_0} + H^k_{Dir}(\mathrm{Hom}(TM,\kb))$, the tangent space at $\nabla_A\in\Ck$ is\
$H^k_{Dir}(\mathrm{Hom}(TM,\kb))$. One can verify that $H_A$ is $$H_A = \{\alpha\in H^k_{Dir}(\mathrm{Hom}(TM,\kb)): d_A^*\alpha = 0\}.$$ This connection is natural in the sense that $H_A$ is the $L^2$ orthogonal complement to the vertical vectors at $\nabla_{A}$. Indeed, one can show that given $\gamma\in
\Lie(\Gk)=H^{k+1}_{Dir}(\mathrm{Hom}(TM,\kb))$, the fundamental vector field associated to $\gamma$ is $d_A \gamma$. Hence the vertical vectors are those vectors of the form $d_A\gamma$ for some $\gamma\in\Lie(\Gk)$ (see [@Gross] or [@NR]). By the same reasoning as the proof of Lemma 7.1 in [@NR], the Coulomb connection is indeed a connection on $\Ck\to\Ck/\Gk$. The principal tool of our study of the holonomy group of the Coulomb connection is the image of the curvature form of the Coulomb connection. Let $\mathcal{R}_A$ be the curvature form of the Coulomb connection at $\nabla_A$. By the same calculation in the proof of Lemma 7.2 in [@NR], we have $$\label{curveq}
\mathcal{R}_A(\alpha,\beta)=-2G_A([\alpha\cdot\beta]),\mbox{ for } \alpha,\beta\in H_A.$$
Coordinates at the Boundary of $\partial M$ {#2.1}
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In this investigation, certain types of coordinates at the boundary are useful. Consider the following system of coordinates at the boundary that satisfy the following:
1. $\partial/\partial x_n$ is orthogonal to $\partial/\partial x_1,\ldots,\partial/\partial x_{n-1}$ on the boundary.\[A1\]
2. $\partial/\partial x_n$ has norm 1 everywhere.\[A2\]
3. $\partial/\partial x_n$ is the inward pointing unit normal vector on the boundary.\[A3\]
We describe such a coordinate system as *Type A*. They have been constructed in [@Milnor1] and [@RS]. We also will use a similar coordinate system such that
1. $\partial /\partial y_n$ is orthogonal to $\partial /\partial y_1,\ldots\partial /\partial y_{n-1}$ everywhere.
2. $\partial /\partial y_n$ is a positive (perhaps nonconstant) multiple of the inward pointing unit normal vector on the boundary.
We call such coordinates *Type B*. A detailed contruction of these coordinates can be found in [@GrycThesis]. In what follows, we will use $\{x_1,\ldots,x_n\}$ to denote Type A coordinates and $G=(g_{ij})$ to denote the associated metric tensor. For Type B coordinates, we use $\{y_1,\ldots,y_n\}$ and $\tilde{H}=(h_{ij})$, respectively (we use $\tilde{H}$ to distinguish this matrix from the mean curvature of the boundary, which we will denote $H_{}$).
Mean Curvature of the Boundary and Coordinates {#2.2}
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We will see that the mean curvature $H_{}$ of the immersion $\iota:\partial M\to M$ will come into play in our investigation of the image of the curvature form. (For the definition of mean curvature, see, for example, [@doCarmo]).) It will be useful to have a characterization of $H_{}$ in our Type A and Type B coordinates.
\[taux\] Consider the mean curvature $H_{}$ of the immersion $\iota:\partial M\to M$. Let $\{x_1,\ldots,x_n\}$ be Type A coordinates with metric tensor $G=(g_{ij})$. Then $H_{}$ can be written as $$H_{}(x_1,\ldots,x_{n-1}) = \frac{1}{n-1}\frac{\partial a}{\partial x_n}(x_1,\ldots,x_{n-1},0)\cdot\frac{1}{a(x_1,\ldots,x_{n-1},0)},$$ where $a:=\sqrt{\det(g_{ij})}$.[^2]
Let $\Gamma^m_{ij}$ be the Christoffel symbols corresponding to $\{x_1, \ldots, x_n\}$. Since the connection we are considering is the Levi-Civita connection, we have $$\Gamma^m_{ij}=\frac{1}{2}\sum_{k=1}^n \left(\frac{\partial}{\partial x_i}(g_{jk})+\frac{\partial}{\partial x_j}(g_{ik})- \frac{\partial}{\partial x_k}(g_{ij})\right)g^{km}.$$ (see, for example, [@doCarmo]). By our choice of coordinate system, we have $$g_{1n}=\ldots=g_{(n-1)n}=g^{1n}=\ldots=g^{(n-1)n}=0$$ on the boundary and $\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_{n-1}}$ are tangent to the boundary. So on the boundary we have for $i<n$ $$\begin{aligned}
\Gamma^i_{in}=\frac{1}{2}\sum_{k=1}^{n-1}\left(\frac{\partial}{\partial x_n}(g_{ik})g^{ki}\right)\label{tau1}.\end{aligned}$$ Using a Laplace expansion on the bottom row of $G$, we also have $$\label{laplace}
\det(G) = \sum_{k=1}^n (-1)^{n+k} g_{nk}\cdot\det(G(n|k)).$$ Above and in what follows, $G(i|j)$ is the $(n-1)$-by-$(n-1)$ matrix obtained from $G$ by deleting the $i^\mathrm{th}$ row and $j^\mathrm{th}$ column. On the boundary, for $k<n$ we have $g_{nk}=0$ and $\det(G(n|k))=0$, since the last column of $G(n|k)$ is all zeros. Also, $g_{nn}\equiv 1$ everywhere by our choice of Type A coordinagtes. Using these preceeding facts, we apply the product rule to to get $$\label{tau4}
\frac{\partial}{\partial x_n}\det(G) = \frac{\partial}{\partial x_n}\det(G(n|n))
\mbox{ on $\partial M$.}$$ Define a set of permuations $S^{i,j}_{n-1}$ as $$S^{i,j}_{n-1} = \{\sigma\in S_{n-1}: \sigma(i)=j\}.$$ Then $S^{i,j}_{n-1}$ is isomorphic to $S_{n-2}$, and if $\sigma\in S^{i,j}_{n-1}$ corresponds to $\tilde{\sigma}\in S_{n-2}$, then one can show that $$\label{tau5}
\mathrm{sgn}(\sigma)=(-1)^{(i+j)}\mathrm{sgn}(\tilde{\sigma}).$$ Indeed, one can prove by considering the permuation matrix of $\sigma$ (the determinant of which is $\mathrm{sgn}(\sigma)$), and using a Laplace expansion down the $i^{th}$ row. Define the $(n-1)$-by-$(n-1)$ matrix $C:=G(n|n)$. Combining ) and we have $$\begin{aligned}
H_{} &=& -\frac{1}{n-1}\sum_{i=1}^{n-1}\left(\nabla_{\frac{\partial}{\partial x_i}}(-\frac{\partial}{\partial x_n})\right)_i\\
&=& \frac{1}{n-1}\sum_{i=1}^{n-1} \Gamma^i_{in} = \frac{1}{n-1}\sum_{i=1}^{n-1}\frac{1}{2}\sum_{j=1}^{n-1}\frac{\partial}{\partial x_n}(g_{ij})g^{ji}\\
&=& \frac{1}{2(n-1)}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\frac{\partial}{\partial x_n}(g_{ij})c^{ji}\\
&=& \frac{1}{2(n-1)\det(C)}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\frac{\partial}{\partial x_n}(g_{ij})(-1)^{(i+j)}\det(C(i|j))\\
&=& \frac{1}{2(n-1)\det(C)}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\sum_{\tilde{\sigma}\in S_{n-2}} \frac{\partial}{\partial x_n}(g_{ij})(-1)^{(i+j)}\mathrm{sgn}(\tilde{\sigma})c_{1\tilde{\sigma}(1)}\cdot\ldots\cdot\widehat{c_{ij}}\cdot\\
& & \ldots\cdot c_{(n-1)\tilde{\sigma}(n-1)}. \\\end{aligned}$$ Inserting to the above yields $$\begin{aligned}
H&=& \frac{1}{2(n-1)\det(C)}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\sum_{\sigma\in S^{i,j}_{n-1}}\mathrm{sgn}(\sigma) \frac{\partial}{\partial x_n}(g_{ij})c_{1\sigma(1)}\cdot\ldots\cdot\widehat{c_{ij}}\cdot\\
& & \ldots\cdot c_{(n-1)\sigma(n-1)} \\
&=& \frac{1}{2(n-1)\det(G(n|n))}\sum_{i=1}^{n-1}\sum_{\sigma\in S_{n-1}} \frac{\partial}{\partial x_n}(g_{i\sigma(i)})g_{1\sigma(1)}\cdot\ldots\cdot\widehat{g_{i\sigma(i)}}\cdot\\
& & \ldots\cdot g_{(n-1)\sigma(n-1)} \\
&=& \frac{1}{2(n-1)\det(G(n|n))}\sum_{\sigma\in S_{n-1}} \frac{\partial}{\partial x_n}(g_{1\sigma(1)}\cdot\ldots\cdot g_{(n-1)\sigma(n-1)}) \\
&=& \frac{1}{2(n-1)\det(G(n|n))}\frac{\partial}{\partial x_n}(\det(G(n|n))) = \frac{1}{2(n-1)\det(G)}\frac{\partial}{\partial x_n}(\det(G)) \\
&=& \frac{1}{(n-1)}\frac{\partial a}{\partial x_n}\frac{1}{a},\end{aligned}$$ as desired.
We can also write $H_{}$ in terms of Type B coordinates:
\[tauy\] Consider the mean curvature $H_{}$ of the immersion $\iota:\partial M\to M$. Consider Type B coordinates $\{y_1,\ldots,y_n\}$ with metric tensor $\tilde{H}=(h_{ij})$. Then $H_{}$ can be written as $$H_{} = \frac{1}{n-1}\left({\sqrt{h_{nn}}}\cdot d\left(\frac{1}{\sqrt{h_{nn}}}\right)(\nu) + \frac{db(\nu)}{b}\right),$$ where $b:=\sqrt{\det(h_{ij})}$ and $\nu$ is the unit inward pointing normal vector.
One can use a coordinate change between Type A and Type B coordinates (see the $n=3$ case in [@GrycThesis]). Or, one can proceed similarly as the proof of Proposition \[taux\]. Indeed, we have a similar chain of equations from the proceeding proof. Below we write a shortened list of equations where the steps that are the same in the Type A case are omitted: $$\begin{aligned}
H_{} &=& -\frac{1}{n-1}\sum_{i=1}^{n-1}\left(\nabla_{\frac{\partial}{\partial y_i}}(-\frac{1}{\sqrt{h_{nn}}}\frac{\partial}{\partial y_n})\right)_i\\
&=& \frac{1}{\sqrt{h_{nn}}(n-1)}\sum_{i=1}^{n-1}\frac{1}{2}\sum_{j=1}^{n-1}\frac{\partial}{\partial y_n}(h_{ij})h^{ji}\\
&=& \frac{\sqrt{h_{nn}}}{2(n-1)\det(\tilde{H})}\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\frac{\partial}{\partial y_n}(h_{ij})C(i|j)\\
&=& \frac{\sqrt{h_{nn}}}{2(n-1)\det(\tilde{H})}\frac{\partial}{\partial y_n}(\det(\tilde{H}(n|n)))\\
&=& \frac{\sqrt{h_{nn}}}{2(n-1)\det(\tilde{H})}\frac{\partial}{\partial y_n}\left(\frac{\det(\tilde{H})}{h_{nn}}\right)\\
&=& \frac{h_{nn}}{2(n-1)\det(\tilde{H})}d\left(\frac{\det(\tilde{H})}{h_{nn}}\right)(\nu)\\
&=& \frac{1}{n-1}\left({\sqrt{h_{nn}}}\cdot d\left(\frac{1}{\sqrt{h_{nn}}}\right)(\nu) + \frac{db(\nu)}{b}\right),\end{aligned}$$ as desired. Twice above (once in the beginning and once at the end), we used the fact that $\frac{1}{\sqrt{h_{nn}}}\frac{\partial}{\partial y_n}=\nu$.
The Image of $\mathcal{R}_A$ {#2.3}
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We now use mean curvature of $\partial M$ in the following lemma, which relates $H$ and the image of the curvature form.
\[bct\] Suppose $M$ is a $n$ dimensional manifold with boundary, $k>n/2+1$ where $k$ is an integer, $\alpha,\beta\in H_A$ and $\nabla_{A}\in\Ck$. Then $$\label{thecondition1}
d_A[\alpha\cdot\beta](\nu)= -2(n-1)H_{}[\alpha\cdot\beta]\mbox{ on $\partial M$,}$$ where $\nu$ is the unit inward pointing vector field and $H$ is the mean curvature of the boundary.
Since $k-n/2>1$, note that $[\alpha\cdot\beta]$ is $C^1$, and thus $d_A[\alpha\cdot\beta]$ is continuous. Hence, the above equality is true not just in the trace sense, but as an equality of two continuous functions.
We will use Type A coordinates $\{x_1,\ldots,x_n\}$, and assume that the vector bundle $\k_P$ is also trivialized in this neighborhood. Recall that the metric tensor in this coordinate system has the feature that $g_{in}=\delta_{in}$ on the boundary, and $g_{nn}=1$ everywhere. Thus, $g^{in}=\delta_{in}$ also on the boundary. Also, $\frac{\partial}
{\partial x_n}$ is the inward pointing unit normal vector on the boundary. Take $\alpha,\beta$ as above and define $\alpha_i$ and $\beta_i$ so that $\alpha = \sum_{i=1}^n\alpha_i dx_i$ and $\beta = \sum_{i=1}^n\beta_i dx_i$. Since we are assuming $\k_P$ has a fixed trivialization in our neighborhood, we can view the $\alpha_i$ and $\beta_i$ as $\k$-valued functions. Also, since $\frac{\partial}{\partial x_n}$ is the inward pointing unit normal vector and $\alpha,\beta$ satisfy Dirichlet boundary conditions, we have $$\label{bc1}
\alpha_i=\beta_i=0\mbox{ for $i<n$ on $\partial M$.}$$ Let $d$ be the flat connection with respect to our fixed trivialization of $\k_P$ and define a $\k$-valued $1$-form $A$ so that $d_A = d + A$. Define $A_i$ so that $A=\sum_{i=1}^n A_idx_i$. On this coordinate patch, we have $$\begin{aligned}
\label{bc2}
[\alpha\cdot\beta] &=& \sum_{j,k=1}^n [\alpha_j,\beta_k](dx_j\cdot dx_k)
= \sum_{j,k=1}^n [\alpha_j,\beta_k]g^{jk}.\end{aligned}$$ Taking the derivative $d_A$ yields $$\begin{aligned}
d_A([\alpha\cdot\beta])&=& \sum_{j,k=1}^n d_A([\alpha_j,\beta_k]g^{jk})\\
&=& \sum_{j,k=1}^n [d_A(\alpha_j),\beta_k]g^{jk} + [\alpha_j,d_A(\beta_k)]g^{jk} + [\alpha_j,\beta_k]d(g^{jk})\end{aligned}$$ By the properties of Type A coordinates and by , on $\partial M$ the above reduces to $$\begin{aligned}
\label{bc2.5}
d_A([\alpha\cdot\beta])|_{\partial M} = [d_A\alpha_n,\beta_n]+
[\alpha_n,d_A\beta_n] + [\alpha_n,\beta_n]d(g^{nn}).\end{aligned}$$ Using the adjoint matrix, we see that $$g^{nn}=\frac{\det(G(n|n))}{\det(G)}.$$ Combining the fact that $\det(G(n|n))=\det(G)$ on $\partial M$ and yields $$\begin{aligned}
\label{bc2.75}
\frac{\partial g^{nn}}{\partial x_n} &=& \frac{\left(\frac{\partial}{\partial x_n}\det(G(n|n)) \right)\det(G)-
\det(G(n|n))(\frac{\partial}{\partial x_n}\det(G))}{\det(G)^2}\\
&=& 0 \mbox{ on $\partial M$.}\end{aligned}$$ Hence, we have $$\label{bc3}
d_A([\alpha\cdot\beta])|_{\partial M}(\frac{\partial}{\partial x_n}) = [d_A\alpha_n(\frac{\partial}{\partial x_n}),\beta_n]+
[\alpha_n,d_A\beta_n(\frac{\partial}{\partial x_n})].$$
We will leave $d_A([\alpha\cdot\beta])|_{\partial M}$ for the moment and investigate what $d^{*}_A\alpha = d^{*}_A\beta = 0$ means in our coordinate system. We will calculate $d^{*}$ by using the Hodge star operator. Set $a:=\sqrt{\det(G)}=1/\sqrt{\det(G^{-1})}$. Using the boundary properties of the $\alpha_i$’s, $\beta_i$’s and $g^{ij}$’s, we have $$\begin{aligned}
-d^* \alpha &=& *d*\left(\sum_{i=1}^n\alpha_i dx_i\right) \\
&=& *d\left(a\sum_{i,j=1}^n(-1)^{j+1}\alpha_i g^{ji} dx_1\wedge\ldots\wedge\widehat{dx_j}\wedge\ldots\wedge dx_n \right)\\
&=& *(da)\left(\sum_{i,j=1}^n(-1)^{j+1}\alpha_i g^{ji} dx_1\wedge\ldots\wedge\widehat{dx_j}\wedge\ldots\wedge dx_n \right) + \\
& & \left(\sum_{i,j=1}^n \frac{\partial}{\partial x_i}(\alpha_ig^{ji})\right)*(adx_1\wedge\ldots\wedge dx_n) \\
&=& *[(da)(-1)^{n+1}\alpha_ng^{nn}dx_1\wedge\ldots\wedge dx_{n-1}] + \frac{\partial}{\partial x_n}(\alpha_n)g^{nn} + \\ & & \alpha_n\frac{\partial}{\partial x_n}(g^{nn})\\
&=& \frac{\partial a}{\partial x_n}\frac{1}{a}*(adx_1\wedge\ldots\wedge dx_n) + \frac{\partial}{\partial x_n}(\alpha_n)g^{nn} + \alpha_n\frac{\partial}{\partial x_n}(g^{nn})\\
&=& \frac{\partial a}{\partial x_n}\frac{1}{a} + \frac{\partial \alpha_n}{\partial x_n}g^{nn},\end{aligned}$$ where we used on the last line. Since $d_A^*\alpha =d^*\alpha - [A\cdot\alpha]$, we have, $$\begin{aligned}
\label{bc4}
-d^*_A\alpha|_{\partial M}&=& \frac{\partial \alpha_n}{\partial x_n} +
\frac{1}{a}\frac{\partial a}{\partial x_n}\alpha_n + [A\cdot\alpha]\\
&=& \frac{\partial \alpha_n}{\partial x_n} +
\frac{1}{a}\frac{\partial a}{\partial x_n}\alpha_n + [A_n, \alpha_n]
\label{bc5},\end{aligned}$$ where we used (\[bc1\]) and (\[bc2\]) (replacing $\beta$ with $A$) in the last line. Of course, an analogous statement holds for $\beta$ replacing $\alpha$.
We now revisit (\[bc3\]) and insert (\[bc5\]): $$\begin{aligned}
d_A([\alpha\cdot\beta])|_{\partial M}(\nu) &=&
[d_A\alpha_n,\beta_n](\partial/\partial x_n)+
[\alpha_n,d_A\beta_n](\partial/\partial x_n) \\
&=& [\frac{\partial\alpha_n}{\partial x_n},\beta_n]+[[A_n,\alpha_n],\beta_n]
\\ & & + [\alpha_n,\frac{\partial\beta_n}{\partial x_n}] +
[\alpha_n,[A_n,\beta_n]] \\
&=& -[\frac{1}{a}\frac{\partial a}{\partial x_n}\alpha_n + [A_n,\alpha_n]+d_A^*\alpha,
\beta_n]+[[A_n,\alpha_n],\beta_n] \\
& & - [\alpha_n,\frac{1}{a}\frac{\partial a}
{\partial x_n}\beta_n + [A_n,\beta_n]+d_A^*\beta] + [\alpha_n,[A_n,\beta_n]] \\
&=& -\frac{2}{a}\frac{\partial a}{\partial x_n}[\alpha_n,\beta_n]-[d^*_A\alpha,\beta_n]-[\alpha_n, d^*_A\beta] \\
&=& -2(n-1)H_{}[\alpha\cdot\beta]|_{\partial M}-[d^*_A\alpha,\beta(\nu)]-[\alpha(\nu), d^*_A\beta]\\
&=& -2(n-1)H_{}[\alpha\cdot\beta]|_{\partial M}.\end{aligned}$$ where we again used (\[bc2\]) on the second to last line, as well as the fact that $\alpha,\beta\in H_A$. The lemma is thus proven.
For future reference, we rewrite the second to last equation above $$d_A([\alpha\cdot\beta])|_{\partial M}(\nu) = -2(n-1)H_{}[\alpha\cdot\beta]|_{\partial M}-[d^*_A\alpha,\beta(\nu)]-[\alpha(\nu), d^*_A\beta],\label{nohor}$$ and note that it holds even if neither $\alpha$ nor $\beta$ is horizontal.
Define a linear map $$T_A: \Lie(\Gk)\to L^2_{k-\frac{5}{2}}(\kb|_{\partial M})$$ given by $$T_A(g)= d_A\Delta_A g(\nu) + 2(n-1)H_{}\Delta_A g.$$ To justify the target space, recall that $\Lie(\Gk)=H^{k+1}_{Dir}(\kb)$. So, for $g\in\Lie(\Gk)$, we have $d_A\Delta_A g\in H^{k-2}(\kb)$. By Theorem 9.3 in [@Palais], the restriction map from $H^{k-2}(\kb)=L^2_{k-2}\to L^2_{k-\frac{5}{2}}$ is continuous if $k-\frac{5}{2}>0$. From our assumption that $k>\frac{n}{2}+1$, we have for $n \ge 3$ $$k-\frac{5}{2} > \frac{n-3}{2}\ge 0.$$ If $n=2$, since $k$ is integral, the condition $k>\frac{2}{2}+1$ forces $k\ge 3$, and thus $k-\frac{5}{2}\ge\frac{1}{2}>0$. Thus, for $n\ge 2$, we see that $T_A$ is well-defined and a continuous operator. Define a set $\mathcal{L}_A\subseteq\Lie(\Gk)$ as $$\mathcal{L}_A:=\Span\{\mathcal{R}_A(\alpha,\beta):\alpha,\beta\in H_A\}$$ The previous lemma yields
\[bigcor\] The set $\mathcal{L}_A$ is contained in $\mathrm{ker}(T_A)$. In particular, since $T_A$ is not identically $0$, we have that $\overline{\mathcal{L}_A}$ is a proper subset of $\Lie(\Gk)$ (where the closure is taken in the $H^{k+1}$ norm).
For $g\in\mathcal{L}_A$, we have $g\in\mathrm{ker}(T_A)$ by the equation and Lemma \[bct\].
This theorem shows that the image of the curvature form of the Coulomb connection *at one fixed connection* $\nabla_A$ can never be dense in the gauge algebra, unlike the case in [@NR].
The Smooth $\mathcal{R}_A$ and $\mathrm{ker}(T_A)$ {#2.4}
--------------------------------------------------
While Theorem \[bigcor\] shows that $\overline{\mathcal{L}_A}$ cannot equal $\Lie(\Gk)$, the closure of the algebra generated by $\mathcal{L}_A$ still may equal $\Lie(\Gk)$. Indeed, equation is not closed under brackets as we will show in Proposition \[smooth1\]. In investigating the algebra generated by $\mathcal{L}_A$, we will restrict our attention to $C^\infty$ functions. Since $C^\infty$ functions are dense in our Sobolev spaces, we do not lose much generality in this restriction, although it will give a denseness result rather than a full Sobolev space result. So our goal will be to show that $\mathcal{L}_A\cap C^\infty$ algebraically generates $\Lie(\Gk)\cap C^\infty$. If this is the case, then the closure of the algebra generated by $\mathcal{L}_A$ will be all of $\Lie(\Gk)$.
We will show that $\mathcal{L}_A\cap C^\infty$ does algebraically generate $\Lie(\Gk)\cap C^\infty$, and thus the closure of the algebra generated by $\mathcal{L}_A$ will be all of $\Lie(\Gk)$, in the certain case where $P$ is the trivial bundle $\bar{O}\times K\to \bar{O}$ for a bounded open set $O\subseteq\mathbb{R}^n$ with smooth boundary and where the base connection $\nabla_{A_0}$ is the flat connection. In this case, $\Kb$ is isomorphic to $\bar{O}\times K\to \bar{O}$, and $\kb$ is isomorphic to $\bar{O}\times\mathfrak{k}\to \bar{O}$. We can view gauge transformations $g$ as $K$-valued functions on $\bar{O}$, gauge algebra elements $\psi$ as $\mathfrak{k}$-valued functions, and $\kb$-valued forms as $\mathfrak{k}$-valued forms.
As in Section \[1.1\], we denote the flat connection as $\nabla_0$. This means we should denote exterior differentiation by $d_0$, but since $\nabla_0 = d$ (as asserted in Section \[1.1\]), we will instead simply use $d$ without a subscript. Similiarly, we denote $d^*_0$ by $d^*$.
Our first step in showing that $\mathcal{L}_0\cap C^\infty$ algebraically generates $\Lie(\Gk)\cap C^\infty$ is to prove a converse to Lemma \[bct\]; that is, we will show that $$\mathrm{ker}(T_0)\cap C^\infty = \mathcal{L}_0\cap C^\infty.$$
To do this, we will consider slightly different sets than $\mathrm{ker}(T_0)$ and $\mathcal{L}_0$. Consider the operator $\tilde{T}_0:H^{k-1}(\kb)\to L^2_{k-\frac{5}{2}}(\kb|_{\partial M})$ given by $$\tilde{T}_0(f)= d f + 2(n-1)H_{} f.$$ Also, consider the set $\tilde{\mathcal{L}}_0$ defined as $$\tilde{\mathcal{L}}_0:=\Span\{[\alpha\cdot\beta]:\alpha,\beta\in H_0\}.$$ If $f=\Delta g$, note that $f\in\mathrm{ker}(\tilde{T}_0)$ if and only if $g\in\mathrm{ker}(T_0)$, and $f\in\tilde{\mathcal{L}}_0$ if and only if $g\in\mathcal{L}_0$ since $\Delta:H^{k+1}_{Dir}\to H^{k-1}$ is an isomorphism. Thus, we have $\mathrm{ker}(T_0)\cap C^\infty = \mathcal{L}_0\cap C^\infty$ if and only if $\mathrm{ker}(\tilde{T}_0)\cap C^\infty = \tilde{\mathcal{L}}_0\cap C^\infty$. We will prove the latter.
First we look at neighborhoods of the boundary of $O$ and show that all the smooth $\Psi$ that satisfy the boundary condition of Lemma \[bct\] are in $\tilde{\mathcal{L}_0}\cap C^\infty$.
\[lbo\] Let $O\subset\mathbb{R}^n$ be open and bounded with a smooth boundary and suppose $P=\overline{O}\times K\to \overline{O}$. Let $U\subseteq\overline{O}$ be open in the subset topology. Suppose that $U$ includes a part of the boundary $\partial O$, admits the Type B coordinates $\{y_1,\ldots,y_n\}$, and is the cube $(0,\delta)^{n-1}\times[0,\delta)$ under these coordinates. Let $\Psi\in \mathrm{ker}(\tilde{T_0})\cap C^\infty_c(U;\k)$. Then $\Psi\in\tilde{\mathcal{L}_0}\cap C^\infty_c(U;\k)$.
In what follows, we shorten “Dirichlet boundary conditions” to *DBC*. Also, viewing $U$ as the cube $(0,\delta)^{n-1}\times[0,\delta)$, a function $f\in C^\infty_c(U)$ has its support contained in $(\epsilon,
\delta-\epsilon)^{n-1}\times [0,\delta-\epsilon)$ for some $\epsilon>0$. The point is that $f$ need not vanish on the boundary $\{y_n=0\}$. Lastly, the set $C^\infty_c(U;\k))$ above is the set of all $\k$-valued smooth functions on $U$ with compact support.
Let $\{v_i\}$ be basis of $\k$. Then we can write $\Psi = \sum \psi_i\cdot v_i$. Since the basis elements are independent and $\Psi\in \mathrm{ker}(\tilde{T_0})$, we have that $d\psi_i(\nu)=-2(n-1)H_{}\psi_i$. Since $\mathfrak{k}$ is semisimple, each basis element $v_i$ can be written as a sum of commutators $v_i=\sum_{j=1}^{\alpha(i)}[f_j^i,g_j^i]$. Hence, we can write $\Psi$ as $$\Psi = \sum_i \sum_{j=1}^{\alpha(i)}\psi_i [f_j^i,g_j^i].$$ So without loss of generality we can assume $\Psi = \psi\cdot[A,B]$, where $A,B$ are fixed elements of $\k$ and $\psi\in C^\infty_c(U)$ and $$\label{psieq}
d\psi(\nu)=-2(n-1)H_{}\psi.$$
Coordinatize $U$ using Type B coordinates $\{y_1,\ldots,y_n\}$ under which the domain is the cube $(0,\delta)^{n-1}\times[0,\delta)$. Again let $b:=\sqrt{\det(h_{ij})}$, where $h_{ij}$ is the metric tensor of our chart.
Choose $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6\in\mathbb{R}$ so that $supp(\psi)\subset(\gamma_4,\gamma_5)^{n-1}
\times[0,\gamma_5)$ and $0<\gamma_1<\gamma_2<\gamma_3<\gamma_4<\gamma_5<\gamma_6<\delta$. We define a function $\phi$ by setting $$\begin{aligned}
\phi(y_1,\ldots,y_n) &:=& -I(y_1,\ldots,y_{n-2}, y_n)\eta(y_{n-1}) + \\
& & h^{nn}(y_1,\ldots,y_n)b(y_1,\ldots,y_n)^2\psi(y_1,\ldots,y_n),\end{aligned}$$ where $\eta$ is a bump function with $\eta\in C_c^{\infty}(\gamma_2,\gamma_3)$ and $\int_{\gamma_2}^{\gamma_3} \eta(s) ds =1$, and $I$ is defined by $$\begin{aligned}
I(y_1,\ldots,y_{n-2}, y_n)&=&\int_{\gamma_4}^{\gamma_5} h^{nn}(y_1,\ldots,y_{n-2},s, y_n)b(y_1,\ldots,y_{n-2},s, y_n)^2\\
& & \psi(y_1,\ldots,y_{n-2},s, y_n)ds.\end{aligned}$$ Then $\phi$ is smooth with compact support. However $h$ also has an additional property. Using Proposition \[tauy\], Equation , and noting $h^{nn}={1}/{h_{nn}}$, we have $$\begin{aligned}
\frac{\partial (h^{nn}b^2\cdot\psi)}{\partial y_n} &=&
2\sqrt{h^{nn}}\frac{\partial\sqrt{h^{nn}}}{\partial y_n}b^2\psi +
h^{nn} 2b\cdot\frac{\partial b}{\partial y_n}\cdot\psi +
h^{nn}b^2\cdot\frac{\partial\psi}{\partial y_n}\\
&=& h^{nn}b^2\left[2\left(\frac{1}{\sqrt{h^{nn}}}\frac{\partial\sqrt{h^{nn}}}
{\partial y_n} + \frac{1}{b}\cdot\frac{\partial b}{\partial y_n}\cdot
\right)\psi + \frac{\partial\psi}{\partial y_n}\right] \\
&=& 0 \mbox{ on $\partial O\cap U$}.\end{aligned}$$ Hence, differentiating under the integral yields $$\label{bco4}
\frac{\partial \phi}{\partial y_n} = 0\mbox{ on $\partial O\cap U$}.$$
Define $F:U\to \mathbb{R}$ as $$F(y_1,\ldots,y_n)=\int_0^{y_{n-1}} \phi(y_1,\ldots,y_{n-2},s,y_n)ds.$$ Then $F$ is smooth and $supp(F)\subset(\gamma_4,\gamma_5)\times(\gamma_1,\gamma_5)\times[0,\gamma_5)$ by our construction of $\phi$. In particular, the term $-I(y_1,\ldots,y_{n-2}, y_n)\eta(y_{n-1})$ was included in the definition of $\phi$ to make $F$ have compact support in the $y_{n-1}$ variable.[^3] Also, note that $F_{n-1} = \phi$ (where the subscript $n-1$ denotes we are taking the partial derivative of $F$ with respect to $y_{n-1}$). Also, by (\[bco4\]), differentiating under the integral sign yields $$\label{bco5}
F_n = 0\mbox{ on $\partial O\cap U$}.$$
We now construct another function $G:[0,\delta]^n\to\mathbb{R}$ by setting $$G(y_1,\ldots,y_n)=\prod_{i=1}^n v_i(y_i),$$ where each $v_i:[0,\delta]\to\mathbb{R}$ is constructed as follows: for $i \le n-2$, $v_i\in C^\infty_c(0,\delta)$, $v_i|_{[\gamma_4,\gamma_5]}\equiv 1$, and $supp(v_i)\subset (\gamma_3,\gamma_6)$; $v_{n-1}\in C^\infty_c(0,\delta)$, $v_{n-1}|_{[\gamma_4,\gamma_5]}(x)=x$, and $supp(v_{n-1})\subset (\gamma_3,\gamma_6)$; $v_n\in C^\infty_c([0,\delta))$, $v_n|_{[0,\gamma_5]}\equiv 1$, and $supp(v_n)\subset
[0,\gamma_6)$. Then $$G_{n-1}|_{(\gamma_4,\gamma_5)^{n-1}\times[0,\gamma_5]}\equiv 1,$$ and has compact support in $U$. One can verify that the support of the product $F_{n-1}\cdot G_{n-1}$ lies in $(\gamma_4,\gamma_5)^{n-1}
\times[0,\gamma_5)$, just like the support of $\psi$. Furthermore, we have on $(\gamma_4,\gamma_5)^{n-1}\times[0,\gamma_5)$ $$F_{n-1}\cdot G_{n-1}=h^{nn}b^2\psi,$$ and thus the equation holds everywhere. Now define 2-forms $\omega_1, \omega_2$ as $$\begin{aligned}
\omega_1 &=& F\cdot A(*^{-1}(dy_1\wedge dy_2\wedge\ldots\wedge dy_{n-2})) \\
\omega_2 &=& G\cdot B (*^{-1}(dy_1\wedge dy_2\wedge\ldots\wedge dy_{n-2}))\end{aligned}$$ for $n\ge 3$, and $$\begin{aligned}
\omega_1 &=& F\cdot A(*^{-1}(b^{-1}) \\
\omega_2 &=& G\cdot B (*^{-1}(b^{-1}))\end{aligned}$$ for $n=2$. Let $\alpha:=d^*\omega_1$ and $\beta:=d^*\omega_2$. Since $(d^*)^2=0$, we have $d^*\alpha = d^*\beta =0$. One can check that $$*(dy_1\wedge\ldots\wedge\widehat{dy_j}\wedge\ldots dy_n) = (-1)^{n+j}\frac{1}{b}\sum_{i=1}^nh_{ij}dx_i.%(-1)^{n-k}(-1)^{i+k}=(-1)^{n+i}
%*(dy_1\wedge\ldots\wedge\widehat{dy_i}\wedge\ldots dy_n) = b\sum_{k=1}^n(-1)^{n-k}|H^{-1}(i|k)|dx_k.$$ We have for $n\ge 2$ $$\begin{aligned}
\alpha &=& d^*(\omega_1) = (-1)^{2n+n+1}*d*(\omega_1) \\
&=& (-1)^{n+1}*d(F\cdot Ady_1\wedge dy_2\wedge\ldots\wedge dy_{n-2})\\
&=& (-1)^{n+1}*((-1)^{n-2}F_{n-1} A dy_1\wedge dy_2\wedge\ldots\wedge dy_{n-2}\wedge dy_{n-1} +\\
& & (-1)^{n-2} F_n A dy_1\wedge dy_2\wedge\ldots\wedge dy_{n-2}\wedge dy_{n}) \\
&=& -\frac{1}{b}\cdot\sum_{i=1}^n (F_{n-1}h_{in} - F_n h_{i(n-1)})dy_iA.
%&=& -(b\cdot\sum_{k=1}^n (-1)^{n+k}(F_{n-1}|H^{-1}(n|k)|+ F_n |H^{-1}(n-1|k)|)dy_i)A.\end{aligned}$$ Note that in our Type B coordinates we have $h_{in}=0$ for $i<n$ everywhere. So since $F_n = 0$ on the boundary by (\[bco5\]), $\alpha$ satisfies DBC Similarly, $$\beta = -\frac{1}{b}\cdot\sum_{j=1}^n (G_{n-1}h_{jn} - G_n h_{j(n-1)})dy_jB
%\beta &=& -(b\cdot\sum_{k=1}^n (-1)^{n+k}(G_{n-1}|H^{-1}(n|k)|+ G_n |H^{-1}(n-1|k)|)dy_i)B$$ and $\beta$ also satisfies DBC. Indeed, as above $h_{jn}=0$ for $j<n$ everywhere. Also, since $v_n(y_n)$ is constant on $[0,\gamma_5]$, we have $\left.G_n\right|_{(0,\delta)^{n-1}\times[0,\gamma_5)}=0$, and thus in particular $\left.G_n\right|_{\partial O}=0$.
To calculate $[\alpha\cdot\beta]$, we first note that by the definition of matrix inverses, we have $$\label{inverses}
\sum_{j=1}^n h^{kj}h_{ji} = \delta_{ik}.$$ Using the above, we have $$\begin{aligned}
[\alpha\cdot\beta] &=& \frac{1}{b^2}\left(\sum_{i,j=1}^n(F_{n-1}h_{in} - F_nh_{i(n-1)})\cdot(G_{n-1}h_{jn}- G_nh_{j(n-1)})h^{ij}\right)[A,B] \\
&=& \frac{1}{b^2}\left(\sum_{i,j=1}^n F_{n-1}G_{n-1}h_{in}h_{jn}h^{ij}-F_{n-1}G_n h_{in}h_{j(n-1)}h^{ij} -\right.\\
& & F_n G_{n-1}h_{i(n-1)}h_{jn}h^{ij}+ F_n G_n h_{i(n-1)}h_{j(n-1)}h^{ij}\Bigg)[A,B] \\
&=& \frac{1}{b^2}\left(\sum_{i=1}^n F_{n-1}G_{n-1}h_{in}\delta_{in}-F_{n-1}G_n h_{in}\delta_{i(n-1)} -\right.\\
& & F_n G_{n-1}h_{i(n-1)}\delta_{in}+ F_n G_n h_{i(n-1)}\delta_{i(n-1)}\bigg)[A,B] \\
&=& \frac{1}{b^2}\left( F_{n-1}G_{n-1}h_{nn}-F_{n-1}G_n h_{(n-1)n} -\right.\\
& & F_n G_{n-1}h_{n(n-1)}+ F_n G_n h_{(n-1)(n-1)}\bigg)[A,B] \\
&=& \frac{1}{b^2}\left( F_{n-1}G_{n-1}h_{nn}+ F_n G_n h_{(n-1)(n-1)}\right)[A,B],\end{aligned}$$ where the last line is justified by the fact that $h_{(n-1)n}=0$ everywhere in Type B coordinates. As noted previously, we have $\left.G_n\right|_{(0,\delta)^{n-1}\times
[0,\gamma_5)} = 0$. Since $supp(F)\subset(\gamma_4,\gamma_5)\times(\gamma_1,\gamma_5)\times[0,\gamma_5)$, we have $F_n|_{(0,\delta)^{n-1}\times[\gamma_5,1]} = 0$. Hence, $F_nG_n\equiv 0$. So, continuing the above, we have $$\begin{aligned}
[\alpha\cdot\beta] &=& \frac{1}{b^2}(F_{n-1}G_{n-1}h_{nn})[A,B] \\
%&=& (F_{n-1}G_{n-1}\frac{\det(H^{-1})}{h^{nn}})[A,B] \\
&=& \frac{1}{b^2}(h^{nn}b^2\psi h_{nn})[A,B] \\
&=& \psi[A,B],\end{aligned}$$ where on the last line we used the fact that $h_{kn}=h^{kn}=0$ for Type B coordinates and thus $h_{nn}h^{nn}=1$ by . In sum, $\alpha,\beta\in H_0$, and $[\alpha\cdot\beta]=\psi[A,B]$, proving $\Psi=\psi[A,B]\in \tilde{\mathcal{L}}_0\cap C^\infty$, as desired.
Next we check that the previous result holds for functions $\Psi$ with compact support.
\[noboundary2\] Let $O\subset\mathbb{R}^n$ be a bounded open set with a smooth boundary and suppose $P=\overline{O}\times K\to \overline{O}$. Then $C^\infty_c(O;\k)=\Span\{[\alpha\cdot\beta]: \alpha,\beta\in C^\infty_c(\Lambda^1(O;\k)), d^*\alpha =d^*\beta = 0\}\subset\tilde{\mathcal{L}}_0\cap C^\infty$.
Given $f\in C^\infty_c(O;\k)$, one can cover the support of $f$ with finitely many cubes, and reconstruct $f$ as a product $[\alpha\cdot\beta]$ on each cube in a fashion similar to the process of Lemma \[lbo\]. The construction here is simpler since boundary conditions do not come into play. In particular, one can use the standard coordinates of $\mathbb{R}^n$ whose metric tensor $\{g_{ij}\}$ is of course the identity matrix, greatly simplifying the work. Details for the $n=3$ case can be found in [@GrycThesis].
We now combine Lemma \[lbo\] and Propostion \[noboundary2\] to get our desired global result.
\[curve\_im\] Let $O\subset\mathbb{R}^n$ be a bounded open set with a smooth boundary and suppose $P=\overline{O}\times K\to \overline{O}$. Then $\mathrm{ker}(\tilde{T_0})\cap C^\infty=\tilde{\mathcal{L}}_0\cap C^\infty$, and thus $\mathrm{ker}(T_0)\cap C^\infty=\mathcal{L}_0\cap C^\infty$.
The backward direction has already been shown in Lemma \[bct\]. For the forward direction, suppose $f\in\mathrm{ker}(\tilde{T_0})\cap C^\infty$, and thus satisfies $df(\nu) = -2(n-1)H_{} f \mbox { on
$\partial O$}$. There exists a finite cover $\{U_k\}_{k=0}^m$ of $\overline{O}$ that satisfies the following: $\overline{U}_0\subseteq O$, $\{U_k\}_{k=1}^m$ covers the boundary $\partial O$ and each $U_k$ for $k\ge 1$ is a cube in Type B coordinates, and there is a partition of unity $\{\lambda_k\}_{k=0}^m$ subordinate to $\{U_k\}_{k=0}^m$ so that $d\lambda_k(\nu)=0$ on the boundary. A construction of such a partition of unity can be found in [@GrycThesis] for $n=3$ which can easily be generalized for $n\ge 2$.
With such a partition of unity, we have $d(\lambda_k f)(\nu) = -2(n-1)H_{}\lambda_k f \mbox { on $\partial O$}$. So, by Lemma \[lbo\] and Proposition \[noboundary2\] there exists $\{\alpha_i^k\}, \{\beta_i^k\}$ such that each $\alpha_i^k,\beta_i^k\in C^\infty_c(\Lambda^1(U_k\otimes\k))$, $d^*\alpha_i^k=
d^*\beta_i^k=0$, $\alpha_i^k,\beta_i^k$ satisfy DBC, and $\lambda_k\cdot f = \sum_{i=1}^{n}[\alpha_i^k\cdot\beta_i^k]$ on $U_k$. Extending the $\alpha_i^k$’s and $\beta_i^k$’s by zero, we have $\lambda_k\cdot f = \sum_{i=1}^{n}[\alpha_i^k\cdot\beta_i^k]\in \tilde{\mathcal{L}}_0\cap C^\infty$. Since $\tilde{\mathcal{L}}_0$ is a span, $f=\sum_{k=1}^m (\lambda_k\cdot f)\in\tilde{\mathcal{L}}_0$ also, as desired. So, $\mathrm{ker}(\tilde{T_0})\cap C^\infty=\tilde{\mathcal{L}}_0\cap C^\infty$, and thus $\mathrm{ker}(T_0)\cap C^\infty=\mathcal{L}_0\cap C^\infty$ by the note in the beginning of this subsection.
The Generation of the Smooth Gauge Algebra {#2.5}
------------------------------------------
In this section we will use brackets of the image of the curvature form to get every smooth function in $\Lie(\Gk)$ for the special case $P=\overline{O}\times K\to \overline{O}$. The main tool will be Lemma \[curve\_im\]. The first thing we must do is see how the equation $$\label{mainderivative2}
d(\Delta g)(\nu)=-2(n-1)H_{}\Delta g.$$ changes when we introduce brackets. More specifically, recall that if $g\in\mathcal{L}_0$, then Lemma \[curve\_im\] says $g$ satisfies above. We want to know how changes if $g$ is replaced by $[g_1,g_2]$, for $g_i\in \mathcal{L}_0$. Indeed, we have
\[smooth1\] Suppose $g_1,g_2\in\mathcal{L}_0$. Then we have $$d(\Delta([g_1,g_2]))(\nu)=-2(n-1)H_{}\Delta[g_1,g_2] + 3[\Delta g_1, d g_2(\nu)]
+3[d g_1(\nu), \Delta g_2].$$
We state the above proposition for all elements of $\mathcal{L}_0$, not just the smooth elements because the proposition holds in the general case. However, the use of the proposition in this paper will be just for the smooth case.
First note that $$\begin{aligned}
\Delta([g_1,g_2]) = [\Delta g_1,g_2]+[g_1,\Delta g_2]-2[d g_1\cdot d g_2].\end{aligned}$$ So we have $$\begin{aligned}
d(\Delta([g_1,g_2]))(\nu) &=& d([\Delta g_1,g_2]+[g_1,\Delta g_2]-2[d g_1\cdot d g_2])(\nu)\nonumber\\
&=& [d(\Delta g_1)(\nu),g_2] + [\Delta g_1,dg_2(\nu)] + [dg_1(\nu),\Delta g_2] \nonumber\\
& & + [g_1,d(\Delta g_2)(\nu)] - 2d([d g_1\cdot d g_2])(\nu).\label{brack1}\end{aligned}$$ By Lemma \[curve\_im\], we have $$\label{brack2}
d(\Delta g_i)(\nu) = -2(n-1)H_{}\Delta g_i .$$ By equation which follows the proof of Lemma \[bct\], we see that if $\alpha,\beta\in H^k_{Dir}(\kb)$ but are not necessarily in $H_A$, we have $$d_A([\alpha\cdot\beta])(\nu) = -2(n-1)H_{}[\alpha\cdot\beta] - [d_A^*\alpha,\beta(\nu)] - [\alpha(\nu), d_A^*\beta].$$ Setting $\alpha=d g_1$ and $\beta=d g_2$ above yields $$\label{brack3}
-2d([d g_1\cdot d g_2])(\nu) = -2(-2(n-1)H_{}[d g_1\cdot d g_2] - [\Delta g_1, d g_2(\nu)] - [d g_1(\nu),\Delta g_2]).$$ Inserting and into , we have $$\begin{aligned}
d(\Delta([g_1,g_2]))(\nu) &=& -2(n-1)H_{}[\Delta g_1,g_2] + [\Delta g_1,dg_2(\nu)] + [dg_1(\nu),\Delta g_2] + \\
& & - 2(n-1)H_{}[g_1,\Delta g_2] + \\ & &-2(-2(n-1)H_{}[d g_1\cdot d g_2] - [\Delta g_1, d g_2(\nu)] - [d g_1(\nu),\Delta g_2])\\
&=& -2(n-1)H_{}([\Delta g_1,g_2]+[g_1,\Delta g_2]-2[d g_1\cdot d g_2]) + \\
& & 3[\Delta g_1, d g_2(\nu)] + 3[d g_1(\nu),\Delta g_2] \\
&=& -2(n-1)H_{}\Delta([g_1,g_2]) + 3[\Delta g_1, d g_2(\nu)] + 3[d g_1(\nu),\Delta g_2],\end{aligned}$$ as desired.
We will now show that the new term in Proposition \[smooth1\] is actually very general.
\[smooth2\] Let $F$ be a smooth $\mathfrak{k}$-valued function on $\partial O$. Then there exists smooth $\k$-valued functions $g_i,h_i\in\mathcal{L}_0$ such that $$d(\Delta(\sum_i [g_i,h_i]))(\nu)+2(n-1)H_{}\Delta(\sum_i[g_i,h_i])= F.$$
Since $\mathfrak{k}$ is semi-simple, there exists $A_i,B_i,C_i\in
\mathfrak{k}$ that $$F =\sum_i f_i[[A_i,B_i],C_i]$$ for some real valued smooth functions $f_i$. So, without loss of generality, assume that $F=f[[A,B],C]$ for some $A,B,C\in\k$.
Take any non-negative, nonzero, real-valued $\phi\in C^\infty_c(O)$. By the Strong Maximum Principle, we have $G\phi > 0$ for the interior of each connected component of $O$, and thus on the whole interior of $O$ (note that our definition of the Laplacian as $\Delta=d^*d$ means that in local coordinates $\Delta=-\sum_i \frac{\partial}{\partial x_i}$). Thus, we can apply Lemma 3.4 of [@GT] to get $$\frac{\partial (G\phi)}{\partial \nu} > 0. %see 2/17/08 notes for all this$$ In particular, $d(G\phi)(\nu)$ never vanishes. We set $h:= G\phi\cdot C$. Since $\Delta h = \phi\cdot C$ has compact support, $h\in\mathcal{L}_0$ by Lemma \[curve\_im\].
Let $\{U_k\}_{k=0}^{m}$ be an open cover of $O$ such that $\{U_k\}_{k=1}^{m}$ covers $\partial O$ and $U_k$ are cubes in Type A coordinates for $k\ge 1$. Let $\{\lambda_k\}_{k=1}^m$ be the corresponding partition of unity for the cover $\{U_k\cap\partial O\}_{k=1}^m$ of the boundary. We set $$f_k:=\lambda_k\cdot \frac{f}{3d(G\phi)(\nu)}.$$ In the cube of $U_k$, suppose the $x_n$ interval is $[0,a]$. Choose a $C^
\infty$ function $\eta:[0,a]\to[0,1]$ such that $\eta|_{[0,a/4]}\equiv
1$ and $supp(\eta)\subseteq([0,a/2])$. We can extend $f_k$ to a function $\tilde{f}$ on $U_k$ by $$\tilde{f}(x_1,\ldots,x_n) = f_k(x_1,\ldots,x_{n-1})\eta(x_n)\exp(-2(n-1)H_{}(x_1,\ldots,x_{n-1})x_n).$$ Note that the support of $\tilde{f}$ lies in $U_k$, so $\tilde{f}$ is a function on all of $\overline{O}$. On $U_k$, we have $$\begin{aligned}
d\tilde{f}(\nu) &=& \frac{\partial}{\partial x_n}\bigg|_{x_n=0}f_k(x_1,\ldots,x_{n-1})\eta(x_n)\exp(-2(n-1)H_{}(x_1,\ldots,x_{n-1})x_n) \\
&=& -2(n-1)H_{}(x_1,\ldots,x_{n-1})f_k(x_1,\ldots,x_{n-1})\eta(x_n)\cdot \\
& & \exp(-2(n-1)H_{}(x_1,\ldots,x_{n-1})x_n)\\
&=& -2(n-1)H_{} \tilde{f}.\end{aligned}$$ By Lemma \[curve\_im\], the above shows that $G\tilde{f_k}[A,B]\in\mathcal{L}_0$. Let $g=\sum_{k=1}^m G\tilde{f_k}[A,B]$. We now verify that $g$ and $h$ were well-chosen. By Proposition \[smooth1\] and since $\Delta h|_{\partial O}=\phi C|_{\partial O}\equiv 0$, $$\begin{aligned}
d(\Delta([g,h]))(\nu) + 2(n-1)H_{}\Delta[g,h] &=& 3[\Delta g, d h(\nu)]
+3[d g(\nu), \Delta h] \\
&=& 3[\Delta g, d h(\nu)] \\
&=& 3[\sum_k f_k[A,B], dG\phi(\nu)\cdot C] \\
&=& 3[(\sum_k \lambda_k)\frac{f}{3dG\phi(\nu)}[A,B],dG\phi(\nu)C] \\
&=& f[[A,B],C]]=F,\end{aligned}$$ proving the proposition.
We are now at the point where we can prove the key lemma for our main theorem. Let $\mathcal{F}$ be the linear space spanned by $\mathcal{L}_0$ and $[\mathcal{L}_0,\mathcal{L}_0]$.
\[mainthm\] Suppose our principal bundle is $P=\overline{O}\times K\to \overline{O}$, where $O\subseteq\mathbb{R}^n$ for $n\ge 2$ is open, bounded and has smooth boundary. Suppose $g\in\mathrm{Lie}(\Gk)$ and is $C^\infty$. Then $g\in\mathcal{F}\cap C^\infty$.
Let $g\in\mathrm{Lie}(\Gk)\cap C^\infty$. Recall our linear map $T_0:C^\infty(O;\mathfrak{k})\to C^\infty(\partial O;\mathfrak{k})$ defined as $$T_0(f) = d(\Delta f)(\nu) + 2(n-1)H_{}\Delta f.$$ Set $u:=T_0(g)$. By Proposition \[smooth2\], there exists a smooth function $f\in\mathcal{F}$ such that $T_0(f)=u$. Since $T_0$ is linear, we have that $T_0(g-f)=0$. By Lemma \[curve\_im\], we know that $g-f\in\mathrm{Span(Im}(\mathcal{R}_0))\subseteq\mathcal{F}$. Hence, $g=f+(g-f)\in\mathcal{F}$, as we desired.
The preceeding lemma gives us our main result.
\[lastcor\] Consider the trivial principal bundle $\overline{O}\times K\to \overline{O}$, where $O\subseteq\mathbb{R}^n$ for $n\ge 2$ is open, bounded, and has smooth boundary. Let $k>n/2+1$, and suppose $\nabla_{A_0}=\nabla_0$. The restricted holonomy group $(\Hko)^0(\nabla_0)$ with base point $\nabla_0$ of the Coulomb connection of the associated bundle $\Ck\to\Ck/\Gk$ is dense in the connected component of the identity of $\Gk$.
Before we prove this theorem, we should mention what we mean by “holonomy group.” We define $\Hko(\nabla_0)$ the the same way it would be definied in finite dimensions. That is $g\in\Hko(\nabla_0)$ if and only if $\nabla_0\cdot g$ can be connected to $\nabla_0$ by a horizontal path in $\Ck$. It has been shown that with this definition, $\Hko(\nabla_0)$ is a Banach Lie group, and the restricted holonomy group $(\Hko)^0(\nabla_0)$ is also a Banach Lie group (for the statement of this theorem, see [@Vas]).
This follows directly from Lemma 7.6 and Proposition 7.7 in [@NR]. Specifically, Lemma 7.6 and the beginning of the proof of Proposition 7.7 of [@NR] imply that every element of $\mathcal{F}$ is the tangent vector to a curve in $(\Hko)^0(\nabla_0)$. Then Proposition 7.7 of [@NR] tells us that $(\Hko)^0(\nabla_0)$ is dense in the connected component of the identity of $\Gk$ since $\mathcal{F}$ is dense in $\mathrm{Lie}(\Gk)$, completing the proof.
In sum, we used the image of the curvature $\mathcal{R}$ of the Coulomb connection to tell us about the Lie algebra of the holonomy group $\Hko$. The fact that this image generates the entire holonomy group is a well-known theorem in finite dimensions. A version of this theorem also holds in the infinite-dimensional case, as proved in [@Magnot].
[^1]: *Present address:* Department of Mathematics, Morehouse College, 830 Westview Dr, Atlanta, GA 30314
[^2]: For those also reading [@GrycThesis] the $H_{}$ defined here is $1/2$ times the $\tau$ defined in [@GrycThesis].
[^3]: I thank my advisor, Prof. Leonard Gross, for his ideas in making the integral function $F$ have compact support.
|
---
abstract: 'We study the properties of a mobile hole in the $t-J$ model on the square lattice by means of variational Monte Carlo simulations based on the entangled-plaquette ansatz. Our energy estimates for small lattices reproduce available exact results. We obtain values for the hole energy dispersion curve on large lattices in quantitative agreement with earlier findings based on the most reliable numerical techniques. Accurate estimates of the hole spectral weight are provided.'
author:
- Fabio Mezzacapo
title: 'Variational study of a mobile hole in a two-dimensional quantum antiferromagnet using entangled-plaquette states'
---
Introduction
============
Since the fundamental suggestion made by Anderson in 1987, the two-dimensional (2D) $t-J$ model ($tJ$M) has been regarded as an effective hamiltonian description of the basic properties of the superconducting copper oxides.[@And87; @Z88] Essential physical features of the insulating copper-oxide planes of these compounds are successfully reproduced by the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model (AHM),[@Man91] to which the $tJ$M reduces when holes are absent. On the other hand, such insulating planes can turn superconducting if doped with mobile holes, whose presence is accounted for in the $tJ$M by adding to the AHM a nearest-neighbor hole-hopping term. Hence, the $t-J$ Hamiltonian is $$H=-t\sum_{\langle i,j \rangle,s}(\overline{c}^+_{i,s}\overline{c}_{j,s}+h.c.)+J\sum_{\langle i,j \rangle}(\mathbf{S}_i \cdot \mathbf{S}_j - \frac{1}{4}\hat{n}_i\hat {n}_j)
\label{eq:ham}$$ where the summations run over nearest-neighbor sites on the square lattice, $t$ is the hole hopping amplitude, and $J$ the antiferromagnetic coupling between site $i$ and $j$; $\overline{c}^+_{i,s}$ creates an electron, at site $i$, of spin projection (e.g., along the $z$ axis) $s=\pm\frac{1}{2}$ avoiding double occupancy. Therefore, in terms of the usual fermionic creation operator $\overline{c}^+_{i,s}=\hat{c}^+_{i,s}(1- \hat{n}_{i,-s})$; $\mathbf{S}_i$ and $\hat{n}_i=\hat{n}_{i,s}+\hat{n}_{i,-s}$, with $\hat{n}_{i,\alpha}=\hat{c}^+_{i,\alpha}\hat{c}_{i, \alpha}$, are the spin-$\frac{1}{2}$ and the number operator respectively.
Despite the simple form of the above Hamiltonian, its study constitutes, even in presence of a single hole, a very difficult problem. Exact diagonalization (ED), possible only for small system sizes (i.e., up to $N\sim26$ sites),[@Dag91; @Bar92; @Poi93] approximated analytical treatments,[@Kan89; @Mar91; @Man91a; @Hor91] as well as numerical approaches of various type,[@Mas91; @Mas92; @Mas94; @Mur00] have been largely employed to gain insight into the physics of the one-hole $tJ$M. Specifically, quantum Monte Carlo (QMC) techniques based on imaginary time projection furnish, even for large systems, accurate (exact in principle) results at half filling (i.e., no-hole case) or for $t$=0 (i.e., static-hole case).[@Mas94; @Mur00] However, when $t \neq 0$ these methods are affected by the sign problem, for which only essentially uncontrolled workarounds (e.g., fixed-node approximation) exist. Furthermore, being a 2D one, the model of our interest cannot be tackled, due to the unfavorable scaling of the required computational resources, by using the Density matrix renormalization group (DMRG),[@Whi92] the accepted method of choice for 1D lattice models. Variational Monte Carlo (VMC) schemes do not suffer from the sign problem; on the other hand they are not strictly exact, their accuracy depending on the particular choice of the trial wave function (WF). For the $tJ$M with a single hole, specific variational WF’s [@Mas91; @Mas92] have been designed affording estimates of the hole dispersion relation in reasonable agreement with those, generally more accurate, obtained by QMC.
Recently, the suggestion was made of combining DMRG and Monte Carlo in order to design novel variational algorithms for tensor-network based ansatze applicable to systems of spatial dimension higher than one.[@San07; @Sch0810] The WF introduced by us,[@eps] founded on entangled-plaquette states (EPS), has been successfully employed to study ground-state (GS) properties of lattice spin models with open or periodic boundary conditions (PBC). Such ansatz has been proven to yield estimates of physical observables of accuracy at least comparable to that obtainable with the best alternative techniques or trial WF’s.[@eps; @cps; @eps1; @cps1] The EPS WF can be efficiently optimized for plaquettes of a given size, as well as systematically improved by adding more plaquettes and/or increasing their size.[@eps; @cps]
In this paper we apply the EPS ansatz to investigate the dynamics of a hole in a quantum antiferromagnet described in Eq. (\[eq:ham\]). Specifically, we compute the hole energy dispersion curve $E(\k)$, as well as the spectral weight of the quasi-hole state, for lattice sizes up to $N = 256$ sites in the parameter-range $0.4 \le \frac{J}{t} \le 2.0$ (the $tJ$M is believed to be relevant to the cuprate superconductors for a value of $\frac{J}{t} \sim 0.4$). From the methodological standpoint, this study is considerably more challenging than that of the various (un)frustrated spin models to which the EPS ansatz has been recently applied with success.[@eps; @eps1]
The main goal of this work is to demonstrate the accuracy of the EPS WF in describing the property of a mobile hole in the $tJ$M. This is achieved by a systematic comparison of our results for the energy dispersion curve and the hole spectral weight with estimates computed by different approaches. Furthermore, besides the mentioned methodological aspects we offer extremely accurate results for the hole spectral weight in the so called “physical region” (i.e.,$ \frac{J}{t}\sim 0.4$) where the accuracy of QMC data, the most reliable ones obtained so far, becomes less satisfactory.[@Mur00] Assessing the reliability of the EPS ansatz in a case (i.e., the $tJ$M with one mobile hole) for which a large number of results obtained with alternative trial states, or other methodologies are available in literature, is a fundamental step also in view of future applications of our WF to the long-standing problem of understanding the phase diagram of the $tJ$M at finite hole concentration. The accuracy of the estimates presented here is clearly not a proof of the reliability of our variational choice when more holes are present, it constitutes, however, the necessary condition to consider the EPS WF, with no need of essential adjustments, a promising option for the investigation of the finite hole concentration scenario. A direct study of the latter problem, given the paucity of reliable results obtained in the past, would render the assessment of the trustability of our ansatz problematic leaving, on the other hand, room for dubitative, definitely well-posed, questions such as: [*Why results at finite hole concentration should be taken seriously if the reliability of the EPS WF has not even been proved in the simpler single hole case ?*]{}
On using a minimal EPS ansatz based on $N$ $2\times 2$ entangled plaquettes, we obtain estimates of the hole dispersion curve in qualitative and semi-quantitative agreement with those computed by QMC. We achieve quantitative agreement, also for other observables, with an EPS WF based on plaquettes comprising $9$ lattice sites.
The accuracy of our findings for one hole confirms the flexibility of the EPS ansatz, which can be applied with no substantial modifications to a large class of lattice models. More importantly, the EPS WF used in this work is easily generalizable to the case of a finite hole concentration, where the sign problem renders QMC inapplicable, and the generalization of different trial states employed for one or two holes appears quite complicated.[@Mas91; @Mas92]
Variational wave function
=========================
Let us consider a square lattice of $N$ sites comprising one electron per site, a generic WF can be written as $$|\Phi\rangle=\sum_{\mathbf{n}}W(\mathbf{n})|\mathbf{n} \rangle$$
where $|\mathbf{n} \rangle = |n_1,n_2, \ldots , n_N\rangle$, $n_i $ is the eigenvalue of $
\sigma^z_i $ and $W(\mathbf{n})$ is the weight of a configuration of the system. The EPS ansatz is constructed as follows:
i\) Cover the systems with $M$ plaquettes in such a way that $l$ sites labelled by $n_{1,P},n_{2,P},\ldots,n_{l,P}=\n_P$ belong to the $P_{th}$ plaquette.
ii\) Express the weight of a global spin configuration as a product of variational coefficients $C_1^{\n_1},C_2^{\n_2},\ldots,C_M^{\n_M}$ in biunivocal correspondence to the particular configuration (e.g., spin state along the $z$ axis of the plaquette sites) of the plaquettes. Hence, taking into account explicitly the bipartite nature of the spin-$\frac{1}{2}$ square Heisenberg antiferromagnet at half filling: $$\langle \n | \Phi \rangle=W(\n)=(-1)^{F(\n)}\mathcal{P}(\n)
\label{coeffh}$$ being $F(\n)$, according to the Marshall-sign rule, the number of “down” spins in one of the two sublattices[@Mar55] and $$\mathcal{P}(\n)=\prod_{P=1}^M C_P^{\n_{P}}$$ the crucial part of the ansatz.
Analogously, in presence of a single hole, our EPS WF is defined via $$\langle \m | \psi_{\mathbf{k}} \rangle=W(\m)=%\sum_{\mathbf{R}_h}
e^{-i\mathbf{k}\mathbf{R}_h}(-1)^{F(\m)}\mathcal{P}(\m);
\label{coeffk}$$ here $\m$ refers to the generic configuration containing one hole and $N-1$ electrons, $\mathbf{k}$ and $\mathbf{R}_h$ are the hole momentum and the spatial position of the empty site respectively. It is worth mentioning that the trial states adopted in this work naturally incorporate, for plaquettes including more than one site, electron-electron as well as electron-hole correlations. We estimate GS energies of the Hamiltonian in Eq. (\[eq:ham\]) via variational optimization of the states (\[coeffh\]) at half filling and (\[coeffk\]) in presence of a single hole. Independent optimizations are carried on for each $\mathbf{k}$ value by Monte Carlo sampling both the energy and its derivatives with respect to the varational parameters (i.e., the $C^{\n_{P}}_P$’s). Further details concerning the variational family of EPS and the numerical technique adopted to minimize the energy are given in Refs. and . Although the EPS ansatz is exact when a single plaquette as large as the system is employed, accurate results are already obtainable by using $N$ entangled (i.e., overlapping) plaquettes comprising $4$ sites, the accuracy of the estimates being sensibly improvable increasing the plaquette size. Specifically, we compute the hole energy dispersion curve $$E(\mathbf{k})=E^{tJ\text{M}}(\mathbf{k})-E^{\text{AHM}},
\label{eq:disp}$$ defined by the difference between the $\k$-dependent GS energy of the one-hole $tJ$M and that of the AHM, and the hole spectral weight $$Z(\mathbf{k})=|\langle\psi_{\mathbf{k}}|\overline{c}_{\mathbf{k}}|\Phi\rangle|^2,
\label{eq:zeta}$$ where $|\psi_{\mathbf{k}}\rangle$ ($|\Phi\rangle$) is the normalized GS of the one (zero) -hole system.
We compare the EPS results with exact ones for a $4\times4$ lattice and with the most accurate available in literature, for larger systems. In this case the main features of the hole dispersion relation, namely the characteristic shape with a global minimum at $\mathbf{k}=(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ and a nearly flat region around $\mathbf{k}=(\pm \pi,0)$, are very well reproduced with an ansatz based on $2\times2$ plaquettes, while the increasing of the plaquette size to $3\times3$ gives rise to an approximately rigid shift, towards lower energies, of the band and an almost perfect agreement (not reachable, to our knowledge, with other variational WF’s) with previous QMC studies for all the $\frac{J}{t}$ values considered in this work. Moreover, with the $3\times3$ EPS, accurate estimates of the hole spectral weight are obtained, on the same footing, via the Monte Carlo method.
Results
=======
The GS energies of the one-hole $tJ$M on a $4\times4$ lattice can be obtained, given the small system size, by using a single plaquette comprising all the lattice sites. In this case the EPS ansatz is exact, the error bar of our estimates being only due to the limited number of configurations sampled, and reproduces, as expected, ED results.[@Bar92]
Clearly, for larger systems, the use of a single plaquette is not feasible, therefore, it is important to asses the accuracy of our WF for smaller plaquette size. The error relative to the exact result: $E^{tJ\text{M}}(\pm\frac{\pi}{2},\pm\frac{\pi}{2})=-21.161$ of our estimate of the global energy minimum at $\frac{J}{t}=0.5$ is $\sim 3$% for the $2\times2$ EPS ansatz, decreasing down to less than $0.2$% when $3\times3$ plaquettes are used. For this plaquette size, our results compare favorably to those obtained with different trial states, even when larger lattices are considered. For example, the EPS energy when $\mathbf{k}=(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ on a $8\times8$ lattice at $\frac{J}{t}=0.4$ is $-78.463(7)J$, while the WF including both spin-spin and momentum dependent spin-hole correlations adopted in Ref. yields a value of $-76.753(14)J$ (i.e., $\sim 2$% higher).
[![(color online). Hole energy dispersion curve for a lattice of $256$ sites with PBC at $\frac{J}{t}=2.0$. EPS estimates (triangles) are computed using square plaquettes of 9 sites. QMC results (crosses) are also shown for comparison.[@Mur00][]{data-label="fig:1"}](fig1 "fig:")]{}
[![(color online). Hole energy dispersion curve for a lattice of $256$ sites with PBC at $\frac{J}{t}=0.8$. EPS estimates (triangles) are computed using square plaquettes of 9 sites. QMC results (crosses) are also shown for comparison.[@Mur00][]{data-label="fig:2"}](fig2 "fig:")]{}
[![(color online). Hole energy dispersion curve for a lattice of $256$ sites with PBC at $\frac{J}{t}=0.4$. EPS estimates are computed using square plaquettes of 4 (circles) or 9 (triangles) sites. GFMC results (crosses)[@Mas94] are also shown for comparison.[]{data-label="fig:3"}](fig3 "fig:")]{}
As opposed to the case in which more holes are present and the sign problem is particularly severe, QMC techniques based on imaginary time projection, for the one-hole $tJ$M, provide extremely accurate results, to our knowledge the most accurate in literature, when, due to the large system size, the ED of the Hamiltonian is not possible. Hence, a comparison of the EPS WF against these approaches is in order.
The hole energy dispersion curve, defined in Eq. (\[eq:disp\]), is shown in Fig. \[fig:1\] for the $16\times16$ lattice at $\frac{J}{t}=2.0$. Our estimates (triangles), computed with the $3\times3$ EPS ansatz, are indistinguishable (taking into account the error bars) from available QMC ones[@Mur00] (crosses). Around $\k=(\pi,0)$ the curve is nearly flat, in agreement with other studies.[@Mas91; @Ham98] It has to be mentioned that the degeneracy observed on the $4\times4$ lattice between $E(\pi,0)$ and $E(\frac{\pi}{2}, \frac{\pi}{2})$ is due to geometrical reasons[@Bar92] and is no longer present when the lattice size increases.
The hole dispersion relation for lower values of $\frac{J}{t}$ is plotted in Figs. \[fig:2\] and \[fig:3\]. The main qualitative features of the curve (e.g., the position of the minimum) appear independent of the value of $\frac{J}{t}$. To illustrate further the accuracy of the EPS WF, estimates of $E(\k)/t$ obtained with $2\times2$ plaquettes (circles in Fig. \[fig:3\]) are also shown for $\frac{J}{t} = 0.4$. Despite the relatively small number of variational coefficients utilized for this ansatz, the resulting dispersion relation is in close agreement with that found with $3\times3$ plaquettes as well as with Green function Monte Carlo (GFMC) results[@Mas94] (triangles and crosses respectively in the same figure) and QMC ones.[@Mur00] Specifically, the hole band predicted by using such a minimal EPS ansatz differs from the more accurate prediction achievable with $3\times3$ plaquettes only by a rigid energy shift towards lower energy values. This energy shift (albeit quantitatively different) occurs for all the three values of $\frac{J}{t}$ considered here, pointing out how the EPS ansatz, even in its simplest and economical implementation (i.e., when $2\times2$ plaquettes are used), can provide reliable estimates of observable such as the effective mass, the band width, and generally all those obtainable from band energy differences.
$\frac{J}{t}$ W G
--------------- --------- ---------
2.0 0.53(3) 0.15(3)
0.8 1.54(3) 0.39(3)
0.4 2.63(4) 0.74(8)
: \[tab:2\]Band width ($W$) and energy difference ($G$) between $E(\pi,0)$ and $E(\frac{\pi}{2},\frac{\pi}{2})$ , in units of $J$, for a system of $256$ sites with PBC. Error bars are in parenthesis.
$\frac{J}{t}$ EPS VMC QMC SCBA
--------------- ---------- ---------- --------- -------
2.0 0.596(2) 0.663(3) 0.58(4) -
0.8 0.427(3) - 0.40(4) 0.504
0.4 0.340(2) 0.375(2) 0.32(2) 0.34
: \[tab:3\]Hole spectral weight at $\k=(\frac{\pi}{2},\frac{\pi}{2})$ for a $16\times16$ lattice with PBC. Estimates obtained with VMC based on a different WF,[@Mas92] QMC,[@Mur00] and SCBA[@Hor91] are also shown for comparison. Error bars are in parenthesis.
Values of the band width $W$ and the energy gap $G$ respectively defined as $W=E(0,0)-E(\frac{\pi}{2},\frac{\pi}{2})$ and $G=E(\pi,0)-E(\frac{\pi}{2},\frac{\pi}{2})$ are reported, as a function of $\frac{J}{t}$, for a $16\times16$ lattice in Tab. \[tab:2\]. Our results of the band width are in excellent agreement with QMC[@Mur00] and GFMC[@Mas94] calculations. At $\frac{J}{t}=2.0$ our estimate differs from the VMC one: $2.11(3)$ reported in Ref. . Moreover we find that $G$, at least in the cases examined here, varies linearly with $t$ according to the relation $\frac{G}{t} \sim 0.3$.
Next we discuss our findings for the hole spectral weight $Z$ (see Eq. (\[eq:zeta\])). On a $4\times4$ lattice we obtain, for $\frac{J}{t}=0.4$, $Z(\frac{\pi}{2},\frac{\pi}{2})=0.3996(5)$, in agreement with the ED value[@Dag91] of $0.4$. EPS values of $Z(\frac{\pi}{2},\frac{\pi}{2})$ as a function of $\frac{J}{t}$ on a $16\times16$ lattice are shown in Tab. \[tab:3\]. Our estimates, computed with plaquettes of $9$ sites, are in agreement with QMC ones[@Mur00] for all the values of $\frac{J}{t}$; agreement is found with self consistent Born approximation (SCBA) calculations[@Hor91] at $\frac{J}{t}=0.4$, this approach providing, as well as the WF used in Ref. , higher values of the spectral weight when $\frac{J}{t}$ increases.
The momentum dependence of the spectral weight for $\frac{J}{t}=0.4$ is shown in Fig. \[fig:4\] along particular cuts of the Brillouin zone. $Z(\k)$ is approximately $13$% at $\k=(0,0)$ and, going in the $(1,0)$ direction, reaches $\sim 0.4$ at $\k=(\pi,0)$; it then decreases, slightly, down to $\sim 0.34$ at $\k=(\frac{\pi}{2},\frac{\pi}{2})$, and more pronouncedly, in the $(1,1)$ direction, beyond this $\k$ value.
[![(color online). Hole spectral weight for a $16\times16$ lattice with PBC at $\frac{J}{t}=0.4$. The dashed line is a guide to the eye.[]{data-label="fig:4"}](fig4 "fig:")]{}
Conclusions and outlook
=======================
The two dimensional $t-J$ model in presence of a single mobile hole has been investigated by means of variational Monte Carlo simulations based on the EPS trial wave function. For systems comprising up to $N=256$ sites (i.e., the maximum size considered in this work), an EPS ansatz consisting of $N$ entangled plaquettes of $4$ sites provides estimates of the dispersion curve in qualitative and semi-quantitative agreement (up to a rigid energy shift) with the most accurate alternative numerical approaches. By increasing the plaquette size to $9$ sites, with a minimal additional computational effort, quantitative agreement is easily recovered and accurate results for the hole spectral weight obtained.
Given the accuracy of the results presented in this work, as well as the simplicity of the variational algorithm used to minimize the energy, the EPS ansatz appears a very promising option for future investigations of the $t-J$ model in the case of finite hole concentration, a problem of great physical interest to which exact quantum Monte Carlo techniques based on imaginary time projection are not applicable due to the sign problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author acknowledges discussions with J. I. Cirac. This work has been supported by the DFG (FOR 635) and the EU project QUEVADIS.
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|
---
abstract: 'We have measured the dependence of the relative integral cross section of the reaction Li+HF$\to$LiF+H on the collision energy using crossed molecular beams. By varying the intersection angle of the beams from 37$^o$ to 90$^o$ we covered the energy range 25meV$\leq E_{\rm tr} \leq 131\,$meV. We observe a monotonous rise of the cross section with decreasing energy over the entire energy range indicating that a possible translational energy threshold to the reaction is significantly smaller than 25meV. The steep rise is quantitatively recovered by a Langevin-type excitation function based on a vanishing threshold and a mean interaction potential energy $\propto R^{-2.5}$ where $R$ is the distance between the reactants. To date all threshold energies deduced from *ab-initio* potentials and zero-point vibrational energies are at variance with our results, however, our findings support recent quantum scattering calculations that predict significant product formation at collision energies far below these theoretical thresholds.'
author:
- 'Rolf Bobbenkamp, Hansjürgen Loesch'
- 'Marcel Mudrich, Frank Stienkemeier'
title: '**[The excitation function for Li+HF$\to$LiF+H at collision energies below 80meV]{}**'
---
Introduction
============
The rapid development of techniques for cooling, trapping and manipulating cold and ultracold molecules opens the opportunity of studying chemical reactivity in the low temperature regime [@Balakrishnan:2004; @Krems:2008; @Bell:2009; @Hutson:2010]. Cold reactive collisions are not only important limitations to stable trapping of molecules but also present fascinating aspects of their own interest. In the cold collision regime, reactions are governed by quantum dynamics involving tunneling and scattering resonances [@Krems:2008; @Weck:2005]. For our study of cold collision phenomena we have selected the title reaction due to both the availability of theoretical results and the applicability of a variety of experimental methods.
Attractive from the theoretical point of view is the small number of electrons together with three chemically very different but light atomic constituents that make the slightly exoergic ($\approx$0.16meV) reaction Li+HF$\to$LiF+H an ideal prototype system for developing methods to calculate reliable ab-initio potential energy surfaces (PESs) [@Balint-Kurti:1977; @Zeiri:1978; @Shapiro:1979; @Chen:1980; @Carter:1980; @Palmieri:1989; @Parker:1995; @Aguado:1997; @Jasper:2002; @Bobbenkamp:2005; @Aguado_unpublished] and to calculate quantum scattering phenomena [@Parker:1995; @Walker:1981; @Lagana:1988; @Baer:1989; @Lagana:1991; @Baer:1994; @BaerJCP:1994; @Balint-Kurti:1993; @Gogtas:1996; @Aguado:1997; @Lara:1993; @Lara:1998; @Lagana:1993; @Zhu:1997; @Lagana:2000; @LaganaCPL:2000; @Hobel:2004; @Weck:2005; @Zanchet:2009]. Particularly challenging for a quantitative prediction of observables is the low energy range near threshold where quantum effects dominate and a precise knowledge of the PES around the transition state is essential. Crucial for the assessment of the preciseness and reliability of the applied theoretical methods is the comparison of experimental and computational results. Very qualified for this purpose are certainly angular and velocity distributions of products [@Bobbenkamp:2005; @HobelDiss] but also the dependence of the relative integral reaction cross section (IRCS) on the translational collision energy, $E_{\rm tr}$, the excitation function [@BaerJCP:1994; @Lara:1998; @LaganaCPL:2000; @Qiu:2006; @Hobel:2001]. At low collision energies the excitation function bears direct information on the existence and size of a translational threshold that allows a sensitive examination of the shape of the PES near the transitions state and the height of a possible potential energy barrier.
An inspection of computational results shows that all PESs [@Balint-Kurti:1977; @Zeiri:1978; @Shapiro:1979; @Chen:1980; @Carter:1980; @Palmieri:1989; @Parker:1995; @Aguado:1997; @Jasper:2002; @Bobbenkamp:2005; @Aguado_unpublished] known to date feature a significant barrier at a bent transition state with substantial height. However including zero point energies of both the reactants and the transition state the semi-classical translational threshold energies deduced from these barrier heights turn out to be markedly smaller. For example, for the most frequently used recent PESs, the semi-classical thresholds and barriers amount to $\approx$27 and 182meV [@Parker:1995], $\approx$ 68meV and 233meV [@Aguado:1997], $\approx$56 and 221meV [@Bobbenkamp:2005; @Aguado_unpublished], respectively. Neglecting quantum scattering phenomena the semi-classical thresholds manifest the lowest collision energies required for product formation from ground state reactants. An observation of products at energies below these calculated thresholds could be a clue to marked quantum phenomena or to insufficient accuracy of the PESs or to a combination of both. Recently, it has been demonstrated in time-dependent wave packet scattering calculations [@Weck:2005; @Zanchet:2009] based on the PES of ref. [@Bobbenkamp:2005; @Aguado_unpublished] that resonances play an important role for Li+HF. They lead to product formation at a significant rate at collision energies far below the semi-classical threshold. One interesting experimental aspect of the title reaction is the availability of methods that allow to investigate scattering phenomena in a wide range of collision energies from hot (several 100meV) to cold (20meV) and ultra-cold ($<1$meV). In an earlier experimental study we have measured the excitation function within the range 82meV $\leq E_{\rm tr}\leq$ 376meV [@Hobel:2001] using molecular beams intersecting at a 90$^o$ angle. The energy was varied by applying the seeded beam technique. At energies above $\approx$ 120meV the function turns out to be roughly constant but below it rises steeply with decreasing $E_{\rm tr}$ and assumes at the lowest energies the largest values within the entire range. Consequently, a threshold to reaction, if existing at all, has to be located below 82meV.
In the present paper we report the results of another crossed beam study on the excitation function designed to lower the range of translational energy to 25meV. Variation of the collision energy is achieved by varying the intersection angle of the beams (see below). We find that in the extended energy range the IRCS continues to rise monotonously with decreasing $E_{\rm tr}$ thus shifting the upper boundary of a threshold to below 25meV.
It should be noted that a novel experimental setup using new techniques for preparing samples of cold atoms and molecules is about to be completed. It will allow to continue our search for the threshold of this process at energies below 1meV.
Experimental Methods
====================
![Schematic view of the molecular beams arrangement. The setup is basically the same as used in our previous experiments [@Hobel:2001] except for the HF beam that can now be rotated around the intersection volume to vary the intersection angle $\Gamma$ in the range $37^o \leq \Gamma \leq 90^o$. An additional HF-beam modulator placed between nozzle and skimmer is not shown for clarity.\[fig:setup\]](Fig1_Setup-eps-converted-to.pdf "fig:"){width="\columnwidth"}\
To achieve translational energies below 82meV (the lowest energy accessible in ref. [@Hobel:2001]) we abandon the usual 90$^o$ geometry and cross the beams at a variable intersection angle [@Buntin:1987; @Hsu:1997; @Naulin:1999]. Fig. \[fig:setup\] shows a sketch of the molecular beams arrangement. It deviates from the one used in earlier experimental studies [@HobelDiss; @Hobel:2001; @Loesch:1993] only by the setup of the HF-beam. The nozzle is now mounted within a differentially pumped vacuum chamber (not shown) that can be rotated around the scattering volume. In this way the intersection angle $\Gamma$ of the two beams and thus also the mean relative velocity $$\bar v=[v_{\rm Li}^2 +v_{\rm HF}^2-2 v_{\rm Li}v_{\rm HF}\cos\Gamma]^{0.5}
\label{eq:relv}$$ and the translational collision energy $$E_{\rm tr}(\Gamma)=\frac{1}{2}\mu\bar v^2$$ can be varied continuously. $\mu$ denotes the reduced mass of the colliding reactants and $v_{\rm Li}, ~ v_{\rm HF}$ are their nominal velocity. Depending on $\Gamma$, significantly smaller or larger collision energies can be achieved compared to the one obtained for the usual perpendicular intersection. In this study the two vacuum chambers housing the beam sources are designed such that $\Gamma$ can be varied in the range $\rm 37^o\leq \Gamma \leq
90^o$.
The experimental method and data acquisition follows closely the ones discussed in ref. [@Hobel:2001]. Briefly, the neat HF beam is created by a nozzle heated to around 500K to avoid clustering. Stagnation pressure (250mbar) and nozzle temperature are always kept constant. The elevated temperature leads to a mean rotational energy of 8.9meV [@Hobel:2001] and a population of the first vibrationally excited state of $<0.25\%$. The beam is chopped with a frequency of a few Hz using a tuning-fork like modulator or a chopper wheel at a duty cycle of 50% and the intensity is monitored by a quadrupole mass spectrometer detector equipped with an electron bombardment ion source. The Li beam is diluted with Ne and its intensity is measured by surface ionization on a hot rhenium (Re) ribbon. During an experimental run only the intersection angle $\Gamma$ is varied while the operational conditions of the beams are kept constant.
Run $u_{\rm Li}$ $\alpha_{\rm Li}$ $u_{\rm HF}$ $\alpha_{\rm HF}$
----- -------------- ------------------- -------------- -------------------
A/B 1530 221 1205 185
C 1750 290 1215 195
: Beam velocity parameters, velocities in m/s[]{data-label="tab1"}
The velocity of both beams is determined by conventional time-of-flight (TOF) arrangements. The (density) velocity distributions, $n(v)$, are extracted from the measured TOF profiles by fitting the parameters $u,~ \alpha$ of the expression $$n(v)= const~ v^2~ \exp\{-[(v-u)/\alpha]^2\}$$ to the data. The parameters $u$ and $\alpha$ of the various data sets are compiled in Table 1. The velocity spreads of the Li and HF beams $\alpha$ result in an experimental relative uncertainty of the collision energy $E_{tr}$ of about 25%.
The products created in the intersection volume of the beams are detected via surface ionization on a hot Re ribbon mounted in a separately pumped ultra high vacuum chamber (residual gas pressure below $10^{-8}\,$mbar). A channeltron converts the ions desorbing from the Re surface to electron pulses which are counted by a two channel scaler synchronized with the HF beam modulator. One scaler counts the signal and background, the other only the background pulses. The detected scattering intensity (signal) $I_{\rm tot}$ is then derived as difference of the two scaler contents. Angular distributions of the signal $I_{\rm
tot}(\Theta)$ are obtained by rotating the main detector automatically around the intersection volume in the plane of the beams within a wide range of laboratory (LAB) scattering angles $\Theta$. Crucial for the data analysis (see below) is the knowledge of time-of-flight (TOF) distributions of scattered particles at various scattering angles. They are measured employing a fast spinning chopper wheel with 8 equally spaced slots (2mm wide) mounted between the skimmer of the HF beam and the scattering volume (see Fig. \[fig:setup\]). The length of the flight path is 254mm.
![Typical TOF profiles measured at $\Gamma=37^o$ and $50^o$ for the indicated scattering angles. The profiles are used to separate elastic (smooth curves) from the detected scattering intensity (data points). See text for more details.\[fig:tof\]](Fig2_Tof-eps-converted-to.pdf "fig:"){width="0.9\columnwidth"}\
The surface ionization is not specific with respect to alkali atoms or alkali compounds and detects both species with roughly the same efficiency. Therefore, the detected scattering intensity $I_{\rm tot}$ is proportional to the sum of fluxes of both the elastically scattered Li atoms $I_{\rm Li}$ and the products LiF, $I_{\rm LiF}$. To separate both components we employ the TOF distributions; examples are displayed in Fig. \[fig:tof\] for $\Gamma=37^o$ and $50^o$. All TOF profiles feature a peak at short flight times and a broad shoulder or even a second peak at longer times. At $\Gamma=90^o$ and higher Li velocities the peaks are always well separated [@Hobel:2001].
![Newton diagrams illustrating the kinematics for three beam intersection angles. The radius of the outer circle around the tip of the centroid velocity vector $\bf C$ corresponds to the velocity of the elastically scattered Li atoms in the CM system, the radius of the inner filled circle to the maximal product velocity.\[fig:newton\]](Fig3_NewtonDiagrams-eps-converted-to.pdf "fig:"){width="0.5\columnwidth"}\
The collision kinematics are illustrated in Newton-diagrams (Fig. \[fig:newton\]) relating the LAB and center-of-mass (CM) frame velocities of reactants and products for beam intersection angles $\Gamma=37^o,~50^o$, and $90^o$. The light arrows represent the laboratory frame velocity vectors ${\bf v}_{\rm Li}$ and ${\bf v}_{\rm HF}$ of the reactants Li and HF which include the intersection angle $\Gamma$. The radius of the outer circle around the tip of the centroid velocity vector $\bf C$ corresponds to the velocity of the elastically scattered Li atoms in the CM system, the radius of the inner shaded circle represents the maximal product velocity. The light straight lines that are tangent to the inner circle indicate the range of scattering angles $\Delta\Theta$ where reaction products LiF are expected to occur in addition to elastically scattered Li atoms. Elastically scattered Li atoms along $\bf C$ have higher speeds than the LiF products and will therefore reach the detector at shorter flight times. Li atoms that are elastically scattered in the opposite direction in the CM frame have low speeds in the LAB frame and will spatially disperse on their way to the detector such that no significant contribution to the signal is expected.
![Angular distributions of the detected (upper diagrams) and of the product (lower diagrams) scattering intensity measured at $\Gamma=37^o$ and $50^o$. The open circles in the upper diagrams mark the elastic intensity deduced from the TOF distributions of Figure 2. The solid line is the best fit interpolation used to deduce the reactive scattering intensity (lower diagrams) by subtracting the elastic from the detected scattering intensity.\[fig:angledistr\]](Fig4_AngleDistr-eps-converted-to.pdf "fig:"){width="\columnwidth"}\
Thus, in Fig. \[fig:tof\] the peak at short flight times (fast particles) is attributed to elastically scattered Li atoms while the one at later times (slow particles) is associated with reaction products LiF. The solid line represents a numerically simulated TOF-distribution of elastically scattered Li atoms whose reliability was confirmed by a comparison with TOF profiles measured for the pure elastic system Li+Ne that features practically the same kinematics as Li+HF (open circles in Fig. \[fig:tof\]). The elastic TOF profile is then fitted to the fast slope of the data. The area below this scaled elastic peak over the area below the entire TOF profile eventually provides the fraction of the detected scattering intensity associated with elastically scattered atoms. For a given $\Gamma$ this fraction has been determined at scattering angles for which TOF distributions are available and used to isolate the angular distribution of the elastic component $I_{\rm Li}(\Theta)$ from the detected scattering intensity $I_{\rm tot}(\Theta)$. The result of this procedure is shown in Fig. \[fig:angledistr\] (upper panel). The open circles represent the angular distribution of the elastic component $I_{\rm Li}(\Theta)$ and the solid line is a best fit curve through these points. The difference between the detected scattering intensity and the elastic component eventually yields the angular distribution of the product flux $I_{\rm
LiF}(\Theta)$ (lower panel in Fig. \[fig:angledistr\]). For more details see [@Hobel:2001].
In principle the signal contribution identified with reactive scattering could also be generated by rotationally inelastic collisions. At $\Gamma = 50^o$ inelastic collisions Li+HF$(j=0)\rightarrow$Li+HF$(j=5)$ would yield Li atoms with similar CM speeds as the LiF products. At $\Gamma = 37^o$, Li atoms could be generated at speeds similar to those of LiF in inelastic collisions Li+HF$(j=0)\rightarrow$Li+HF$(j=3)$. Such selective rotation-changing collisions appear quite implausible, though, given the common scaling laws that predict a fast decay of the rotational inelastic collision cross section with increasing level spacing [@Raghavan:1985].
Results
=======
The IRCS, $\rm \sigma_r(E_{\rm tr})$, is proportional to the total flux of products $\dot
N_{\rm LiF}^{\rm total}$ generated in the scattering volume ${\cal V}$ and defined by the expression $$\sigma_r(E_{\rm tr})=\dot N_{\rm LiF}^{\rm total}/(n_{\rm Li}~n_{\rm HF}~\bar v~{\cal V}).
\label{eq:ircs}$$ Here, $n_{\rm Li},~ n_{\rm HF}$ and $\bar v$ denote the number densities of the indicated beams at the intersection volume and the mean relative velocity, respectively. Deviating from our earlier study [@Hobel:2001], we leave the operational conditions of the beam sources constant and vary only the intersection angle $\Gamma$. Thus both densities $n_{\rm HF}$ and $n_{\rm Li}$ are constant, $\bar v$ can be easily deduced from the most probable beam velocities (eq. \[eq:relv\]), and an inspection of the intersection geometry shows that ${\cal V} ={\cal V}_{90^o} /\sin \Gamma$ holds approximately.
The crucial quantity $\dot N_{\rm LiF}^{\rm total}$ is not directly accessible in the present in-plane scattering experiment but can be deduced from the measured total in-plane product flux $I_{\rm LiF}^{\rm in-plane}$ using the formal expression $$\dot N_{\rm LiF}^{\rm total}={\dot N_{\rm LiF}^{\rm total} \over I_{\rm LiF}^{\rm in-plane}}
I_{\rm LiF}^{\rm in-plane}.
\label{eq:formalflux}$$ Inserting eq. \[eq:formalflux\] into eq. \[eq:ircs\] and suppressing all constant quantities we obtain the expression $$\sigma_r(E_{\rm tr})\propto{\dot N_{\rm LiF}^{\rm total} \over I_{\rm LiF}^{\rm in-plane}}
~\sin\Gamma / \bar v ~ I_{\rm LiF}^{\rm in-plane}
\label{eq:inplaneflux}$$ relating the measured quantities and the relative integral reaction cross section. The ratio of fluxes in eqs. \[eq:formalflux\] and \[eq:inplaneflux\] corrects for the fraction of products that miss the detector; it can be readily deduced from the relative differential reaction cross section (DRCS) in the center-of-mass frame. In a previous study [@HobelDiss] using perpendicularly intersecting beams we have measured the relative DRCS at 6 energies within the range 82meV$\leq E_{\rm tr} \leq$ 376meV and found that the ratio is constant with respect to the energy within an error margin of $\pm 4\%$ [@Hobel:2001]. Unfortunately, a comparable extensive investigation of DRCSs is not yet available for the present low energy range. However, a preliminary analysis of the product angular distributions measured at $\Gamma=37^o$ (24meV) and 50$^o$ (45meV) indicates that from 119meV [@HobelDiss] ($\Gamma=90^o)$ to 24meV ($\Gamma=37{^o}$) a transition from the forward/backward to a preferred sideways type DRCS occurs. Taking this into account we find for all data sets that the product of factors left to the in-plane flux in eq. \[eq:inplaneflux\] is constant within the band width $\pm 5\%$. In view of a forthcoming more sophisticated determination of the DRCS we suppress these small corrections and derive the relative IRCS from the data using the simplified relation $$\sigma_r(E_{\rm tr})\propto I_{\rm LiF}^{\rm in-plane}.
\label{eq:sigmainplane}$$
We have measured three sets of angular distributions A, B, and C at various intersection angles for the beam parameters given in Table 1. The constancy of the operational conditions was checked carefully by measuring a reference angular distribution repeatedly during one run. The intersection angles range from 37$^o$ to 90$^o$ (A,B) and 48$^o \leq \Gamma \leq 90^o$ (C) corresponding to the energy range 25meV $\leq E_{\rm tr} \leq$ 108meV (A,B) and 50meV $\leq
E_{\rm tr} \leq$ 131meV (C). For each angular distribution of a given set we measured between four and six TOF-profiles to separate elastic and reactive scattering.
![Product angular distributions of data set A measured at the intersection angles 90$^o$ (open square), 82$^o$ (open circle), 75$^o$ (open triangle), 68$^o$ (open diamond,) 62$^o$ (solid square), 55$^o$ (solid circle), 45$^o$ (solid triangle) and 37$^o$ (solid diamonds). Note the shift of the curves to smaller angles with decreasing $\Gamma$ caused by the kinematics (Fig. \[fig:newton\]) and the strong increase of the peak intensity. The area below the curves is the total product intensity used to determine the IRCS.\[fig:inplane\]](Fig5_InPlane-eps-converted-to.pdf "fig:"){width="0.8\columnwidth"}\
As an example we show in Fig. \[fig:inplane\] the product angular distributions of set A. With decreasing $\Gamma$ the curves shift to smaller LAB angles according to the changing kinematics (Fig. \[fig:newton\]) and their peak intensities rise dramatically. The total in-plane intensity $I_{\rm LiF}^{\rm in-plane}$ or $\sigma_r(E_{\rm tr})$ (eq. \[eq:sigmainplane\]) is given by the sum over all product intensities multiplied by the Lab-angle increment (area below the curves), $$I_{\rm LiF}^{\rm in-plane}=\sum I_{\rm LiF}(\Theta_{\rm i}) \Delta \Theta_{\rm i}.$$
![The excitation function deduced from data set A (circles), B (triangles) and C (diamonds). The data sets are mutually normalized to obtain the best agreement within the overlap. The solid line through the data points is given by the simple power law $\sigma_r(E_{\rm tr})\propto( 1/E_{\rm tr})^{0.8}$.\[fig:sigmapowerfit\]](Fig6_SigmaPowerFit-eps-converted-to.pdf "fig:"){width="0.8\columnwidth"}\
[lcccc]{}\
$E_{\rm tr}$/meV &$\Gamma$/degree & &$I_{\rm LiF}^{\rm in-plane}$&\
& & A & B & C\
25 & 37 & 83411 & 110104 &\
35 & 45 & 68510 & 93050 &\
48 & 55 & 45930 & 60628 &\
50 & 48 & & & 395200\
59 & 62 & 39040 & 55431 &\
64 & 65 & & 51909 &\
69 & 68 & 31774 & 46050 &\
71 & 60 & & & 335300\
80 & 65 & & & 289750\
81 & 75 & 31774 & 48590 &\
86 & 68 & & & 253200\
94 & 82 & 30517 & 44170 &\
100 & 75 & & & 238650\
108 & 90 & 27465 & 39540 &\
114 & 82 & & & 239200\
131 & 90 & & & 199300\
The results for set A are compiled in Table 2 together with those for B and C and displayed in Fig. \[fig:sigmapowerfit\] as a function of the collision energy. The data points are normalized such that they agree optimally within the overlapping energy range. Their statistical error is not included in Fig. \[fig:sigmapowerfit\] for clarity. It can be estimated from the scatter of the points and amounts to about $\pm 5\%$ essentially due to uncertainties occurring in the process of separating elastic and reactive scattering. The solid line through the data points represents the simple power law $\sigma_r(E_{\rm
tr})\propto( 1/E_{\rm tr})^{0.8}$.
Discussion
==========
A strong motivation for performing the present experiments was the search for an answer to the question raised in the earlier study [@Hobel:2001]: How does the excitation function continue below a collision energy of 82meV? The previous results indicated two possibilities: either the excitation function assumes a maximum followed by a decline towards a threshold as one would expect for a reaction featuring a non-vanishing translational threshold energy or it continues to increase as expected for a reaction without threshold. The results of the present investigation strongly support the assumption that no threshold hinders the reaction of Li with HF. The excitation function continues to ascend monotonously at least down to 25meV and is likely to continue this way [@ArSeededLiBeam]. Due to the elevated nozzle temperature of 500K, required to suppress dimerisation, low rotational states of HF are populated with a mean rotational energy of 8.9meV corresponding to a mean rotational quantum number of $\bar j=1.4$ [@Hobel:2001]. Thus the indicated threshold energies refer on average to the reaction Li+HF ($v=0$,$\bar j$=1.4). This internal energy may increase the threshold of the ground state reaction Li+HF($v=0$,$j=0$) on average by the mean rotational energy.
![Excitation function including the results of our previous study [@Hobel:2001] (solid squares). The new and earlier results are mutually normalized to obtain the best agreement within the overlap. The curves at energies $\leq$ 200meV are Langevin cross sections based on potential energy functions $\propto R^{-6}$ (dotted), $\propto R^{-4}$ (dashed) and $\propto R^{-2.5}$ (solid). The solid smooth line for energies $\geq 200$meV refers to a constant hard sphere collision model matched to the Langevin functions at 200meV. The light solid line represents the result of recent wave packet calculations [@Zanchet:2009].\[fig:sigmapowerconst\]](Fig7_Roncero-eps-converted-to.pdf "fig:"){width="\columnwidth"}\
In Fig. \[fig:sigmapowerconst\] we have combined the results of the present and earlier experiments and obtain an excitation function that covers now a wide energy range of more than one order of magnitude from 25meV to 376meV. The curve can be subdivided into three sections: above 200meV it is more or less constant, below 130meV the function rises monotonously with decreasing $E_{\rm tr}$, and in between it gradually changes the shape correspondingly. This behavior suggests the following classical reaction model.
\(a) At energies $>200$meV a rigid sphere collision mechanism [@Qiu:2006; @Levine] prevails where the reaction occurs with a constant, energy independent probability $\cal P$ whenever the impact parameter $b$ is smaller than the maximal one $b_{\rm max}$. The resulting constant IRCS is then $\sigma_{\rm r}={\cal P} \pi b_{\rm max}^2$. The mechanism neglects long range forces and thus all trajectories of the approaching reagents are straight lines.
\(b) Below 130meV long range forces start getting important. The force curbs the trajectories of the colliding reagents toward the center of the force and thus collisions with impact parameters larger than $b_{\rm max}$ may react. With decreasing energy the influence of the forces grows and thus the maximal impact parameter and the IRCS increase. Provided there is a vanishing threshold the IRCS continues to rise and eventually diverges classically for $E_{\rm
tr}\to 0$. Assuming that (i) the interaction potential energy is $\propto 1/R^s$ where $R$ is the distance between the reagents, and (ii) the reaction occurs with a constant probability whenever the reagents overcome the effective potential’s maximum, then the IRCS is described by the “Langevin” power law $\sigma_{\rm r}(E_{\rm tr})\propto (1/E_{\rm
tr})^{2/s}$ [@Qiu:2006; @Levine]. The best fit curve in Fig. \[fig:sigmapowerfit\] corresponds accordingly to an interaction potential $\propto 1/R^{2.5}$. In Fig. \[fig:sigmapowerconst\] the $s=2.5$ result (solid line) is compared with curves referring to $s=6$ (dotted line) and $s=4$ (dashed line) for energies below 200meV. The corresponding potentials describe the long range forces of an atom-molecule ($s=6$) and of an ion-molecule system ($s=4$). Both curves feature the typical steep rise but are clearly at variance with our data. The best fit power $s=2.5$ indicates that in our low energy range the chemical forces near the hard sphere radius are responsible for the rise of the IRCS rather than the asymptotic long range forces with $s=6$.
\(c) Between 130 and 200meV the transition between the hard sphere and the potential energy dominated mechanism occurs.
The prediction of this classical model is illustrated in Fig. \[fig:sigmapowerconst\] as solid line where we have omitted the transition region and matched directly the hard sphere and Langevin functions at 200meV. The curve recovers the data nearly quantitatively over this wide energy range.
The above classical model suggests a vanishing threshold but the experimental results provide only an upper boundary and a small threshold may well be in agreement with our findings. According to classical mechanics such a translational energy threshold is tightly related to the height of the potential energy barrier $V_{\rm b}$ separating reagents and products. Provided the approaching molecules are in their vibrational and rotational ground states ($v=0$, $j=0$) the formation of products requires that the translational energy of the approaching particles exceeds $V_{\rm b}$. A non vanishing reaction threshold is then the consequence of an existing barrier.
Quantum mechanics relaxes the tight classical relation between threshold and barrier height. The colliding particles possess zero-point vibrational energy that varies during the approach adiabatically from the zero-point energy of the free molecule $E^0_{\rm vib}$ to the one of the triatomic aggregate at the barrier $E_{\rm b}^0$. Taking this into account, product formation for ground state molecules ($v=0$,$j=0$) is then allowed in a classical sense, if the sum of the collision and zero-point energy of the reagent molecule exceeds the sum of the potential and zero-point energy at the barrier. The resulting semi-classical threshold energy $$E_{\rm th}^0=V_{\rm b}+E_{\rm b}^0-E^0_{\rm vib}
\label{eq:eth}$$ includes in addition to the barrier height $V_{\rm b}$ also the relevant zero-point energies $E_{\rm b}^0$ and $E^0_{\rm vib}$. However, in contrast to classical mechanics, eq. \[eq:eth\] constitutes no rigorous lower limit for the translational energy leading to product formation. Due to quantum phenomena such as tunneling and resonances products may be formed also at collision energies far below this semi-classical threshold (see below).
To date all PESs available for Li+HF predict a significant barrier height $V_{\rm
b}$ [@Balint-Kurti:1977; @Zeiri:1978; @Shapiro:1979; @Chen:1980; @Carter:1980; @Palmieri:1989; @Parker:1995; @Aguado:1997; @Jasper:2002; @Bobbenkamp:2005; @Aguado_unpublished]. Including the zero point energy of HF (256meV) and of the transition state (95$\pm 10$meV depending on the geometric structure of the PES) the semi-classical threshold (eq. \[eq:eth\]) for the most recent PESs amounts to 27meV [@Parker:1995], 56meV [@Bobbenkamp:2005; @Aguado_unpublished] and 68meV [@Aguado:1997]. The present experiments provide as upper boundary for the threshold energy 25meV and, considering the significant product flux observed for the Ar seeded Li experiment [@ArSeededLiBeam], 17meV. Furthermore, the excellent fit of the data predicted by a model based on a vanishing threshold suggests a threshold located at energies $\ll 17$meV. Thus our results suggest that the experimental threshold energy lies markedly below the predicted semi-classical values. This discrepancy could be due to insufficient accuracy of the computational methods used to calculate the PESs or to quantum effects that promote reactions at energies far below the semi-classical threshold. The latter is supported by recent wave packet calculations based on the PES of [@Bobbenkamp:2005; @Aguado_unpublished] that predict a threshold for the total reaction cross section at about 10meV which is qualitatively reconcilable with our measurements [@Zanchet:2009]. The theoretical excitation function [@Zanchet:2009] is displayed in Fig. \[fig:sigmapowerconst\]. The curve exhibits only a slight modulation around the value $\sigma_r\approx 0.5\,$Å as the translational energy is reduced down from $E_{tr}=250\,$meV to a threshold value of about 10meV (see Fig. \[fig:sigmapowerconst\]). The shape of the predicted excitation function departs significantly from the experimental one but the threshold energy is in accord with the present result. As the main source for product formation near threshold collisions with small $J$ (total angular momentum) have been identified. The reaction probabilities for $J=0$ exhibit a rich spectrum of oscillations and resonances as a function of energy [@Weck:2005; @Zanchet:2009] but these structures disappear if realistic IRCSs are calculated by summing over all $J$.
Conclusion
==========
The benchmark reaction Li+HF$\to$LiF+H has been studied at translational energies down to 25meV using a new crossed-beam apparatus with variable scattering angle between Li and HF beams. This arrangement allows to tune the translational energy while keeping the beam source conditions constant. The integral reactive scattering rate (excitation function) is deduced from angle-resolved scattering as well as from time-of-flight traces in comparison with purely elastic Li+Ne scattering. The resulting excitation function, which extends earlier measurements to lower energies, steeply rises as the collision energy falls below about 150meV. This behavior is consistent with a barrier-less Langevin-type reactive process with $R^{-2.5}$-scaling of the atom-molecule interaction potential. Alternatively, an energy threshold below $\sim 25$meV may be present as predicted by recent wave packet simulations [@Zanchet:2009].
Future efforts to further reduce the collision energy for conclusively disclosing or ruling out the existence of a reaction threshold will demand more sophisticated experimental approaches. To this end, a magneto-optical trap for preparing an ultracold Li scattering target is currently being set up and will be combined with a source for slow and cold molecules based on a rotating nozzle and electrostatic guiding [@Strebel:2010].
Support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
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---
abstract: |
Recent observations suggest that the abundance of carbon in the interstellar medium is only $\sim 60$% of its solar value, and that other heavy elements may be depleted by a similar amount. Furthermore, more than half of the interstellar carbon is observed to be in the gas in the form of C$^+$, leaving less than $\sim 40$% of the solar carbon abundance available for the dust phase. These observations have created the so-caled interstellar “carbon crisis”, since traditional interstellar dust models require about twice that value to be tied up in graphite grains in order to explain the interstellar extinction curve.
Recently, Mathis (1996) suggested a possible solution to this crisis. In his newly-proposed dust model the majority of the interstellar carbon is contained in composite and fluffy grains that are made up of silicates and amorphous carbon grains, with $45\%$ of their volume consisting of vacuum. Per unit mass, these grains produce more UV extinction, and can therefore account for the interstellar extinction curve with about half the carbon required in traditional dust models.
This paper presents an detailed assessment of the newly-proposed dust model, and concludes that it falls short in solving the carbon crisis, in providing a fit to the UV-optical interstellar extinction curve. It also predicts a far-infrared emissivity in excess of that observed with the [*COBE*]{}/DIRBE and FIRAS instruments from the diffuse interstellar medium. This excess infrared emission is a direct consequence of the lower albedo of the composite fluffy dust particles, compared to that of the traditional MRN mixture of bare silicate and graphite grains. The failure of the new model highlights the interrelationships between the various dust properties and their observational consequences, and the need to satisfy them all simultaneously in any comprehensive interstellar dust model.
In light of these problems, the paper examines other possible solutions to the carbon crisis.
author:
- Eli Dwek
title: 'CAN COMPOSITE FLUFFY DUST PARTICLES SOLVE THE INTERSTELLAR CARBON CRISIS?'
---
INTRODUCTION
============
The most commonly used interstellar dust model, first introduced by Mathis, Rumpl & Nordsieck (1977; hereafter MRN), consists of two distinct populations of bare silicate and graphite particles with an $a^{-3.5}$ power law distribution in grain sizes extending approximately from 0.0050 to 0.25 . With the optical constants of Draine & Lee (1984) and Draine (1985) the model is very succesful in reproducing the average intestellar extinction curve, the 9.7 and 18 silicate extinction features, and the average interstellar albedo and polarization. The model requires that about 75% of the solar abundance of carbon be locked up in graphite grains, and essentially all the solar abundance of Mg, Si, and Fe and about 12% of the solar abundance of oxygen be locked up silicate dust (assuming a silicate composition of {MgSiFe}O$_4$).
There have recently been various lines of evidence for the case that the abundances of various heavy elements in the interstellar medium (ISM) gas are less than their nominal solar values. Concentrating on carbon, its solar abundance, normalized to a hydrogen abundance of 10$^6$ H-atoms, is $Z_{C,\odot}\approx$ 355 (Grevesse, Noels, & Sauval 1996). In the ISM, the abundance of this element is significantly lower, only 225$\pm$50 C-atoms ([@sw95]). This value is based on observations of young stars whose photospheric carbon abundances reflect those of the interstellar medium (gas plus dust phases) at the time of their formation. A significant amount of this interstellar carbon abundance is in the gas phase. Using the Goddard High Resolution Spectrograph on board the [*Hubble Space Telescope*]{}, Cardelli et al. (1996) inferred a C abundance of 140$\pm$20 atoms from C II\] $\lambda 2325$ Å absorption measurements in the direction of six stars. This leaves a total of 85$\pm$55 C-atoms available for the dust phase, compared to the $\sim$ 300 atoms required by the MRN and various other dust models to be in solids. This has created what Kim & Martin (1996) termed the “C/H crisis”. Furthermore, there is evidence that the ISM abundances of other heavy element, such as nitrogen and oxygen, are depleted by as much as 60-70% compared to their solar value (see review by Mathis 1996). The crisis is therefore not confined to carbon alone, since dust models use up essentially all the available solar abundance of refractory elements such as Mg, Si, Ca, Ti, and Fe.
This carbon crisis has motivated Mathis (1996) to propose a new interstellar dust model in which most of the dust is bonded together in the form of a loose aggregate of dust grains. A previous version of this model, and the motivations behind it were presented by Mathis & Whiffen (1989). Simply put, these fluffy aggregates can produce more extinction per unit mass than their combined individual constituent dust grains. An interstellar dust model with fluffy aggregates as a major dust component will therefore require a minimal amount of carbon and silicate dust, and perhaps be consistent with current interstellar abundances constraints.
In addition to the abundance constraints, any interstellar dust model should be able to explain the observed interstellar extinction, albedo, polarization, and infrared (IR) emission. In this paper we concentrate on the abundance, extinction, albedo, and IR emission constraints placed on the composite fluffy dust (CFD) model presented by Mathis. The detection of 3.5 - 1000 emission from high lattitude cirrus clouds by the Diffuse Infrared Background Experiment (DIRBE) and the Far Infrared Absolute Spectrophotometer (FIRAS) instruments on board the [*COBE*]{} satellite (Bernard et al. 1994; Weiland et al. 1996; Arendt et al. 1997, [@d97]) provide strong constraints on interstellar dust models. The emission in the 3.5 - 12 regime can be used to derive the abundance of the carriers of the 3.3, 6.2, 7.7, 8.6, and 11.3 features, commonly referred to as the unidentified infrared emission bands (UIBs). The amount of carbon in these carriers has to be added to any interstellar dust model. The [*COBE*]{} observations also provide an estimate of the fraction of the total IR radiation that is emitted by these particles. They must therefore be responsible for a significant fraction of the integrated UV-optical extinction. Finally, changes in the UV-optical properties of the dust can alter their IR emissivities and temperatures, effects that can be detected in the observed spectrum of the diffuse ISM.
THE OPTICAL PROPERTIES OF COMPOSITE FLUFFY DUST (CFD) PARTICLES
===============================================================
The new Mathis model consists of very small graphite grains which are required to produce the [2200 Å]{} extinction feature, very small silicates, and composite particles consisting of loosely packed amorphous carbon and silicates particles. In order to minimize the amount of carbon needed to be in the solid phase, the model uses only 55 C-atoms in the form of graphite, just enough to produce the 2200 Å bump. The remaining model parameters are then varied in order to produce the best fit to the interstellar extinction curve of Cardelli, Clayton, & Mathis (1989) with the minimum amount of carbon and silicate dust. For amorphous carbon, Mathis found that the samples of type Be (see Rouleau & Martin 1991) provided the best fit to the extinction data. The silicate dust is assumed to be composed of {MgSiFe}O$_4$ with Ca, Al, and Ti oxides as additional minor constitutents. We characterize the silicate abundance by the number of silicate atoms locked up in the dust, with the understanding that a comparable number of Mg- and Fe-atoms, and about 4 times as many O- atoms are locked up in the dust as well.
We reproduced the Mathis CFD model with dust particles with the following characteristics (abundances quoted are normalized to 10$^6$ H-atoms): (1) graphite grains with a C abundance of 55 ; (2) silicate particles with a Si abundance of 6.5; and (3) composite particles containing 105 C-atoms in amorphous form, 26 Si-atoms in silicates, and vacuum. The total dust phase abundance of silicate is thus 32.5, about 85-92% of its solar abundance of $\sim$ 35.5-38. The fractional volume occupied by these three composite grain constituents are 17, 38, and 45%, respectively. The mass density of the three grain constituents are 2.25 g/cm$^3$ for the graphite particles; 3.3 g cm$^{-3}$ for the silicates; and, adopting a value of 1.85 g cm$^{-3}$ for the amorphous carbon, 1.55 g cm$^{-3}$ for the composite fluffy particles. Interstellar grains are not spherical. Consequently, we followed Mathis’ procedure, and multiplied the extinction calculated for the spherical bare silicates and the composite particles by a factor of 1.09, to account for the increased extinction due to the oblateness of the dust particles.
The dielectric properties of composite particles can be derived by averaging those of their individual dust constituents. Two of the most common rules used to derive the dielectric properties of inhomogeneous particles are the Maxwell-Garnett, and the Bruggemann rule (see Bohren & Huffman 1983). The former rule regards the composite particle as a matrix with embedded inclusions, and is not invariant under the interchange of matrix and inclusions. The Bruggeman rule is, and for a composite particle consisting of three constituents (amorphous carbon, silicates, and vacuum), the average dielectric constant $\epsilon_{av}$ is given by the solution of a cubic equation (see Bohren & Huffman eq. \[8.51\]):
$$\sum^3_{j=1} {f_j {{\epsilon_j-\epsilon_{av}}\over
{\epsilon_j+2\epsilon_{av}}} }$$
where $\epsilon_j$ is the dielectric constant of the j$^{th}$ constituent, and where the sum of the volume filling factors, $f_j$, of the constituent material is unity. Figure 1 compares the dielectric constants of the composite particle with those of its constituents as a function of wavelength. The dielectric constants of the silicate grains were taken from Draine & Lee (1984), and those for amorphous carbon from Rouleau & Martin (1991).
The size distribution of the composite fluffy particles is given by (Mathis 1996): $$f(a)=a^{-\alpha_0}\ exp[-(\alpha_1a + \alpha_2/a +
\alpha_3 a^2)]$$ where $a$ is the grain radius, and the parameters {$\alpha_0, \alpha_1, \alpha_2, \alpha_3$} are equal to {3.5, 0.0033, 0.437, 50}, respectively (Mathis 1996, private communication). The size distribution is concentrated around 0.1 , dropping to 10% of its peak value at $a\approx
0.05$, and $0.2\mu$m.
Two factors contribute to the enhancement in the extinction of the composite grains at infrared wavelength. This can be seen from the extinction optical depth at wavelength $\lambda$ per H-column density which is given by: $$\begin{array}{lll}
\tau_{ext}(\lambda)/N_H & = & {\cal M}_d\ <\kappa_d(\lambda)> \\
& = & ({\cal M}_d/\rho) <3Q_{ext}/4a>
\end{array}$$ where ${\cal M}_d$ is the dust mass column density, and $<\kappa_d(\lambda)>\equiv
<3Q_{ext}/4\rho a>$ is the size-averaged dust mass extinction coefficient at wavelength $\lambda$, where Q$_{ext}$ is the extinction efficiency of a grain of radius $a$ and mass density $\rho$. The mass column density of the composite grains is equal to the sum of the mass column densities of its grain constituents. However, because of its porosity its mass density, $\rho$, is smaller than theirs. As a result, the extinction of the composite grains is larger than the sum of the extinction of its consituent particles, which is the main reason for the enhanced extinction at UV-optical wavelengths. At long wavelengths ($\lambda\gtrsim 1\ \mu$m), an additional factor plays a role in increasing the extinction of the composite particles, namely an increase in their value of Q$_{ext}$ compared to that of the amorphous carbon and silicate grains. In the Rayleigh limit, when $2\pi a/\lambda \ll\ 1$, Q$_{ext}/a$ can be written in terms of $\epsilon_1$, and $\epsilon_2$, the real and imaginary parts of the dielectric constant as: $$Q_{ext}/a = {24\pi\over\lambda} {\epsilon_2 \over (\epsilon_1+2)^2+\epsilon_2^2}$$ The value of Q$_{ext}/a\rightarrow 0$ when either $\epsilon_2\ll \epsilon_1$, or $\epsilon_2\gg 1$. It has a maximum when $\epsilon_2\approx\epsilon_1+2$. Figure 1 shows that at long wavelengths ($\lambda \gtrsim \ 100 \mu$m), $\epsilon_2 \ll
\epsilon_1$ for silicates, and $\epsilon_2 \gg 10$ for amorphous carbon. For the composite grains $\epsilon_2\approx \epsilon_1+2\ \approx 10 $, yielding a larger value of Q$_{ext}$/a over that of amorphous carbon or silicate grains. So at far-IR wavelengths, the averaging of the optical properties plays an important role in increasing the absorptivity/emissivity of the composite grains. As a result, the composite fluffy grains are somewhat cooler than either graphite or silicate grains of identical radius.
THE EFFECTS OF THE INCLUSION OF PAHs ON THE COMPOSITE FLUFFY DUST MODEL
=======================================================================
The Infrared Emission
---------------------
An important test for the validity of the CFD model is if it can reproduce the observed [*COBE*]{}/DIRBE and FIRAS 3.5 - 1000 emission from the diffuse ISM ([@ber94], [@was96], [@ar97], [@d97]). The Mathis model does not include carriers of the UIBs which are necessary to account for the emission in the 3.5, 4.9, and 12 DIRBE bands ([@d97]). So a priori, the CFD model needs to be ammended to include UIB carriers. For calculational purposes we identified these carriers with polycyclic aromatic hydrocarbons (PAHs), with extinctions and IR properties as given by Désert et al. (1990). With this PAH model, most of the 12 $\mu$m and all shorter wavelength diffuse emission is produced by these particles. The abundance of PAHs can therefore be directly determined from the near-IR [*COBE*]{} observations of the diffuse ISM (see Dwek et al. 1997 for details of the model). The PAH abundance required to reproduce the [*COBE*]{} data was added to the CFD model.
We note here that whereas the diffuse ISM spectrum obtained by the [*Infrared Astronomical Satellite*]{} ([*IRAS*]{}) could be reproduced by a simple extension of the graphite or silicate grain size distribution to very small sizes (Draine & Anderson 1985; Weiland et al. 1986), the DIRBE spectrum [*cannot*]{} be fitted by the same method. The DIRBE extends the [*IRAS*]{} 12, 25, 60, and 100 observations to shorter (and longer) wavelengths, and the near-IR 3.5 and 4.5 colors observed by DIRBE from the diffuse ISM are inconsistent with those calculated for stochastically-heated graphite or silicate grains (see Dwek et al. 1997; Figure 2). They are consistent, however, with those produced by PAHs. The addition of PAHs is therefore necessary in order to reproduce the short wavelength diffuse IR emission observed by the [*COBE*]{}.
Calculation of the IR emission requires knowledge of the composition, abundances, and size distribution of the dust particles, and the spectrum and intensity of the ambient radiation field. In the CFD model, all the dust parameters are determined by the fit to the interstellar extinction curve, except for the size distribution of the bare silicate and graphite grains. Their size distribution was not specified by Mathis (1996) since it does not affect the extinction as long as the dust grains are Rayleigh particles. For the purpose of our calculations we adopted an MRN power law distribution, $dn(a)/da\sim a^{-3.5}$ for the bare silicate and graphite grains in the 0.005 to 0.015 radius interval. The upper limit corresponds to the largest grain radius for which the value of $Q_{ext}/a$ is still independent of grain size.
Dust in the diffuse high latitude clouds is heated by the local interstellar radiation field (LISRF), and for calculational purposes we adopted the LISRF spectrum and intensity from the studies of Mathis, Mezger, & Panagia (1983). With the dust model and interstellar radiation field specified, we calculated the emerging IR emission without further adjustments of any model parameters. Our calculations include the effects of the stochastic heating and temperature fluctuations on the IR dust spectrum (see Dwek et al. 1997 for details).
Figure 3 compares the IR emission predicted by the CFD model to the observations. With the addition of PAHs, the model gives a very good fit to the observed intensity at $\lambda \lesssim 140\ \mu$m. PAHs radiate about 25% of the total diffuse IR emission (Dwek et al. 1997). However, at longer wavelengths the model produces a large excess of emission over that detected by the [*COBE*]{} satellite. As shown below, the enhanced IR emission is an unavoidable consequence of the UV-optical properties of the CFD particles.
The Interstellar Extinction and Dust Albedo
-------------------------------------------
Figure 4 presents the extinction predicted by the CFD model in the various wavelength regimes. Figure 4a depicts the UV-optical portion of the curve as a function of $x \equiv 1/\lambda(\mu m)$. The thin line in the figure shows the extinction presented by Mathis (1996) without PAHs. The model provides a very good fit to the data, deviating by less than 10% over the given range of $x$. The fit would have been better if, following Mathis (1996), we had modified the optical constants of the silicate grains in the 6 - 8 $\mu$m$^{-1}$ region to elliminate an artificial kink in their extinction curve at $\sim 7\
\mu$m$^{-1}$. The bold line in the figure depicts the total extinction when PAHs are included in the dust model. The calculated extinction is now significantly higher than the observed one, which should not be surprising, since PAHs radiate about 25% of the total diffuse IR emission. The [*shape*]{} of the PAH extinction is highly uncertain. Here we adopted the extinction properties of the PAHs from the work of Désert et al. (1990), which were chosen to reproduce the general interstellar extinction curve with their dust model. However, regardless of the exact form of their UV-optical cross sections, PAHs (or any other carriers of the UIBs) must be responsible for about $25(1-A)$% of the integrated extinction in the $1/\lambda=1 - 10\ \mu m^{-1}$ interval, where $A$ is the average albedo at UV-optical wavelength.
Figure 4b compares the extinction produced by the CFD model (including PAHs) to the observed infrared extinction (Mathis 1990; Table 1). The model produces higher extinction at wavelengths $\gtrsim 10 \mu$m, but it is nevertheless consistent with the observations considering the fact that uncertainties in the data are at least a factor of two at $\lambda \gtrsim 15 \mu$m.
Figure 5 depicts the albedo of the CFD model as a function of $x \equiv
1/\lambda(\mu m)$ (bold line). For comparison, we also plotted the albedo predicted by the MRN model with Draine-Lee optical constants (dotted line). In particular, the visual albedo of the MRN model is about 0.6, consistent with the value of $0.61\pm 0.07$ suggested by Witt (1989). The albedo of the CFD model is lower, about $0.5$, a fact already pointed out by Mathis (1996). So in spite of the fact that both, the MRN and the CFD, models reproduce the observed interstellar extinction equally well, the CFD model has a lower albedo over the whole UV-optical wavelength range. The composite fluffy dust particles therefore absorb more energy from the LISRF than the bare silicate graphite grains in the MRN model. This energy has to be reradiated at IR wavelengths, producing an excess far-IR emission compared to that produced by an MRN distribution of bare graphite and silicate grains.
The Carbon Abundance
--------------------
The main motivation for the composite dust model was to solve the interstellar carbon crisis. The original MRN model requires a C abundance of about 270 atoms to be in the dust (using Draine-Lee optical constants), significantly above the value of 85$\pm$55 implied from the recent ISM values. Kim & Martin (1996) optimized the grain size distribution, and produced a similar C abundance of 270$\pm$50 atoms in carbon dust. The CFD model of Mathis takes therefore a large step towards easing the carbon crisis, requiring only $\sim$160 C-atoms to be in the dust phase. However, the model ignores the amount of carbon locked up in PAHs which was recently estimated from the [*COBE*]{} data to be 70$\pm$20 C-atoms ([@d97]). This PAH abundance is higher by about a factor of two compared to previous estimates (e.g. Désert et al. 1990; Siebenmorgen & Krügel 1992). However, recent observations of the 6.05 absorption feature in the ISM with the [*Infrared Space Observatory*]{} ([*ISO*]{}; Schutte et al. 1996) suggest a similarly large value. Solid H$_2$O provides the bulk of the absorption in the 6.05 band. However, an excess absorption in the red wing of the feature is due to the C-C stretching mode, and the authors estimate that about 20% of the solar carbon abundance (they adopt (C/H)$_\odot
= 10^{-4}$) must be in this bond to produce the observed absorption. If this bond is in PAHs, then these observations confirm the PAH abundance estimates of Dwek et al. (1997). If we add PAHs to the composite dust model we get that it will require a total C abundance of 225$\pm$20 atoms (55 in small graphite grains, 105 as amorphous carbon in composite grains, and 70 in PAHs). This carbon abundance is smaller than any other dust model \[see Kim & Martin (1996) or Snow & Witt (1995) for a summary of the carbon abundance requirements of the various dust models\], but still significantly larger that the maximum available for the dust in the ISM.
PAHs as Carriers of the 2200 Å extinction feature?
--------------------------------------------------
One way to ease the carbon abundance constraint is to assume that PAHs are the carriers of the 2200 Å extinction bump as well as the UIBs. This assumption will reduce the C abundance by 55 atoms, the amount locked up in graphite grains. There are several arguments in favor of such an identification: (1) Laboratory measurements show that small PAHs have an enhanced UV absorptivity around 2200 Å (Joblin, Léger, & Martin 1992). Peak cross sections are about $8\ 10^{-18}\ cm^2$ per C-atom. For an opacity of $\tau_{2200}/N_H=5.27\
10^{-22}\ cm^{-2}$, and a Drude profile for the emission bump (e.g. Draine 1989), the required C abundance in PAHs is $\sim$ 65 atoms, comparable with the amount required to produce the UIB emission features; (2) extinction measurements along several lines of sight through the Chamaeleon cloud ([@bpg94]) show a corelation between the strength of the 2200 Å bump and the magnitude of the IRAS 12 emission. Since most of the 12 emission originates from PAHs, the correlation suggests that PAHs can be significant contributors to the UV extinction bump.
The problem with PAHs is that the peak wavelength of the UV emission bump and its profile seems to vary with PAH composition. Furthermore, the PAH cross sections exhibit a feature around 3000 Å (Joblin et al. 1992) that is not observed in the ISM. It yet remains to be seen if a suitable mixture of PAHs will match the interstellar 2200 Åextinction profile and smooth out the unobserved UV feature.
Even if PAHs can be substituted for graphite, the elimination of these particles will create an additional problem for the CFD model. The very small graphite particles are not only responsible for the 2200 Å extinction, but also contribute to the observed 25 - 60 excess emission, since they are stochastically heated by the LISRF. A population of very small particles will therefore be needed to account for the observed mid-IR emission, which is currently absent in the size distribution of the composite grains.
SUMMARY
=======
In this paper we showed that composite interstellar dust particles consisting of amorphous carbon and silicate dust with vacuum comprising 45% of their volume cannot be a major interstellar dust component in the diffuse ISM. Heated by the local interstellar radiation field, these particles produce a significantly higher IR emission than observed by the [*COBE*]{}/DIRBE and FIRAS instruments for the diffuse high latitude emission. The excess IR emission is the result of the increased absorptivity of the composite fluffy dust particles at UV-optical wavelengths. The robustness of this conclusion depends on the validity of the calculations of the optical properties of composite dust particles. A different rule for producing the average optical properties of the composite particles may produce an IR emissivity that is consistent with the [*COBE*]{} data, but it will certainly affect the UV-optical extinction as well, requiring a whole new assesment of the composite dust model.
The neglect of the composite fluffy dust model (or any other dust model) to include the carriers of the unidentified IR emission bands (UIBs) will result in an underestimate of the carbon abundance by $\sim$20% of its solar value, and will introduce a $f(1-A)$% uncertainty in the UV-optical part of the interstellar extinction curve, where $A\approx
0.6$ is the average UV-optical albedo of the dust, and $f\approx 0.25$ is the fraction of the total IR radiation emitted by these particles. Since the shape of the UV-optical absorptivity of the coal or PAHs that are responsible for the UIBs is unknown, it introduces a major uncertainty in any interstellar extinction model.
We point out that the inferred abundance of the UIB carriers is inversely proportional to their wavelength-averaged UV-optical absorption cross section. Lowering the amount of carbon in these particles will require an increase in their UV-optical extinction, and conversely, minimizing their effects on the extinction will require a high C abundance to be tied up in these carriers. To summarize, the composite fluffy dust model does not reproduce the UV-optical extinction to better than about 12%, and with the addition of UIB carriers it uses up twice the amount of the carbon believed to be available in the ISM for the dust.
Since the composite dust model falls short in solving the carbon crisis it is worth examining other possible solutions to this crisis. The major assumption leading to this crisis is that the dust abundance should reflects that of the current local ISM. This assumption requires that the ISM gas and dust have identical spatial and temporal histories. However, there is evidence to suggest that this may not be the case: (a) observations of a large sample of stars in the solar neighborhood (e.g. Edvardsson et al. 1993) shows an intrinsic scatter in their observed abundances as a function of time and metallicity; (b) the presence of isotopic anomalies in meteorites suggests that the solar neighborhood may have been polluted by the nucleosynthetic products of a neigboring stars and supernovae (e.g. Zinner 1996); (c) additional inhomogeneities may be introduced by dynamical fractionation between gas and dust in star formation processes. Numerical simulations of protostellar collapses (Ciolek & Mouschovias 1996) show that ambipolar diffusion can “strain” the infalling material, leaving the grains behind and reducing their abundance in the collapsed core which is destined to become a star.
We conclude that inhomogeneous chemical mixing and fractionation effects in the ISM may be responsible for the differences in metallicities between the sun and the local ISM, and perhaps between that inferred from the B stars and the extinction as well.
I acknowledge the Institut d’Astrophysique Spatiale in Orsay, and the Institut d’Astrophysique de Paris for providing a stimulating atmosphere and environment for the duration of my research. During my work I benefitted from many stimulating discussions with Francois Boulanger, Xavier Désert, Ant Jones, Alain Léger, Alain Omont, Renauld Papoular, Jean-Loup Puget, and Bill Reach. Special thanks are due to John Mathis who provided stimulating discussions, some intermediate results of his model which were used to check current calculations, and comments on a preliminary version of the manuscript. Comments by an anonymous referee led to further improvement in the manuscript. Finally, I acknowledge the NASA/GSFC Research and Study Fellowship Program for providing financial assistance during my stay at these institutes.
Arendt, R. G. et al. 1997, in preparation Bernard, J. -P., Boulanger, F., Désert, F.,-X., Giard, M., Helou, G.,& Puget, J. -L. 1994, , 291, L5 Bohren, C. F., & Huffman, D. R. 1983, Absorption and Scattering of Light by Small Particles (Wiley: New York) Boulanger, F., Prévot, M. L., & Gry, C. 1994, , 284, 956 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 245, 345 Cardelli, J. A., Meyer, D. M., Jura, M., & Savage, B. D. 1996, , 467, 334 Ciolek, G. E., & Mouschovias, T. Ch. 1996, ,468, 749 Désert, F. -X., Boulanger, F., & Puget, J. L.1990, , 237, 215 Draine, B. T., & Lee, H. M. 1984, , 285, 89 bibitem\[Draine & Anderson 1985\][da85]{} Draine, B. T., & Anderson, N. 1985, , 292, 494 Draine, B. T. 1985, , 57, 587 Draine, B. T. 1989, in IAU Symp. No. 135, Interstellar Dust, eds. L. J. Allamandola & A. G. G. M. Tielens (Boston: Kluwer), p. 313 Dwek, E. et al. 1997, , 475, 000 Edvardsson, B., Andersen, J., Gustafsson, B., Lambert, D. L., Nissen, P. E., & Tomkin, J. 1993, , 275, 101 Grevesse, N., Noels, A., Sauval, A. J. 1996, in Cosmic Abundances, ASP Conference Series, eds. S. S. Holts, & G. Sonneborn (San Francisco: ASPCS), p. 117 Joblin, C., Léger, A., & Martin, P. 1992, , 393, L79 Kim, S-H, Martin, P. G. 1996, , 462, 296 Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, , 217, 425, MRN Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, , 128, 212 Mathis, J. S., Whiffen, G. 1989, , 341, 808 Mathis, J. S. 1990, , 28, 37 Mathis, J. S. 1996, ApJ, 472, 643 Rouleau, F., & Martin, P. G. 1991, , 377, 526 Schutte, W. A. 1996, , 315, L333 Siebenmorgen, R., & Krügel, E. 1992,, 259, 614 Snow, T. P., & Witt, A. N. 1995, Science, 270, 1455 Weiland, J. L., Blitz, L., Dwek, E., Hauser, M. G., Magnani, L.,& Rickard, L. J. 1986, , 306, L101 Weiland, J. L., Arendt, R. G., & Sodroski, T. J. 1996, in AIP Conf. Proc. No. 348, Unveiling the Cosmic Infrared Background, ed. E. Dwek (New York: AIP), p. 74 Witt, A. N. 1989, in Interstellar Dust, eds. L. J. Allamandola & A. G. G. M. Tielens (Dordrecht: Kluwer), p. 87 Zinner, E. 1996, in Cosmic Abundances, ASP Conference Series, eds. S. S. Holts, & G. Sonneborn (San Francisco: ASPCS), p. 147
|
[**Neutrino Oscillations at High Energy by MACRO** ]{}
[**F Ronga$^{1}$ for the MACRO collaboration**]{}\
[*$^{1}$INFN Laboratori Nazionali di Frascati P.O. Box 13 Frascati ITALY\
*]{}
[**Abstract\
**]{}
We present updated results of the measurement of upward-going muons produced by neutrino interactions in the rock below the MACRO detector. These data support MACRO’s previously published results. They favor a neutrino oscillation explanation of the atmospheric neutrino anomaly.
Introduction: {#intro.sec}
=============
The interest in precise measurements of the flux of neutrinos produced in cosmic ray cascades in the atmosphere has been growing over the last years due to the anomaly in the ratio of contained muon neutrino to electron neutrino interactions. The past observations of Kamiokande, IMB and Soudan 2 are now confirmed by those of SuperKamiokande, MACRO and Soudan2 (with higher statistics) and the anomaly finds explanation in the scenario of $\nu_{\mu}$ oscillation (Fukuda 1998a). The effects of neutrino oscillations have to appear also in higher energy ranges. The flux of muon neutrinos in the energy region from a few GeV up to a few TeV can be inferred from measurements of upward throughgoing muons (Ahlen 1995,Ambrosio 1998b,Hatakeyama 1998, Fukuda 1998b). As a consequence of oscillations, the flux of upward throughgoing muons should be affected both in the absolute number of events and in the shape of the zenith angle distribution, with relatively fewer observed events near the vertical than near the horizontal due to the longer path length of neutrinos from production to observation.
Here an update of the measurement of the high energy muon neutrino flux is presented. The new data are in agreement with the old data. The MACRO low energy data are presented in another paper at this conference (Surdo 1999).
\[t\]
Upward Throughgoing Muons:
==========================
The MACRO detector is described elsewhere (Ahlen 1993, Ambrosio 1998b). Active elements are streamer tube chambers used for tracking and liquid scintillator counters used for the time measurement. The direction that muons travel through MACRO is determined by the time-of-flight between two different layers of scintillator counters. The measured muon velocity is calculated with the convention that muons going down through the detector are expected to have 1/$\beta$ near +1 while muons going up through the detector are expected to have 1/$\beta$ near -1.
Several cuts are imposed to remove backgrounds caused by radioactivity or showering events which may result in bad time reconstruction. The most important cut requires that the position of a muon hit in each scintillator as determined from the timing within the scintillator counter agrees within $\pm$70 cm with the position indicated by the streamer tube track.
When a muon hits 3 scintillator layers, there is redundancy in the time measurement and 1/$\beta$ is calculated from a linear fit of the times as a function of the pathlength. Tracks with a poor fit are rejected. Other minor cuts are applied for the tracks with only two layers of scintillator hit.
It has been observed that downgoing muons which pass near or through MACRO may produce low-energy, upgoing particles. These could appear to be neutrino-induced upward throughgoing muons if the down-going muon misses the detector (Ambrosio 1998a). In order to reduce this background, we impose a cut requiring that each upgoing muon must cross at least 200 g/cm$^2$ of material in the bottom half of the detector. Finally, a large number of nearly horizontal ($\cos \theta > -0.1$), but upgoing muons have been observed coming from azimuth angles corresponding to a direction containing a cliff in the mountain where the overburden is insufficient to remove nearly horizontal, downgoing muons which have scattered in the mountain and appear as upgoing. We exclude this region from both our observation and Monte-Carlo calculation of the upgoing events.
\[ht\]
Figure 1A) shows the $1/\beta$ distribution for the throughgoing data from the full detector running. A clear peak of upgoing muons is evident centered on $1/\beta=-1$.
There are 561 events in the range $-1.25 < 1/\beta < -0.75$ which we define as upgoing muons for this data set. We combine these data with the previously published data (Ahlen, 1995) for a total of 642 upgoing events. Based on events outside the upgoing muon peak, we estimate there are $12.5 \pm 6$ background events in the total data set. In addition to these events, we estimate that there are $10.5 \pm 4$ events which result from upgoing charged particles produced by downgoing muons in the rock near MACRO. Finally, it is estimated that $12 \pm 4$ events are the result of interactions of neutrinos in the very bottom layer of MACRO scintillators. Hence, removing the backgrounds, the observed number of upgoing throughgoing muons integrated over all zenith angles is 607.
In the upgoing muon simulation we have used the neutrino flux computed by the Bartol group (Agrawal 1996). The cross-sections for the neutrino interactions have been calculated using the GRV94 (Glück,1995) parton distributions set, which varies by +1% respect to the Morfin and Tung parton distribution that we have used in the past. We estimate a systematic error of 9% on the upgoing muon flux due to uncertainties in the cross section including low-energy effects (Lipari 1995). The propagation of muons to the detector has been done using the energy loss calculation (Lohmann 1985) for standard rock. The total systematic uncertainty on the expected flux of muons adding the errors from neutrino flux, cross-section and muon propagation in quadrature is $\pm17\%$. This theoretical error in the prediction is mainly a scale error that doesn’t change the shape of the angular distribution. The number of events expected integrated over all zenith angles is 824.6, giving a ratio of the observed number of events to the expectation of 0.74 $\pm0.031$(stat) $\pm0.044$(systematic) $\pm0.12$(theoretical).
Figure 1 B) shows the zenith angle distribution of the measured flux of upgoing muons with energy greater than 1 GeV for all MACRO data compared to the Monte Carlo expectation for no oscillations and with a $\nu_{\mu} \rightarrow\nu_{\tau}$ oscillated flux with $\sin^2 2 \theta = 1$ and $\Delta m^2 = 0.0025$ eV$^2$ (dashed line).
The shape of the angular distribution has been tested with the hypothesis of no oscillation excluding the last bin near the horizontal and normalizing data and predictions. The $\chi^2$ is $22.9$, for 8 degrees of freedom (probability of 0.35% for a shape at least this different from the expectation). We have considered also oscillations $\nu_{\mu} \rightarrow\nu_{\tau}$. The best $\chi^2$ in the physical region of the oscillations parameters is 12.5 for $\Delta m^2$ around $0.0025 eV^2$ and maximum mixing (the best $\chi^2$ is 10.6 , outside the physical region for an unphysical value of $\sin^2 2 \theta =1.5$).
To test the oscillation hypothesis, we calculate the independent probability for obtaining the number of events observed and the angular distribution for various oscillation parameters. They are reported for $\sin^2 2 \theta = 1$ in Figure 2 A) for $\nu_{\mu}
\rightarrow\nu_{\tau}$ oscillations. It is notable that the value of $\Delta m^2$ suggested from the shape of the angular distribution is similar to the value necessary in order to obtain the observed reduction in the total number of events in the hypothesis of maximum mixing. Figure 2 B) shows the same quantities for sterile neutrinos oscillations (Akhmedov 1993,Liu 1998).
Figure 3 A) shows probability contours for oscillation parameters using the combination of probability for the number of events and $\chi^2$ of the angular distribution. The maximum of the probability is 36.6% for oscillations $\nu_{\mu} \rightarrow\nu_{\tau}$. The best probability for oscillations into sterile neutrinos is 8.4%. The probability for no oscillation is 0.36%.
Figure 3 B) shows the confidence regions at the 90% and 99% confidence levels based on application of the Monte Carlo prescription in (Feldman 1998). We plot also the sensitivity of the experiment. The sensitivity is the 90% contour which would result from the preceding prescription when the data are equal to the Monte Carlo prediction at the best-fit point.
Conclusions:
============
The upgoing throughgoing muon data set is in favor of $\nu_{\mu} \rightarrow\nu_{\tau}$ oscillation with parameters similar to those observed by Superkamiokande with a probability of 36.6% against the 0.36% for the no oscillation hypothesis. The probability of oscillations from the angular distributions only is 13%. The probabilities are higher than the ones of the old data (Ambrosio 1998b). The neutrino sterile oscillation hypothesis is slightly disfavored.
[**References**]{}
Agrawal V. et al 1996 Phys. Rev. D53 1314\
Ahlen S. et al.(MACRO collaboration) 1995, Phys. Lett. B 357 481\
Ahlen S. et al.(MACRO collaboration) 1993, Nucl.Instrum.Meth.A324:337-362\
Akhmedov E. , Lipari P. Lusignoli M. 1993, Phys.Lett. B300:128-136\
Ambrosio M. et al.(MACRO collaboration) 1998a, Astropart.Phys.9:105-117\
Ambrosio M. et al.(MACRO collaboration) 1998b, Phys Lett. B. 434 451\
Feldman G. and Cousins R. 1998 Phys. Rev. D57 3873\
Fukuda Y. et al. (SuperKamiokande collaboration) 1998a Phys.Rev.Lett.81:1562-1567\
Fukuda Y. et al. (SuperKamiokande collaboration) 1998b, e-Print Archive hep-ex/9812014\
Glück M., Reya E. and Stratmann M.1995, Z. Phys. C67, 433\
Hatakeyama S. et al. (Kamiokande collaboration) 1998, Phys Rev Lett 81 2016\
Lipari P. Lusignoli M. and Sartogo F. 1995, Phys. Rev. Lett. 74 4384\
Liu Q.Y. and Smirnov A.Yu. 1998, Nucl.Phys. B524 505\
Lohmann H. Kopp R.,Voss R. 1985, CERN-EP/85-03\
Surdo A. (MACRO collaboration) 1999, HE4.1.06 in this conference
|
---
abstract: 'We explore *folded* spinning string configurations over torsional Newton Cartan (TNC) geometry with $ R\times S^2 $ topology within the semiclassical approximation. To start with, considering zero temperature strong coupling limit, we compute the so called *Hamiltonian spectrum* from spinning strings on $R \times S^2 $. We further extend our analysis considering quantum fluctuations over the classical space of solutions and compute one loop string ($ \alpha'' $) correction to the energy spectrum in the gauge theory sector that is dual to TNC spinning strings on $R \times S^2 $. Finally, we consider the large $ c $ and/or nonrelativistic (NR) limit associated with the world-sheet d.o.f. and compute the one loop string correction to the energy spectrum in the dual Spin Matrix Theory (SMT) theory at strong ($ \mathtt{g}\gg 1 $) coupling and low temperatures.'
author:
- |
[**[Dibakar Roychowdhury]{}$
$[^1]**]{}\
[Department of Physics, Indian Institute of Technology Roorkee,]{}\
[Roorkee 247667, Uttarakhand, India]{}\
title: '[**[Semiclassical dynamics for nonrelativistic strings]{}**]{}'
---
Overview and Motivation
=======================
The quest for a consistent (UV finite) low energy description of relativistic string theory had been one of the active areas of theoretical investigations for last couple of decades [@Gomis:2000bd]-[@Batlle:2017cfa]. Apparently there seems to be existing two parallel formulations of nonrelativistic (NR) string theories over curved target space geometries. One of these formulations is based on taking $ 1/c $ limit of General Relativity (GR) in the first order formalism that eventually leads towards curved manifold structures (known as *string Newton-Cartan* (SNC) geometry) together with a flux-less auxiliary two form field [@Andringa:2012uz]-[@Bergshoeff:2019pij]. It turns out that under such circumstances one could in fact define a NR quantum consistent 2D sigma model that is invariant under transformations generated by SNC generators [@Bergshoeff:2018yvt]-[@Bergshoeff:2018vfn]. In other words, NR closed string spectra on SNC background may be obtained through large $ c $ expansion of the relativistic Nambu-Goto (NG) action.
The other approach is based on the formulation of 2D sigma models on curved manifolds called torsional Newton-Cartan (TNC) geometries [@Harmark:2017rpg]-[@Roychowdhury:2019olt]. TNC strings are obtained through target space null reduction of Poincare invariant 2D string sigma models while keeping the string momentum along the null isometry direction fixed[^2]. Taking a zero tension limit of TNC strings finally leads towards an emerging new sector in the celebrated gauge/string duality. For example, in the case of $ AdS_5 \times S^5 $ (super)strings such scaling results in the so called NR 2D sigma model/Spin Matrix Theory (SMT) correspondence [@Harmark:2017rpg]-[@Harmark:2018cdl]. The NR 2D sigma model thus obtained has been found to possess an underlying Galilean Confromal Algebra together with a NR Weyl invariance. On the other hand, in the case of $ AdS_5 \times S^5 $ strings, the SMT limit corresponds to taking a NR limit of the relativistic magnon dispersion relation in $ \mathcal{N}=4 $ SYM [@Harmark:2008gm]-[@Harmark:2014mpa].
The central idea that lives at the heart of the gauge/string duality is the identification of the energy spectra associated with gauge invariant operators (in the dual gauge theory) to that with the stringy excitations over curved background geometry. In other words, the duality conjecture drives us towards a natural identification of *heavy* operators and/or gauge theory states with large quantum numbers (in the strongly coupled QFTs) to that with the solitonic excitations associated to certain 2D sigma model in the supergravity approximations [@Berenstein:2002jq]-[@Gubser:2002tv]. An immediate example along this line of argument is the celebrated correspondence between the energy spectra ($ \sim E $) associated to $ AdS_5\times S^5 $ super-strings in 10D type IIB supergravity and the operator spectrum ($ \sim \Delta $) corresponding to $ \mathcal{N}=4 $ SYM in 4D [@Berenstein:2002jq]-[@Frolov:2003tu]. The purpose of the present analysis is therefore to pursue similar questions in the context of NR strings/SMT correspondence where (considering certain specific limits [@Harmark:2014mpa]) we explore gauge theory states (with large quantum numbers) in the strongly coupled SMT sector by probing solitonic (stringy) excitations associated to NR 2D sigma model on $ R\times S^2 $.
The set of gauge invariant operators (in the dual SMT theory at strong coupling and low temperatures) that we choose to work with are those with large R charge ($ \tilde{\mathtt{J}}_{\varphi} \gg1$) and large energy quantum numbers ($ \tilde{\Delta}_{NR}\gg 1 $). On the string theory side, we realize these states as solitonic excitations [@Harmark:2014mpa] associated to closed NR folded spinning string configurations over $ R\times S^2 $. These strings are supposed to be stretched along the polar coordinate of $ S^2 $ and spinning around their center of mass that is happened to be coincident with the north pole of $ S^2 $ [@Gubser:2002tv]. A straightforward computation on the stringy side reveals (\[e87\]), $$\begin{aligned}
\tilde{\Delta}_{NR}-\tilde{\mathtt{J}}_{\varphi}=\tilde{\tilde{\mathtt{f}}}(\bar{\mathtt{g}})\tilde{\mathtt{J}}_{\varphi}
\label{anomalous}\end{aligned}$$ where we compute the function, $$\begin{aligned}
\tilde{\tilde{\mathtt{f}}}(\bar{\mathtt{g}})=\mathfrak{q}_1\bar{\mathtt{g}}\left( 1+\frac{\mathfrak{q}_2}{\sqrt{\tilde{\mathtt{J}}_{\varphi}}}\frac{1}{\bar{\mathtt{g}}^{1/4}}+..\right) \end{aligned}$$ upto one loop in the string corrections. Here, $ \bar{\mathtt{g}}(=\frac{\mathtt{g}}{\tilde{\mathtt{J}}_{\varphi}^{2}} )\ll 1$ should be regarded as being the *effective* expansion parameter in the dual SMT in the regime of strong $ (\mathtt{g}=c^2 \lambda\gg1 )$ coupling where, $ \lambda \ll 1 $ is the standard t’Hooft coupling in the $ \mathcal{N}=4 $ SYM theory and $ c(\rightarrow \infty) $ is the speed of light [@Harmark:2017rpg]-[@Harmark:2018cdl]. The above result (\[anomalous\]) is therefore a non-perturbative effect from the perspective of the dual SMT and is the key finding of the present analysis.
The rest of the paper is organised as follows. We start our analysis in Section 2, with a formal computation of the folded spinning TNC (closed) string spectra on $ R\times S^2 $ using NG formulation of the 2D sigma model [@Roychowdhury:2019olt]. We further extend this analysis in Section 3 by incorporating one loop stringy effects on the operator spectrum in the dual gauge theory at strong couplings. Our analysis also reveals that in the presence of stringy fluctuations the spectrum is analytically tractable only in the limit of *short* strings. However, the most significant part of this paper seems to be contained within Section 4 where we take $ 1/c $ expansion of the world-sheet d.o.f. which finally results in NR 2D sigma models over TNC geometry with $ R\times S^2 $ topology. From the perspective of $ \mathcal{N}=4 $ SYM theory such a tensionless limit of (super)strings propagating over (sub-space of) $ AdS_5\times S^5 $ geometry corresponds to zooming into a specific sub-sector of the full operator spectrum near its unitarity bound [@Harmark:2017rpg]-[@Harmark:2018cdl] known as the Spin Matrix Theory (SMT). In our analysis, considering a the limit of strong ($ \mathtt{g}\gg 1 $) coupling as well as low temperatures we find an effective way of expressing the energy eigenvalues (associated with the dual operator spectrum with large R charge) from solitonic excitations [@Harmark:2014mpa] over $ R\times S^2 $. Going one step further, we also explore NR stringy effects beyond leading order approximation and compute one loop stringy corrections to the energy eigenvalues at strong coupling. Finally, we summarise and conclude in Section 5.
TNC strings on $ R \times S^2 $
===============================
Our analysis starts with a formal construction of the spinning string sigma model over torsional Newton-Cartan (TNC) geometry [@Harmark:2017rpg]-[@Harmark:2018cdl] with $ R \times S^2 $ topology [@Grosvenor:2017dfs], $$\begin{aligned}
ds_{TNC}^2 = 2 \tau (d \mathfrak{u}-\mathfrak{m})+\mathfrak{h}_{\mu \nu}dx^{\mu}dx^{\nu}\label{e1}\end{aligned}$$ where each of the individual entities could be formally expressed as [@Roychowdhury:2019olt], $$\begin{aligned}
\tau &=&\tau_{\mu}dx^{\mu}=\frac{1}{2}d\psi +dt-\frac{1}{2}\cos\theta d\varphi ~;~ \mathfrak{u}=\frac{\psi}{4}-\frac{t}{2}\nonumber\\
\mathfrak{m}&=&\mathfrak{m}_{\mu}dx^{\mu} =\frac{1}{4}\cos\theta d\varphi ~;~
\mathfrak{h}_{\mu \nu}dx^{\mu}dx^{\nu} =\frac{1}{4}\left[ d\theta^{2}+\sin^2 \theta d\varphi^{2}\right].\label{e2}\end{aligned}$$
Notice that, here $ \mathfrak{u} $ and $ \varphi $ are the *isometry* directions associated with the target space geometry [@Grosvenor:2017dfs],[@Roychowdhury:2019olt]. The relativistic action corresponding to the *bosonic* sector of the 2D string sigma model could be formally expressed as[^3], $$\begin{aligned}
\mathcal{S}_{NG} =\int d^2\sigma \mathcal{L}_{NG}=- \frac{\sqrt{\lambda}}{4 \pi}\int d\tau d\sigma \tilde{\mathcal{L}}_{NG}~;~\sqrt{\lambda}=\frac{L^2}{\alpha'}
\label{e4}\end{aligned}$$ where we identify the corresponding sigma model Lagrangian as[^4] [@Harmark:2018cdl],[@Roychowdhury:2019olt] $$\begin{aligned}
\tilde{\mathcal{L}}_{NG}=\frac{\varepsilon^{\alpha \alpha'}\varepsilon^{\beta \beta'}}{\varepsilon^{\alpha \alpha'}\chi_{\alpha}\partial_{\alpha'}\zeta}(\partial_{\alpha'}\zeta \partial_{\beta'}\zeta -\chi_{\alpha'}\chi_{\beta'})\left( \partial_{\alpha}\theta \partial_{\beta}\theta +\sin^{2}\theta\partial_{\alpha}\varphi \partial_{\beta}\varphi\right)-\varepsilon^{\alpha \beta}\cos\theta \partial_{\alpha}\varphi \partial_{\beta}\zeta.
\label{E4}\end{aligned}$$
Notice that, here $ \zeta $ is the additional compact direction associated with the target space geometry along which the string has non zero windings [@Harmark:2017rpg]-[@Harmark:2018cdl]. Moreover, here $ \varepsilon^{01}=-\varepsilon_{01}=1 $ is the Levi-Civita symbol in 2D together with [@Roychowdhury:2019olt], $$\begin{aligned}
\chi_{\alpha}=2\partial_{\alpha}t+\partial_{\alpha}\psi - \cos\theta \partial_{\alpha}\varphi.\end{aligned}$$
The BMN limit
-------------
Before getting into the spinning string dynamics, it is customary first to consider the BMN like limit for semiclassical strings propagating over TNC geometry with $ R \times S^2 $ topology. We start with a particular ansatz for the semiclassical string whose centre of mass is considered to be rotating within $ S^2 $ (at an angle $ \theta =\theta_{0}= $const.) with an angular velocity along the azimuthal direction ($\varphi $). The most natural ansatz in this case turns out to be, $$\begin{aligned}
t=\tau ~;~ \varphi =\varpi \tau ~;~\psi = const ~;~\zeta = \mathfrak{n}~\sigma\end{aligned}$$ where, $\mathfrak{n} $ is the corresponding winding number of the string along the additional compact dimension ($ \zeta $) of the target space geometry.
The corresponding Lagrangian density (\[E4\]) turns out to be[^5], $$\begin{aligned}
\tilde{\mathcal{L}}_{NG}=\frac{\sin^2\theta_{0}\dot{\varphi}^{2}}{(2 \dot{t}-\cos\theta_{0}\dot{\varphi})}-\cos\theta_0 \dot{\varphi}.
\label{e7}\end{aligned}$$
The integrals of motion associated with the 2D sigma model (\[e7\]) could be formally expressed as, $$\begin{aligned}
\label{e8}
\mathcal{E}&=&\frac{\sqrt{\lambda}\varpi^{2}\sin^2 \theta_0}{(2 - \varpi \cos \theta_0 )^2}\\
\mathcal{J}_{\varphi}&=&\frac{\sqrt{\lambda}}{2}\left(\cos\theta_0 - \varpi \sin^2\theta_0\frac{(4 -\varpi \cos\theta_0)}{(2-\varpi \cos\theta_0)^2} \right).
\label{e9}\end{aligned}$$
The parameters of the theory, on the other hand, are constrained by the so called Virasoro constraint(s) [@Roychowdhury:2019olt], $$\begin{aligned}
\label{e10}
T_{\tau \tau}&=&\frac{\varpi}{4}(\varpi - 2 \cos\theta_0)+(2-\varpi \cos\theta_0)\left( \frac{\varpi}{4}\cos\theta_0 + \ell_{-}(2-\varpi \cos\theta_0)\right) \approx 0\\
T_{\tau \sigma}&\approx &0\end{aligned}$$ where $ \ell_{-} $ is the so called Lagrange multiplier.
Combining (\[e8\]) and (\[e9\]) we find, $$\begin{aligned}
\Delta_{NR} -\mathcal{J}_{\varphi}=\mathfrak{f}(\lambda)
\label{e12}\end{aligned}$$ where $ \Delta_{NR}\sim \mathcal{E} $ is the energy eigenvalue associated with the operator (spectrum) (in the strongly correlated QFT) dual to that of the (*relativistic*) stringy excitations over TNC geometry. The above (\[e12\]) is a valid dispersion relation provided both $\Delta_{NR}\sim |\mathcal{E}|\gg 1 $ and $ |\mathcal{J}_{\varphi}|\gg 1$ such that the ratios like $ \frac{|\mathcal{E}|}{\sqrt{\lambda}} $ and $ \frac{|\mathcal{J}_{\varphi}|}{\sqrt{\lambda}} $ are finite in the strong coupling ($ \lambda \gg 1 $) limit of the dual QFT. In other words, one essentially computes the spectrum associated with the so called *heavy* operators in the dual gauge theory. The function, $$\begin{aligned}
\mathfrak{f}(\lambda)=\frac{\sqrt{\lambda}}{2}~\varsigma (\varpi , \theta_{0})~;~\varsigma (\varpi , \theta_{0})=\varpi \sin^2\theta_0 \frac{(2\varpi +4 -\varpi \cos\theta_0)}{(2-\varpi \cos\theta_0)^2}-\cos\theta_0\end{aligned}$$ corresponds to the leading order correction to the energy eigenvalues ($ \Delta_{NR} $) of the heavy (BMN) operators in the dual gauge theory.
Considering the centre of mass of the string to be rotating along the equator of $ S^2 $ at a speed of light amounts of setting $ \theta_{0}=\frac{\pi}{2} $ and $ \varpi =1 $ which finally yields, $$\begin{aligned}
\Delta_{NR}\approx \mathcal{J}_{\varphi}\sqrt{1- \frac{3}{4}\tilde{\lambda}}= \mathcal{J}_{\varphi}\left(1-\frac{3}{8}\tilde{\lambda} +..\right)
\label{e17}\end{aligned}$$ which is the pp wave analogue of [@Berenstein:2002jq] where one could think of $ \Delta_{NR} $ as being the dimension associated to BMN (like) operators in the near BPS sector whose energy eigenvalue gets corrected due to an *effective* coupling, $ \tilde{\lambda}=\frac{\lambda}{\mathcal{J}^{2}_{\varphi}} $ such that $ | \tilde{\lambda}| <1 $ in the limit when both $ |\lambda| \gg 1 $ and $ |\mathcal{J}_{\varphi}| \gg 1 $.
Comparing (\[e17\]) to that with the corresponding (semi)classical stringy excitations over $ AdS_5 \times S^5 $ [@Frolov:2003qc]-[@Frolov:2003xy] we propose the following “BMN like" expression for the semi-classical stringy excitations over TNC geometry with $ R\times S^2 $ topology, $$\begin{aligned}
\mathcal{E}=k_{0}\mathcal{J}_{\varphi}\left( 1+ k_1 \tilde{\lambda}+..\right) \end{aligned}$$ where we identify $ k_{0}=1 $ and $ k_1 = -\frac{3}{8} $ which are clearly distinct numbers as compared to that with the corresponding relativistic scenario [@Frolov:2003qc]. We further notice that, like in the $ AdS_5 \times S^5 $ example [@Berenstein:2002jq], the non-perturbative (quantum) corrections (to the classical NR stringy excitations) are indeed suppressed in the limit of large $ \mathcal{J}_{\varphi} $.
More general ansatz
-------------------
We now proceed towards establishing the dispersion relation corresponding to spinning string configurations over $ R\times S^2 $. Unlike the previous example, the centre of mass of the string does not move on $ S^2 $ rather the string itself is considered to be spinning around its centre of mass and is stretched along the direction of the polar coordinate ($ \theta $). We choose the centre of mass of the spinning string to be coincident to that with the north pole of $ S^2 $. Therefore, we choose to work with an ansatz that corresponds to *folded* spinning strings inside an $ S^2 $ which is extended along the polar angle ($ \theta $) with its end points rotating along the compact azimuthal ($ \varphi $) direction [@Gubser:2002tv], $$\begin{aligned}
t=\tau ~;~ \varphi =\varpi \tau ~;~\psi = const ~;~\zeta = \sigma ~;~\theta = \theta (\sigma).\end{aligned}$$
The corresponding Lagrangian density could be formally expressed as, $$\begin{aligned}
\tilde{\mathcal{L}}_{NG}=\frac{\dot{\varphi}^{2}\sin^2\theta}{(2\dot{t}-\dot{\varphi}\cos\theta)}-(2\dot{t}-\dot{\varphi}\cos\theta)\theta'^2 -\dot{\varphi}\cos\theta\end{aligned}$$ where prime stands for derivative w.r.t. $ \sigma $ and dot corresponds to derivative w.r.t. $ \tau $.
The equation of motion corresponding to $ \theta $ could be found as, $$\begin{aligned}
2\theta'' (2-\varpi \cos\theta)+\varpi \sin\theta (1+\theta'^2)+ \frac{\varpi^{2}\sin\theta}{(2-\varpi \cos\theta)^2}(4\cos\theta -\varpi (1+\cos^2 \theta)) = 0.
\label{e18}\end{aligned}$$
On the other hand, the Virasoro constraint(s) [@Roychowdhury:2019olt] yield, $$\begin{aligned}
\label{e19}
T_{\tau \tau}&=&T_{\sigma \sigma}\nonumber\\
&=&\theta'^2 +(2-\varpi \cos\theta)(4\ell_- (2-\varpi \cos\theta)+\varpi \cos\theta)+\varpi (\varpi -2 \cos\theta)\approx 0\\
T_{\tau \sigma}&=& \ell_+ \chi_{\tau} \approx 0\end{aligned}$$ which thereby identically sets one of the Lagrange multipliers [@Harmark:2018cdl],[@Roychowdhury:2019olt] namely, $ \ell_+ =0 $. Furthermore, in order to simplify our analysis, and without loss of any generality we set $ \varpi =1 $ which thereby yields, $$\begin{aligned}
\theta'^2 (\sigma)=\vert(2\cos\theta -1)-(2- \cos\theta)(\mathfrak{a} +\mathfrak{b} \cos\theta)\vert\equiv \varrho (\theta)
\label{e21}\end{aligned}$$ where $ \mathfrak{a} =8 \ell_- $ and $ \mathfrak{b} = 1-4\ell_- $ are two real constants for our problem.
In the following we note down the corresponding conserved charges[^6], $$\begin{aligned}
\label{e22}
\mathcal{E}- \frac{1}{2}|\mathtt{J}_{\varphi}|=\frac{\sqrt{\lambda}}{ 2\pi}\int_{0}^{\theta_{0}} \frac{d\theta}{\sqrt{|\varrho|}}\left[ \cos\theta \left( \frac{(5 -4 \cos\theta)}{(2- \cos\theta)^2}- \varrho\right)+4\varrho\right] =\frac{\sqrt{\lambda}}{ 2\pi}\Lambda (\theta_0)\end{aligned}$$ where we identify the (semi)classical energy of the string[^7], $$\begin{aligned}
\mathcal{E}= \frac{\sqrt{\lambda}}{ \pi}\int_{0}^{\theta_{0}} \frac{d\theta}{\sqrt{|\varrho|}}\left( \frac{2\sin^2 \theta}{(2- \cos\theta)^2}+2 \varrho\right)= \frac{\sqrt{\lambda}}{ \pi}\Theta (\theta_{0}).\end{aligned}$$
A straightforward computation further reveals that both $ \Lambda (\theta) $ and $\Theta(\theta) $ are pretty complicated functions of various elliptic integrals. However both of them vanish exactly at the pole of $ S^2 $ namely for $ \theta =0 $. Therefore, the corresponding values associated with these conserved charges are solely fixed by the upper bound of the polar angle $ \theta \sim \theta_0 $.
### Short strings
We first consider the limit in which the folded strings are considered to be spinning close to the north pole. The short strings are therefore defined in the limit $| \theta_{0}|\ll 1 $ which finally yields, $$\begin{aligned}
\Lambda (\theta_{0})\approx \frac{\theta_{0} (-3 \mathfrak{a}-3 \mathfrak{b}+4)}{\sqrt{|- \mathfrak{a}- \mathfrak{b}+1|}}+\frac{\theta_{0} ^3 \left(7 \mathfrak{a} \mathfrak{b}+ \mathfrak{a} (5 \mathfrak{a}-2)+2 \mathfrak{b}^2\right)}{12 |- \mathfrak{a}- \mathfrak{b}+1|^{3/2}}+..
\label{e24}\end{aligned}$$
Using (\[e22\]) and (\[e24\]) it is now straightforward to show, $$\begin{aligned}
\Delta_{NR}\sim \mathcal{E}\approx \frac{1}{2}\mathtt{J}_{\varphi}\left(1+\frac{\tilde{\lambda}}{\pi^2} \mathfrak{g}(\theta_{0}) \right) \end{aligned}$$ where we introduce the function, $$\begin{aligned}
\mathfrak{g}(\theta_0)=\theta_{0} ^2 \left(\frac{1}{ \mathfrak{a}+ \mathfrak{b}-1}-3 \mathfrak{a}-3 \mathfrak{b}+1\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\frac{1}{6} \theta_{0} ^4 \left(-\frac{6}{ \mathfrak{a}+ \mathfrak{b}-1}-\frac{ \mathfrak{a}+2}{( \mathfrak{a}+ \mathfrak{b}-1)^2}- \mathfrak{a}+2 \mathfrak{b}+12\right)+..\end{aligned}$$ Here, $ \tilde{\lambda}=\frac{\lambda}{\mathtt{J}^{2}_{\varphi}} \ll 1$ is the effective coupling in the dual gauge theory in the semiclassical approximation, $ \mathtt{J}_{\varphi}\gg 1 $.
For short strings, it is also possible to find the embedding function $ \theta(\sigma) $ to certain approximations. Using (\[e18\]) and (\[e21\]) and after some simple steps it is quite straightforward to find, $$\begin{aligned}
\theta'' (\sigma)+ \kappa \theta \approx 0\end{aligned}$$ which thereby yields[^8], $$\begin{aligned}
\theta (\sigma)\equiv \theta_s (\sigma)=\sin \sqrt{\kappa}\sigma \approx \sqrt{\kappa}\sigma +..
\label{E28}\end{aligned}$$ where, $ \kappa = 2-\frac{1}{2}(\mathfrak{a}+ \mathfrak{b}) $ and $ -\theta_0 \leq \sigma \leq \theta_0 $ .
With this short string approximation (\[E28\]), it is now straightforward to compute, $$\begin{aligned}
\mathcal{E}- \frac{1}{2}\mathtt{J}_{\varphi} \approx\frac{2 \sqrt{\lambda}}{3\pi} \theta _0 \left(3-8 \theta _0^2\right)
\label{E29}\end{aligned}$$ where we set $ \mathfrak{a}=2 $ and $ \mathfrak{b}=0 $ for the sake of simplicity.
Using (\[E29\]), after few simple steps one finally arrives at the dispersion relation of the following form, $$\begin{aligned}
\Delta_{NR}\sim \mathcal{E}\approx \frac{1}{2}\mathtt{J}_{\varphi}\left(1+\hat{\gamma}_{S} (\tilde{\lambda})\right) \end{aligned}$$ where we identify the function, $$\begin{aligned}
\hat{\gamma}_{S}(\tilde{\lambda})\approx \mathfrak{h}_1\tilde{\lambda}+..
\label{E31}\end{aligned}$$ together with the coefficient, $$\begin{aligned}
\mathfrak{h}_1 \approx \frac{128\theta _0^4}{3\pi^2}+\mathcal{O}(\theta _0^5)\end{aligned}$$ Our goal in the next section would be to compute the one loop string correction to the above expression in (\[E31\]). This would be achieved considering a specific embedding for fluctuating TNC strings over $ R\times S^2 $.
### Extended strings
We now consider the *extended* string limit which essentially corresponds to folded spinning strings whose end points reach the equator namely, $| \theta_{0}|\sim \frac{\pi}{2} $. This finally yields the dispersion relation of the following form, $$\begin{aligned}
\mathcal{E}- \frac{1}{2}\mathtt{J}_{\varphi}\approx \frac{\sqrt{\lambda}}{ 2\pi}\Lambda \left( \frac{\pi}{2}\right).
\label{e27}\end{aligned}$$ It turns out that $ \Lambda \left( \frac{\pi}{2}\right) $ is a quite cumbersome function unless we fix the Lagrange multiplier(s) (associated to TNC string sigma models [@Harmark:2018cdl],[@Roychowdhury:2019olt]) which in turn fixes the constants $ \mathfrak{a} $ and $ \mathfrak{b} $ in the problem.
A careful analysis further reveals that for extended strings, $$\begin{aligned}
\Delta_{NR} \sim \mathcal{E} \approx \frac{1}{2}\mathtt{J}_{\varphi}\left( 1+\hat{\gamma}_L(\tilde{\lambda})\right)\end{aligned}$$ where we identify, $$\begin{aligned}
\hat{\gamma}_L (\tilde{\lambda})\approx \mathfrak{f}_1\tilde{\lambda}+..\end{aligned}$$ together with the coefficient, $$\begin{aligned}
\mathfrak{f}_1=\frac{\Lambda \left( \frac{\pi}{2}\right)}{\pi^2}\left( 2\Theta \left( \frac{\pi}{2}\right)-\Lambda \left( \frac{\pi}{2}\right)\right).\end{aligned}$$
As an example, setting $ \mathfrak{a}=2 $ and $ \mathfrak{b}=0 $ we note, $$\begin{aligned}
\Theta \left( \frac{\pi}{2}\right)=\frac{1}{3} \left(\sqrt{5}-6 F\left(\left.\frac{\pi }{4}\right|-8\right)+14 E\left(\left.\frac{\pi }{4}\right|-8\right)\right)
\label{e31}\end{aligned}$$ $$\begin{aligned}
\Lambda \left( \frac{\pi}{2}\right)=\frac{1}{6} \Im \left(-156 F\left(\sin ^{-1}\left(\sqrt{5}\right)|\frac{1}{9}\right)+171 E\left(\sin ^{-1}\left(\sqrt{5}\right)|\frac{1}{9}\right)\right)\nonumber\\
+\frac{8}{3}\Im\left( \Pi \left(-\frac{1}{3};-\sin ^{-1}\left(\sqrt{5}\right)|\frac{1}{9}\right)\right) -\frac{\sqrt{5}}{3}
\label{e32}\end{aligned}$$ where, $ \Im $ stands for the imaginary part of the complex *elliptic* integrals [@abramh] above in (\[e32\]).
Stringy corrections
===================
Below we develop a general algorithm to find one loop string corrections to the energy eigenvalue ($ \Delta_{R} $) corresponding to TNC spinning string solitons on $ R\times S^2 $. This is in spirit is quite similar to that with the earlier analysis done in the context of $ AdS_5 \times S^5 $ string solitons [@Frolov:2002av]. Our aim would be to find a specific *solvable* sector within the TNC string solitonic sigma models and calculate one loop correction to the corresponding energy eigenvalue.
The formalism
-------------
We now build up a general formalism to compute quantum corrections to the dispersion relation for semiclassical strings propagating over TNC geometry. In order to do so, we consider the effects of quantum fluctuations upto leading order in the t’Hooft coupling, $$\begin{aligned}
t= \tau +\frac{\tilde{\tau}(\sigma^{\alpha})}{\lambda^{1/4}}~;~\psi =\frac{\tilde{\psi}(\sigma^{\alpha})}{\lambda^{1/4}} ~;~ \varphi = \tau +\frac{\tilde{\varphi}(\sigma^{\alpha})}{\lambda^{1/4}}~;~\theta = \theta (\sigma) +\frac{\tilde{\theta}(\sigma^{\alpha})}{\lambda^{1/4}}~;~\zeta = \sigma + \frac{\tilde{\zeta}(\sigma^{\alpha})}{\lambda^{1/4}}
\label{e34}\end{aligned}$$ where, $ \sigma^{\alpha}=\lbrace \tau , \sigma \rbrace $ stands for the coordinates on the sting world-sheet. The above perturbative expansion (\[e34\]) naturally yields, $$\begin{aligned}
\chi_{\alpha}=\chi^{(0)}_{\alpha}+\frac{1}{\lambda^{1/4}}\chi^{(1)}_{\alpha}+\frac{1}{\sqrt{\lambda}}\chi^{(2)}_{\alpha}+..\end{aligned}$$ where, the individual entities could be formally expressed as, $$\begin{aligned}
\chi^{(0)}_{\alpha}&=&2\partial_\alpha \tau - \cos\theta \partial_\alpha \tau \\
\chi^{(1)}_{\alpha}&=&2 \partial_\alpha \tilde{\tau}+\partial_\alpha\tilde{\psi}-\cos\theta \partial_\alpha\tilde{\varphi}+\tilde{\theta}\partial_\alpha\tau \sin\theta \\
\chi^{(2)}_{\alpha}&=&\tilde{\theta}\sin\theta \partial_\alpha\tilde{\varphi}.\end{aligned}$$
Substituting (\[e34\]) into (\[e4\]) and expanding the action upto quadratic order in the fluctuations we find, $$\begin{aligned}
\mathcal{S}_{NG}=\mathcal{S}_{NG}^{(0)}+\mathcal{S}_{NG}^{(1)}+\mathcal{S}_{NG}^{(2)}+\mathcal{O}(1/\sqrt{\lambda})\end{aligned}$$ where, the effective Lagrangian density in the *quadratic* fluctuations could be formally expressed as[^9], $$\begin{aligned}
\mathcal{L}_{NG}^{(2)}=\frac{\mathcal{M}^{(0)}}{\chi^{(0)}_{\tau}}\left(\left(\tilde{\zeta}' +\frac{\chi_{\tau}^{(1)}}{\chi_{\tau}^{(0)}} \right)^2 -\frac{\Phi (\sigma^{\alpha})}{\chi_{\tau}^{(0)}} \right)-\frac{\mathcal{M}^{(1)}}{\chi_{\tau}^{(0)}}\left(\tilde{\zeta}' +\frac{\chi_{\tau}^{(1)}}{\chi_{\tau}^{(0)}} \right)+\frac{\mathcal{M}^{(2)}}{\chi_{\tau}^{(0)}}-\mathcal{K}^{(2)}
\label{e40}\end{aligned}$$ where the individual entities above in (\[e40\]) are given by, $$\begin{aligned}
\Phi (\sigma^{\alpha})&=& \varepsilon^{\alpha \beta}\chi^{(1)}_{\alpha} \partial_{\beta}\tilde{\zeta}+\chi_{\tau}^{(2)}\\
\mathcal{M}^{(0)}&=&\sin^2\theta -(\chi^{(0)}_{\tau})^2 \theta'^2 \\
\mathcal{M}^{(1)}&=&-2\chi^{(0)}_{\tau} \theta' (\chi^{(1)}_{\tau}\theta' +\chi^{(0)}_{\tau} \tilde{\theta}')+2(\dot{\tilde{\varphi}}+\tilde{\zeta}')\sin^2\theta +2\tilde{\theta}\sin\theta \cos\theta\end{aligned}$$ $$\begin{aligned}
\mathcal{M}^{(2)}=\dot{\tilde{\theta}}^2 -2\theta' \dot{\tilde{\theta}}\dot{\tilde{\zeta}}+\dot{\tilde{\zeta}}^2 \theta'^2 - \tilde{\theta}'^2 (\chi_{\tau}^{(0)})^2-2\chi_{\tau}^{(0)} \chi_{\tau}^{(1)}\theta' \tilde{\theta}' -2 \theta'^2 \chi_{\tau}^{(0)}\chi_{\tau}^{(2)}-\theta'^2(\chi_{\tau}^{(1)})^2 \nonumber\\
+\tilde{\theta}^2 \cos^2\theta +4 \tilde{\theta}(\dot{\tilde{\varphi}}+\tilde{\zeta}')\sin\theta \cos\theta +(\dot{\tilde{\varphi}}^2 +\tilde{\zeta}'^2 +2\dot{\tilde{\varphi}}\tilde{\zeta}'+2\varepsilon^{\alpha \alpha'}\partial_{\alpha}\tilde{\varphi}\partial_{\alpha'}\tilde{\zeta})\sin^2\theta \nonumber\\
+2\chi_{\tau}^{(0)}\theta' \varepsilon^{\alpha \alpha'}\partial_{\alpha}\tilde{\theta}\chi_{\alpha'}^{(1)}+(2\chi^{(0)}_{\tau}\chi^{(1)}_{\sigma}\tilde{\varphi}' -(\chi^{(0)}_{\tau})^2 \tilde{\varphi}'^2 -(\chi^{(1)}_{\sigma})^2)\sin^2\theta\end{aligned}$$ and, $$\begin{aligned}
\mathcal{K}^{(2)}=-\varepsilon^{\alpha \beta}\cos\theta \partial_{\alpha}\tilde{\varphi}\partial_{\beta}\tilde{\zeta}+\tilde{\theta} \sin\theta (\tilde{\zeta}' +\dot{\tilde{\varphi}}).\end{aligned}$$
Next, we note down the equations of motion corresponding to different fluctuations which we present categorically here in the following. These equations should be understood as being solved order by order as a perturbation in $ \frac{1}{\sqrt{\lambda}} $ in the semiclassical approximation. However, in our analysis, we would be interested in solving these equations only at the level of *leading* and/or *zeroth* order in $ \frac{1}{\sqrt{\lambda}} $ which are thereby obtained by varying the action quadratic in fluctuations namely, $ \mathcal{S}_{NG}^{(2)}=\int d^2\sigma \mathcal{L}_{NG}^{(2)} $.
An example: Short strings
-------------------------
### Solving fluctuations
The dynamics of stringy fluctuations is indeed quite difficult to deal with until and unless one takes some special limits. In order to solve the dynamics of stringy fluctuations, we therefore choose to work with the so called short string limit namely, $ |\theta_0|\ll 1 $ together with a particular string embedding of the following form, $$\begin{aligned}
\tilde{\tau}(\tau , \sigma)=\xi \tau ~;~\tilde{\theta} = \tilde{\theta}(\sigma)~;~\tilde{\psi}(\tau , \sigma)=\gamma_{2} \tau ~;~\tilde{\varphi}(\tau , \sigma)=\gamma_{1} \tau ~;~\tilde{\zeta}(\tau , \sigma)=\eta \sigma.
\label{e51}\end{aligned}$$
The resulting Lagrangian could be approximated as, $$\begin{aligned}
\mathcal{L}_{NG}^{(2)}\approx \left( \left(\eta + \upsilon \right)^2 -\eta \upsilon\right) (\theta^{2}_s -\theta'^2_{s}) -\left( \eta + \frac{\upsilon}{2}\right) \theta_{s} \tilde{\theta}\nonumber\\
-(\eta + \upsilon)\left( 2 \theta_s \tilde{\theta}+2(\gamma_{1}+\eta)\theta^{2}_s -2 \upsilon \theta'^2_{s}-2\theta'_s \tilde{\theta}'\right) \nonumber\\
-\tilde{\theta}'^2 -2\upsilon \theta_s' \tilde{\theta}' -\upsilon^2 \theta'^2_s +\tilde{\theta}^2 +4 (\eta + \gamma_1)\tilde{\theta}\theta_s +\epsilon_{1} \theta^2_s \nonumber\\
-2\upsilon \theta'_s \tilde{\theta}' + \eta \gamma_1 -(\eta +\gamma_1)\tilde{\theta}\theta_s +..\end{aligned}$$ where $\theta (\sigma)\equiv \theta_s(\sigma)$ (with $ \kappa =1 $) is the *classical* background solution (\[E28\]) in the limit of short strings together with, $ \upsilon = 2\xi +\gamma_{2}-\gamma_{1} $ and $ \epsilon_{1}=\gamma^{2}_1 +\eta^2 +4\gamma_{1}\eta $. Here we drop all those terms and their derivatives which are $ \sim \mathcal{O}(\theta^{3}) $ and higher order in the fluctuations.
The resulting equation of motion could be formally expressed as, $$\begin{aligned}
\tilde{\theta}'' (\sigma)+\tilde{\theta}(\sigma) +\jmath \sin\sigma \approx 0\end{aligned}$$ where, $ \jmath = \eta -\frac{9\upsilon}{4}+\frac{3 \gamma_1}{2} $. The corresponding solution turns out to be, $$\begin{aligned}
\tilde{\theta}(\sigma) &=&\cos \sigma \left(\mathfrak{c}_1+\frac{\jmath \sigma }{2}\right)+\mathfrak{c}_2 \sin \sigma \nonumber\\
& \approx & \mathfrak{c}_1+\sigma \left(\mathfrak{c}_2+\frac{\jmath}{2}\right)-\frac{1}{2} \mathfrak{c}_1 \sigma ^2 +..\end{aligned}$$
Finally, we note down the Virasoro constraints which for the present example yields, $$\begin{aligned}
\label{e55}
T_{\tau \tau}&=&T_{\sigma \sigma}\approx 1+\theta_{s}'^2 +\frac{1}{\lambda^{1/4}}\left(\xi +\frac{\gamma_2}{2}-\frac{\gamma_1}{2} +\frac{\theta_{s}' \tilde{\theta}'}{2}\right) +..\approx 0\\
T_{\tau \sigma}&\approx & 0.
\label{e56}\end{aligned}$$
### Conserved charges
With the above machinery in hand, we are now in a position to compute the one loop ($ \alpha' \sim \frac{1}{\sqrt{\lambda}} $) quantum corrections to the energy ($ \mathcal{E} $) and R charge ($ \mathtt{J}_{\varphi} $) of NR strings propagating over TNC geometry. To start with, the formula we derive below is valid for *generic* TNC strings spinning over $ R \times S^2 $. A straightforward computation reveals, $$\begin{aligned}
\label{e57}
\mathcal{E}&=&\frac{\sqrt{\lambda}}{2\pi}\int_{0}^{4\theta _0}d\sigma \left[ \frac{\zeta' \dot{\varphi}^2 \sin^2\theta}{(2\dot{t}+\dot{\psi}-\cos\theta \dot{\varphi})^2}+\frac{\theta'^2}{\zeta'}\right]\\
|\mathtt{J}_{\varphi}| &=&\frac{\sqrt{\lambda}}{4\pi}\int_{0}^{4\theta _0}d\sigma \left[\zeta' \sin^2\theta \dot{\varphi}\frac{\left( 4\dot{t}+2\dot{\psi}-\cos\theta \dot{\varphi}\right) }{(2\dot{t}+\dot{\psi}-\cos\theta \dot{\varphi})^2}+\frac{\cos\theta}{\zeta'}\left( \theta'^2 -\zeta'^2\right) \right].
\label{e58}\end{aligned}$$
Using (\[e51\]), one could further simplify the above expressions (\[e57\])-(\[e58\]) for short strings, $$\begin{aligned}
\label{e59}
\mathcal{E} \approx \frac{\sqrt{\lambda}}{2\pi}\left( 4\theta_0 +4\tilde{\theta}_0(2 \mathfrak{c}_2-\eta +\jmath)-\frac{64}{3} \left(-4 \gamma _1+2 \gamma _2-2 \eta +\jmath +4 \xi \right)\tilde{\theta}_0\theta_0^2\right) +..\end{aligned}$$ $$\begin{aligned}
\mathtt{J}_{\varphi} \approx \frac{\sqrt{\lambda}}{4\pi}\left( \frac{128\theta_0^3}{3}+4\tilde{\theta}_0(2\mathfrak{c}_2-2 \eta +\jmath)+32\mathfrak{c}_1 \tilde{\theta}_0\theta_0 \right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+\frac{32\sqrt{\lambda}}{12\pi}(16 \gamma_1 -8 \gamma_2 +\jmath + 10 \eta - 16 \xi + 6 \mathfrak{c}_2)\tilde{\theta}_0\theta_0^2 +..~~~~~~~~~~~~~~~~~~~~~~~~~
\label{e60}\end{aligned}$$ where, we introduce the entity $ \tilde{\theta}_0=\frac{\theta_0}{\lambda^{1/4}} \ll 1$ . Using (\[e59\]) and (\[e60\]), it is quite straightforward to recover (\[E29\]) in the strict classical limit namely, $ \tilde{\theta}_0\rightarrow 0 $. However, for our present purpose it is desired to find the dispersion relation of the following form, $$\begin{aligned}
\mathcal{E}-\mathtt{J}_{\varphi} =\frac{\sqrt{\lambda}}{4 \pi}\left(8\theta_0 -\frac{128 \theta_0^3}{3}\right) +\frac{\sqrt{\lambda}\tilde{\theta}_0}{4\pi}\left( 2(\tilde{a}-\theta_0^{2}\tilde{b})-\tilde{c}\right) -\frac{\sqrt{\lambda}\tilde{\theta}_0\theta_0 }{4\pi}\left( \tilde{d}+\theta_0 \tilde{e}\right)
\label{e61}\end{aligned}$$ where we use the following shorthand notation $$\begin{aligned}
\tilde{a}&=&4(2 \mathfrak{c}_2-\eta +\jmath)~;~\tilde{b}=\frac{64}{3} \left(-4 \gamma _1+2 \gamma _2-2 \eta +\jmath +4 \xi \right)\nonumber\\
\tilde{c}&=&4(2\mathfrak{c}_2-2 \eta +\jmath)~;~\tilde{d}=32\mathfrak{c}_1~;~\tilde{e}=\frac{32}{3}(16 \gamma_1 -8 \gamma_2 +\jmath + 10 \eta - 16 \xi + 6 \mathfrak{c}_2).\end{aligned}$$
The above equation (\[e61\]) could be messaged further to obtain, $$\begin{aligned}
\Delta_{NR}\sim \mathcal{E}=\mathtt{J}_{\varphi}\left( 1+\mathtt{f}(\tilde{\lambda})\right) \end{aligned}$$ where we identify, $$\begin{aligned}
\mathtt{f}(\tilde{\lambda})\approx \frac{\tilde{\lambda}\theta_0^2}{2\pi^2}\left( \frac{128 \theta_0^2}{3}+\frac{\sqrt{\alpha'}}{L}\left( \tilde{c}+\theta_0 \tilde{d}+\theta_0^2\left( \frac{32}{3}\left(\tilde{a}-\tilde{c} \right) +\tilde{e}\right) \right) +..\right) \end{aligned}$$ as being the one loop ($ \alpha' $) corrected *anomalous* dimension associated to folded (short) spinning TNC strings over $ R\times S^2 $.
NR spinning strings on $ R\times S^2 $
======================================
We conclude our analysis with a detailed discussion on the Hamiltonian spectrum corresponding to NR folded strings spinning over TNC geometry with $ R \times S^2 $ topology[^10]. This is achieved by taking the so called *scaling limit* [@Harmark:2017rpg]-[@Harmark:2018cdl] corresponding to 2D sigma model. The resulting sigma model action could be formally expressed as[^11], $$\begin{aligned}
\mathcal{S}_{NR}=\frac{\sqrt{\mathtt{g}}}{4\pi}\int d^2\sigma \mathcal{L}_{NR}\end{aligned}$$ where we identify the corresponding Lagrangian density as [@Roychowdhury:2019olt], $$\begin{aligned}
\mathcal{L}_{NR}=\frac{\varepsilon^{\alpha \alpha'}\varepsilon^{\beta \beta'}}{\dot{t}}\partial_{\alpha'}t\partial_{\beta'}t(\partial_{\alpha}\theta \partial_{\beta}\theta + \sin^2\theta \partial_{\alpha}\varphi \partial_{\beta}\varphi)+\varepsilon^{\alpha \beta}\cos\theta \partial_{\alpha}\varphi \partial_{\beta}\zeta +\mathcal{O}(1/c^2)
\label{e66}\end{aligned}$$ where, $ c(\rightarrow \infty) $ is the speed of light.
Classical solutions
-------------------
We start with the folded spinning string ansatz of the following form, $$\begin{aligned}
t=\tilde{\kappa}\tau ~;~ \varphi =\tilde{\varpi} \tau ~;~\zeta =\tilde{\ell} \sigma ~;~\theta = \theta (\sigma).
\label{e67}\end{aligned}$$
Substituting (\[e67\]) into (\[e66\]) we find, $$\begin{aligned}
\mathcal{L}_{NR}=\tilde{\kappa}\theta'^2 +\tilde{\varpi}\tilde{\ell}\cos\theta.\end{aligned}$$
The corresponding equation of motion turns out to be, $$\begin{aligned}
\theta''(\sigma) +\frac{\tilde{\varpi}\tilde{\ell}}{2\tilde{\kappa}}\sin\theta = 0
\label{e69}\end{aligned}$$ which has the most general solution of the following form[^12], $$\begin{aligned}
\theta (\sigma)\equiv \theta_{cl}=2 \mathcal{C}_1\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1} \left(\sigma +c_2\right)}{2 \sqrt{\tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1}\right)\nonumber\\
-2 \mathcal{C}_2\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1} \left(\sigma +c_2\right)}{2 \sqrt{\tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1}\right)
\label{e70}\end{aligned}$$ where, $ \mathcal{C}_{1,2} $ are real coefficients. However, for our subsequent analysis, without any loss of generality we set, $ \mathcal{C}_1=1 $ and $ \mathcal{C}_2=0 $.
Next, we note down the corresponding conserved charges namely the energy ($ \tilde{\mathcal{E}} $) and the R charge ($ \tilde{\mathtt{J}}_{\varphi} $) associated with the stringy dynamics, $$\begin{aligned}
\tilde{\mathcal{E}}&=&\frac{\sqrt{\mathtt{g}}}{4\pi}\int_{0}^{2\pi} \theta'^2 d\sigma\nonumber\\
&=&\frac{\sqrt{\mathtt{g}}}{\pi} \left(\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{\tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1}\right)^2-\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1} c_2}{2 \sqrt{\tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+\tilde{\kappa} c_1}\right)^2\right)\nonumber\\
&\equiv &\frac{\sqrt{\mathtt{g}}}{\pi}~ \mathtt{E}
\label{e71}\end{aligned}$$ and[^13], $$\begin{aligned}
\tilde{\mathtt{J}}_{\varphi}&=&\frac{\sqrt{\mathtt{g}}\tilde{\ell}}{4\pi}\int_{0}^{2\pi} \cos\theta d\sigma =\frac{\sqrt{\mathtt{g}}\tilde{\ell}}{4\pi}\int_{0}^{2\pi } \sqrt{1-\frac{4\tilde{\kappa}^2}{\tilde{\varpi}^2 \tilde{\ell}^2}(\theta'')^2}d\sigma \nonumber\\
&=&\frac{\sqrt{\mathtt{g}}\tilde{\ell}\sqrt{\tilde{\kappa}}}{4\pi}\frac{\mathtt{N}}{\mathtt{D}}.
\label{e72}\end{aligned}$$
Using (\[e71\]) and (\[e72\]) it is now straightforward to obtain[^14], $$\begin{aligned}
\tilde{\Delta}_{NR}\sim \tilde{\mathcal{E}}=\tilde{\mathtt{J}}_{\varphi}\left(1+\tilde{\mathtt{f}}(\bar{\mathtt{g}}) \right) \end{aligned}$$ where we identify, $$\begin{aligned}
\tilde{\mathtt{f}}(\bar{\mathtt{g}}) =\frac{\bar{\mathtt{g}}}{4\pi^2}\frac{\tilde{\ell}\sqrt{\tilde{\kappa}}\mathtt{N}}{\mathtt{D}}\left( \mathtt{E}-\frac{\tilde{\ell}\sqrt{\tilde{\kappa}}}{4}\frac{\mathtt{N}}{\mathtt{D}}\right)\equiv \mathfrak{q}_{1} \bar{\mathtt{g}}
\label{e74}\end{aligned}$$ as being the leading order correction to the Hamiltonian spectrum in the limit of strong ($ \mathtt{g}\gg 1 $) coupling. Here we identify, $ \bar{\mathtt{g}}=\frac{\mathtt{g}}{\tilde{\mathtt{J}}^{2}_{\varphi}} $ as an effective coupling constant in the dual SMT theory at low temperatures.
One loop corrections
--------------------
Our aim now is to compute quantum $(\sqrt{\mathtt{g}} )^{-1} $ corrections to the above function in (\[e74\]). In order to do so, we choose to work with the string embedding of the following form, $$\begin{aligned}
t=\tilde{\kappa}\tau +\frac{\hat{t}(\sigma^{\alpha})}{\mathtt{g}^{1/4}}~;~ \varphi =\tilde{\varpi} \tau +\frac{\hat{\varphi}(\sigma^{\alpha})}{\mathtt{g}^{1/4}} ~;~\zeta =\tilde{\ell} \sigma +\frac{\hat{\zeta}(\sigma^{\alpha})}{\mathtt{g}^{1/4}}~;~\theta =\theta_{cl} (\sigma)+\frac{\hat{\theta} (\sigma)}{\mathtt{g}^{1/4}}.
\label{e75}\end{aligned}$$
Substituting (\[e75\]) into (\[e66\]) we arrive at the quadratic Lagrangian of the following form, $$\begin{aligned}
\mathcal{L}^{(2)}_{NR}=\tilde{\kappa}\hat{\theta}'^2 +2\hat{\theta}' \theta_{cl}'+\mathfrak{w}\mathfrak{l}\cos\theta_{cl}-(\tilde{\varpi}\mathfrak{l}+\tilde{\ell}\mathfrak{w})\hat{\theta}\sin\theta_{cl}\end{aligned}$$ where we choose to work with the following ansatz, $$\begin{aligned}
\hat{t}(\sigma^{\alpha})=\tau ~;~\hat{\varphi}(\sigma^{\alpha})=\mathfrak{w}\tau ~;~\hat{\zeta}(\sigma^{\alpha})=\mathfrak{l}\sigma \end{aligned}$$ together with the fact that $ \mathfrak{w} $ and $ \mathfrak{l} $ are some real positive integers.
The equation of motion corresponding to $ \hat{\theta}(\sigma) $ could be formally expressed as, $$\begin{aligned}
2\tilde{\kappa}\hat{\theta}''(\sigma)+2\theta_{cl}''(\sigma)+(\tilde{\varpi}\mathfrak{l}+\tilde{\ell}\mathfrak{w})\sin\theta_{cl}=0.
\label{e78}\end{aligned}$$
The above equation (\[e78\]) has a remarkably simple solution once we set, $$\begin{aligned}
\tilde{\kappa}=\tilde{\varpi}=\tilde{\ell}=\mathfrak{w}=\mathfrak{l}=1\end{aligned}$$ which by means of (\[e69\]) yields, $$\begin{aligned}
\hat{\theta}(\sigma)=\theta_{cl}(\sigma)+\mathfrak{s}_1\sigma +\mathfrak{s}_2
\label{e80}\end{aligned}$$ where $ \mathfrak{s}_{1,2} $ are integration constants.
Using (\[e80\]), it is now straightforward to compute one loop stringy correction to the energy spectrum[^15], $$\begin{aligned}
\tilde{\mathcal{E}}=\frac{\sqrt{\mathtt{g}}}{\pi}~\left( \mathtt{E}+\frac{\sqrt{\hat{\alpha}'}}{L}\Delta \mathtt{E}\right)
\label{e81}\end{aligned}$$ where we identify one loop correction to the NR string spectrum as[^16], $$\begin{aligned}
\Delta \mathtt{E}=-\text{gd}\left(\frac{1}{\sqrt{2}}\right)+\text{gd}\left(\frac{1+2 \pi }{\sqrt{2}}\right)+\sqrt{2} \sinh \left(\sqrt{2} \pi \right) \text{sech}\left(\frac{1}{\sqrt{2}}\right) \text{sech}\left(\frac{1+2 \pi }{\sqrt{2}}\right).\end{aligned}$$
Finally, we compute one loop correction to the R charge, $$\begin{aligned}
\tilde{\mathtt{J}}_{\varphi}=\frac{\sqrt{\mathtt{g}}}{4\pi}~\left( \frac{\mathtt{N}}{\mathtt{D}}+\frac{\sqrt{\hat{\alpha}'}}{L}\Delta \mathtt{J}\right) \label{e83}\end{aligned}$$ where, $$\begin{aligned}
\Delta \mathtt{J}=\frac{4 \sqrt{2} \left(2 e^{\frac{1+2 \pi }{\sqrt{2}}} \left(\text{gd}\left(\frac{1+2 \pi }{\sqrt{2}}\right)+\pi \right)+3\right)}{1+e^{\sqrt{2} (1+2 \pi )}}
-\frac{4 \sqrt{2} \left(2 e^{\frac{1}{\sqrt{2}}} \text{gd}\left(\frac{1}{\sqrt{2}}\right)+1\right)}{1+e^{\sqrt{2}}}\nonumber\\
-4 \sqrt{2}+2 \pi +6+8 \tan ^{-1}\left(e^{\frac{1}{\sqrt{2}}}\right)-8 \tan ^{-1}\left(e^{\frac{1+2 \pi }{\sqrt{2}}}\right)-\sqrt{2} \cosh ^{-1}(3).\end{aligned}$$
Combining (\[e81\]) and (\[e83\]) we finally obtain, $$\begin{aligned}
\tilde{\Delta}_{NR}\sim \tilde{\mathcal{E}}=\tilde{\mathtt{J}}_{\varphi}(1+\tilde{\tilde{\mathtt{f}}}(\bar{\mathtt{g}}))
\label{e87}\end{aligned}$$ where we identify the function, $$\begin{aligned}
\tilde{\tilde{\mathtt{f}}}(\bar{\mathtt{g}})=\tilde{\mathtt{f}}(\bar{\mathtt{g}})(1+\Delta \tilde{\mathtt{f}})\end{aligned}$$ that include one loop stringy correction of the form, $$\begin{aligned}
\Delta \tilde{\mathtt{f}}&=&\frac{\sqrt{\hat{\alpha}'}}{L}\left(\frac{\Delta \mathtt{E}+\Delta \mathtt{S}\frac{\mathtt{D}}{\mathtt{N}}\left(\mathtt{E}-\frac{\mathtt{N}}{2\mathtt{D}} \right) }{\mathtt{E}-\frac{\mathtt{N}}{4\mathtt{D}} } \right) +\mathcal{O}(\hat{\alpha}'/L^2)\nonumber\\
&=&\frac{\mathfrak{q}_2}{\sqrt{\tilde{\mathtt{J}}_{\varphi}}}\frac{1}{\bar{\mathtt{g}}^{1/4}}+\mathcal{O}(1/(\tilde{\mathtt{J}}_{\varphi}\sqrt{\bar{\mathtt{g}}}))
\label{e89}\end{aligned}$$ where, $ \mathfrak{q}_{2}=\left(\frac{\Delta \mathtt{E}+\Delta \mathtt{S}\frac{\mathtt{D}}{\mathtt{N}}\left(\mathtt{E}-\frac{\mathtt{N}}{2\mathtt{D}} \right) }{\mathtt{E}-\frac{\mathtt{N}}{4\mathtt{D}} } \right) $.
Finally, substituting (\[e89\]) into (\[e87\]) we find, $$\begin{aligned}
\tilde{\Delta}_{NR}=\left( 1+\mathfrak{q}_1\bar{\mathtt{g}}+\frac{\mathfrak{q}_1\mathfrak{q}_2}{\sqrt{\tilde{\mathtt{J}}_{\varphi}}}\bar{\mathtt{g}}^{3/4}+..\right) \tilde{\mathtt{J}}_{\varphi}.
\label{e90}\end{aligned}$$
Summary and final remarks
=========================
The present paper is an attempt towards understanding the NR sigma model/SMT correspondence through some explicit computations of the energy spectrum in the dual gauge theory at strong ($ \mathtt{g}\gg 1 $) coupling. In our analysis, we consider specific example of NR folded closed string configurations on $ R\times S^2 $ spinning around the north pole of $ S^2 $. By exploring solitonic excitations associated with these NR 2D sigma models we finally compute the Hamiltonian spectrum ($ \tilde{\Delta}_{NR} $) corresponding to *heavy* states ($ \mathtt{\tilde{J}}_{\varphi}>\sqrt{\mathtt{g}}\gg 1 $) in the dual gauge theory at strong coupling. In other words, on the gauge theory side we consider states with large length ($ L \sim \mathtt{\tilde{J}}_{\varphi}$) which thereby implies an effective planar limit at low temperatures. On the dual stringy sector this limit corresponds to *free* strings propagating over $ S^2 $ [@Harmark:2014mpa]. The key observation of our analysis turns out to be the identification of an effective expansion parameter $ \bar{\mathtt{g}}(=\frac{\mathtt{g}}{\tilde{\mathtt{J}}_{\varphi}^{2}} )\ll 1$ for the gauge theory which allows us to compute quantum corrections associated with the Hamiltonian spectra (around its classical value) in the limit of strong ($ \mathtt{g}\gg 1 $) coupling and low temperatures. An interesting question along this line of arguments turns out to be how does the spectrum changes (at strong coupling) as temperature increases above the critical transition temperature ($ T_c $) where the classical stringy picture breaks down and one therefore needs to take into account the so called *non planar* effects [@Harmark:2014mpa]. The other question that might be of worth exploring is whether it is possible to match the spectrum at *finite* (or, *small*) coupling ($ \mathtt{g} $). In other words, to check the spin chain (or, near planar $ \mathcal{N}=4 $ SYM)/ NR sigma model correspondence using the notion of effective coupling constant ($\bar{\mathtt{g}} $) discussed in this paper. We hope to address some of these questions in the near future.\
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[**[Acknowledgements :]{}**]{} The author is indebted to the authorities of IIT Roorkee for their unconditional support towards researches in basic sciences.\
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Here, in the Appendix, we provide detailed expressions for the functions $\mathtt{N}$ and $ \mathtt{D} $ those which appear in the expression for the R charge (\[e72\]) associated to NR TNC strings spinning over $ R\times S^2 $. Below we enumerate their individual expressions as, $$\begin{aligned}
\mathtt{N}=-\left(c_1 c_2 \sqrt{ \tilde{\kappa}} \sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}-2 \left(c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}\right) E\left(\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)\right)~~~~~~~~~~~~~\nonumber\\
\times \text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2 \text{dn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)~~~~~~~~~~~~\nonumber\\
\times \sqrt{\frac{-2 \tilde{\varpi}\tilde{\ell} \text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2+c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}}~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
+c_1 c_2 \sqrt{ \tilde{\kappa}} \sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}-2 \left(c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}\right) E\left(\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)~~~~~~~~~~~\nonumber\\
\times \text{dn}\left(\frac{\left(c_2+2 \pi \right) \sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\times \left(\text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2-\text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2\right)~~~~~~~~~~~~~~~~~~~~~~~~~\nonumber\\
\times \sqrt{\frac{-2 \tilde{\varpi}\tilde{\ell} \text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2+c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}} \text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2~~~~~~~~~~~~~~\nonumber\\
+\left( c_1 \left(c_2+2 \pi \right) \sqrt{ \tilde{\kappa}} \sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}-2 \left(c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}\right) E\left(\text{am}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)\right) ~~~\nonumber\\
\times \text{dn}\left(\frac{c_2 \sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}\right)
\left( \text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2-\text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2\right)~~~~~~~ \nonumber\\
\times \left( \text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2-\text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2\right) ~~~~~~~~~~~~~~~~~~\nonumber\\
\times \sqrt{\frac{-2 \tilde{\varpi}\tilde{\ell} \text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2+c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{aligned}$$ and, $$\begin{aligned}
\mathtt{D}=\tilde{\varpi}\tilde{\ell}\sqrt{c_1 \tilde{\kappa}+\tilde{\varpi}\tilde{\ell}} \text{dn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right) \text{dn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)\nonumber\\
\times \left( \text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2\tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2-\text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} c_2}{2 \sqrt{ \tilde{\kappa}}}|\frac{2\tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2\right) \nonumber\\
\times\left( \text{cn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2-\text{sn}\left(\frac{\sqrt{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1} \left(c_2+2 \pi \right)}{2 \sqrt{ \tilde{\kappa}}}|\frac{2 \tilde{\varpi}\tilde{\ell}}{\tilde{\varpi}\tilde{\ell}+ \tilde{\kappa} c_1}\right)^2\right).\end{aligned}$$
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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 589-590, 1972.
[^1]: E-mail: dibakarphys@gmail.com, dibakarfph@iitr.ac.in
[^2]: Recently, the authors in [@Harmark:2019upf] have shown that under certain specific assumptions on the compact longitudinal spatial direction, the TNC strings (in the presence of background NS-NS fluxes) can be mapped to NR strings propagating over SNC geometry.
[^3]: Here $ L $ is the radius of the 2 sphere.
[^4]: This is the Lagrangian that represents *relativistic* strings propagating over TNC geometry in the semiclassical approximation. The world-sheet theory is nonrelativisitc (NR) only from the perspective of the target space geometry given in the problem [@Harmark:2017rpg]-[@Harmark:2018cdl]. However, it is always possible to consider a large $ c(\rightarrow \infty) $ limit [@Harmark:2017rpg]-[@Harmark:2018cdl] associated to 2D sigma models that eventually leads towards NR strings propagating over TNC geometry. This will be our subject of discussion in Section 4.
[^5]: We have rescaled the Lagrangian by an overall factor of $ \mathfrak{n} $.
[^6]: Taking into account the full four segments of the folded string configuration [@Gubser:2002tv], the actual energy ($ \mathcal{E} $) and R charge ($ \mathtt{J}_{\varphi} $) should be four times the value computed for each segment.
[^7]: The semiclassical limit is defined to be the limiting case in which these conserved charges take sufficiently large values such the the ratios like $ \frac{\mathcal{E}}{\sqrt{\lambda}} $ are finite. This therefore corresponds to setting the corresponding t’Hooft coupling, $ \lambda \gg 1 $.
[^8]: In the short string approximation one may therefore choose to work with the *static* gauge $ \theta = \sigma $ [@Frolov:2003qc]. Here $ 4\theta_0 $ stands for the periodicity along $ \sigma $.
[^9]: We rescale the Lagrangian by an overall factor of $ -\frac{1}{4 \pi} $.
[^10]: Unlike the earlier example, here we perform an *exact* evaluation of $ 0\leq\sigma \leq 2\pi $ integral without any approximation on the string length and compute the corresponding energy eigenvalue ($ \tilde{\Delta}_{NR} $) and its one loop stringy correction. Following the original arguments of [@Harmark:2014mpa], one should remember that the present computation is strictly valid in the limit of strong ($ \mathtt{g}\gg 1 $) coupling and low temperatures.
[^11]: Here, $ \mathtt{g} (=c^2 \lambda)$ is the string tension associated to TNC strings in the large $ c(\rightarrow \infty) $ limit.
[^12]: The entity introduced on the r.h.s. of (\[e70\]) is known as the *Jacobi amplitude* ($ \phi(u,k)\equiv \text{am}(u|k)=\int_{0}^{u} dn(u',k)du'$ where, $ dn(u',k)du' $ is known as *Jacobi elliptic function* with elliptic modulus) which is defined as being the inverse of the elliptic integral function of first kind. For more details on different types of Jacobi elliptic functions the enthusiastic reader is encouraged to see [@abramh].
[^13]: See Appendix for details.
[^14]: Here $ \tilde{\Delta}_{NR} $ is the energy eigenvalue associated with the corresponding dual operator spectrum in the SMT theory at strong coupling and low temperatures.
[^15]: For the sake of computational simplicity we set, $ c_1=c_2=\mathfrak{s}_1=1 $ and $ \mathfrak{s_2}=0 $.
[^16]: Notice that here, $ \text{gd}(x)=\int_{0}^{x}\frac{dt}{\cosh t} $ is the so called *Gudermannian* function [@abramh].
|
---
abstract: 'We use moderate-resolution optical spectrophometry and the new MARCS stellar atmosphere models to determine the effective temperatures of 74 Galactic red supergiants (RSGs). The stars are mostly members of OB associations or clusters with known distances, allowing a critical comparison with modern stellar evolutionary tracks. We find we can achieve excellent matches between the observations and the reddened model fluxes and molecular transitions, although the atomic lines Ca I $\lambda 4226$ and Ca II H and K are found to be unrealistically strong in the models. Our new effective temperature scale is significantly warmer than those in the literature, with the differences amounting to 400 K for the latest-type M supergiants (i.e., M5 I). We show that the newly derived temperatures and bolometric corrections give much better agreement with stellar evolutionary tracks. This agreement provides a completely independent verification of our new temperature scale. The combination of effective temperature and bolometric luminosities allows us to calculate stellar radii; the coolest and most luminous stars (KW Sgr, Case 75, KY Cyg, HD 206936=$\mu$ Cep) have radii of roughly 1500$R_\odot$ (7 AU), in excellent accordance with the largest stellar radii predicted from current evolutionary theory, although smaller than that found by others for the binary VV Cep and for the peculiar star VY CMa. We find that similar results are obtained for the effective temperatures and bolometric luminosities using only the de-reddened $V-K$ colors, providing a powerful demonstration of the self-consistency of the MARCS models.'
author:
- 'Emily M. Levesque, and Philip Massey'
- 'K. A. G. Olsen'
- Bertrand Plez and Eric Josselin
- Andre Maeder and Georges Meynet
title: 'The Effective Temperature Scale of Galactic Red Supergiants: Cool, But Not As Cool As We Thought'
---
Introduction {#Sec-intro}
============
Red supergiants (RSGs) are an important but poorly characterized phase in the evolution of massive stars. As discussed recently by Massey (2003) and Massey & Olsen (2003), stellar evolution models do not produce RSGs that are as cool or as luminous as those observed. Such a discrepancy is not surprising, given the tremendous challenge RSGs present to evolutionary calculations. The RSG opacities are uncertain because of possible deficiencies in our knowledge of molecular opacities. The atmospheres of these stars are highly extended, but in general the models assume plane-parallel geometry. In addition, the velocities of the convective layers are nearly sonic, and even supersonic in the atmospheric layers, giving rise to shocks (Freytag et al. 2002). This invalidates the mixing-length assumptions, making the star’s photosphere very asymmetric and its radius poorly defined, as demonstrated by recent high angular resolution observations of Betelgeuse (Young et al. 2000).
While considering these challenges, we must, however, recognize that the “observed" location of RSGs in the H-R diagram is also highly uncertain, as it requires a sound knowledge of the effective temperatures of these stars. For stars this cool (roughly 3000 to 4000 K), the bolometric corrections (BCs) are quite significant (-4 to -1 mags), and these BCs are a steep function of effective temperature. This makes an accurate effective temperature scale doubly necessary, as a 10% error in $T_{\rm eff}$ would lead to a factor of 2 error in bolometric luminosity computed from $V$, according to the Kurucz (1992) model atmospheres as described by Massey & Olsen (2003). While interferometric data has provided a good fundamental calibration of red [*giants*]{} (see, for example, Dyck et al. 1996), there are not enough nearby red supergiants to employ this method in determining an effective temperature scale. Instead, previous scales have relied upon using broad-band colors to assign temperatures based on the few red supergiants with measured diameters (Lee 1970, following Johnson 1964, 1966), or upon “observed" bolometric corrections (from IR measurements) combined with the assumption of a blackbody distribution for the continuum (see Flower 1975, 1977). However, $(B-V)_o$ is highly sensitive to the surface gravity of the star due to the increased effects of line blanketing with lower surface gravities; the effect is particularly pronounced at $B$ due to the multitude of weak metal lines in the region; see discussion in Massey (1998). In point of fact, the true continuum at these temperatures (that produced by continuous absorption) is probably never seen, as noted by Aller (1960). White & Wing (1978) attempt to get around this problem by a novel scheme involving an 8-color narrow band filter set, which was fit by a blackbody curve and iteratively corrected to determine uncontaminated continua. However, in all such continuum fits there is always some degeneracy between changes in the effective temperature and changes in the amount of reddening to be applied to the models; this is particularly important for the RSGs, as they can be heavily reddened.
An alternative approach would be to make use of the incredibly rich TiO molecular bands which dominate the optical spectra of M-type stars. Atmospheric models, however, have not always included an accurate opacity, especially for molecular transitions. The problem largely stems from the fact that any molecule found in a stellar atmosphere—even that of an M-type star—must be considered “high temperature", while most laboratory data have been obtained at much lower temperatures and do not include high-excitation transitions. The situation has decided improved in the past years, in large part due to the great efforts by a few groups to compute ab initio line lists (e.g., Partridge & Schwenke 1997, Harris et al. 2003), and it now seems quite satisfactory regarding oxygen-rich mixtures, such as TiO (see reviews by Gustafsson & Jorgensen 1994 and Tsuji 1986).
The new generation of MARCS models (Gustafsson et al. 1975; Plez et al. 1992) now includes a much-improved treatment of molecular opacity (see Plez 2003, Gustafsson et al. 2003). Using absolute spectrophotometry (both continuum fluxes and the strengths of the G-band for K stars and the TiO bands for M stars), these models can now be used to make a far more robust determination of $T_{\rm eff}$ and the reddening. Since band strengths are used to determine the spectral type of RSGs, these new models could serve as a definitive connection between spectral type and $T_{\rm eff}$.
Here we present spectrophotometry of 74 Galactic K- and M-type supergiants (Sec. \[Sec-obs\]), and use fits of the MARCS models to determine effective temperatures (Sec. \[Sec-analysis\])[^1]. We begin the analysis by reclassifying the stars (Sec. \[Sec-types\]), and then construct a new $T_{\rm eff}$ scale for Galactic RSGs (Sec. \[Sec-teff\]). Most of these stars are members of associations and clusters with known distances, allowing us to also place the stars on the H-R diagram for comparison with the latest generation of stellar evolutionary models, which include the effects of stellar rotation (Sec. \[Sec-evolution\]). In Sec. \[Sec-K\], we compare the physical parameters of these stars derived in the optical to those found from K-band photometry in order to test the self-consistency of the MARCS models.
In future work, we will extend this study to the lower metallicity Magellanic Clouds, where the distribution of spectral subtypes is considerably earlier than in the Milky Way (Elias et al. 1985; Massey & Olsen 2003). The lower abundance of TiO may by itself lead to a lower $T_{\rm eff}$ for stars of the same spectral subtype, as suggested by Massey & Olsen (2003).
Observations {#Sec-obs}
============
The Sample
----------
Our stars are listed in Table \[tab:stars\]. The sample was selected in order to cover the full range of spectral subtypes from early K through the latest M supergiants. The sample was originally chosen to contain only stars with probable membership in OB associations and clusters with known distances (Humphreys 1978, Garmany & Stencel 1992), but we supplemented this list with spectral standards from the list of Morgan & Keenan (1973) in order to help refine the spectral classification. We indicate both cluster membership and/or use as a spectral standard in the table. The photometry comes from a variety of sources, as indicated; one should keep in mind that most (if not all) of these stars are variable at the level of several tenths or more in $V$. We have included the $V-K$ colors as well, where the $K$ comes from Josselin et al. (2000) and references therein.
Distances of OB associations are notoriously uncertain, as most are far too distant for reliable trigonometric parallaxes; instead, spectral parallaxes need to be used, often resulting in a large dispersion due to the large scatter in $M_V$ for a given spectral subtype (see Conti 1988). In Table \[tab:dists\] we give the distances from several sources, when possible, for each OB association or cluster in our sample; in general, the agreement is within a few tenths of a magnitude. We also include in Table \[tab:dists\] the “average" reddenings determined from the values for [*early*]{}-type members listed by Humphreys (1978) for the OB associations. For the clusters, we list the average reddening values given by Mermilliod & Paunzen (2003). We note that in general membership in an OB association is never perfectly well established, and therefore there is an additional uncertainty connected with the distance of any particular star.
Spectrophotometry
-----------------
The spectrophotometry data were obtained during three observing runs: two with the KPNO 2.1-meter telescope and GoldCam Spectrograph (17-24 March 2004 and 28 May - 1 June 2004), and one with the CTIO 1.5-meter telescope and Cassegrain CCD spectrometer (7-12 March 2004). Similar resolutions and wavelength coverage were obtained in both hemispheres. Detailed information on the observing parameters is given in Table \[tab:obs\].
The Kitt Peak observations were taken under sporadically cloudy conditions in March, while all of the nights in May/June were photometric. Before observing each object, the spectrograph was manually rotated so that the slit was aligned with the parallactic angle. We aimed for a S/N of 50 per spectral resolution element (4 pixels) at the bluest end of the spectrum, with care being taken not to saturate the detector at the reddest end. For the brightest stars ($V<6$), we employed a 5.0 mag or 7.5 mag neutral density filter. Observations of the flat-field lamps were obtained both with and without these filters in order to correct for the wavelength dependence of the transmission of these “neutral" filters. Throughout each night we also observed a set of spectrophotometric standards. The seeing for these observations was 1.2” to 3”, with a typical value of 2”. Exposures of both dome flats and projector flats were obtained at the beginning of each night; for the red nights, we also obtained projector flats throughout the night to monitor any shifting of the fringe pattern that affects the CCD at longer wavelengths. Wavelength calibration exposures of a He-Ne-Ar lamp were obtained at the beginning and end of each night.
At CTIO, it was not practical to always observe at the parallactic angle; instead, each star was observed with a single exposure through a very wide slit (for good relative fluxes), followed by a series of shorter exposures obtained through a narrow slit (for good resolution). Projector flats were obtained for all wavelength regions through both the wide and narrow slits. The exposures were typically obtained in conjunction with a wavelength calibration exposure, although in practice we found there was little flexure. Observations of spectrophotometric standards were obtained throughout the night.
We reduced the data using IRAF[^2] packages CCDRED, KPNOSLIT and CTIOSLIT. We used dome flats to flatten the blue Kitt Peak data, and the projector lamps to flatten the red Kitt Peak data. We found that the only shift in the fringe pattern in the red occurred during the thirty minutes following a refill of the dewar with liquid nitrogen. After that the fringe pattern was stable, so we simply combined these projector flats. The spectra were all extracted using an optimal-extraction algorithm. When reducing the CTIO data the narrow slit width observations were combined and divided into the wide slit observations. The resulting division was fit with a low-order function and used to correct the fluxes of the narrow-slit observations. In a few cases the wide-slit observations included the presence of an additional star in the extraction aperture; these stars were eliminated from further consideration, and do not appear in Table \[tab:stars\].
The observations of spectrophotometric standards were used to create sensitivity functions; typically a grey-shift was applied to each night’s data, and the resulting scatter was 0.02 mag. In addition, the different wavelength regions were grey-shifted to agree in the regions of overlap.
When we began our analysis we found significant discrepancies between the reddened model atmospheres and the observed spectra in the near-UV region ($<$4000Å for the reddest stars. Careful investigations suggest that, despite the excellent agreement of the spectrophotometric standards, there could be contamination by a grating ghost in the near-UV, in which a small fraction of the red light contaminates the data. The expected flux ratio $F_{\lambda 7000} / F_{\lambda 3500}$ is roughly 10,000 for the most reddened M supergiants, a regime seldom encountered in astronomical spectrophotometry. We do not know if this contamination is a small fraction of the near-UV light, or if it dominates, and we have therefore restricted our study to the data longwards 4100Å, where our tests show that the ghosting effect is negligible.
Analysis {#Sec-analysis}
========
Reclassification of Spectral Type {#Sec-types}
---------------------------------
We reclassified each of our stars by visually comparing each spectrum to the spectral standards. In order to avoid questions of normalizing these rich and complicated spectra (with uncertain continuum levels), we did this comparison in terms of log flux. Given that many of these stars can be variable in spectral type, we were unsurprised to find that we needed to reclassify a few of the spectral standards for consistency. For the late-K and all M type stars, the classification was based upon the depths of the TiO bands, which are increasingly strong with later spectral type. The reclassification of the early and mid K supergiants was based primarily on the strengths of the G-band plus the strength of Ca I $\lambda 4226$. These features all get weaker with later spectral types (Jaschek & Jaschek 1987), a temperature effect which we confirmed with the MARCS models. We list our revised spectral types in Table \[tab:results\].
Modeling the Stars: Effective Temperatures and Reddenings\[Sec-teff\]
---------------------------------------------------------------------
We compared the observed spectral energy distribution (SED) of each star to a series of MARCS stellar atmosphere models. The models used in our comparisons ranged from 3000 K to 4300 K in increments of 100 K, with $\log g$ from +1.0 to -1.0 in increments of 0.5 dex. The choice of surface gravity was derived iteratively; we began by adopting the $\log
g=-0.5$ models for all of the fits, as this surface gravity was generally what was expected with the old effective temperature scale and the resulting placement in the HRD. However, our new temperatures were a bit warmer than that predicted by the old scale, so we re-evaluated all of the fits using $\log g=0.0$, as this was more consistent with the revised locations in the HRD. The effective temperatures remained unchanged with the use of these higher surface gravities (except for the occasional star), but the values of $E(B-V)$ increased slightly. Finally, we used the revised temperature scale and bolometric luminosities with the evolutionary tracks (Sec. \[Sec-evolution\]) to compute $\log g$ star-by-star for the RSGs with distances. This confirmed that our choice of the $\log g=0.0$ models was appropriate for most of the stars. We refit the stars using the appropriate $\log g$ if the distance was known; otherwise, we adopted $\log g=0.0$. In practice the choice of $\log g$ affected the derived values of $E(B-V)$ by $<0.1$ mag, and had no effect on the derived effective temperatures.
In making the fits, we reddened each model using a Cardelli et al. (1989) reddening law with the standard ratio of total-to-selective extinction $R_V=3.1$[^3]. Our initial guess for $E(B-V)$ was based upon the average value for the cluster, i.e., $E(B-V)_{\rm cluster}$ from Table \[tab:dists\]. The temperature was determined primarily from the strengths of the TiO bands (M supergiants) and the G-band (early-to-mid K supergiants), with $E(B-V)$ then adjusted to produce the best fit to the continuum. In Fig. \[fig:plots\] we show a sample of spectra and fits, covering a range of spectral types; the complete set is available in the electronic edition of the journal. The fitting was all done in log units of flux in order to facilitate comparisons of line intensities without the uncertain process of normalization, as mentioned previously.
As described in the previous section, our initial modeling revealed a significant discrepancy between the models and the data in the near-UV region (3500-4000Å) for the reddest stars, which we first attributed to a peculiar reddening law, caused, we argued, by circumstellar dust. However, since we cannot exclude the possibility that the near-UV data are affected by an instrumental problem, we will revisit this issue at a later time when we have better data in the near-UV.
Otherwise, the agreement between the SEDs of the reddened models and the data is extremely good, both in terms of the continua and molecular band depths. The only significant problem we encountered was for the early-to-mid K stars, where we expected to use both the Ca I $\lambda 4226$ line as well as the G-band in modeling the stars, as these lines are among the primary classification lines for the early K’s. However, we found that these Ca I line was stronger in the MARCS models’ synthetic spectra than in our stars, usually by a factor of 3 or so. The Ca I line show the same qualitative behavior with effective temperature as expected (see above discussion), but the absolute line strengths in the models were too strong. Since Ca has a very low ionization potential (6.11 eV), most of the Ca is in the form of Ca II: $\sim$99% in the model at 4000 K and 98% in the model at 3600 K (at an optical depth of 0.01 for the continuum). Thus, a small over-ionization will strongly impact the line strength, due to either the effects of non-LTE or a slight error in the model’s temperature structure. The occurrence of cool or hot spots on the stellar surface could also lead to regions where the ionization equilibrium is strongly affected.
In addition to the problem with the Ca I line, the models also produce Ca II H and K lines that are significantly stronger than those observed, even in those stars for which the fluxes of the dereddened stars and the stars were in good agreement. This is not hard to understand, as RSGs are expected to have chromospheric activity, and this is not accounted for in the models. The presence of a chromosphere would lead to emission in the H and K lines, resulting in weaker observed lines than would be the case if the lines were purely photospheric in origin. This explanation could be further investigated by means of high-dispersion spectroscopy.
We relied instead upon the G-band for the fitting of the early and mid-Ks. We substantiated that we obtained similar temperatures using the G-band and the TiO bands for the late Ks. The resulting temperature fits for the early and mid Ks are therefore less certain, probably $\pm 100$ K.
We list the effective temperatures in Table \[tab:results\]. In general, we found that the data could be matched very well by the models, [*both*]{} in line strength and in continuum shape, and we expect that the effective temperatures of the M supergiants have been obtained to a precision of 50 K. The A$_V$ values are determined to a precision of about 0.15 mag. We note with interest that the derived A$_V$ values of about one-third our sample are significantly higher than the average found from the OB stars in the same associations and clusters using the data from Humphreys (1978); indeed, the same conclusion could have been drawn from the older data given in that paper. A possible interpretation is that this extra extinction is due to circumstellar dust. We include the $\Delta$A$_V$ values in Table \[tab:results\]. We will revisit this issue in a subsequent paper, once our analysis of the Magellanic Cloud data (where the foreground reddening is low, and relatively uniform) is complete.
Our new effective temperature scale is given in Table \[tab:NewT\], where we include the number of stars and standard deviation of the mean ($\sigma_\mu$) at each spectral type. We compare this scale to that of Humphreys & McElroy (1984) and Massey & Olsen (2003) in Fig. \[fig:tscale\]. Both of the latter are “averages" from the literature. We see that the new scale agrees well for the K supergiants, but is progressively warmer than past scales for later spectral types, with the differences amounting to 400 K by the latest M supergiants (M5 I). The overall progression of the temperature with later spectral type is more gradual than in past studies.
We include in Table \[tab:NewT\] the bolometric corrections to the V-band corresponding to the new effective temperature scale; these values are the linear interpolations of the BCs from the MARCS models given below. The values are for $\log g=0.0$, but there is little change with surface gravity ($<0.05$ mag over 0.5 dex in $\log g$ for $3500\le T_{\rm eff}\le 4300$). Note that we have referenced the BCs to the system advocated by Bessell et al. (1998), i.e., that the Sun has a BC of -0.07 mag. This results in values less luminous (by 0.12 mag) than the historical one; on this system the Sun is [*defined*]{} to have an $M_{\rm bol}$ of 4.74.
Comparison to Evolutionary Models {#Sec-evolution}
---------------------------------
In order to compare these stars to the evolutionary models, we must convert the absolute visual luminosities to bolometric luminosities. In Table \[tab:BCs\] we give the bolometric corrections (BCs) as a function of effective temperature determined from the MARCS models. We list the bolometric magnitudes for each star with a known distance in Table \[tab:results\][^4].
In Fig. \[fig:HRD\](a) we show the solar-metallicity evolutionary tracks of Meynet & Maeder (2003) compared to the location of the Galactic RSGs taken from Humphreys (1978) using the effective temperature and bolometric corrections of Humphreys & McElroy (1984). The disagreement is not as bad as that shown by Massey (2003), as the new tracks extend further to the right at higher luminosities than did the older tracks of Schaller et al. (1992). Nevertheless, it is clear that there are significant differences between theory and “observation". In Fig. \[fig:HRD\](b) we now compare the same tracks to the Galactic RSGs using the new effective temperatures and bolometric models found by using the MARCS models. We have marked with filled symbols those five stars for which the $K$-band data suggests that our $M_{\rm bol}$ values are too luminous (Sec. \[Sec-K\]), and in Fig. \[fig:HRD\]c show the location of these stars based upon the K-band data. In (b) and (c) the disagreement between theory and location in the HRD has now disappeared, giving us some confidence in the accuracy of our new calibration. A few stars have luminosities significantly higher than the evolutionary tracks would predict (Fig. \[fig:HRD\]b), but these are invariably the stars whose $M_{\rm bol}$ values derived from $V$ are at odds with those derived from $K$ (Fig. \[fig:HRD\]c), presumably due to mistakes in the correction for reddening, and hence their value must be considered poorly determined; Fig. \[fig:HRD\]c is the best determined.
The fact that we have both M$_{\rm bol}$ and $T_{\rm eff}$ allows us to determine the stellar radii $R/R_\odot$ from the formal definition of $T_{\rm eff}$; i.e., $(R/R_\odot)=(L/L_\odot)^{0.5}(T_{\rm
eff}/5781)^2$, where the numerical quantity is the effective temperature of the sun (see discussion in Bessell et al. 1998), and $L/L_\odot=10^{-(M_{\rm bol}-4.74)/2.5}$. We include these values in Table \[tab:results\]. The stars with the largest radii in our sample, KW Sgr, Case 75, KY Cyg, and HD 206936 (=$\mu$ Cep)[^5] all have radii of roughly 1500$R_\odot$ (7 AU), making these the largest normal stars known. For comparison, Betelgeuse has a radius (measured by interferometry) of 645 $R_\odot$ (Perrin et al. 2004). We include in Fig. \[fig:HRD\](c) lines of constant radii. These four large stars are right at what current evolutionary theory predicts is the maximum radius for Galactic RSGs, as the largest radius reached by the tracks is found at roughly $M_{\rm bol}=-9$ and $\log T_{\rm eff}=3.57$ (3715 K).
Two peculiar RSGs are known with significantly larger radii. The M2 I primary in the interacting binary VV Cep has a radius that has been variously estimated as 1200$R_\odot$ (Hutchings & Wright 1971) to 1600$R_\odot$ (Wright 1977) and beyond, with a reasonable upper limit of 1900$R_\odot$ determined by Saito et al. (1980); see discussion in Bauer & Bennett (2000). VV Cep consists of a RSG primary and a hotter companion orbiting within a common envelope. These estimates of the radii are complicated by the uncertainties in the orbital inclination in Section 3 of Saito et al. 1980), with the “definition" of radius determined by the eclipse method leading to a further ambiguity. In any event, gravitational interactions are certainly taking place in this system (Hutchings & Wright 1971), and thus may not have applications to normal, single stars.
The other star with a humongous radius is VY CMa. Using interferometry with Keck, Monnier et al. (2004) find a photospheric radius of 14 AU ($3020 R_\odot$) for this star, where the distance of 1.5 kpc appears fairly certain from maser proper motions (Richards et al. 1998; see discussion in Monnier et al. 1999). The properties of this intriguing object have been recently discussed by Monnier et al. (1999) and Smith et al. (2001). With a luminosity of 2 to 5 $\times 10^{5} L_\odot$ ($M_{\rm bol}=-8.5$ to $-9.5$) well established from the IR (Monnier et al. 1999, Smith 2001), the star’s temperature would have to be extremely cool, (2225 K!) to have such a large radius. Using K-band photometry and a simple model, Monnier et al. (2004) suggest an effective temperature of 2600 K, similar to the 2800 K value found by Le Sidaner & Le Bertre (1996); again, see discussion in Monnier et al. (1999, 2004). We note that none of these stellar properties are in accord with stellar evolutionary theory (Fig. \[fig:HRD\]c), and, indeed, based upon its inferred mass-loss history, Humphreys et al. (2005) describe the star as “perplexing", and argue that it may be in a “unique evolutionary state." Such an object may provide important insight into a previously unrecognized avenue of normal stellar evolution, or its peculiarities may be the product of (for instance) binary evolution, as is the case for VV Cep. Smith et al. (2001) state that any hot, massive companion in VY CMa would have been previously detected spectroscopically, but we believe that further searching, particularly in the UV, is warranted. This star was not included in our sample, but we hope to perform an analysis in the near future.
Comparison with $(V-K)_o$ {#Sec-K}
-------------------------
In Sec. \[Sec-teff\] we derived a precise relationship between spectral subtype and effective temperature based primarily on the strengths of the TiO band strengths. One test of this result’s accuracy is to check for consistency with other temperature indicators. Josselin et al. (2000) have emphasized the usefulness of K-band photometry in deriving bolometric luminosities of RSGs, as the bolometric correction to the K-band is relatively insensitive to effective temperature and surface gravities, while RSGs themselves are less variable in the K band than at $V$. In addition, correction for interstellar reddening is minor at $K$. At the same time, the effective temperature is a very sensitive function of $(V-K)_o$. Since over half of our sample have $K$-band photometry (Table \[tab:stars\]), we can perform some exacting tests of the models to see if we obtained similar physical parameters by very different techniques.
We have derived synthetic $(V-K)_o$ colors for our models following the procedure and assumptions of Bessell et al. (1998). The $V$ bandpass comes from Bessell (1990), while the $K$ bandpass comes from Bessell & Brett (1988). Note that the latter is similar to the “standard" K system of Elias et al. (1982) and Johnson (1965), and a good approximation to the convolution of detector and filter used at UKIRT and with most ESO and NOAO instrumentation; however, this differs considerably from the $K_s$ filter employed in other modern instruments, such as the 2Mass survey and the VLT ISAAC instruments, and the transformations given below will not apply to $K_s$.
We found that we could approximate the relationship between $(V-K)_o$ color as a simple power-law: $$T_{\rm eff}=7741.9-1831.83 (V-K)_o + 263.135 (V-K)_o^2 -13.1943
(V-K)_o^3,$$ over the range $2.9 <
(V-K)_o < 8.0$, (3200-4300 K) with a dispersion of 11K, where the dispersion comes from considering the full range of appropriate surface gravities ($\log g = -1$ to 1); see Fig. \[fig:K1\]a. The bolometric correction at K is an almost linear relationship with effective temperature over the range $3200<T_{\rm eff}<4300$: $${\rm BC}_K=5.574-0.7589(T_{\rm eff}/1000),$$ with a dispersion of 0.01 mag (Fig. \[fig:K1\]). We have included these BC$_K$ values in Table \[tab:BCs\].
We de-reddened the $V-K$ colors in Table \[tab:stars\] using the extinction values derived from the model fits, and assuming $E(V-K)=2.73 E(B-V)$, based on Schlegel et al. (1998); this does not account for a modest change due to the shifts of the effective wavelenths of the band-passes for very red stars. We compare the effective temperatures derived by this method with those obtained from the spectral types in Fig. \[fig:K2\]a. The scatter is large, as one might expect given the strong functional dependence of $T_{\rm eff}$ on $(V-K)_o$, the variability of $V$, and the fact that the effective temperatures now depend strongly upon the assumed reddenings. Note that our uncertainty of 0.15 in A$_V$ translates to an uncertainty of 0.13 in $E(V-K)$, and hence 50 K in $T_{\rm eff}$. However, in general the agreement is good, with the $(V-K)_o$ colors yielding a temperature whose median difference is 60 K warmer than our adopted scale.
How well do the bolometric luminosities then agree? In Fig. \[fig:K2\]b we show the relationship between the bolometric luminosities derived from $V$ and our $T_{\rm eff}$ values (Table \[tab:results\]) and those found [*purely from the de-reddened $V-K$ colors*]{}. The agreement here is excellent, with only one significant outlier, KY Cyg, the most luminous star shown.
What if instead we had derived bolometric luminosities without reference to the $V-K$ colors at all, but rather simply used the extinction values from the fits to derive the absolute magnitude in the K-band \[$M_K=K-0.37E(B-V)-(m-M)_o$\], and then determined the bolometric correction at K using the $T_{\rm eff}$ of the models fits? We give that comparison in Fig. \[fig:K2\]c. Clearly there is excellent agreement.
In Fig. \[fig:K2\]d we show the comparison between the bolometric magnitudes derived from the K-band in the same manner as in (c) and the bolometric magnitudes derived from the visible. This plot is similar to that of Fig. \[fig:K2\]b, with excellent agreement in general. There are five significant outliers, whose differences are greater than 1 mag using the two methods (KY Cyg, CD-31 4916, BD+35 4077, BD+60 2613, and HD 14528), all in the sense that the bolometric luminosity derived from $V$ may be overly luminous. We flagged all five stars in Fig. \[fig:HRD\] as well as in Table \[tab:results\].
The fact that these five outliers were all more luminous based upon $V$ than upon $K$ raised the the question as to whether our method was systematically overestimating A$_V$. Recall that we did find that our A$_V$ values tend to be higher than that of OB stars in the same clusters and associations. Following a suggestion offered by the referee, we derived the bolometric luminosities at $K$ based instead based upon the $J-K$ colors alone, ignoring our A$_V$ values. We used the effective temperatures derived from our model fits both to determine the bolometric corrections, and to compute the intrinsic $(J-K)_o$ colors from a relationship we found using the MARCS models: $$(J-K)_o=3.10-0.547 (T_{\rm eff}/1000).$$ We adopt the broad-band extinction terms suggested by Schlegel et al. (1998), i.e., $A_K=0.70 E(J-K)$. Again, the numerical factor here does not take into account the shift of $R_K$ due to the change in the effective wavelengths of the broad-band filters, but it should be good enough for the test we intend. The $J-K$ photometry comes from Josselin et al. (2000) and references therein. We show this comparison between the two ways of deriving $M_{\rm bol}$ from $K$ in Fig. \[fig:K2\]e. There is excellent agreement between the two methods. Thus, there cannot be anything systematically wrong with our A$_V$ values in general. Interestingly, the two stars with the largest deviations in this figure are KY Cyg and KW Sgr. The $J-K$ correction suggests that if anything the $M_{\rm bol}$ values should be intermediate between what we derive above from $V$ and from $K$ using our values for the extinction, and that perhaps our extinction values for these two stars are too low, rather than too high. A comparison between the lumionsities derived from $K$ and $E(J-K)$ and those derived from $V$ and the extinctions derived from our model fits is shown in Fig. \[fig:K2\]f; this is very similar to what we found in Fig. \[fig:K2\]d.
We summarize the conclusions from these comparisons as follows. First, the MARCS models yield consistent results (to within 100 K) both from the molecular band strengths and from the $V-K$ colors. Thus if one’s goal is simply to derive bolometric luminosities, $V$ and $K$ band data will suffice for most purposes, if one has an estimate of the reddenings. However, for accurate placement in the H-R diagram (requiring $T_{\rm eff}$) then there is as yet no substitute for spectroscopy and the use of molecular bands. It is possible that other colors, such as $V-R_c$, might prove more effective in determining the effective temperatures than $V-K$, given the shorter baseline and therefore lower sensitivity to reddening, i.e. $E(V-R_c)=0.64 E(B-V)$. We will explore this further when we consider the Magellanic Cloud sample, as these stars have simultaneous $V$ and $R$ measures, and the reddenings are low and uniform (Massey et al. 1995).
Conclusions and Summary
=======================
We have determined a new effective temperature scale for Galactic RSGs by fitting moderate resolution (4-6Å) spectrophotometry with the new MARCS stellar atmosphere models. Our effective temperature scale is significantly warmer than previous scales, particularly for the later M supergiants, where the differences amount to 400 K. However, our new results give excellent agreement with the evolutionary tracks of Meynet & Maeder (2003), resolving the issue posed by Massey (2003) that the evolutionary models do not produce RSGs as cool and as luminous as “observed".
Our fitting showed excellent agreement between the models and data for the majority of the stars. The Ca I $\lambda 4226$ and Ca II H and K atomic lines appear to be too strong in the models, but the molecular transitions agree well at this dispersion. When we compare the physical properties derived from the model fits to our optical spectrophotometry with those found from the models using only the dereddened $V-K$ colors, we find good agreement, providing an exacting demonstration of the self-consistency of the MARCS models.
Extension of these studies to the Magellanic Clouds is underway, which should help us understand the effect that metallicity has on the effective temperature scale of RSGs. In addition, the fact that the reddenings are small and uniform and the distances are known (van den Bergh 2000) will allow further investigation of colors as probes of the physical properties of these interesting massive stars.
This work was supported by NSF Grant AST 00-093060 to PM, and and the NSF’s Research Experience for Undergraduates program at Northern Arizona University (AST 99-88007). We are very grateful for the excellent hospitality and support provided by the staff at KPNO and CTIO, and also thank Nat White and Kathy Eastwood for their encouragement and guidance. Hank Levesque pointed us towards some discussions of previous “record holders" of the largest stars known, and we also had correspondence with Jim Kaler on this topic. John Monnier kindly called his interesting results on VY CMa to our attention. We also gratefully acknowledge correspondence with Geoff Clayton. Comments by an the referee, Roberta Humphreys, led to improvements in the presentation of our arguments in several places.
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[l c c r r r r r l l l l]{} BD+59 38 &00 21 24.29 &+59 57 11.2 & 9.13 &2.49 & & 1&&M2 Iab &M2 I &Cas OB4 &MZ Cas\
Case 23 &00 49 10.71 &+64 56 19.0 &10.72 &2.77 & 7.80 & 2&&M1 Iab &M3 I &Cas OB7 &\
HD 236697 &01 19 53.62 &+58 18 30.7 & 8.65 &2.16 & 5.41 & 3&&M2 Ib &M1.5 I &NGC 457 &V466 Cas\
BD+59 274 &01 33 29.19 &+60 38 48.2 & 8.55 &2.09 & 5.24 & 1&&M0 Ib &M1 I &Cas OB8/NGC581 &\
BD+60 335 &01 46 05.48 &+60 59 36.7 & 9.15 &2.34 & & 1&&M3 Iab &M4 I &Cas OB8/NGC663 &\
HD 236871 &01 47 00.01 &+60 22 20.3 & 8.74 &2.27 & & 2&&M3 Iab &M2 I &Cas OB8 &\
HD 236915 &01 58 28.91 &+59 16 08.7 & 8.30 &2.20 & & 2&&M2 Iab &M2 I &Per OB1-A &\
BD+59 372 &01 59 39.66 &+60 15 01.7 & 9.30 &2.28 & & 2&&K5-M0 I &K5-M0 I &Per OB1-A &\
BD+56 512 &02 18 53.29 &+57 25 16.7 & 9.23 &2.47 & 6.99 & 1&&M4 Ib &M3 I &Per OB1-D &BU Per\
HD 14469 &02 22 06.89 &+56 36 14.9 & 7.63 &2.17 & 6.24 & 1&&M3-4 Iab &M3-4 I &Per OB1-D &SU Per\
HD 14488 &02 22 24.30 &+57 06 34.3 & 8.35 &2.27 & 6.62 & 1&&M4 Iab &M4 I &Per OB1-D/NGC884&RS Per\
HD 14528 &02 22 51.72 &+58 35 11.4 & 9.23 &2.65 & 7.78 & 1&&M4e I &M4.5 I &Per OB1-D &S Per\
BD+56 595 &02 23 11.03 &+57 11 58.3 & 8.18 &2.23 & 5.42 & 1&&M0 Iab &M1 I &Per OB1-D &\
HD 14580 &02 23 24.11 &+57 12 43.0 & 8.45 &2.27 & 5.43 & 2&&M0 Iab &M1 I &Per OB1-D &\
HD 14826 &02 25 21.86 &+57 26 14.1 & 8.24 &2.32 & 6.28 & 2&&M2 Iab &M2 I &Per OB1-D &\
HD 236979 &02 38 25.42 &+57 02 46.1 & 8.20 &2.35 & 6.17 & 4&&M2 Iab &M2 I &Per OB1-D? &YZ Per\
W Per &02 50 37.89 &+59 59 00.3 & 10.39 &2.53 & 8.30 & 4&&M3 Iab &M4.5 I &Per OB1-D? &HD 237008\
BD+57 647 &02 51 03.95 &+57 51 19.9 & 9.52 &2.74 & & 4&&M2 Iab &M2 I &Per OB1-D? &HD 237010\
HD 17958 &02 56 24.65 &+64 19 56.8 & 6.24 &2.03 & & 1&&K3 Ib &K2 I &Cam OB1 &\
HD 23475 &03 49 31.28 &+65 31 33.5 & 4.48 &1.88 & & 5&&M2+ IIab &M2.5 II &&\
HD 33299 &05 10 34.98 &+30 47 51.1 & 6.72 &1.62 & & 6&&K1 Ib &K1 I &Aur OB1 &\
HD 35601 &05 27 10.22 &+29 55 15.8 & 7.35 &2.20 & 5.61 & 2&&M1 Ib &M1.5 I &Aur OB1 &\
HD 36389 &05 32 12.75 &+18 35 39.2 & 4.38 &2.07 & & 1&&M2 Iab-Ib &M2 I &&\
HD 37536 &05 40 42.05 &+31 55 14.2 & 6.21 &2.09 & 5.28 & 3&&M2 Iab &M2 I &Aur OB1 &\
$\alpha$ Ori &05 55 10.31 &+07 24 25.4 & 0.50 &1.85 & & 1&&M1-2 Ia-Ib &M2 I &&\
HD 42475 &06 11 51.41 &+21 52 05.6 & 6.56 &2.25 & 5.70 & 3&&M0-1 Iab &M1 I &Gem OB1 & TV Gem\
HD 42543 &06 12 19.10 &+22 54 30.6 & 6.39 &2.24 & 5.46 & 3&&M1-2 Ia-Iab &M0 I &Gem OB1 & BU Gem\
HD 44537 &06 24 53.90 &+49 17 16.4 & 4.91 &1.97 & & 1&&K5-M0 Iab-Ib&M0 I &&\
HD 50877 &06 54 07.95 &-24 11 03.2 & 3.86 &1.74 & & 6&&K2.5 Iab &K2.5 I &Coll 121 &\
HD 52005 &07 00 15.82 &+16 04 44.3 & 5.68 &1.63 & & 3&&K3 Ib &K5 I &&\
HD 52877 &07 01 43.15 &-27 56 05.4 & 3.41 &1.69 & 3.90 & 6&&K7 Ib &M1.5 I &Coll 121 &$\sigma$ CMa\
CD-31 4916 &07 41 02.63 &-31 40 59.1 & 8.91 &2.16 & 5.20 & 1&&M2 Iab &M2.5 I &NGC2439 &\
HD 63302 &07 47 38.53 &-15 59 26.5 & 6.35 &1.78 & & 5&&K1 Ia-Iab &K2 I &&\
HD 90382 &10 24 25.36 &-60 11 29.0 & 7.45 &2.21 & 6.05 & 4&&M3 Iab &M3-4 I &Car OB1-D &CK Car\
HD 91093 &10 29 35.37 &-57 57 59.0 & 8.31 &2.21 & 6.65 & 4&&M2 Iab &M2 I &Car OB1-A &\
CPD-57 3502 &10 35 43.71 &-58 14 42.3 & 7.44 &2.02 & 5.17 & 4&&M1.5 Iab-Ib &M1.5 I &Car OB1-B/NGC329&\
HD 303250 &10 44 20.04 &-58 03 53.5 & 8.92 &2.51 & & 4&&M3 Iab &M2 I &Car OB1-B? &\
CD-58 3538 &10 44 47.15 &-59 24 48.1 & 8.36 &2.31 & 6.54 & 4&&M2+ Ia-0 &M2 I &Car OB1-E &RT Car\
HD 93420 &10 45 50.63 &-59 29 19.5 & 7.55 &1.87 & 6.15 & 4&&M4 Ib &M4 I &Car OB1-E &BO Car\
HD 94096 &10 50 26.30 &-59 58 56.5 & 7.38 &2.24 & 5.64 & 4&&M2 Iab &M2 I &Car OB1-E &IX Car\
HD 95687 &11 01 35.76 &-61 02 55.8 & 7.35 &2.12 & 5.81 & 4&&M2 Iab &M3 I &Car OB2 &\
HD 95950 &11 03 06.15 &-60 54 38.6 & 6.75 &2.04 & 5.18 & 4&&M2 Ib &M2 I &Car OB2 &\
HD 97671 &11 13 29.97 &-60 05 28.8 & 8.39 &2.52 & 7.42 & 4&&M3 Ia &M3-4 I &Car OB2 &\
CD-60 3621 &11 35 44.96 &-61 34 41.0 & 7.27 &1.92 & 4.74 & 4&&M0 Ib &M1.5 I &NGC3766 &\
HD 100930 &11 36 26.22 &-61 19 10.0 & 7.78 &1.95 & 5.68 & 4&&M2.5 Iab-Ib &M2.5 I &&\
CD-60 3636 &11 36 34.84 &-61 36 35.1 & 7.62 &1.81 & & 4&&M0 Ib &M0 I &NGC3766 &\
V396 Cen &13 17 25.05 &-61 35 02.3 & 7.85 &2.15 & 6.74 & 4&&M4 Ia-Iab &M3-4 I &Cen OB1-D &HD 115283\
CPD-53 7344 &16 12 56.91 &-54 13 13.8 & 8.79 &1.78 & & 4&&K2 Ib &K2 I &NGC6067 &\
CPD-53 7364 &16 13 04.01 &-54 12 21.2 & 9.13 &1.86 & & 4&&K4 Ib &K2 I &NGC6067 &\
HD 160371 &17 40 58.55 &-32 12 52.1 & 6.14 &1.82 & & 1&&K2.5 Ib &K2.5 I &M6 &BM Sco\
$\alpha$ Her &17 14 38.86 &+14 23 25.2 & 3.06 &1.45 & & 3&&M5 Ib-II &M5 I &&\
KW Sgr &17 52 00.73 &-28 01 20.5 & 9.35 &2.78 & 7.98 & 4&&M3 Ia &M1.5 I &Sgr OB5 &HD 316496\
HD 175588 &18 54 30.28 &+36 53 55.0 & 4.30 &1.67 & & 5&&M4 II &M4 II &&$\delta^2$ Lyr\
HD 181475 &19 20 48.31 &-04 30 09.0 & 6.96 &2.14 & & 6&&M0 II &M1 II &&\
HD 339034 &19 50 11.93 &+24 55 24.2 & 9.36 &3.05 & & 1&&M1 Ia &K3 I &Vul OB1 &Case 15\
BD+35 4077 &20 21 12.37 &+35 37 09.8 & 9.72 &2.93 & 8.11 & 3&&M3 Iab &M2.5 I &Cyg OB1 &\
BD+36 4025 &20 21 21.88 &+36 55 55.7 & 9.33 &2.49 & 8.75 & 3&&M3 Ia &M3-4 I &Cyg OB1 &BI Cyg\
BD+37 3903 &20 21 38.55 &+37 31 58.9 & 9.97 &3.26 & 9.75 & 3&&M3.5 Ia &M3 I &Cyg OB1 &BC Cyg\
KY Cyg &20 25 58.08 &+38 21 07.0 &10.57 &3.64 &10.40 & 4&&M3 Ia &M3-4 I &Cyg OB1 &Case 66\
BD+39 4208 &20 28 50.59 &+39 58 54.4 & 8.69 &2.87 & 8.21 & 1&&M3-4 Ia-Iab &M3 I &Cyg OB9 &RW Cyg\
HD 200905 &21 04 55.86 &+43 55 40.2 & 3.70 &1.65 & & 5&&K4.5 Ib-II &K4.5 I &&\
HD 202380 &21 12 47.25 &+60 05 52.8 & 6.62 &2.39 & & 2&&M2- Ib &M2 I &Cep OB2-A &\
HR 8248 &21 33 17.89 &+45 51 14.4 & 6.23 &1.78 & & 7&&K4 Ib &K1 I &Cyg OB7 &HD 205349\
HD 206936 &21 43 30.46 &+58 46 48.1 & 4.08 &2.35 & 5.96 & 1&&M2 Ia &M1 I &Cep OB2-A &$\mu$ Cep\
HD 210745 &22 10 51.28 &+58 12 04.5 & 3.35 &1.55 & & 5&&K1.5 Ib &K1.5 I &&\
BD+56 2793 &22 30 10.73 &+57 00 03.1 & 8.09 &2.28 & 6.22 & 3&&M2 Ia &M3 I &Cep OB2-B &HD 239978, ST Cep\
Case 75 &22 33 35.0 &+58 53 45 &10.67 &3.18 & & 4&&M1 Ia &M2.5 I &Cep OB1 &V354 Cep\
Case 78 &22 49 10.8 &+59 18 11 &10.76 &2.30 & & 4&&M2 Ib &M2 I &Cep OB1 &V355 Cep\
HD 216946 &22 56 26.00 &+49 44 00.8 & 4.94 &1.77 & & 5&&M0- Ib &M0 I &Lac OB1 &\
Case 80 &23 10 10.90 &+61 14 29.9 & 9.72 &2.60 & & 4&&M2 Iab &M3 I &Cas OB2 &GU Cep\
Case 81 &23 13 31.50 &+60 30 18.5 & 9.92 &2.70 & & 4&&M2 Ia &M2 I &Cas OB2 &V356 Cep?\
HD 219978 &23 19 23.77 &+62 44 23.2 & 6.77 &2.27 & & 2&&K5 Ib &M1 I &Cep OB3 &V809 Cas\
BD+60 2613 &23 44 03.28 &+61 47 22.2 & 8.50 &2.77 & 7.48 & 1&&M4 Ia &M3 I &Cas OB5 &PZ Cas\
BD+60 2634 &23 52 56.24 &+61 00 08.3 & 9.17 &2.51 & 7.22 & 3&&M2 Iab &M3 I &Cas OB5 &TZ Cas\
[l c c r r r r r c c]{} Cas OB4 &119.5&-0.4& 11.0 &1 &12.3 &2 &11.6 &0.74\
Cas OB7 &123.5& 0.9& 12.0 &2 &&&12.0 &0.86\
NGC 457 &126.6&-4.4& 12.0 &2 &11.9 &3 &11.9 &0.47\
Cas OB8/NGC581 &128.0&-1.8& 11.9 &4 &11.7 &3 &11.8 &0.38\
Cas OB8/NGC663 &129.5&-1.0& 11.6 &4 &11.5 &3 &11.5 &0.78\
Cas OB8 &129.4&-0.9& 11.2 &1 &12.3 &2 &11.7 &0.70\
Per OB1-A &131.1&-1.5& 11.0 &1 &11.8 &2 &11.4 &0.66\
Per OB1-D &135.0&-3.5& 11.4 &1 &11.8 &2 &11.4 &0.66\
Per OB1-D/NGC 884 &135.1&-3.6& 12.0 &4 &11.9 &3 &11.9 &0.56\
Cam OB1 &140.4& 1.9& 10.0 &1 &10.0 &2 &10.0 &0.70\
Aur OB1 &174.6& 1.2& 10.7 &1 &10.6 &2 &10.7 &0.53\
Gem OB1 &188.9& 3.4& 10.6 &1 &10.9 &2 &10.7 &0.66\
Coll 121 &237.9&-7.7& 8.9 &5 & 8.4 &3 & 8.4 &0.03\
NGC 2439 &246.4&-4.4& 13.2 &2 &12.9 &3 &13.0 &0.41\
Car OB1-A &284.5&-0.0& 11.9 &1 &12.0 &2 &11.9 &0.49\
Car OB1-B/NGC 3293 &285.9&+0.1& 12.0 &2 &11.8 &3 &11.9 &0.26\
Car OB1-B &286.0& 0.5& 11.7 &1 &12.0 &2 &11.9 &0.26\
Car OB1-D &286.6&-1.8& 11.9 &1 &12.0 &2 &11.7 &0.26\
Car OB1-E &287.6&-0.7& 12.1 &1 &12.0 &2 &12.0 &0.26\
Car OB2 &290.6&-0.1& 11.7 &1 &11.5 &2 &11.6 &0.46\
NGC 3766 &294.1&-0.0& 11.6 &1 &11.7 &6 &11.6 &0.20 &11.2 (3)\
Cen OB1-D &305.5& 1.6& 11.4 &1 &12.0 &2 &11.6 &0.70\
NGC 6067 &329.8&-2.2& 11.6 &2 &10.8 &3 &10.8 &0.38\
M 6 &356.6&-0.7& 8.3: &2 & 8.4 &3 & 8.4 &0.14\
Sgr OB5 & 0.2&-1.3& 12.4 &2 & &&12.4 &0.85\
Vul OB1 & 59.4&-0.1& 12.0 &1 &11.5 &2 &11.8 &0.83 &12.7 (7)\
Cyg OB1 & 75.6& 1.1& 10.7 &1 &11.3 &2 &11.0 &0.97\
Cyg OB9 & 76.8& 1.4& 10.7 &1 &10.4 &2 &10.6 &1.08\
Cep OB2-A & 99.3& 3.8& 9.9 &1 & 9.6 &2 & 9.7 &0.64 & 9.0 (5)\
Cep OB2-B &103.6& 5.6& 9.4 &1 & 9.6 &2 & 9.5 &0.64 & 9.0 (5)\
Cep OB1 &108.5&-2.7& 11.7 &1 &12.7 &2 &12.2 &0.6:\
Cyg OB7 & 90.0& 2.0& 9.5 &2 & 9.6 &8 & 9.5 &0.4:\
Lac OB1 & 96.8&16.1& 9.0 &1 & 8.9 &9 & 8.9 &0.11 & 7.8 (5)\
Cas OB2 &112.0& 0.0& 12.1 &2 & &&12.1 &0.96\
Cep OB3 &110.4& 2.9& 9.6 &1 & 9.7 &2 & 9.7 &0.8:\
Cas OB5 &115.5& 0.3& 11.8 &1 &12.0 &2 &11.9 &0.68\
[l c c c c c c]{} Grating/l mm$^{-1}$ &26/600 &58/400 & &26/600 &58/400 &58/400\
Blocking Filter &none &GG495 & &none &GG495 &OG570\
Wavelength Coverage(Å ) &3200 - 6000 &5000 - 9000 & &3500 -5200 &5000 - 7500 &6300 - 9000\
Slit Width (”/$\mu$m) &3.0/250&2.1/170 & &4.9/270 &3.6/200 &3.6/200\
Dispersion(Å mm$^{-1}$) &1.3 &1.9 & &1.5 & 2.2 &2.2\
Resolution(Å) &3.6 &5.7 & &5.0 &6.4 &6.4\
[l l c c c c c c c c]{} BD+59 38 &M2 I &3650&3.10& 0.0& 0.1& 600& -5.57& -7.17&0.81\
Case 23 &M3 I &3600&3.25& 0.5& 0.3& 410& -4.53& -6.28&0.59\
HD 236697 &M1.5 I &3700&1.55& 0.5& 0.4& 380& -4.80& -6.25&0.09\
BD+59 274 &M1 I &3750&1.55& 0.5& 0.4& 360& -4.80& -6.14&0.37\
BD+60 335 &M4 I &3525&2.63& 0.0& 0.1& 610& -4.99& -7.05&0.22\
HD 236871 &M2 I &3625&2.17& 0.0& 0.2& 520& -5.13& -6.80&0.00\
HD 236915 &M2 I &3650&1.71& 0.0& 0.3& 420& -4.80& -6.40&-0.34\
BD+59 372 &K5-M0 I&3825&2.48& 0.5& 0.6& 290& -4.58& -5.77&0.43\
BD+56 512 &M3 I &3600&3.25& 0.5& 0.1& 620& -5.42& -7.17&1.21\
HD 14469 &M3-4 I &3575&2.01& 0.0& -0.1& 780& -5.78& -7.64&-0.03\
HD 14488 &M4 I &3550&2.63& 0.0& -0.3&1000& -6.18& -8.15&0.90\
HD 14528 &M4.5 I &3500&4.18&-0.5& -0.4/-0.1&1230/780& -6.36& -8.53/-7.53&2.14\
BD+56 595 &M1 I &3800&1.86& 0.0& 0.4& 380& -5.08& -6.31&-0.19\
HD 14580 &M1 I &3800&2.17& 0.5& 0.4& 380& -5.12& -6.35&0.12\
HD 14826 &M2 I &3625&2.48& 0.0& 0.0& 650& -5.64& -7.31&0.43\
HD 236979 &M2 I &3700&2.32& 0.0& 0.2& 540& -5.52& -6.97&0.28\
W Per &M4.5 I &3550&4.03& 0.0& 0.1& 620& -5.13& -7.09&1.98\
BD+57 647 &M2 I &3650&4.03& 0.0& 0.0& 710& -5.91& -7.51&1.98\
HD 17958 &K2 I &4200&2.17& 0.5& 0.5& 360& -5.93& -6.63&0.00\
HD 23475 &M2.5 II&3625&1.08& 0.0& & & & &\
HD 33299 &K1 I &4300&0.77& 0.5& 0.9& 190& -4.76& -5.37&-0.87\
HD 35601 &M1.5 I &3700&2.01& 0.0& 0.2& 500& -5.36& -6.81&0.37\
HD 36389 &M2 I &3650&1.24& 0.0& & & & &\
HD 37536 &M2 I &3700&1.39& 0.0& 0.1& 630& -5.88& -7.33&-0.25\
$\alpha$ Ori&M2 I &3650&0.62& 0.0& & & & &\
HD 42475 &M1 I &3700&2.17& 0.0& -0.1& 770& -6.31& -7.76&0.12\
HD 42543 &M0 I &3800&2.01& 0.0& 0.0& 670& -6.32& -7.55&-0.03\
HD 44537 &M0 I &3750&0.62& 0.0& & & & &\
HD 50877 &K2.5 I &3900&0.16& 0.5& 0.6& 280& -4.69& -5.75&0.06\
HD 52005 &K5 I &3900&0.00& 0.0& & & & &\
HD 52877 &M1.5 I &3750&0.16& 0.5& 0.3& 420& -5.14& -6.48&0.06\
CD-31 4916 &M2.5 I &3600&2.01& 0.0& -0.1/0.2& 850/500& -6.11& -7.85/-6.69&0.74\
HD 63302 &K2 I &4100&0.62& 0.0& & & & &\
HD 90382 &M3-4 I &3550&1.86& 0.0& -0.3&1060& -6.31& -8.27&1.05\
HD 91093 &M2 I &3625&2.01& 0.0& 0.0& 640& -5.60& -7.28&0.50\
CPD-57 3502 &M1.5 I &3700&1.08& 0.0& 0.2& 540& -5.54& -6.99&0.28\
HD 303250 &M2 I &3625&2.94& 0.0& -0.1& 750& -5.92& -7.60&2.14\
CD-58 3538 &M2 I &3625&3.10& 0.0& -0.3&1090& -6.74& -8.41&2.29\
HD 93420 &M4 I &3525&1.08& 0.0& -0.1& 790& -5.53& -7.60&0.28\
HD 94096 &M2 I &3650&1.86& 0.0& -0.2& 920& -6.48& -8.08&1.05\
HD 95687 &M3 I &3625&1.71& 0.0& -0.1& 760& -5.96& -7.63&0.28\
HD 95950 &M2 I &3700&1.24& 0.0& 0.0& 700& -6.09& -7.54&-0.19\
HD 97671 &M3-4 I &3550&2.63& 0.0& -0.2& 860& -5.85& -7.81&1.21\
CD-60 3621 &M1.5 I &3700&0.77& 0.0& 0.3& 440& -5.11& -6.55&0.16\
HD 100930 &M2.5 I &3600&1.08& 0.0& & & & &\
CD-60 3636 &M0 I &3800&0.77& 0.5& 0.5& 320& -4.76& -5.98&0.16\
V396 Cen &M3-4 I &3550&2.48& 0.0& -0.3&1070& -6.33& -8.29&0.31\
CPD-53 7344 &K2 I &4000&0.77& 1.0& 1.3& 100& -2.78& -3.70&-0.40\
CPD-53 7364 &K2 I &4000&1.08& 1.0& 1.3& 100& -2.76& -3.67&-0.09\
HD 160371 &K2.5 I &3900&0.31& 1.0& 1.3& 100& -2.57& -3.63&-0.12\
$\alpha$ Her&M5 I &3450&1.40& 0.0& & & & &\
KW Sgr &M1.5 I &3700&4.65&-0.5& -0.5&1460& -7.70& -9.15&2.01\
HD 175588 &M4 II &3550&0.47& 0.0& & & & &\
HD 181475 &M1 II &3700&1.39& 0.0& & & & &\
HD 339034 &K3 I &4000&5.27& 0.0& -0.2& 980& -7.71& -8.63&2.70\
BD+35 4077 &M2.5 I &3600&5.27& 0.0& -0.3/0.1&1040/620& -6.55& -8.30/-7.18&2.26\
BD+36 4025 &M3-4 I &3575&5.11&-0.5& -0.4&1240& -6.78& -8.64&2.11\
BD+37 3903 &M3 I &3575&5.58&-0.5& -0.3&1140& -6.61& -8.46&2.57\
KY Cyg &M3-4 I &3500&7.75&-1.0& -0.9/-0.5&2850/1420& -8.18&-10.36/-8.84&4.74\
BD+39 4208 &M3 I &3600&4.49& 0.0& -0.2& 980& -6.41& -8.15&1.15\
HD 200905 &K4.5 I &3800&0.00& 0.0&& & & &\
HD 202380 &M2 I &3700&2.63& 0.0& 0.1& 590& -5.72& -7.16&0.65\
HR 8248 &K1 I &4000&0.93& 1.0& 0.9& 200& -4.20& -5.12&-0.31\
HD 206936 &M1 I &3700&2.01&-0.5& -0.5&1420& -7.63& -9.08&0.03\
HD 210745 &K1.5 I &4000&0.00& 0.0& & & & &\
BD+56 2793 &M3 I &3600&2.32& 0.5& 0.6& 290& -3.73& -5.48&0.34\
Case 75 &M2.5 I &3650&6.05&-0.5& -0.5&1520& -7.57& -9.17&4.18\
Case 78 &M2 I &3650&4.65& 0.0& -0.1& 770& -6.09& -7.69&2.79\
HD 216946 &M0 I &3800&0.31& 0.5& 0.7& 260& -4.27& -5.50&-0.03\
Case 80 &M3 I &3625&2.94& 0.0& 0.1& 570& -5.32& -7.00&-0.03\
Case 81 &M2 I &3700&3.56& 0.0& 0.1& 590& -5.74& -7.19&0.59\
HD 219978 &M1 I &3750&2.17& 0.5& 0.4& 410& -5.10& -6.44&-0.31\
BD+60 2613 &M3 I &3600&4.49&-0.5& -0.7/-0.4&1940/1190& -7.89& -9.64/8.57&2.39\
BD+60 2634 &M3 I &3600&3.25& 0.0& -0.1& 800& -5.98& -7.73&1.15\
[l c l l c]{} K1-1.5 &4100 &100 &3 &-0.79\
K2-3 &4015 &40 &7 &-0.90\
K5-M0 &3840 &30 &3 &-1.16\
M0 &3790 &13 &4 &-1.25\
M1 &3745 &17 &7 &-1.35\
M1.5 &3710 &8 &6 &-1.43\
M2 &3660 &7 &17&-1.57\
M2.5 &3615 &10 &5 &-1.70\
M3 &3605 &4 &9 &-1.74\
M3.5 &3550 &11 &6 &-1.96\
M4-4.5 &3535 &8 &6 &-2.03\
M5 &3450 &&1 &-2.49\
[c c c]{} 3200 &-4.58 & 3.16\
3300 &-3.66 & 3.08\
3400 &-2.81 & 3.00\
3500 &-2.18 & 2.92\
3600 &-1.75 & 2.84\
3700 &-1.45 & 2.76\
3800 &-1.23 & 2.68\
3900 &-1.06 & 2.61\
4000 &-0.92 & 2.53\
4100 &-0.79 & 2.47\
4200 &-0.70 & 2.40\
4300 &-0.61 & 2.33\
![\[fig:plots\] A comparison between our spectrophotometry (solid black) and the MARCS models (dotted red). The data are plotted on a $\log F_\lambda$ scale to facilitate comparison between the size of the molecular transitions. The models have been reddened by the indicated amount using the standard $R_V=3.1$ reddening law of Cardelli et al. (1989). The stars shown have spectral types K1 I (HR 8248), M1 I (BD+59 274), M2 I (HD 14826), and M4.5 I (HD 14528). The complete set of comparisons is available electronically. ](f1a.eps "fig:") ![\[fig:plots\] A comparison between our spectrophotometry (solid black) and the MARCS models (dotted red). The data are plotted on a $\log F_\lambda$ scale to facilitate comparison between the size of the molecular transitions. The models have been reddened by the indicated amount using the standard $R_V=3.1$ reddening law of Cardelli et al. (1989). The stars shown have spectral types K1 I (HR 8248), M1 I (BD+59 274), M2 I (HD 14826), and M4.5 I (HD 14528). The complete set of comparisons is available electronically. ](f1b.eps "fig:") ![\[fig:plots\] A comparison between our spectrophotometry (solid black) and the MARCS models (dotted red). The data are plotted on a $\log F_\lambda$ scale to facilitate comparison between the size of the molecular transitions. The models have been reddened by the indicated amount using the standard $R_V=3.1$ reddening law of Cardelli et al. (1989). The stars shown have spectral types K1 I (HR 8248), M1 I (BD+59 274), M2 I (HD 14826), and M4.5 I (HD 14528). The complete set of comparisons is available electronically. ](f1c.eps "fig:") ![\[fig:plots\] A comparison between our spectrophotometry (solid black) and the MARCS models (dotted red). The data are plotted on a $\log F_\lambda$ scale to facilitate comparison between the size of the molecular transitions. The models have been reddened by the indicated amount using the standard $R_V=3.1$ reddening law of Cardelli et al. (1989). The stars shown have spectral types K1 I (HR 8248), M1 I (BD+59 274), M2 I (HD 14826), and M4.5 I (HD 14528). The complete set of comparisons is available electronically. ](f1d.eps "fig:")
![\[fig:tscale\] The effective temperature scale for Galactic RSGs. The error bars reflect the standard deviation of the means from Table \[tab:NewT\]. For comparison, we show the scales of Humphreys & McElroy (1984) and Massey & Olsen (2003). ](f2.eps)
[^1]: We are making both the observed spectra and models available to others via data files at the Centre de Donnees Astronomiques de Strasbourg (CDS).
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: $R_V$ is typically taken to be $\sim 3.6$ for RSGs (Lee 1970), due to the increase of the effective wavelength of broad-band filters with redder stars; see McCall (2004) for a recent discussion. Our own calculations from the models suggest that $R_V\sim 4.4$ would be more appropriate for $BV$ photometry of RSGs. Note, however, that in the absence of a peculiar reddening law, $R_V=3.1$ is an appropriate choice for our study, since our analysis is not based upon broad-band filter photometry, but rather upon moderate-resolution (5Å) SEDs; i.e., neither the $B$ nor $V$ filters were involved in our extinction determinations. In order to prevent confusion, we use the equivalent A$_V$ values rather than $E(B-V)$, as our A$_V$ values will be directly comparable to those of others, while our $E(B-V)$ values would be larger than those determined from broad-band photometry for RSGs.
[^4]: We note that Josselin et al. (2003) have proposed that the star HD 37536 is an AGB based upon the detection of Tc and Li. On the other hand, the star is seen projected against Aur OB1, and if membership is assumed, a sensible $M_V$ is derived. In addition, the reddening is similar to that of the early-type members of Aur OB1.
[^5]: Also known as Herschel’s “Garnet Star", this star is often cited as the largest known normal star; see http://www.astro.uiuc.edu/$\sim$kaler/sow/garnet.html.
|
---
abstract: 'Jets are defined as impulsive, well-collimated upflows, occurring in different layers of the solar atmosphere with different scales. Their relationship with coronal mass ejections (CMEs), another type of solar impulsive events, remains elusive. Using the high-quality imaging data of AIA/SDO, here we show a well-observed coronal jet event, in which part of the jets, with the embedding coronal loops, runs into a nearby coronal hole (CH) and gets bounced towards the opposite direction. This is evidenced by the flat-shape of the jet front during its interaction with the CH and the V-shaped feature in the time-slice plot of the interaction region. About a half-hour later, a CME initially with a narrow and jet-like front is observed by the LASCO C2 coronagraph, propagating along the direction of the post-collision jet. We also observe some 304 [Å]{} dark material flowing from the jet-CH interaction region towards the CME. We thus suggest that the jet and the CME are physically connected, with the jet-CH collision and the large-scale magnetic topology of the CH being important to define the eventual propagating direction of this particular jet-CME eruption.'
author:
- 'Ruisheng Zheng, Yao Chen, Guohui Du, and Chuanyang Li'
title: 'Solar jet-coronal hole collision and a related coronal mass ejection'
---
Introduction
============
Solar coronal jets, first observed in X-rays with the Soft X-ray Telescope (XRT, Tsuneta et al. 1991) on board the Yohkoh satellite, represent a group of impulsive events characterized by well-collimated upflows with different scales developing in different layers of the solar atmosphere (e.g., Shibata et al. 1992; Savcheva et al. 2007; Chen et al. 2012). They are generally believed to be energized by magnetic reconnection, often associated with an inverse Y-shaped, anemone-like configuration involving open field lines in coronal holes (CHs) or open-like large-scale closed loops extending from an active region (Shibata et al. 1992, 2007; Schmieder et al. 1995; Rachmeler et al. 2010; Pariat et al. 2009, 2015). These open or open-like field lines are important in collimating jets.
Coronal mass ejections (CMEs) are another type of impulsive energy release events in the solar atmosphere, with a much larger scale and stronger impact on nearby coronal structures such as streamers and CHs. There exist a number of studies examining the strong CME disturbance to coronal streamers (e.g., Hundhausen et al. 1987; Sheeley et al. 2000; Tripathi & Raouafi 2007; Chen et al. 2010). In the meantime, both streamers and CHs have been suggested to have effects on the propagating direction of CMEs, a crucial factor determining the CME geo-effectiveness. For instance, Gopalswamy et al. (2009) reported events with sources very close to the solar disk center that are unexpectedly *NOT* associated with interplanetary CMEs (yet accompanied by interplanetary shocks), and they attributed this to possible interaction and further deflection of CMEs by nearby CH(s). Nevertheless, a direct observation of this CME-CH interaction process remains absent. Neither do we know how and where the deflection takes place.
While both jet and CME represent impulsive ejection of plasmas to upper levels of the solar atmosphere, their relationship remains obscure. Can a relatively large-and-fast jet drive a CME or can the CME actually trigger some jets, by, for example, opening initially-closed magnetic field? Different scenarios have been developed (e.g., Pariat et al. 2009, 2015), and actual answers may differ to different event, depending on specific circumstances[[^1]]{}. It is also very interesting to ask, considering the above-mentioned possibility of strong CME-CH interaction, can a jet, if moving along large-scale active region loops, actually interact with a nearby CH? None of this kind of event has been reported ever. In this study, we present unambiguous evidence of such an event, revealing the collision between a set of coronal jets and a nearby CH with high quality data from the Atmospheric Imaging Assembly (AIA: Lemen et al. 2012) onboard the Solar Dynamics Observatory (SDO: Pesnell et al. 2012). It turns out that part of the jets is reflected towards the opposite direction, and this dynamical jet-CH interaction may have led to a successful eruption along the same direction.
Observations and Data Analysis
==============================
We mainly analyzed the AIA/SDO data that provides the essential observations of the event. The AIA instrument has ten EUV and UV wavelengths, covering a wide range of temperatures. The AIA observes the full disk (4096 $\times$ 4096 pixels) of the Sun and up to 0.5 $R_\odot$ above the limb, with a pixel resolution of 0.6$"$ and a cadence of 12 s. The eruption is visible in all AIA EUV channels. The passbands of interest here are 131 [Å]{} (Fe XXI, $\sim$10 MK), 211 [Å]{} (Fe XIV, $\sim$2.0 MK), 171 [Å]{} (Fe IX, $\sim$0.6 MK), and 304 [Å]{} (He II, $\sim$0.05 MK). Magnetograms and intensity maps from the Helioseismic and Magnetic Imager (HMI: Scherrer et al. 2012), with a cadence of 45 s and pixel scale of 0.6$"$, were used to check the magnetic field configuration of the source region. The CME evolution in the high corona was captured by the Large Angle and Spectrometric Coronagraph (LASCO) C2 (Brueckner et al. 1995).
The kinematics of the jets and associated mass flow were analyzed with the time-slice approach. The speeds were determined by linear fits, with error bars given by the measurement uncertainty that is assumed to be 4 pixel ($\sim1.74$ Mm) for AIA data. We also used the Potential Field Source Surface (PFSS: Schrijver [&]{} De Rosa 2003) model to extrapolate the HMI photospheric field measurement to describe the large-scale magnetic field geometry.
Results
=======
Coronal jets
------------
The event occurred at the eastern boundary of the NOAA Active Region (AR) 12403 on 25 August 2015. Upper panels of Fig.1 show the AR image observed at AIA 211 [Å]{}, and the intensity map and magnetogram of HMI. The small white boxes ($\sim$S14E13) present the source area in which jets originated. It can be seen that the AR consists of a positive-polarity leading sunspot and a negative-polarity following sunspot. In the region given by white boxes, there exists a small parasitic positive polarity. An elongated low-latitude elephant-trunk CH exists eastward of the AR (white arrows in Fig.1a).
In Fig.1d-h, we present the sequence of HMI magnetogram from 03:30 UT to 11:30 UT to examine the magnetic evolution of the jet source region. As a result of earlier magnetic flux emergence, a small positive patch was embraced by negative dominant polarities and became the parasitic polarity. Comparing these magnetograms, we see that significant flux cancellation took place. This is further confirmed by the temporal changes of positive and negative magnetic fluxes in the FOV of Fig.1d-h, as plotted in Fig.1i. Before 07:12 UT (dashed vertical line), the negative flux increased continuously, while the positive counterpart did not change much. After 07:12 UT, both fluxes started to decrease with the positive one changing at a much steeper gradient. At about 10:20 UT (dotted vertical line), the positive flux presented an even faster declining rate.
The later time was consistent with the onset of the jet event. In the meantime, the GOES soft X-ray (SXR) profiles started to increase after 10:23 UT. Two SXR peaks were recorded in the following hour, corresponding to a C1.7 flare (peaking at 10:34 UT) and a C2.2 flare (peaking at 10:44 UT), respectively. Both flares were associated with jet activities. This is consistent with the general picture that jets are energized by magnetic reconnection, as evidenced here by significant flux cancellation and flare occurrence (e.g., Wang et al. 1998; Chae et al. 1999; Chifor et al. 2008; Yang et al. 2011; Liu et al. 2011; Pariat et al. 2009, 2015).
In Fig.2, we present the dynamical evolution of coronal jets observed at AIA 171 [Å]{}. As seen from Fig.2a-c and the accompanying animation, the jet started from the southern end of the bright flaring loops, exhibiting a gradual footpoint migration towards the northern end. The migration indicates an apparent motion of the main flaring reconnection site. For the convenience of description, we separate the jets into three subsequent episodes, consisting of the initial relatively weak part (J1, starting at 10:26 UT), the middle part which is the strongest one and of particular interest to this study (J2, starting at 10:30 UT), and the third part which is basically confined by underlying loops (J3, starting at 10:38 UT). Note that similar confining mass flows have been used to trace the twisted structure internal of a flux rope (e.g., Li & Zhang 2013; Yang et al. 2014). These episodes of jets have been pointed out in Fig.2a-c. As mentioned, the jets mainly emanated from the FOV of Fig.1d-h, above the parasitic polarity. This is a general source property prescribed in jet modelling (e.g., Pariat et al. 2009, 2015). Because J1 was relatively weak and J3 was mostly confined, here we focus our study on J2.
Using the time-slice approach along the dotted line (S1) in Fig.2c, the derived velocity of J2 is close to 500 km s$^{-1}$ (Fig.2f), much faster than the statistical average speed of $\sim$200 km s$^{-1}$ for jets (Shimojo et al. 1996). It is clear that J2 lasted for $\sim$ 30 mins with continuous mass ejection. J2 initially moved along its associated AR loops, and carried the loops to extend. It is interesting to see that the forward extension of the jet-loop structure was suddenly stopped. The curved side of the jet-loop structure became flat-shaped with kinks at both ends, and part of J2 was clearly bounced towards the opposite direction while the left part returned to the solar surface (best seen in the online animation). The first sign of bounced-back material was present around 10:38 UT as seen from the online animation. The flat-shaped feature appeared around the interface between the nearby CH and the east edge of the AR, indicating that the jet carrying the loop ran into the AR-CH boundary and got reflected there. The reflection is also seen from the height-time plot along slices S2 and S3 (short lines in Fig.2d-e), from which we see a distinct V-type structure (Fig.2g-h). The speeds of the jet-loop structure along S2 and S3 before and after the reflection are nearly the same ($\sim$90 km s$^{-1}$).
Note that, after the jet-CH interaction, part of the jet material was stagnated (black arrows in Fig.2g-h) while the left part presented a signature of continuation of mass flows towards west (white arrows in Fig.2d-e, see also the online animation), at a fast speed of $\sim$400 km s$^{-1}$. See Fig.2i for the time-slice plot along S4. This is only slightly slower than the pre-collision jet. Yet, the jet front faded away shortly. So, it is not known, at this time, whether the reflected jet flows have escaped the corona or not.
CME and its relation with the jet
---------------------------------
A weak CME feature appeared in the LASCO C2 FOV at 11:14 UT, with a hardly identifiable narrow front, developing into a much clearer CME structure in 10-20 mins (see Fig.3 and the accompanying animation). The central position angle of the ejecta at 11:26 UT was 238$^\circ$ (the angle increases counterclockwise with 0$^\circ$ along north). Initially, the CME presented a narrow jet-like morphology and later became diffusive without a clear flux-rope signature. It is thus difficult to determine the exact type of this eruption (see Vourlidas et al. 2013). The appearance time of a clear CME signature in C2 is about 50 mins later than the first sign of the jet-CH collision ($\sim$ 10:38 UT). In addition, the continuation part of the reflected jet flow is basically toward the CME direction. This close temporal-spatial correlation suggests that the jet may be associated with the CME.
It is crucial to further figure out whether the jet front continued its westward motion toward the solar limb to become a part of the CME or it actually moved downwards along a curved loop path and confined there. For short, we refer these suggestions to be the eruptive picture and the confining picture, respectively. In the following we present observational facts that, from our point of view, favor the first possibility.
Firstly, from Fig.4 and accompanying animations, in which a set of AIA images at 171, 211, and 131 [Å]{} are presented, we see that after 10:55 UT the jet front seemed to have moved beyond the associated AR loop system before its eventual fading-away, as pointed out by the arrows in Fig.4a-e, rather than returned to the solar surface along a curved loop-like path if being confined.
Secondly, following the jet front, since 11:06 UT a systematic westward motion of a set of loops started to appear, as pointed by arrows in Fig.4f-h, with a speed of 10-20 km s$^{-1}$. This is best seen from the online animations. The motion lasted for more than 30 mins, possibly an effect of continuous stretching exerted by the westward mass motion. Again, this is not inconsistent with the eruptive picture.
Thirdly, from the 304 [Å]{} data we observed an obvious outflow of filament-like dark material after 11:20 UT (see the white box in Fig.5a-c). It seems that the material corresponded to part of the reflected jet, being stagnated and accumulated around the jet-CH interaction region (best seen from the accompanying animation). They became dark possibly due to a cooling process. Note that there was no filament eruption observed during the event, so these dark material was not due to any filament eruption. Its outflow speed ($\sim$239 km s$^{-1}$) can be derived using the distance-time analysis along the slice S5 (Fig.5d). The 304 [Å]{} material moved out of the AR, along the direction pointing to the CME. This traces the open path from the jet-CH collision region to the CME, providing additional support to the first picture. Note that due to the time delay, the 304 [Å]{} material could not become the CME front yet they may provide some mass supply to the eruption.
The last observational fact worthy of mention is that no any other detectable eruptive activities were present on the solar disk according to all passband data of AIA, and on the back side according to the Extreme Ultraviolet Imager (EUVI) (Howard et al. 2008) on board the twin spacecraft of Solar-Terrestrial Relations Observatory (STEREO: Kaiser et al. 2008) with separation angles from the Earth $172.348^\circ$ (STEREO-A) and $175.495^\circ$ (STEREO-B) at the time.
In summary, the above observational facts favor the first picture, i.e., the post-collision jet further evolves into a part of the CME. The distance from the jet-CH interaction region to the CME front is $\sim$ 3 R$_\odot$, indicating an average projected propagation speed of $\sim$ 700 km s$^{-1}$ if assuming the post-collision jet front later becoming the CME front. This is faster than the AIA-measured projection speed of the jet, suggesting that either the jet gets further accelerated during its outward propagation or the jet is not the counterpart of the CME front and there exists other or earlier eruptive magnetic structures ahead of the jet. It should be pointed out that how the jet evolves into the CME and exactly which part of the CME corresponds to the jet front remains not resolved with available data set, partly due to the absence of CME signatures in the AIA FOV.
Further examining the PFSS results of the CH-AR magnetic field lines (Fig.3 c-d), we see that the CH open field lines are of negative polarity and lying next to the closed loop system that is rooted at the large negative polarity of the AR. The CH field lines, with a strong non-radial expansion, occupy the space above the closed AR loops. This magnetic configuration helps understand how the observed initially-collimated jet (along the eastern edge of the AR) runs into field lines of the nearby CH and then flows outward along the specific trajectory.
Summary
=======
Here we present a first-of-its-kind observational study on a jet-CH colliding process showing that the post-collision jet was reflected towards the opposite direction. We also present compelling evidence supporting that the jet activity may have developed into a successful eruption (i.e., a CME). The jet-CH collision is evidenced by the flat morphology of the jet front observed by AIA, while the jet-CME relation is supported by their close temporal-spatial correlation, the observed outflow at 304 [Å]{}, the large-scale CH-AR magnetic field configuration given by PFSS, and the fact that no other identifiable eruptive activities on the solar surface including the backside, among other observations. The presumed jet-CME route basically follows the over-expanding trend of the CH open field lines above the AR according to PFSS extrapolation, indicating a strong role played by the CH structure in defining the CME propagating direction. This is consistent with earlier studies, which were however not based on direct observation of CME-CH interaction, that CHs are important in affecting the CME propagating direction and thus the consequent geo-effectiveness. The study is possible because of the unprecedented high-quality data of AIA/SDO.
SDO is a mission of NASA’s Living With a Star Program. The authors thank the SDO team for providing the data. This work is supported by grants NSBRSF 2012CB825601, NNSFC 41274175, and 41331068, and Yunnan Province Natural Science Foundation 2013FB085.
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[^1]: Note that during the peer-review process of our manuscript, a study reporting a CME event likely triggered by a coronal jet was published (Liu et al., 2015), which presents evidence supporting the close relation between a jet and a CME, with the jet pushing some overlying blob-like magnetic structure which later becomes the CME front and the jet likely evolves into the CME core.
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abstract: 'We resolve the old riddle related to the critical behavior of the heat capacity near the smectic A – hexatic second order phase transition. Experiment suggests a “large” specific heat critical exponent $\alpha=0.5 \div 0.7$ inconsistent with the universality class for this phase transition implying the very small negative exponent $\alpha\approx-0.01$. We show that essential features of the heat capacity for the smectic A – hexatic phase transition can be rationalized in the framework of a theoretical model treating jointly fluctuations of the hexatic orientational order and of the positional (translational) order parameters. Assuming that the positional (translational) correlation length $\xi_{tr}$ is larger than the hexatic correlation length $\xi_h$, we calculate a temperature dependence of the specific heat in the critical region near the smectic A – hexatic phase transition. Our results are in a quantitative agreement with the calorimetric experimental data.'
author:
- 'E.I.Kats$^{1}$, V.V.Lebedev$^{1}$, and A.R.Muratov$^2$'
title: 'Why the smectic A – hexatic phase transition does not follow its universality class?'
---
Introduction
============
Smectic liquid crystals are remarkable layered phases possessing an astonishingly rich variety of structures. The simplest smectic structure is the smectic A liquid crystal, which is solid-like in the direction perpendicular to the layers and fluid-like within the layers. By other words, the smectic A can be thought as a stack of liquid layers. Smectics C possess nematic-like orientationally ordered layers. In the late seventies, first as a theoretical suggestion [@BL78; @BN81], and later on by experimental observations [@PM81; @BA86; @JO03; @ZK15] some smectic phases have been identified as stacks of layers with hexagonal orientational order, the phases are termed as hexatic smectics or hexatics. The hexatics possess a long-range hexagonal orientational order, however, they have no long-range positional (translational) order. It is worth to note that the hexatic phases are not merely a funny state of a few liquid-crystalline materials. They appear in many biological systems (see, e.g., Refs. [@NT00; @MR05]), and even in planetary or astrophysical science as a form of dust plasma [@PV15]. Investigation of the hexatics is a multidisciplinary area including many fundamental physical problems and involving various questions of chemistry and biology. However, even after some decades of investigations of the hexatics, a complete description of this liquid crystalline state is still not available, and a number of phenomena remain to be clarified.
In this work we examine properties of a liquid crystal near the continuous smectic A – hexatic phase transition, such transition is observed in a variety of materials, see Refs. [@PM81; @BA86; @ZK15; @VL83; @HV83; @PN85; @GS91; @SG92; @GS93; @HK97; @RD05; @MP13]. We investigate the phase transition in spirit of the Landau approach in terms of the hexatic orientational order parameter. The order parameter for the phase transition has to be introduced from symmetry reasoning [@CL00]. Say, the nematic order parameter is a second order symmetric traceless tensor $Q_{ik}$ [@GP93]. In the nematic phase the average value of $Q_{ik}$ is nonzero and therefore the complete rotational symmetry of the isotropic liquid is reduced to the quadrupole symmetry of the nematic. An analogous “nematic-like” orientational order parameter can be introduced for the smectics $C$. It is also a second order symmetric traceless tensor, however, its components are within the smectic layers (i.e., the order parameter is a $2 \times 2$ tensor unlike the $3 \times 3$ order parameter in nematics). A non-zero average value of the order parameter means reduction of the uniaxial rotational symmetry $D_{\infty h}$, characteristic of the smectic layers in the phase A, to the biaxial rotational symmetry $D_{2h}$ in the phase C. By other words, in the smectics C the layers are invariant under rotation around the second order axis whereas in the smectics A the layers are isotropic.
Similarly to the smectic A – smectic C phase transition the smectic A – hexatic phase transition means a reduction of the rotational symmetry of the smectic layers. However, the symmetry of the smectic layers in the hexatics is $D_{6h}$ instead of $D_{2h}$ in the smectics C. By other words, in the hexatics the layers are invariant under rotations around the axis of the sixth order. The corresponding order parameter is a six-order symmetric irreducible tensor $Q_{injklm}$ (its irreducibility means $Q_{iijklm}=0$), having components solely within the layers. In the smectic A phase the average value of the tensor $Q_{injklm}$ is zero (the layers are isotropic) whereas in the hexatic phase its average value becomes non-zero. As a result, the smectic layers possess a hexagonal symmetry, i.e., they are invariant under rotation around the six-order rotation axis. The tensor $Q_{injklm}$ has two independent components [@GKL91]. Apart of small fluctuations, almost parallel smectic layers are well-defined. If we direct the $Z$-axis perpendicular to the layers, then the two independent components of the tensor are $Q_{xxxxxx}$ and $Q_{xxxxxy}$. It is convenient to deal with the scalar complex field $$\Psi =Q_{xxxxxx}+iQ_{xxxxxy},
\label{complex}$$ instead of the tensor $Q_{injklm}$. At rotation by an angle $\chi$ around the $Z$-axis the order parameter $\Psi$ is transformed as $$\Psi \to \exp(6i\chi) \Psi .
\label{rotate}$$ The order parameter $\Psi$ is equivalent to the traditional hexatic order parameter introduced in [@BL78; @BN81] in terms of the molecular bond orientations (see also the textbooks [@CL00; @GP93]).
The next step in constructing our theory is deriving the Landau functional for the smectic A – hexatic phase transition. Due to the rotational invariance the Landau functional ${\cal F}_{La}$ (determining an energy excess associated with the order parameter $\Psi$) contains only even in $\Psi $ terms in its expansion. Indeed, just the combinations $\Psi \Psi ^\ast$, $(\Psi \Psi ^\ast)^2$ etc are invariant under the transformation (\[rotate\]). Therefore the smectic A – hexatic phase transition should be of the second order (see the textbooks [@LL80; @CL00; @WK74; @ST87]), in accordance with experiment. Since we deal with the two-component order parameter, the phase transition belongs to the same universality class as the superfluid one. Particularly, the heat capacity exponent $\alpha$ for the smectic $A$ – hexatic phase transition should be small and negative $\alpha \approx - 0.01$ (see results of Monte Carlo simulations and experimental data presented in [@ST87; @AN91; @PV02]). Note to the point that the second order $\epsilon$-expansion [@GZ80] gives a positive but also very small (near $0.01$) value for the exponent $\alpha$. In a dramatic contradiction to this theoretical expectation, all known calorimetric data for the smectic A – hexatic phase transition [@VL83; @HV83; @PN85; @GS91; @SG92; @GS93; @HK97; @RD05; @MP13] show large and positive exponent $\alpha = 0.5\div 0.7$. Thus one encounters an obvious problem, and the main motivation of this work is to find where is a catch.
We are not the first trying to resolve the contradiction. Some attempts have been performed in order to provide a rational basis for such behavior of the heat capacity. Two different suggestions can be found in literature. Having in mind that the exponent $\alpha$ is near $0.5$, it is tempting to assume that the smectic A – hexatic phase transition occurs in a vicinity of a tricritical point [@AB86]. Then one could expect a sort of crossover behavior, in spirit of of the nematic – smectic A – smectic C tricritical point [@AV85]. However, it is hard to believe that all known hexatic liquid-crystalline materials, irrespective to the width of the stability region for the hexatic phase (ranging from a few to 50 degrees [@GS93]) are always near the tricritical point (and in the continuous phase transition side). Another suggestion as far as we aware advocated first in the works [@RH82; @BB89; @GC94] is based on an observation that measured experimentally critical exponents for the hexatic – smectic $A$ phase transition are close to those predicted by the $q$-state Potts model [@BK76] (with $q=3$ or $q=4$ in 2- or 3-dimensional space). However it is not clear at all, how (and why) the Potts model can be mapped to the physics of the phase transition with the two-component order parameter. In our opinion, this approach is not consistent to be confronted with the entire array of the experimental data.
We propose another way to reconcile the well established phase transition universality conception and the massive of experimental data for the smectic A – hexatic phase transition. A recent progress in experimental X-ray scattering techniques [@ZK15] reveals a rather unusual feature of the hexatics. Namely, very narrow peaks are observed in the X-ray scattering data there in the hexatic phase, that is the hexatics are almost “ready” to crystallize. The correlation length $\xi_{tr}$ of the short-range density fluctuations $\delta\rho$ (that is the positional order parameter), extracted from the X-ray data [@ZK15] for 3(10)OBC, is anomalously large, ranging from about $3\, nm$ in the vicinity of the transition point up to $20\, nm$ deeply in the hexatic phase. Then because of a relevant coupling between the hexatic and positional order parameters the universal critical behavior, characteristic of the superfluid helium universality class, should be observed only at the condition $\xi_h\gg \xi_{tr}$, where $\xi_h$ is the correlation length of the hexatic order parameter. In the opposite limit, at $\xi_h \leq \xi_{tr}$, the self-interaction of the hexatic order parameter mediated via the translational order parameter fluctuations is effectively non-local and, consequently, some non-standard critical behavior could be expected. It is the intermediate behavior which results in difficulties when developing theory tool. We examine just this possibility.
Of course, at the condition $\xi_h \gg \xi_{tr}$ one has to recover the standard universal behavior. However, to achieve such relatively large value of the orientational correlation length $\xi _h$ one has to probe a very narrow vicinity of the phase transition point. It would be fairly to say that from the available experimental data we are not in the position to estimate how narrow is this region of temperatures with the standard universal behavior. The textbook wisdom about second order phase transitions [@LL80; @CL00] suggests only that $\xi _h \simeq \xi _0 |(T-T_h)/T_h|^{-\nu }$ where $\xi _0$ is the bare (microscopic) correlation length, $T_h$ is the second order smectic A – hexatic transition temperature, and $\nu = 0.76$ is the critical exponent of the correlation length for the universality class with the two-component order parameter (in three-dimensional space). Assuming $\xi _0 \simeq 0.1\, nm$ we find that $\xi _h$ achieves the value a few times larger than $\xi_{tr} \simeq 3\, nm $ in the vicinity of the transition point $\Delta T/T_h \lesssim 10^{-3}$. Beyond this region, one has to deal with the intermediate asymptotic behavior caused by the interaction of the hexatic order parameter $\Psi$ with the long-correlated translational order parameter fluctuations $\delta\rho$. The asymptotic behavior is characterized by a scaling behavior, and the main goal of our work is to construct a theoretical scheme for this case and to compare the theoretical predictions with known calorimetric experimental data.
Our paper is divided into the following sections. In the foregoing section \[sec:mod\] we introduce ingredients and basic notions of our model and present its theoretical treatment. In the section \[sec:fluct\] we examine fluctuation effects in the framework of our model. The section \[sec:exp\] is devoted to confronting the theoretical results and experimental data. The section \[sec:con\] summarizes our main findings.
Model {#sec:mod}
=====
Our model is based on involving into consideration two strongly fluctuating fields: the hexatic order parameter $\Psi$ and the short-range density modulation $\delta\rho$ (that plays a role of the order parameter for the crystallization phase transition). We consider a vicinity of the smectic A – hexatic phase transition where fluctuations of $\Psi$ are strong and one would expect the standard critical behavior with universal exponents characteristic of a two-component order parameter. However, as we noted, the observed behavior of the heat capacity near the smectic A – hexatic phase transition does not follow its universality class. We explain this unusual behavior by an interaction of the order parameter $\Psi$ with fluctuations of the short-range density modulation $\delta\rho$. Note to the point that the interaction is relevant also in the hexatic phase (out of the critical region), where effects related to the non-zero mean value of the order parameter $\Psi$ have to be included into the consideration. We defer the investigation of the region for a future work.
The cornerstone assumptions of our model are based on the experimental X-ray scattering patterns. It is well known, that the X-ray scattering is produced by the short-range electron density fluctuations proportional to the mass density fluctuations $\delta\rho$. Quantitatively, it is determined by the pair correlation function of $\delta\rho$. Fourier transform of the correlation function is known as the structure factor $S(\bm q)$: $$S(\bm q)= \int d^3r\ \exp(-i \bm q \bm r)
\langle \delta\rho(\bm r_1)
\delta\rho(\bm r_1+\bm r) \rangle.
\label{structure}$$ For the smectic phases, the structure factor $S(\bm q)$ has quasi-Bragg peaks at $q_x,q_y=0,q_z=2\pi/l$ (where $l$ is the thickness of the smectic layers) and also maxima at $q_\perp=q_0$, $q_\perp^2=q_x^2+q_y^2$, where $q_0^{-1}$ is on the order of a characteristic inter-molecular distance in the smectic layers. A hallmark feature of the hexatics, established experimentally [@ZK15], is that near the cylinder $q_\perp=q_0$ the structure factor $S(\bm q)$ is almost independent of $q_z$ (more precisely, one should deal with the quasi-momentum and with the first Brillouin zone in $Z$-direction). In terms of the positional order parameter $\delta\rho$, the feature (weak dependence of the structure factor on $q_z$) implies that relevant fluctuations of $\delta \rho$ are strongly confined in the central part of the smectic layers.
The X-ray scattering data provide an information on the temperature dependence of the structure function $S({\bm q})$ near the smectic A – hexatic phase transition (see [@ZK15] and references therein). In the smectic A phase the pattern is a bright diffuse ring (cylinder in the three-dimensional reciprocal space) parallel to the smectic layers with the radius $q_\perp=q_0$. In the hexatic phase the ring is split into six spots, according to the rotational six order symmetry axis of the phase. The angular dependence of the structure function $S({\bm q})$ is related to a non-zero average value of the order parameter $\Psi$ in the hexatic phase. Each spot is narrow in the radial direction (along $q_\perp$) and elongated in the angular direction. The radial behavior of the structure function is characterized by the correlation length $\xi_{tr}$. As we already noted, for 3(10)OBC the correlation length $\xi_{tr}$ is ranging from about $3\, nm $ in the vicinity of the transition point up to $20\, nm$ in the hexatic phase. Any case, the correlation length is much larger, than the characteristic molecular size.
We analyze the critical behavior of a liquid crystal near the smectic A – hexatic phase transition where fluctuations of the hexatic order parameter $\Psi$ are strong. In this case the angular dependence of the structure function in the hexatic phase is weak and therefore it will be neglected below. We are interested in the temperature region where the condition $$\xi _{tr} > \xi _h \ ,
\label{corrl}$$ is valid (as above, $\xi _h$ is the hexatic order parameter correlation length, that characterizes correlations of the $\Psi$ fluctuations). As we explained in Introduction, the inequality (\[corrl\]) is violated only in a very narrow vicinity of the smectic $A$ - hexatic transition temperature where the critical behavior characteristic of the superfluid phase transition has to be restored. Beyond the narrow vicinity, that is in the region where the inequality (\[corrl\]) holds, a special theoretical analysis of the critical behavior is needed, that is the subject of our work.
Crystallization of a conventional liquid or of a liquid crystal is as a rule a strong first order phase transition. Particularly, its latent heat per molecule is on the order of (or larger than) $k_B T_m$, where $k_B$ is the Boltzmann constant, and $T_m$ is the crystallization temperature (room temperature in our case). Therefore one naturally expects the positional correlation length $\xi_{tr}$ to be on the order of the molecular scale $q_0^{-1}$. Since the experimental data [@ZK15] suggests $\xi_{tr} \gg q_0^{-1}$, we do believe to the weak first order crystallization phase transition for the hexatics, and therefore the weak crystallization theory, see Ref. [@KLM93], is a natural tool to describe positional fluctuations both, in the hexatic and in the smectic A phases, near the phase transition. Within this theory the characteristic value of the short scale density mass $\delta\rho$ (positional order parameter) is much smaller than the average uniform mass density. Then, according to the Landau approach, the energy, associated with fluctuations of $\delta\rho$ can be expanded into a series over $\delta\rho$. The expansion, known as the Landau functional, determines correlation functions of $\delta\rho$, particularly, the structure factor (\[structure\]).
The Landau functional contains some terms of the expansion in $\delta\rho$, starting from the second order contribution, which can be written as $$\begin{aligned}
{\cal F}_{(2)}=\int \frac{d^3q}{(2\pi)^3}
\left[ \frac{a}{2} |\delta\rho(\bm q)|^2
+\frac{b}{2} (q_\perp-q_0)^2
|\delta\rho(\bm q)|^2\right].
\label{freeweak2}
\end{aligned}$$ Here $\delta\rho(\bm q)$ is Fourier transform of $\delta\rho(\bm r)$ and $a$, $b$ are some coefficients. The coefficient $a$ diminishes as the temperature decreases, as usually in the Landau approach, we assume $a\propto(T-T_\star)$ where $T_\star$ is the bare crystallization temperature, i.e., the mean field stability limit of the hexatic state. The coefficient at $|\delta\rho(\bm q)|^2$ in Eq. (\[freeweak2\]) has a minimum at $q_\perp=q_0$. It corresponds to the maximum of the structure function (\[structure\]) at $q_\perp=q_0$. There is no dependence on $q_z$ in the expression (\[freeweak2\]) in accordance with the discussed above experimental observations [@ZK15] showing no dependence on $q_z$ of the structure function.
The next relevant term in the Landau expansion is of the fourth order over $\delta\rho$, it can be written as $$\begin{aligned}
{\cal F}_{(4)}
=\int \frac{d^3 q_1\, d^3 q_2 d^3 q_3 d^3 q_4}{(2\pi)^9}
\delta(\bm q_1+\bm q_2+\bm q_3 +\bm q_4)
\nonumber \\
\frac{\lambda l}{24}
\delta\rho(\bm q_1)\delta\rho(\bm q_2)
\delta\rho(\bm q_3 )\delta\rho(\bm q_4).
\label{freeweak}
\end{aligned}$$ The contribution (\[freeweak\]) describes the self-interaction of the short-range density fluctuations. Generally, the factor $\lambda$ depends on the wave vectors $\bm q_1 \div \bm q_4$. Below, for the sake of simplicity (but keeping an essential physics), we restrict ourselves to the case $\lambda=\mathrm{const}$. One more note of caution is in order here. Since the positional order parameter $\delta \rho$ is a scalar quantity, a third order over $\delta \rho$ contribution into the Landau functional is not forbidden by the symmetry. We neglect this third order term in what follows based on the experimentally confirmed large value of the positional correlation length $\xi_{tr} $. Indeed, if the third-order term is not small then the crystallization of the hexatic smectic phase would be a strong first order transition and $\xi_{tr}$ would be on the order of the molecular size.
The average value of the short-range density modulation $\delta\rho$ is zero in both, smectic A and hexatic, phases. By other words, there is no long-range positional order in the smectic layers. Let us stress that we are interested in the mass density with the wave vectors near the circle $q_\perp = q_0$ (besides the standard for all smectics quasi-Bragg peaks at $q_z=2\pi/l,q_\perp=0$, reflecting the density modulation in the direction perpendicular to the layers). Near the smectic A – hexatic phase transition (both, above and below the transition temperature $T_h$) the short-range density modulation $\delta\rho$ is a strongly fluctuating quantity, a manifestation of the fact is its large correlation length $\xi_{tr}$. Therefore to find, say, the structure function (\[structure\]) one has to calculate fluctuation corrections to the bare value determined by the second-order term (\[freeweak2\]). A theoretical framework of the corresponding analysis can be found in our survey [@KLM93], here we adopt the method for the smectics near the smectic A – hexatic phase transition.
The interaction between the orientational order parameter and the density fluctuations is described by a crossed term in the Landau functional depending on the both fields, $\delta\rho$ and $\Psi$. The main interaction term in the Landau functional can be written as $${\cal F}_\mathrm{int}=
-\frac{1}{2q_0^6}\mathrm{Re}
\int dV\ \Psi [(\partial_x-i\partial_y)^3 \delta\rho]^2,
\label{inter}$$ where, as above, distortions of the smectic layers are neglected. Note that the transformation law (\[rotate\]) explicitly demonstrates rotational invariance of the interaction term (\[inter\]). The interaction produces, particularly, a hexagonal angular dependence of the density correlations in the hexatic phase, where the mean value of $\Psi$ is non-zero. Note, that fixing the coefficient in Eq. (\[inter\]) (that is equal to unity) we define the normalization of the orientational order parameter $\Psi$.
Fluctuation effects {#sec:fluct}
===================
To find correlation functions of the order parameters $\Psi$ and $\delta\rho$, one has to take into account a self-interaction of the fields and their coupling. The self-interaction of the hexatic order parameter $\Psi$ leads to a universal scaling behavior characterized by a set of critical exponents [@CL00; @WK74; @ST87; @PP79]. The self-interaction of the parameter $\delta\rho$ produces effects that can be examined in the framework of the weak crystallization theory, see Ref. [@KLM93]. In addition, we should involve into consideration the coupling between the order parameters $\Psi$ and $\delta\rho$ that will be examined in the framework of the perturbation theory. The applicability condition of this approach is weakness of the coupling.
An analysis of the fluctuational effects shows that in the smectic A phase the structure function (\[structure\]) can be written as $$S(\bm q)= \frac{T}{\Delta + b(q_\perp-q_0)^2},
\label{gap}$$ where $b$ is the same parameter as in the Landau functional (\[freeweak2\]). Below, the parameter $\Delta$ will be termed gap. As it follows from the expression (\[gap\]), the positional (translational) correlation length is $\xi_{tr} = \sqrt{b/\Delta}$. The expression (\[gap\]) is correct in the smectic A phase, in the hexatic phase the structure function acquires a hexagonal angular dependence. However, in a vicinity of the the smectic A – hexatic phase transition, we are interested in this work, the angular dependence is weak and therefore it will be ignored in our analysis. Thus, we use the expression (\[gap\]) both, for the smectic A and the hexatic phases. Use of the expression (\[gap\]) is justified by the inequality $q_0 \xi_{tr}\gg1$, that is the main applicability condition for our theory.
The gap $\Delta$ possesses an essential temperature dependence. The bare value of the gap is $a$, see Eq. (\[freeweak2\]), the value is renormalized due to self-interaction of the density fluctuations. The main fluctuation contribution to $\Delta$ is determined by the so-called one-loop term depicted by the following Feynman diagram $$\bullet\quad \quad \feyn{fl flu}
\label{feyn1}$$
where the solid line represents the pair correlation function (\[structure\]) and the bullet represents $\lambda$, see Eq. (\[freeweak\]). Adding this fluctuation contribution to the bare value of $\Delta$, that is $a$, one finds the self-consistent equation for the gap $$\Delta = a + \frac{Tq_0 \lambda}{4\sqrt{b\Delta}},
\label{gap2}$$ in a close analogy to the weak crystallization theory of three-dimensional liquids, see details in the survey [@KLM93]. Note that the equation (\[gap2\]) has a solution for both, positive and negative $a$. That implies that the hexatic phase remains metastable even below the crystallization temperature. Note parenthetically that one has to keep in mind this fact discussing equilibration time in the hexatic smectics.
Next, we should take into account the interaction between the orientational order parameter $\Psi$ and the position order parameter $\delta\rho$ (short-range density fluctuations), determined by the term (\[inter\]) in the Landau functional. The interaction between the orientation and the position order parameters modifies correlation functions of both fluctuating quantities, $\Psi$ and $\delta\rho$. We examine the effect using the perturbation theory, that is we take into account only first corrections to the correlation functions, related to the interaction (\[inter\]). The applicability condition of the perturbative approach will be formulated later on in this section.
Let us examine contributions to the gap $\Delta$ caused by the interaction. First contributions to $\Delta$ related to the interaction term (\[inter\]) can be represented by the following Feynman diagrams $$\feyn{g g \quad \quad fl flu}
\label{feyn2}$$ $$\feyn{ f g1 g2 f }
\label{feyn3}$$ where the wavy lines correspond to the pair correlation function of the hexatic order parameter $$F(\bm r_1,\bm r_2)=
\langle \Psi (\bm r_1)\Psi ^\star(\bm r_2)\rangle.
\label{pair}$$ According to the second order phase transitions theory [@CL00; @WK74; @ST87; @PP79], the Fourier transform of the correlation function (\[pair\]) has the following self-similar form $$F(\bm q) = \frac{1}{q^{2 - \eta }} f(q\xi _h),
\label{pair1}$$ where $\eta$ is the so-called anomalous critical exponent. For the superfluid universality class $\eta \approx 0.02$, see Refs. [@ST87; @AN91; @PV02]. The scaling function $f$ in Eq. (\[pair1\]) provides that $F$ depends solely on $|T - T_h|$ ($T_h$ is the smectic A – hexatic transition temperature) for $q \xi _h \ll 1$ and solely on $q$ in the opposite limit $q \xi _h \gg 1$ , see Refs. [@CL00; @WK74; @ST87; @PP79].
In the smectic A phase the contribution (\[feyn2\]) is zero. Indeed, the closed loop in the diagram (\[feyn2\]) corresponds to the single point average $\langle [(\partial_x-i\partial_y)^3 \delta\rho]^2\rangle$, see Eq. (\[inter\]), the average is zero due to isotropy of the smectic A layers. In the vicinity of the phase transition, we are considering in this paper, the corresponding contribution is also negligible in the hexatic phase. Thus, one should take into account solely the term (\[feyn3\]) that gives the following contribution to the gap $\Delta$ $$\delta\Delta=-\frac{1}{2T} \int \frac{d^3q}{(2\pi)^3}
F(\bm q) S(\bm k +\bm q),
\label{gap3}$$ where $\bm k$ is the wave vector of the density fluctuation. Here the wave vector $k$ lies near the ring (cylinder) $k_\perp=q_0$ whereas the wave vector $q$ can be estimated as $q\sim \xi_h^{-1}$. Thus, the inequalities $k\gg q$ and $\Delta \ll b q^2$ are valid. The first inequality is related to the condition that $\xi_h q_0 \gg 1$, which is valid near the phase transition point, and the second one is equivalent to the condition (\[corrl\]). Using the inequalities, one obtains $$\begin{aligned}
\delta\Delta =
-\frac{1}{8\pi^2\sqrt{b\Delta}} \int dq_z dq_\perp F(\bm q),
\label{delta}
\end{aligned}$$ where at the derivation of (\[delta\]) we used the expression (\[gap\]). Thus we end up with the equation $$\Delta = a + \frac{Tq_0 \lambda}{4\sqrt{b\Delta}} +\delta\Delta,
\label{gap4}$$ instead of Eq. (\[gap2\]). The equation (\[gap4\]) determines the temperature dependence of the gap $\Delta$ in the critical region.
In the region $q\xi _h \gg 1$ the pair correlation function $F(\bm q)\propto q^{\eta-2}$ [@CL00; @WK74; @ST87; @PP79]. Therefore there is an ultraviolet contribution to the integral $\int dq_z dq_\perp F$ to be included into a redefinition of the factor $\lambda$, entering Eq. (\[gap4\]). Besides, there is negative critical contribution to the integral $\int dq_z dq_\perp F$ that behaves $\propto |T-T_h|^{\nu\eta}$, as follows from Eq. (\[pair1\]). We are interested just in this term that produces a singular contribution to the gap $\Delta$. Taking a derivative of the equation (\[gap4\]) and keeping in mind that $\Delta$ remains finite at the transition point, we find $$\frac{\partial}{\partial T}\Delta
\propto |T-T_h|^{\nu\eta-1}.
\label{gap7}$$ The singularity is integrable due to $\eta,\nu>0$. Therefore the gap $\Delta$ remains finite at the transition point, indeed, that justifies our approach.
Now we can formulate the applicability condition of our perturbation approach. For the purpose we should compare the contribution depicted by the one-loop diagram (\[feyn3\]) with contributions of higher order, determined by many-loop diagrams. An example of such diagram is illustrated by the figure $$\feyn{ f g1 g2 f g4 g3 f}
\label{feyn33}$$
where a two-loop diagram is presented. Straightforward estimation shows that one can neglect the diagram (\[feyn33\]) in comparison with the diagram (\[feyn3\]) if $$\delta\Delta \ll \Delta,
\label{criterion}$$ where $\delta\Delta$ is determined by Eq. (\[delta\]). If the inequality (\[criterion\]) is not satisfied, we may not restrict ourselves to the one-loop approximation and summation of an infinite series of terms is needed. Such analysis is beyond the scope of this work.
Our theory yields additional contributions to the heat capacity related to the positional degree of freedom $\delta \rho$. The leading contribution is associated with the $T$-dependence of the coefficient $a$ in Eq. (\[freeweak2\]). Namely, we find for the $a$-dependent part of the free energy $$\frac{\partial F_a}{\partial T}
= \frac{V}{2} \frac{\partial a}{\partial T}
\langle (\delta\rho)^2 \rangle
=\frac{\partial a}{\partial T}
\frac{TVq_0}{4 l\sqrt{b\Delta}} ,
\label{gap8}$$ where the thickness $l$ of the smectic layer appears due to integration over $q_z$ within the first Brillouin zone. Taking the $T$-derivative of the expression (\[gap8\]), we find the following critical contribution to the heat capacity $$-T\frac{\partial^2 F_a}{\partial T^2}
=\frac{V}{8}\frac{\partial a}{\partial T}
\frac{T^2 q_0}{l b^{1/2} \Delta^{3/2}}
\frac{\partial\Delta}{\partial T}.
\label{gap9}$$ In accordance with Eq. (\[gap7\]), the contribution (\[gap9\]) diverges near the phase transition with the exponent $1-\nu\eta$, close to unity.
The additional contribution to the heat capacity (\[gap9\]) is crucial for our approach. The main message of our work (see details in the next section) is to claim that a sum of two critical contributions to the heat capacity, with the “small” exponent $\alpha$ and with the “large” exponent $1-\nu\eta$, enables one to describe quantitatively the known experimental data.
Comparison with experiment {#sec:exp}
==========================
Unfortunately in the critical region for the smectic $A$ - hexatic phase transition, the X-ray scattering experimental data are very scarce (only a few experimental points). Certainly in such a situation it is useless to fit the theory to estimate at least 3 unknown parameters entering the theory equation for the gap. On the contrary, calorimetric data in the critical region are quite informative. Thus in this work we focus on the measurable specific heat behavior and take a pragmatic approach to understand calorimetric features of the smectic $A$ – hexatic smectic phase transition, and provide relationships between different physical properties.
Let us turn to comparing our theory with the heat capacity experimental data. Known from the literature experimental calorimetric data for 65OBC liquid crystalline material [@VL83; @HV83; @PN85; @GS91; @SG92; @GS93; @HK97; @RD05; @MP13] manifest that the smectic A – hexatic transition is a second order phase transition with a strong, nearly symmetric singularity of the heat capacity. The value of the critical exponent $\alpha$ is extracted in the works from fitting the experimental data to a single power law dependence in the temperature range $|T-T_h| > 0.1 K$. For 65OBC the effective critical exponent is $\alpha \approx 0.64$, and the critical amplitude ratio is $A^+/A^- \approx 0.84$. Note that the effective critical exponent $\alpha$ depends on the material. Results presented in the work [@PN85] give the values ranging from $0.48$ to $0.67$ for eight different substances. Such dispersion signals about at least some problems of the approach.
Our theory states that the singular part of the heat capacity in the intermediate critical region is a sum of the two terms: the first term with the standard small critical exponent $\alpha$ characteristic of the theory for the two-component order parameter universality class and the second term with the exponent $1-\nu\eta\approx 0.985$, see Eq. (\[gap9\]). The second contribution originates from the interaction of the order parameters $\Psi$ and $\delta\rho$. Thus to fit the experimental data we utilize the following expression for the heat capacity $$\begin{aligned}
C=\frac{p_1}{|x|^{-0.013}}
+\frac{p_3}{|x|}+p_5,
\quad \mathrm{if}\ x<0,
\label{new} \\
C=\frac{p_2}{x^{-0.013}}
+\frac{p_4}{x}+p_5,
\quad \mathrm{if}\ x>0,
\nonumber
\end{aligned}$$ where $x=(T-T_h)/T_h$ is the reduced temperature, and we borrowed the value of the exponent $\alpha = - 0.013$ from the standard data for the two-component order parameter in three dimensions presented in [@ST87; @AN91; @PV02]. The obtained values giving the best fitting are $T_h=341.11 K$, $p_1=-48.09599$, $p_2=-48.19495$, $p_3=0.0008$, $p_4=0.00064$ and $p_5=91.60242$. Although the found dimensionless parameters $p_3$ and $p_4$ are much smaller than $p_1$ and $p_2$, the former ones are meaningful and anyway larger than the numeric accuracy of the fitting procedure. Physically, the small values of the parameters $p_3$ and $p_4$ indicate (consistently with our perturbation theory assumptions) that the coupling term (\[inter\]) is small.
![Calorimetric experimental data [@RD05] for 65OBC (open circles) versus our calculation (solid line)[]{data-label="f2"}](C65close.eps){height="2.5in"}
Figure \[f2\] confronts results of the computations of the specific heat in our model, see Eqs. (\[gap9\]) and (\[new\]), and the experimental data [@RD05] (which almost coincide, up to a regular shift of the sample-dependent transition temperature, with the data presented in Refs. [@VL83] and [@HK97]). We see a reasonable agreement of the experimental data and our theory (valid in the temperature range about $0.6\, -\, 1\, K$ around $T_h$). The exponent $\alpha $ found in the works [@VL83], [@HK97], [@RD05] is based on the single power-law fitting of the experimental data in the temperature range $0.005 < |T-T_h|/T_h < 1.5$. In this more broad temperature region we expect a sort of crossover behavior which can be fitted by a single (but not universal!) power law. Understanding all its limitations, we are confident that our model captures the essential features of the smectic A – hexatic phase transition in the intermediate critical region (with two strongly fluctuating and coupled order parameters).
Conclusion {#sec:con}
==========
The main advance of our work is establishing for the first time the model enabling to match quantitatively theory and calorimetric data for the smectic A – hexatic phase transition, explaining, particularly, the effective “large” value of the heat capacity critical exponent. To avoid a confusion, let us stress that we are not fighting with the well established and widely accepted conception of the second order phase transition universality. There are no doubts that sufficiently close to the smectic A – hexatic transition temperature, the universal behavior of the heat capacity has to be observed. However, we claim that the temperature region where the standard universality holds in a narrow vicinity of the transition point. Outside the region the critical behavior does not correspond to the standard universality, its description needs some additional ingredients, presented in our paper.
Our model involves an interaction between the strongly fluctuating orientational (hexatic) and positional degrees of freedom, $\Psi$ and $\delta\rho$, that offers a new non-trivial scenario for the temperature dependencies of the specific heat and X-ray scattering patterns near the smectic A – hexatic phase transition. Our approach allows one to reconcile well established phase transition universality conception and the calorimetric experimental data [@VL83; @HV83; @PN85; @GS91; @SG92; @GS93; @HK97; @RD05; @MP13] for the smectic A – hexatic smectic phase transition. Thus, we claim, that we solved the old riddle related to the “large” value of the heat capacity exponent for the phase transition. Our theoretical model lays firm foundations for the further studies for biologists, experimental and theoretical physicists.
Our results, reported in the paper, concern the critical region (region of strong fluctuations of the hexatic order parameter $\Psi$) near the transition point. However, our model can be applied to the hexatic phase (outside the critical region) as well. In the region one has to take into account fluctuations of both, $\Psi$ and $\delta\rho$. Certain features of the fluctuations in the hexatic phase are different from those in the critical region due to non-zero mean value of the order parameter $\Psi$. A description of the region is a subject for future investigations.
One should keep in mind that our scheme implies weakness of the interaction between the orientation and the position degrees of freedom. That is why we restrict ourselves to first corrections over the interaction. An intriguing question about the system behavior when the coupling is not small is beyond the scope of our consideration in this publication, it will be analyzed elsewhere.
Our work was funded by Russian Science Foundation (grant 14-12-00475). We acknowledge stimulating discussions with B.I.Ostrovskii, I.A.Vartanyants, and I.A.Zaluzhnyy, inspired this work. This paper has benefited also from exchanges with many colleagues. We thank in particular I.Kolokolov and L.Schur.
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|
---
author:
- 'F. De Paolis, V.G. Gurzadyan, G. Ingrosso, Ph. Jetzer, A.A. Nucita, A. Qadir, D. Vetrugno, A.L. Kashin, H.G. Khachatryan,'
- 'S. Mirzoyan'
date: 'Submitted: XXX; Accepted: XXX'
title: Possible detection of the M31 rotation in WMAP data
---
[ Data on the cosmic microwave background (CMB) radiation by the Wilkinson Microwave Anisotropy Probe (WMAP) had a profound impact on the understanding of a variety of physical processes in the early phases of the Universe and on the estimation of the cosmological parameters. ]{} [Here, the 7-year WMAP data are used to trace the disk and the halo of the nearby giant spiral galaxy M31. ]{} [We analyzed the temperature excess in three WMAP bands (W, V, and Q) by dividing the region of the sky around M31 into several concentric circular areas. We studied the robustness of the detected temperature excess by considering 500 random control fields in the real WMAP maps and simulating 500 sky maps from the best-fitted cosmological parameters. By comparing the obtained temperature contrast profiles with the real ones towards the M31 galaxy, we find that the temperature asymmetry in the M31 disk is fairly robust, while the effect in the halo is weaker. ]{} [An asymmetry in the mean microwave temperature in the M31 disk along the direction of the M31 rotation is observed with a temperature contrast up to $\simeq 130~\mu$K/pixel. We also find a temperature asymmetry in the M31 halo, which is much weaker than for the disk, up to a galactocentric distance of about $10\degr$ ($\simeq 120$ kpc) with a peak temperature contrast of about $40~\mu$K/pixel. ]{} [ Although the confidence level of the signal is not high, if estimated purely statistically, which could be expected due to the weakness of the effect, the geometrical structure of the temperature asymmetry points towards a definite effect modulated by the rotation of the M31 halo. This result might open a new way to probe these relatively less studied galactic objects using high-accuracy CMB measurements, such as those with the Planck satellite or planned balloon-based experiments, which could prove or disprove our conclusions. ]{}
Introduction
============
Galaxy rotation, in particular for the Andromeda galaxy (M31) has been well studied especially in the optical, IR, and radio bands, and it gives important information on the mass distribution not only in galactic disks but also in their halos [@binney]. On the other hand, since they are not directly observable, but their presence is deduced from their effect on galactic dynamics, galactic halos are relatively less studied structures of galaxies. Various populations, such as globular clusters, RR Lyrae, subdwarfs, and other types of stars, have been used to trace the halo of the Galaxy, its vertical structure, and its rotation speed [@Kinman]. Nevertheless, there are still many ambiguities not only in the main halo constituents, but also in the basic properties such as, in particular, in rotation.
The degree to which galactic halos rotate with respect to the disks is difficult to investigate; actually, as stated in the most recent study of M31 [@courteau], testing for the rotation of M31’s halo is still beyond our reach. Naturally, the importance of understanding the galactic halos is closely related to the nature and distribution of the dark matter, which is relevant for the formation and dynamics of galaxies. In this respect, the methodology adopted in the present paper of using WMAP data to probe both the disk and the halo of M31, even if with the limitation of the presently available data, may suggest a novel way of approaching this problem.
The 7-year WMAP analysis
========================
In our analysis we use the seven-year WMAP data [@J] in the three bands W (94 GHz), V (61 GHz), and Q (41 GHz). Using three WMAP bands is important in revealing the possible contribution of the Galactic foregrounds since dust, free-free, and synchrotron emission contributes differently in each band. Here we remind the reader that the band least contaminated by the synchrotron radiation of the Galaxy is the W-band, which also has the highest angular resolution. The CMB map’s general structure in the W-band in the region of M31, with the marked $4\degr$ radius circle (although our analysis extends farther out), is shown in Fig. \[fig1\] (left panel). In our analysis, we also used the maps provided by the WMAP Collaboration with the Galactic disk contribution modeled and removed [@gold]. It is always specified in the text when we considered these data. To reveal the different contributions by the M31 disk and halo, the region of the sky around the M31 galaxy was divided into several concentric circular areas as shown in Fig. \[fig1\] (right panel). In the optical band the total extent of the M31 galaxy along the major axis is slightly more than about $3\degr$ and along the minor axis is about $1\degr$. Radio observations have shown that the M31 HI disk is more extended with respect to the stellar disk [@Chemin; @Corbelli], with a major axis sizes of about $5.6\degr$ and a minor axis size of about $1.2\degr$. In this paper, the adopted M1 and M2 disk regions (Fig. \[fig1\], right panel) have major axis size of $8\degr$ and minor axis size of $1
\degr$; this allows us to retain the warped part of the M31 disk in the M1 and M2 regions. Moreover, we have checked that it is irrelevant, as far as our analysis is concerned, to extend the M31 minor axis to $1.2\degr$. The mean temperature excess per pixel $T_m$, in $\mu$K/pixel, in each region was obtained in each band and is shown in Table 1 with the $1\sigma$ error [^1], along with the number of pixels in each area. For convenience, Table 1 gives the temperature excess in each M31 region up to $8\degr$, even if our analysis was extended to the region around the M31 disk with concentric annuli with radii up to $20\degr$.
R, deg, kpc Region N, pix W, $T_m\pm \sigma$ V, $T_m\pm\sigma$ Q, $T_m\pm\sigma$
----------------- -------------- -------- -------------------- ------------------- -------------------
1.65, 21.4 N1 + M1 + S1 324 $63.1\pm 5.6$ $67.2\pm 5.4$ $90.0\pm 4.2$
N2 + M2 + S2 321 $20.3\pm 4.7$ $ 17.3\pm 4.3$ $ 37.0\pm 3.3$
N1 + S1 205 45.5$~\pm$ 5.7 38.0$~\pm $ 5.3 64.1$~\pm$ 4.0
N2 + S2 205 33.8$~\pm$ 5.9 34.3$~\pm$ 5.3 41.8$~\pm$ 4.1
M1 119 $121.4\pm 19.6$ $117.6\pm 10.0$ $134.3\pm 7.4$
M2 116 $-7.7\pm 7.4$ $-12.7\pm 6.8$ $28.4\pm 5.5$
2.40, 31.1 N1 + M1 + S1 670 $43.7\pm 3.6$ $43.5\pm 3.4$ $66.0\pm 2.8$
N2 + M2 + S2 664 $21.0\pm 3.6$ $19.1\pm 3.3$ $35.6\pm 2.7$
N1 + S1 506 41.0$~\pm$ 3.5 32.2$~\pm$ 3.1 55.6$~\pm$ 2.6
N2 + S2 504 24.5$~\pm$ 4.3 23.6$~\pm$ 3.9 34.8$~\pm$ 3.3
M1 164 $73.0\pm 9.4$ $78.3\pm 9.3$ $98.2\pm 7.4$
M2 160 $11.2\pm 6.3$ $5.8\pm 6.4$ $38.1\pm 4.8$
3.20, 41.5 N1 + M1 + S1 1176 $36.5\pm 2.7$ $38.9\pm 2.4$ $59.6\pm 2.1$
N2 + M2 + S2 1166 $16.0\pm 2.7 $ $11.3\pm 2.5$ $30.3\pm 2.2$
N1 + S1 980 35.3$~\pm$ 2.7 34.0$~\pm$ 2.3 53.9$~\pm$ 2.1
N2 + S2 976 16.4$~\pm$ 3.0 11.5$~\pm$ 2.8 27.7$~\pm$ 2.4
M1 196 $63.8\pm 8.4$ $63.2\pm 8.3$ $88.1\pm 6.6$
M2 190 $12.5\pm 5.8$ $10.3\pm 5.8$ $44.0\pm 4.4$
4.00, 51.9 N1 + M1 + S1 1818 $37.4\pm 2.2$ $39.6\pm 2.0$ $56.7\pm 1.7$
N2 + M2 + S2 1808 $1.7\pm 2.3$ $-2.4\pm 2.1$ $16.9\pm 1.9$
N1 + S1 1610 36.8$~\pm$ 2.2 36.9$~\pm$ 1.9 53.3$~\pm$ 1.7
N2 + S2 1609 $-$ 0.4$\pm$ 2.5 $-$ 3.8$~\pm$ 2.2 13.5$~\pm$ 1.9
M1 208 $64.7.0\pm8.1$ $60.8\pm 7.9$ $83.6\pm 6.4$
M2 200 $12.5\pm 5.6$ $9.4\pm 5.6$ $43.6\pm 4.3$
6.00, 77.8 N1 + S1 3748 29.7$~\pm$ 1.5 27.0$~\pm$ 1.4 44.0$~\pm$ 1.2
N2 + S2 3749 11.3$~\pm$ 1.7 7.1$~\pm$ 1.5 25.8$~\pm$ 1.3
8.00, 103.8 N1 + S1 6606 34.3$~\pm$ 1.2 34.7$~\pm$ 1.1 51.5$~\pm$ 4.0
N2 + S2 6600 19.2$~\pm$ 1.3 15.0$~\pm$ 1.2 38.7$~\pm$ 1.0
Results for the M31 disk
========================
For the M31 disk, our analysis shows that each M1 region is always hotter than the corresponding M2 region, as can be seen from Table 1. Indeed we find a temperature excess contrast (i.e. the difference between the temperature excesses per pixel) between the M1 and M2 regions in all three WMAP bands that turns out to be about $129\pm 21~\mu$K/pixel within 21.4 kpc (in the W band) and then slightly decreases (but remains as large as about $41\pm
10~\mu$K/pixel at about 50 kpc). This effect seems to come from the rotation induced Doppler shift of the gas and dust emission from the M31 disk - indeed, the hotter (M1) region corresponds to the side of the M31 disk that rotates towards us. [^2] If one compares what WMAP data show towards the M1 and M2 regions with the maps of the M31 thick HI disk obtained at 21 cm [@Chemin; @Corbelli] one sees a remarkable superposition of the hot (M1) and cold (M2) regions in both observations.
Even if the temperature asymmetry between the M1 and the M2 regions looks significant, we have to check whether it might be due to a random fluctuation of the CMB signal. It is indeed well known that the CMB sky map has a “patchy” structure characterized by the presence of many hot and cold spots with temperature excesses up to several tents of $\mu$K/pixel on angular degree scales. We therefore considered (Fig. \[f2\] and also the online material) 500 control fields and 500 simulated sky maps (from the best-fitted cosmological parameters as provided in the WMAP web site) [^3] - and evaluated the temperature contrast with the same geometry as was used towards M31. We also give (red curve) the M31 temperature contrast profile in the M31 disk. Due to our chosen geometry, each curve is given up to $4\degr$. As one can see, the contrast temperature profile for the M31 disk is always a nicely smooth curve that is close to the $2\sigma$ curve in the intermediate region of about $1.5-2\degr$. Both the control field and the CMB simulation analyses show that there is a probability of less than about $4\%$ that the temperature asymmetry revealed comes from a random fluctuation of the CMB signal. Actually, if one takes the direction of rotation of the M31 disk into account, such a probability reduces (by using the theorem of the composite probability) by a factor of two. Finally, we mention that we have found that the temperature excess contrast of the two M31 disk regions obtained by dividing the M1+M2 region with respect to the north-west/south-east symmetry axis (the M31 disk axis) turns out to be $0.008\pm 0.012$ mK, which seems to further confirm that the temperature contrast between the M1 and M2 regions is not due to a random fluctuation in the CMB signal.
Results for the M31 halo
========================
The next step was to enlarge our analysis to the region around the M31 disk by considering concentric circular regions of increasing galactocentric radii (see also Fig. 1, right panel). We estimated the difference of the temperature excess in the region N1+S1 in the three WMAP bands with respect to that in the region N2+S2. A temperature contrast between the region N1+S1 with respect to N2+S2 shows up (see Table 1), and the N1+S1 region turns out to always be hotter than the N2+S2 region. The detected effect resembles the one towards the M31 disk, although with less temperature asymmetry. In all three bands, the maximum temperature contrast reaches a maximum at a galactocentric distance of about $4\degr$ and then decreases slightly. It is apparent from the size of the considered regions that a contamination of the M31 disk in the regions N1, N2, S1, and S2 can be completely excluded, and also the Galactic plane emission cannot account for the observed temperature asymmetry since it eventually would make a larger contribution towards the upper regions of M31 (while the opposite is observed in the data). As for the M31 disk, the temperature asymmetry in the M31 halo is indicative of a Doppler shift modulated effect possibly induced by the rotation of the M31 halo.
Also in this case we need to check the robustness of our results; that is, we have to estimate the probability that the temperature asymmetry in the M31 halo is due to a random fluctuation of the CMB signal. In Fig. \[f3\] (see also the online material) we have considered 500 control fields and 500 simulated sky maps (from the best-fitted cosmological parameters as provided on the WMAP website). As one can see, in all three bands, the contrast temperature profile of the M31 halo is close to the $1\sigma$ curve up to about $10\degr$ and goes slightly beyond it at about $50-60$ kpc where the halo effect is maximum. This means that there is less than $30\%$ probability that the temperature contrast we see towards the M31 halo is due to a random fluctuation of the CMB signal.
We also point out that we have verified that the temperature asymmetry towards the M31 halo vanishes if the adopted geometry is rigidly rotated of an angle larger than about $10\degr$ with respect to the M31 major axis, thus giving a further indication that the halo temperature contrast effect might be genuine and not simply a random fluctuation of the CMB.
We also point out that the use of three WMAP bands is useful for revealing the role of the contribution to the Galactic foregrounds since each emission mechanism contributes differently in each band. That the temperature contrast seems present in all three bands and is more or less the same in each band up to about $10\degr-11\degr$ indicates that the foregrounds are far weaker than the effect. This size corresponds to the typical size inferred for the dark matter halos around massive galaxies and might open the possibility of a new way of studying these systems, galactic disks, and halos, by using the microwave band. In any case, a careful analysis of the Planck data that will be released shortly should allow either proving or disproving our results.
Discussion and conclusions
==========================
We have shown that a temperature asymmetry in all WMAP bands may exist both in the M31 disk and halo in the direction of the M31 spin. For the M31 disk, the effect is fairly clear, and there is a probability below about $2\%$ that it is a random fluctuation of the CMB signal. If real, the detected temperature excess asymmetry should be due to the foreground emission of the M31 disk modulated by the Doppler shift of the disk spin. That the present study is really timely is strengthened by considering that the M31 galaxy is already detected by the Planck observatory [@planckm31] [^4], whereas it did not appear in the WMAP list. These are all reasons to expect that the particular effect we discuss here can be studied more accurately with Planck data.
As for the M31 halo, we have shown that, although less evident than for the M31 disk, there is some evidence of a temperature asymmetry between the N1+S1 and the N2+S2 regions that resembles that of a Doppler shift effect induced by the M31 halo rotation. We have shown in the previous section that there is less than about $30\%$ probability that the detected temperature asymmetry at a galactocentric distance $\sim 50$ kpc comes from a random fluctuation of the CMB signal. [^5]
If one assumes that this temperature asymmetry in the M31 halo relies in the M31 itself and is related to the M31 halo rotation, one could speculate about the origin of this effect. In general, four possibilities may be considered: (a) free-free emission, (b) synchrotron emission, (c) Sunyaev-Zel’dovich (SZ) effect, and (d) cold gas clouds populating the M31 halo [^6]. To work, the first three effects, assume the presence of a rather hot plasma in the halo of M31. Although this hot plasma has not been detected yet, one can assume that a certain amount of this plasma can populate the M31 halo (spiral galaxies are believed to have much less hot gas than ellipticals) and may rotate with a certain speed. Free-free emission arises from electron-ion scattering while synchrotron emission comes mostly from the acceleration of cosmic-ray electrons in magnetic fields. Both effects give rise to a thermal emission with a rather steep dependence on the frequency [@bennett] that therefore should give a rather different temperature contrast in the three WMAP bands. The absence of this effect indicates that the contribution from possibilities (a) and (b) should be negligible. And for (c), even for typical galaxy clusters with diffuse gas much hotter than that possibly expected in the M31 halo, the rotational scattering effect would produce a temperature asymmetry of at most a few $\mu$K/pixel, depending on the rotational velocity and the inclination angle of the rotation axis [@cooray]. Actually, a possible temperature asymmetry in the CMB data towards the M31 halo as a consequence of the existence of a population of cold gas clouds in its halo was predicted in [@paperdijqr95] - possibility (d). Indeed, if the halo of the M31 galaxy contains cold gas clouds, we expect them to rotate like the M31 disk (even if, perhaps, more slowly), and thus there should be a Doppler shift inducing a temperature anisotropy $\Delta T$ between one side of the M31 halo and the other with respect to the rotation axis perpendicular to the disk. In the case of optically thin halo clouds, the Doppler induced temperature anisotropy would be ${\Delta T}/{T_r} \simeq
2{v}S~\bar \tau /c~$, where $v$ is the M31 rotation speed, $\bar
\tau$ the averaged cloud optical depth over the frequency range ($\nu_1 \leq \nu \leq \nu_2$) of a certain detection band, and $S$ the cloud filling factor, i.e. the ratio of filled (by clouds) to total projected surface in a given field of view. We emphasize that the fact that the temperature contrast in Fig. \[f3\] looks approximately the same in each band makes a point towards either possibility (d) or a random fluctuation of the CMB sky (but with a probability, if estimated purely statistically, of less of $30\%$ for the last possibility).
The wealth of data especially in the last decade shows that there is good evidence for the presence in the halos of spiral galaxies of gas in all gaseous phases: neutral, warm atomic, and hot X-ray emitting gas [@bregman]. Atomic gas (often identified as HVCs) is observed in the radio band (particularly at 21 cm) and through absorption lines towards field stars and quasars. The hot gas may be detected in X-rays, while searches for cold gas clouds in galactic halos are more problematic as are searches for them by the presence of a gamma-ray halo [@dixon; @depaolis99], stellar scintillations [@moniez03; @habibi], obscuration events towards the LMC [@drakecook], ortho-$H_2D^+$ line at 372 GHz [@ceccarelli], and extreme scattering events in quasar radio-flux variations [@walker] have given no clear indication of their presence.
In conclusion, we showed that our analysis based on seven-year WMAP data suggests there is a temperature excess asymmetry in the M31 disk is likely due to the M31 foreground emission modulated by the Doppler shift induced by the M31 spin. We find that there is less than $\simeq 2\%$ probability that the signal up to about 20 kpc comes from a random fluctuation in the CMB signal. For the M31 halo, we also find a temperature excess asymmetry between the N1+S1 and the N2+S2 regions along the expected spin direction, suggestive of a rotation induced Doppler shift. The effect in the M31 halo is far weaker than for the disk, as obviously expected, and more precise data are necessary before drawing any firm conclusion. In all cases, this research may open a new window into the study of galactic disks and especially the rotation of galactic halos by using the Planck satellite or planned balloon-based experiments.
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[^1]: The standard error of the mean is calculated as the standard deviation of the excess temperature distribution divided by the square root of the pixel number in each region. We have verified that, within the errors, the sigma values calculated in that way are consistent with those evaluated by using the covariance matrix obtained by a best-fitting procedure with a Gaussian to the same distribution.
[^2]: A detailed study of the frequency dependent temperature asymmetry in the CMB arising from different distributions of gas and dust in the M31 disk is left to a forthcoming paper. In any case, although some inhomogeneity in the disk structure is not excluded, there is no reason to assume that it is the sole cause.
[^3]: CMB maps were simulated by assuming $\Delta T(\hat{n}) = \Delta T_{CMB}(\hat{n})\otimes B(\hat{n}) +
N(\hat{n})$, where $\Delta T_{CMB}$ is a realization of the Gaussian CMB field, $N(\hat{n})$ is the pixel noise and $B(\hat{n})$ is the proper beam of the experiment. We made 500 realizations of the CMB sky using [synfast]{} routine of HEALPix with the best-fit power spectrum as given by the WMAP Collaboration. The maps are then convolved with WMAP beams for W, V, and Q bands, respectively. Noise realizations (simulated with $\sigma_0 = 6.549$ mK, $\sigma_0 =3.137$ mK, and $\sigma_0
=2.197$ mK for W, V, and Q-bands, respectively) are added to the beam convolved maps in the end.
[^4]: However, there is no mention of any temperature asymmetry in the M31 disk in that paper.
[^5]: We also mention that the number and the temperature profile of radio sources in CMB maps [@gurzadyan] excludes their significant contribution in the effect under study.
[^6]: We also considered the possible influence of the observed high-velocity clouds, either in the M31 or in our galaxy halos [@Westmeier; @Hulsbosch; @Morras], by removing the pixels in the direction of each cloud from the analysis. The results obtained do not change with respect to those presented here, as expected when also considering the relatively low number of pixels involved. Also the proposed ejecta by the past interaction of M33 and M31 galaxies [@Bekki] cannot play any role in our analysis since it would at most have made hotter some pixels in the S2 region (where the M31-M33 bridge is located), which is instead colder than the S1 one.
|
---
author:
- |
Ziyu Wang$^1$, Nando de Freitas$^{1,2}$\
$^1$University of Oxford\
$^2$Canadian Institute for Advanced Research\
`{ziyu.wang, nando}@cs.ox.ac.uk`
title: |
Theoretical Analysis of Bayesian Optimisation\
with Unknown Gaussian Process Hyper-Parameters
---
Conclusion
==========
Despite the rapidly growing literature on Bayesian optimisation and the proliferation of software packages that learn the kernel hyper-parameters, to the best of our knowledge, only Bull [@Bull:2011] and us have attacked the question of convergence of GP-based Bayesian optimisation with unknown hyper-parameters. Bull’s results focused on deterministic objective functions. Our new results apply to the abundant class of noisy objective functions.
|
[H]{}\_
[Groups of hierarchomorphisms of trees and related Hilbert spaces]{}
[Yurii A. Neretin]{}[partially supported by the grant NWO 047-008-009]{}
[**0.1. Hierarchomorphisms (spheromorphisms).**]{} The Bruhat–Tits tree $\T_p$ is a infinite tree such that any vertex belongs to $(p+1)$ edges. As was observed by Cartier [@Car], the groups $Aut(\T_p)$ of automorphisms of the trees $\T_p$ are analogues of real and $p$-adic groups of rank 1 (as ${\rm SL}_2(\R)$, ${\rm SL}_2(\C)$, ${\rm O}(1,n)$, ${\rm SL}_2(\Q_p)$ etc.). The representation theory of $Aut(\T_p)$ was developed in Cartier’s [@Car] and Olshansky’s [@Ols1] papers. In fact, the group $Aut(\T_p)$ is essentially simpler than the rank 1 groups on locally compact fields, but many nontrivial phenomena related to rank 1 groups survive for the group of automorphisms of Bruhat–Tits trees.
The absolute of the Bruhat–Tits tree is an analogue of the boundaries of rank 1 symmetric spaces, in particular, the absolute is an analogue of the circle. The group of hierarchomorphisms[^1] $\Spher(\T_p)$ (defined in [@Ner1]) is a tree analogue of the group ${\rm Diff} (S^1)$ of diffeomorphisms of the circle. The group $\Spher(\T_p)$ consists of homeomorphisms of the absolute of $\T_p$ that can be extended to the whole Bruhat–Tits tree except a finite subtree. It turns out to be ([@Ner1], [@Ner2]), that the representation theory of ${\rm Diff} (S^1)$ partially survives for the groups $\Spher(\T_p)$.
In fact, the group $\Spher(\T_p)$ contains the group of locally analytic diffeomorphisms of $p$-adic line (see [@Ner2]), and this partially explains the similarity of ${\rm Diff} (S^1)$ and $\Spher(\T_p)$.[^2]
The following facts $1^\circ$-$4^\circ$ are known about the groups $\Spher(\T_p)$. The phenomena $1^\circ$-$3^\circ$ are an exact reflection of the representation theory of ${\rm Diff} (S^1)$, the last phenomenon now does not have a visible real analogue.
$1^\circ$. ([@Ner1], [@Ner2]) Denote by ${\rm O}(\infty)$ the group of all orthogonal operators in a real Hilbert space $H$. Denote by ${\rm GLO}(\infty)$ the group of all invertible operators in $H$ having the form $A=B+T$, where $B\in {\rm O}(\infty)$ and $T$ has finite rank. Denote by $H_\C$ the complexification of $H$. Denote by ${\rm UO}(\infty)$ the group of all unitary operators in $H_\C$ having the form $A=B+T$, where $B\in {\rm O}(\infty)$ and $T$ has finite rank. There exist some series of embeddings $$\Spher(\T_p)\to {\rm GLO}(\infty);\qquad
\Spher(\T_p)\to {\rm UO}(\infty).$$ This allows to apply the second quantization machinery (see [@Ols3], [@Ners], [@Nerb]) for obtaining unitary representations of $\Spher(\T_p)$.
$2^\circ$. Embeddings $\Spher(\T_p)\to {\rm GLO}(\infty)$ allow to develop a theory of fractional diffusions with a Cantor set time (the Cantor set appears as the absolute of the tree). I never wrote a text on this topic, but, on the whole, the picture here is quite parallel to fractional diffusions with real time (see, [@Ner3]).
$3^\circ$.(Kapoudjian, [@Kap1], [@Kap4]) There exists a $\Z/2\Z$-central extension of $\Spher(\T_p)$.
$4^\circ$.(Kapoudjian, [@Kap3]) Consider the dyadic Bruhat–Tits tree $\T_2$. There exists a canonical action of the group $\Spher(\T_2)$ on the inductive limit of the Deligne–Mumford [@DM] moduli spaces $\lim_{n\to\infty}{\cal M}_{0,2^n}$ of $2^n$ point configurations on the Riemann sphere. This construction also has two versions over $\Bbb R$. The first variant is an action on the inductive limit of Stasheff associahedrons ([@Sta]). The second variant is an action on the inductive limit of the spaces constructed by Davis, Janiszkiewicz, Scott ([@Dav]). The last case is most interesting, since this real space has an interesting topology.
[**0.2. The purposes of this paper.**]{} This paper has two purposes. The first aim is to construct a new series of embeddings of the groups of hierarchomorphisms to the group ${\rm GLO}(\infty)$. By the Feldman-Hajek theorem (see [@Sko]), this gives constructions of unitary representations of groups of hierarchomorphisms, but we do not discuss this subject.
There exists the wide and nice theory of actions of groups on trees (see [@Ser], [@Sha1], [@Sha2], [@Kapo]). It is clear that a hierarchomorphism type extension can be constructed for any group $\Gamma$ acting on a tree (and even on an $\R$-tree), it is sufficient to allow to cut a finite collection of edges. The second purpose of this paper[^3] is to understand, is this “hierarchomorphization” of arbitrary group $\Gamma$ a reasonable object?
One example of such “hierarchomorphization” is quite known, this is the Richard Thompson group [@Tho], which firstly appeared as an counterexample in theory of discrete groups. Later it became clear, that this group is not a semipathological counterexample, but a rich and unusual object (see works of Greenberg, Ghys, Sergiescu, Penner, Freyd, Heller and others [@GhS], [@GrS], [@Pen], [@FH], [@BG] see also [@Can]), relation of the hierarchomorphisms and the Thompson group was observed by Sergiescu).
If the group $\Gamma$ is discrete, then the corresponding group of hierarchomorphisms is a discrete Thompson-like group. If the group $\Gamma$ is locally compact, then the group of hierarchomorphisms (see some examples in [@Ner2]) is an “infinite dimensional group” (or, better, “large group”) similar to the group of diffeomorphisms of the circle or diffeomorfisms of $p$-adic line.
[**0.3. The structure of the paper.**]{} Sections 1-2 contain preliminary definitions and examples.
In Section 3, we define the groups of hierarchomorphisms of tree (this definition can be adapted also for $\R$-tree, but nontrivial constructions of Sections 4–6 do not survive in this case).
In Section 4, we discuss a family of Hilbert spaces $\H(J)$, where $0<\lambda<1$, associated with a tree $J$. The space $\H(J)$ contains the (nonorthogonal) basis $e_a$ enumerated by vertices $a$ of the tree, and the inner products of the vectors $e_a$, $e_b$ are given by $$\langle e_a, e_b\rangle=
\lambda^{\{\text{distance between $a$ and $b$}\}}$$ We show that the group of hierarchomorphisms of $J$ acts in $\H(J)$ by operators of the class ${\rm GLO}(\infty)$.
In Section 5, for sufficiently large $\lambda$ we construct an operator of the ’restriction to the absolute’ in the space $\H$. In Section 6, we discuss the action of the group of hierarchomorphisms in spaces of functions (distributions) on the absolute.
The results of Sections 4–5 are ’new’ for the groups of hierarchomorphisms of the Bruhat–Tits trees. The construction of Section 6 in for Bruhat–Tits trees coincides with [@Ner1].
[**Acknowledgment.**]{} I am grateful to V.Sergiescu and C.Kapoudjian for meaningfull discussions. I thank the administration of the Erwin Schrödinger Institute (Wien) and Institute Fourier (Grenoble), where this work was done, for hospitality.
[**1. Notation and terminology**]{}
[**1.1. Symplicial trees.**]{} [*A simplicial tree*]{} $J$ is a connected graph without circuits.
By $\Vert(J)$ we denote the set of vertices of $J$.
By $\Edge(J)$ we denote the set of edges of $J$.
We say that two vertices $a,b\in\Vert(J)$ are [*adjacent*]{}, if they are connected by an edge. We denote this edge by $[a,b]$.
We assume that the sets $\Vert(J)$, $\Edge(J)$ are countable or finite. A simplicial tree is [*locally finite*]{} if any vertex $a$ belongs to finitely many edges. We admit non locally finite trees.
A [*way*]{} in $J$ is a sequence of [*distinct*]{} vertices $$\dots, a_1, a_2, a_3, \dots$$ such that $a_j$, $a_{j+1}$ are adjacent. A way can be finite, or infinite to one side, or infinite to the both sides.
For vertices $a,b$ there exists a unique way $a_0=a, a_1,\dots, a_k=b$ connecting $a$ and $b$. We say that $k$ is the [*symplicial distance*]{} between $a$ and $b$. We denote the simplicial metrics by $$d_{\s}(a,b).$$
A [*subtree*]{} $I\subset J$ is a connected subgraph in the tree $J$.
The [*boundary*]{} $\partial I$ of a subtree $I\subset J$ is the set of all $a\in \Vert(I)$ such that there exists an edge $[a,b]$ with $b\notin\Vert(I)$.
A subtree $I\subset J$ is [*right*]{}, if the number of edges $[a,b]\in\Edge(J)$ such that $a\in I$, $b\notin I$ is finite.
A subtree $I\subset J$ is a [*branch*]{} if there is a unique edge $[a,b]\in\Edge(J)$ such that $a\in \Vert(I)$, $b\not\in\Vert(I)$, see Picture 1. The vertex $a$ is called a [*root*]{} of the branch. If we delete an edge of the tree $J$, then we obtain two branches.
A subtree $I\subset J$ is a [*bush*]{} if its boundary contain only one point $a$ (a [*root*]{}) and number of edges $[a,b]\in\Edge(J)$ such that $b\not\in I$ is finite, see Picture 1.
[Lemma 1.1.]{} a) [*The intersection of a finite family of right subtrees is a right subtree.*]{}
b\) [*For a right subtree $I\subset J$, there exists a finite collection of edges $\ell_1$, …, $\ell_k\in\Edge(I)$ such that $I$ without $\ell_1$, …, $\ell_k$ is a union of bushes.*]{}
[Proof.]{} The statement a) is obvious.
The statement b). Let $a_1$, …, $a_k$ be the boundary of $I$. Let $L\subset I$ be the minimal subtree containing the vertices $a_1$, …, $a_k$. It is sufficient to delete all edges of $L$. $\square$
We say that a tree $J$ is [*perfect*]{} if any vertex of $J$ belongs to $\ge 3$ edges. Obviously, perfect trees are infinite.
[**1.2. Actions of groups on simplicial trees.**]{} A bijection $\Vert(J)\to\Vert(J)$ is an [*automorphism*]{} of a simplicial tree $J$ if the images of adjacent vertices are adjacent vertices.
An action of a group $\Gamma$ on a simplicial tree is an embedding of $\Gamma$ to its group of automorphisms.
[**1.3. Absolute.**]{} The absolute $\Abs(J)$ of a tree is the set of points of the tree on infinity. Let us give the formal definition.
We say that a [*ray*]{} is an infinite way $a_1,a_2,\dots$. We say that rays $a_1,a_2,\dots$ and $b_1,b_2,\dots$ are equivalent if there exist $k$ and a sufficiently large $N$ such that $b_j=a_{j+k}$ far all $j\ge N$.
A point of an absolute is a class of equivalent ways.
[**1.4. Metric trees.**]{} Let $J$ be a simplicial tree. Let us assign a positive number $\rho(a,b)$ to each edge $[a,b]$. Let $a$, $c$ be arbitrary vertices of $J$, let $a_0=a,a_1,\dots,a_k=c$ be the way connecting $a$ and $c$. We assume $$\rho(a,c)=\sum_{j=1}^k \rho(a_{j-1},a_{j})$$ Obviously, $\rho$ is a metric on $\Vert(J)$. We call by [*metric trees*]{} the [*countable*]{} spaces $\Vert(J)$ equipped with the metrics $\rho$.
Obviously, the edges of $J$ can be reconstructed using the metric $\rho$. Hence we prefer to think that the edges are present in a metric tree as an additional (combinatorial) structure.
[Remark.]{} We also can assume that lengths of all edges is 1, and thus a simplicial tree is a partial case of metric trees.
[Remark.]{} In literature, sometimes the term [*metric tree*]{} is used in the quite different sense (for $\R$-trees).
A metric tree $J$ is [*locally finite*]{} if it is locally finite as a simplicial tree and for any $a\in\Vert(J)$ and each $C>0$ the set of vertices $b$ satisfying $\rho(a,b)<C$ is finite.
[**1.5. Actions of groups on metric trees.**]{} Let $J$ be a metric tree. A bijection $\Vert(J)\to\Vert(J)$ is an [*automorphism*]{} of $J$ if it preserves the distance (hence it automatically preserves the sructure of simplicial tree).
An action of group $\Gamma$ on a metric tree $J$ is an embedding of $\Gamma$ to the group of automorphisms of $J$.
[**2. Examples of actions of groups on trees.**]{}
The purpose of this Section is to give a collection of examples for abstract constructions given in Sections 3-6 (all these examples are standard). For algebraic and combinatorial theory of actions of groups on trees, see [@Ser], [@Sha1], [@Sha2].
[**2.1. Bruhat–Tits trees.**]{} The Bruhat–Tits tree ${\cal T}_p$ is the tree, in which each vertex belongs to $(p+1)$ edges. The group $Aut ({\cal T}_p)$ of automorphisms of $\T_p$ is a locally compact group. This group is similar to rank 1 groups over $\R$ and over $p$-adic fields. The representation theory of $Aut ({\cal T}_p)$ and related harmonic analysis are well understood, see [@Car], [@Ols1], [@Fig1], [@Fig2].
[**2.2. The tree $\T_\infty$.**]{} We denote by $\T_\infty$ the simplicial tree, in which each vertex belongs to a countable set of edges. At first sight, the group $Aut(\T_\infty)$ seems pathological. Nevertheless, it is a useful object as one of the simplest examples of infinite-dimensional groups, see [@Ols2], [@Nerb]. This group is an imitation of the group ${\rm O}(1,\infty)$.
[**2.3. The tree of free group.**]{} Denote by $F_2$ the free group with two generators $\alpha$, $\beta$. Vertices of the tree $J(F_2)$ are numerated by elements of the group $F_2$. Vertices $v_p$, $v_q$ are connected by an edge if $$p=q\alpha^{\pm1} \qquad\text{or}\qquad p=q\beta^{\pm1}.$$ Obviously, $J(F_2)$ is the Bruhat–Tits tree $\T_3$. The group $F_2$ acts on the tree $J(F_2)$ by the transformations $$r:\quad v_p\mapsto v_{rp}, \qquad\text{where}\quad r\in F_2.$$
Fix $l_1$, $l_2>0$. Assign the length $l_1$ to any edge $[v_p,v_{p\alpha}]$, and the length $l_1$ to any edge $[v_p,v_{p\beta}]$. Thus we obtain a metric tree with an action of $F_2$.
[**2.4. Another tree of free group.**]{} Let us contract all the edges of the type $[v_p,v_{p\alpha}]$ of the tree $J(F_2)$ described in 2.3. Thus, we obtain the action of $F_2$ on $\T_\infty$.
[**2.5 Dyadic intervals.**]{} Vertices $V_{u;n}$ of the tree $J_2(\R)$ are enumerated by segments in $\R$ having the form $$S_{u;n}=\left[ \frac u{2^n}, \frac {u+1}{2^n}\right], \qquad
\text{where $u\in \Z, n\in \Z$}.$$ We connect $V_{u;n}$ and $V_{w;n-1}$ by an edge if $S_{w;n-1}\supset V_{u;n}$.
Obviously, we obtain the simplicial tree $\T_2$.
[**2.6. Balls on $p$-adic line**]{}. Denote by $\Q_p$ the field of $p$-adic numbers, denote by ${\Bbb Z}_p$ the $p$-adic integers. Denote by $B_{a, k}$ the ball $$|z-a|\le p^{-k}.$$
[Remark.]{} The radius $p^{-k}$ is determined by the ball. But $B_{a,k}=B_{c,k}$ for any $c\in B_{a,k}$.
[**2.7. Tree of lattices.**]{} Consider the $p$-adic plane $\Q_p^2$ equipped with a skew symmetric bilinear form $A(v,w)$. Denote by ${\rm Sp_2(\Q_p)}$ the group of linear transformations preserving the form $A(v,w)$.
A [*lattice*]{} in $\Q_p^2$ is a compact subset $R \subset\Q_p^2$ having the form $$\Q_pv \oplus \Q_pw; \qquad\text{where $v,w$ are not proportional}.$$ We say that a lattice $R$ is [*self-dual*]{} if
1\. $A(v,w)\in{\Bbb Z}_p$ for all $v,w$ in $R$
2\. if $h\in \Q_p^2$ satisfies $A(h,v)\in {\Bbb Z}_p$ for all $v\in R$, then $h\in R$.
Vertices of the tree $\T(\Q_p^2)$ are self-dual lattices. Two vertices $R,S$ are connected by an edge if $$\text{volume of $R\cap S$}=\frac 1p \text{volume of $R$}$$
It can be shown that $\T(\Q_p^2)$ is the Bruhat–Tits tree $\T_p$. Obviously, the group ${\rm Sp}_2(\Q_p)$ acts on our tree by automorphisms.
[**2.8. Modular tree.**]{} Consider the following standard picture from arbitrary textbook on complex analysis. Consider the Lobachevsky plane $L:{\rm Im}\, z>0$ and the triangle $\Delta$ with three vertices $0$, $1$, $\infty$ on the absolute ${\rm Im}\, z=0$. Consider the reflections of $\Delta$ with respect to the sides of $\Delta$. We obtain 3 new triangles $\Delta_1$, $\Delta_2$, $\Delta_3$. Then we consider the reflections of $\Delta_j$ with respect to their sides etc. We obtain a tilling of $L$ by infinite triangles (with vertices in rational points of the absolute ${\rm Im}\, z=0)$.
Vertices of the modular tree are enumerated by the triangles of the tilling. Two vertices are connected by an edge, if the corresponding triangles have a common side.
The group ${\rm SL}_2(\Z)$ acts on the modular tree in the obvious way.
[**2.9 Tree of pants.**]{} Let $R$ be a compact Riemann surface. Fix a collection $C_1$, …$C_k$ of closed mutually disjoint geodesics on $R$. The universal covering of $R$ is the Lobachevsky plane.
The coverings of the cycles $C_j$ are geodesics on $L$. Thus we obtain the countable family of mutually disjoint geodesics on $L$. They divide $L$ into the countable collection of domains.
Now we construct a tree. Vertices of the tree are enumerated by the domains on $L$ obtained above. Two vertices are connected by an edge, if the corresponding domains have a common side.
The fundamental group $\pi_1(R)$ of the surface $R$ acts on this tree in the obvious way.
[**3. Hierarchomorphisms**]{}
[**3.1. Large group of hierarchomorphisms.**]{} Consider a group $\Gamma$ acting on a simplicial (or metric) tree $J$. Consider a partition of $J$ into a finite collection of right subtrees $S_1$, …$S_k$, i.e., the subtrees $S_j$ are mutually disjoint, and $\Vert(J)=\bigcup\Vert(S_j)$. Let $$g_1:S_1\to J,\dots, g_k:S_k\to J$$ be a collection of embeddings such that
1\) the subtrees $g_j(S_j)$ are mutually disjoint;
2\) $\bigcup\Vert(g(S_j))=\Vert(J)$.
Thus we obtain the bijection $$g=\{g_j,S_j\}:\Vert(J)\to\Vert(J)$$ given by $$g(a)=g_j(a)\qquad\text{if}\quad a\in\Vert(S_j)$$ We call such maps [*hierarchomorphisms*]{}, see Picture 2. Denote the group of all such hierarchomorphisms by $\Spher^\circ(J,\Gamma)$.
[**3.2. Action of hierarchomorphisms on absolute.**]{} Consider a hierarchomorphism $g=\bigl\{g_j,S_j\bigr\}$. Let $\omega\in \Abs(J)$. Let $a_1,a_2,\dots$ be a way leading to $\omega$. For a sufficiently large $N$ and for some $S_j$, we have $a_N,a_{N+1},\dots\in S_j$. Hence $g_j(a_N),g_j(a_{N+1}),\dots\in g_j(S_j)$ is a way leading to some point $$\nu\in\Abs\bigl((g_j(S_j)\bigr)\subset\Abs(J).$$
We assume $$\nu=g(\omega).$$
Fix a point $\xi\in\Vert(J)$. Under the previous notation, consider the sequence $$n_M=\rho(\xi,a_M)-\rho(\xi,g_j(a_M)).$$ This sequence becomes a constant after a sufficiently large $M$. We denote this constant (the [*pseudoderivative*]{}) by $$n(g,\omega)=n_\xi(g,\omega).$$ The following statement is obvious.
[Proposition 3.1.]{} [*For $g,h\in\Spher^\circ(J,\Gamma)$, $\omega\in\Abs(J)$,*]{} $$n(g h,\omega)=n(h,\omega)+n(g,h\omega).$$
[**3.3. Small group of hierarchomorphisms.**]{} Denote by $\Spher(J,\Gamma)$ the group of transformations of the absolute induced by elements $g\in\Spher^\circ(J,\Gamma)$.
The kernel of the canonical map $$\Spher^\circ(J,\Gamma) \to \Spher(J,\Gamma)$$ consists of finite permutations of the set $\Vert(J)$.
Obviously, the pseudoderivative $n(g,\omega)$ is well defined for $g\in\Spher(J,\Gamma)$.
[**3.4. A variant: planar hierarchomorphisms.**]{} Assume a simplicial tree $J$ be planar (this means, that for each vertex $a$ we fix the cyclic order on the set of edges containing $a$; it is the case in some of our examples. Then also we have a canonical cyclic order on the absolute.
Now we can consider the group of hierarchomorphisms that preserves the cyclic order on the absolute.
[**4. Hilbert spaces $\H(J)$**]{}
[**4.1. Definition.**]{} Let $J$ be a metric tree, let $0<\lambda<1$. Denote by $\H(J)$ the real Hilbert space spanned by the formal vectors $e_a$, where $a$ ranges in $\Vert(J)$, with inner products given by $$\langle e_a, e_b\rangle=\lambda^{\rho(a,b)},\qquad
\forall a,b\in\Vert(J).$$
We must show that a system of vectors with inner products (4.1) can be realized in a Hilbert space.
[**4.2. Existence of $\H(J)$.**]{} Let $a$ be a vertex of $J$. Let $b_1,b_2,\dots$ be the vertices adjacent to $a$. Consider an arbitrary unit vector $e_a$ in a real infinite dimensional Hilbert space $\cH$. Consider a collection $L_{b_1}$, $L_{b_2}$,…of pairwise perpendicular two-dimensional planes[^4] containing $e_a$. For each plane $L_{b_k}$, we draw a vector $e_{b_k}\in L_{b_k}$ such that $$\langle e_{b_k}, e_a\rangle=\lambda^{\rho(a,b_k)},$$ see Picture 3.
By the perpendicularity, $$\langle e_{b_k},e_{b_l}\rangle=
\langle e_{b_k},e_{a}\rangle \cdot
\langle e_a,e_{b_l}\rangle=\lambda^{\rho(b_k,b_l)}.$$
Then we apply the following inductive process. Assume that for a subtree $S$ the required embedding $\Vert(S)\to \cH$ is constructed, i.e., we have a subspace $\H(S)\subset \cH$. Let $b\in\Vert(J)$, and $c\not\in\Vert(J)$ be adjacent to $b$. Consider the two-dimensional plane $L_c\subset \cH$ that contains $e_b$ and is perpendicular to $\H(S)$. Let us draw a unit vector $e_c\in L_c$ such that $$\langle e_c,e_b\rangle=\lambda^{\rho(b,c)}.$$ Thus we obtained the required embedding $\Vert(S)\bigcup \{b\}\to \cH$.
There is a sufficient place in the Hilbert space, and thus we obtain the embedding $\Vert(J)\to \cH$.
[Remark.]{} This geometric picture is especially pleasant, if lengths of all edges are equal.
[**4.3. More formal description of $\H(J)$.**]{} Consider an [*affine*]{} real infinite dimensional Hilbert space $\cK$, i.e., a Hilbert space, where the origin of coordinates is not fixed. Denote by $\|\cdot\|$ the length in $\cK$. Consider a collection of points $N_a\in \cK$, where $a\in \Vert(J)$, such that
1\) if $[a,b]$, $[c,d]$ are different edges of $J$, then $N_aN_b\bot N_cN_d$;
2\) for $[a,b]\in\Edge(J)$, $$\|N_aN_b\|^2=\lambda\rho(a,b)\|.$$
The existence of such embedding is obvious.
By the Pythagoras theorem, $$\|N_bN_c\|^2=\lambda\rho(b,c)\qquad \forall b,c\in \Vert(J).$$
Now let us apply the following standard Fock–Schoenberg construction ([@Fok], [@Sch]). For an affine Hilbert space $\cK$, there exists a linear Hilbert space ${\rm Exp}(\cK)$ and an embedding $\phi:\cK\to {\rm Exp}(\cK)$ such that for all $X,Y\in \cK$ $$\langle \phi(X,\phi(Y)\rangle=\exp(-\|XY\|^2).$$
Fix any origin of the coordinates in $\cK$. We can assume that ${\rm Exp}(\cK)$ is the direct sum of all symmetric powers of $\cK$ $${\rm Exp}(\cK)=
\R\oplus \cK\oplus S^2 \cK\oplus S^3\cK\oplus \dots,$$ and $$\phi(X)= e^{-\|X\|^2}\Bigl[1\oplus\frac X{1!}
\oplus \frac{X^{\otimes 2}}{2!}
\oplus \frac{X^{\otimes 3}}{3!}\oplus \dots\Bigr].$$
It remains to apply the Fock–Schoenberg construction to the space $\cK$ constructed above. The vectors $\phi(N_a)$ satisfy the relations (4.1).
[Remark.]{} The spaces $\H$ associated with a tree are present in Olshansky’s paper [@Ols2]. In a implicit form, they are present in [@Ism] (without a tree).
[**4.4. Action of the group of hierarchomorphisms in $\H(J)$.**]{} Let a group $\Gamma$ acts on $J$ by isometries. Then $\Gamma$ acts in $\H(J)$ by the orthogonal operators[^5] of the Hilbert space $\H(J)$ by the formula $$U(g)e_a=e_{ga}.$$
Let now $g\in\Spher^\circ(J,\Gamma)$ be a hierarchomorphism. Define the operators $U(g)$ by the same formula (4.2).
[Theorem 4.1.]{} a) [*The operators $U(g)$ are well defined and bounded.*]{}
b\) [*Each operator $U(g)$ can be represented in the form $U(g)=A(1+R)$, where $A$ is an orthogonal operator and $R$ is an operator of finite rank.*]{}
The theorem is proved below in 4.6.
[**4.5. The subspaces $\H(S)$.**]{} Let $S$ be a subtree in $J$. Denote by $\H(S)$ the subspace in $\H(J)$ generated by the vectors $e_c$, where $c\in\Vert(S)$. Denote by $P(S)$ the operator of projection $\H(J)\to\H(S)$.
[Lemma 4.2.]{} [*Let $S_1$, $S_2$ be two disjoint subtrees in $J$. Let $b\in\Vert(S_1)$, $c\in\Vert(S_2)$ be the nearest vertices of the subtrees $S_1$, $S_2$.*]{}
a\) [*The sum $\H(S_1)+\H(S_2)$ is a topological direct sum in $\H(J)$.*]{}
b\) [*Let $Q:\H(S_1)\to\H(S_2)$ be the restriction of the projection operator $P(S_2)$ to $\H(S_1)$. Then the image of $Q$ is the line spanned by $e_c$, and the kernel of $Q$ is the orthocomplement in $\H(S_1)$ to $e_b$.*]{}
[Proof.]{} Let $h_1\in \H(S_1)$, $h_2\in \H(S_2)$ be unit vectors. The both statements are corollaries of the following inequalities $$\langle h_1, h_2\rangle\le \langle h_1, e_c\rangle;\qquad
\langle h_1, h_2\rangle \le \langle e_b, h_2\rangle.$$
[**4.6. Proof of Theorem 4.1.**]{} Let $g=\{g_j,S_j\}\in\Spher^\circ(J)$ be a hierarchomorphism. Without loss of generality (see Lemma 1.1), we can assume that $S_j$ are bushes or single-point sets.
By Lemma 4.2, the decomposition $$\H(J)=\bigoplus_j \H(S_j)$$ is a topological direct sum.
Consider the bilinear form $$Q(h_1,h_2)=\langle U(g)h_1, U(g)h_2\rangle- \langle h_1, h_2\rangle$$ on $\H(J)\times \H(J)$. It is sufficient to prove that $Q$ is a bounded form on $\H(J)\times \H(J)$ and the rank of $Q$ is finite.
The matrix of $Q$ in the basis $e_a$ is $$Q(e_a,e_b)= \langle e_{ga}, e_{gb}\rangle-
\langle e_{a}, e_{b}\rangle =
\lambda^{\rho(ga,gb)}-\lambda^{\rho(a,b)}$$ The matrix $Q(e_a,e_b)$ has the natural block decomposition corresponding to the partition $$\Vert(J)=\bigcup \Vert(S_j)$$ It is sufficient to prove that each block has finite rank.
Thus, let $a$ ranges in $S_i$, $b$ ranges in $S_j$. If $S_i$ is an one-point space, then the required statement is obvious.
Thus, we assume that $S_i$, $S_j$ are bushes (see 1.1). Let $u_i$, $u_j$ be their roots.
If $S_i=S_j$, then $Q(e_a,e_b)$ is the identical zero.
Thus, assume $S_i\ne S_j$. Then $$\begin{aligned}
\rho(a,b)&=\rho(a,u_i)+\rho(u_i,u_j)+\rho(u_j,b);\\
\rho(ga,gb)&=\rho(ga,gu_i)+\rho(gu_i,gu_j)+\rho(gu_j,gb)= \\
&=\rho(a,u_i)+\rho(gu_i,gu_j)+\rho(u_j,b)
.\end{aligned}$$ Thus, $$\begin{aligned}
Q(e_a,e_b)=\bigl[\lambda^{\rho(gu_i,gu_j)}-
\lambda^{\rho(u_i,u_j)}\bigr]
\cdot \lambda^{\rho(a,u_i)}\cdot \lambda^{\rho(b,u_j)} =\\
={\rm const} \cdot \langle e_{u_i}, e_a\rangle
\cdot \langle e_{u_j}, e_b\rangle
.\end{aligned}$$
Thus we obtain that the bilinear form $Q$ on $\H(S_i)\times\H(S_j)$ is given by the formula $$Q(h_1,h_2)={\rm const} \cdot
\langle e_{u_i}, h_1\rangle \cdot
\langle e_{u_j}, h_2\rangle$$ Thus the form $Q$ on $\H(S_i)\times\H(S_j)$ is of rank $\le 1$. This finishes the proof.
[**4.7. Remark. Spaces $\H$ associated with $\R$-trees.**]{} Let we have a countable family $J_1$, $J_2, \dots$ of metric trees and let we have isometric emeddings $\iota_k:J_k\to J_{k+1}$: $$\dots
\stackrel{\iota_{k-1}}\longrightarrow J_k
\stackrel{\iota_{k}}\longrightarrow J_{k+1}
\stackrel{\iota_{k+1}}\longrightarrow J_{k+2}
\stackrel{\iota_{k+2}}\longrightarrow \dots$$ Let $\bold J$ be the direct limit (the union) of $J_k$. Such spaces are called [*$\R$-trees*]{}.[^6]
Obviously, we have the chain of inclusions $$\dots\subset\H(J_k)\subset \H(J_{k+1})
\subset \H(J_{k+2})\subset \dots$$
Denote the inductive limit of this chain by $\H({\bold J})$. Thus the Hilbert space $\H$ survives for $\R$-trees. Nethertheless, the analogue of Theorem 4.1 is wrong.
[**5. Boundary spaces**]{}
In this Section, we construct some spaces $\E$ of ’distributions’ on the absolute of a metric tree. These spaces can be considered as an analogue of the Sobolev spaces on the spheres. For the Bruhat–Tits trees, the spaces $\E$ are well-known, see [@Car]. We also construct the operator $\H\to\E$ of restriction of a “function on tree” to the absolute.
[*In this Section, $J$ is a locally finite perfect metric tree.*]{}
[**5.1. Balls in absolute.**]{} Let $S$ be a branch of $J$. A ball $B[S]\subset \Abs(J)$ is the absolute of the branch $S$. If we delete the root of the $S$ and all edges containing the root, then $S$ will be disintegrated into the finite collection of branches $S^{(1)}$, $S^{(2)}$…, $S^{(k)}$. Hence the ball $B[S]$ admits the canonical partition $$B[S]=B[S^{(1)}]\cup \dots\cup B[S^{(k)}]
.$$ into the balls $B[S^{(k)}]$.
We define the topology on $\Abs(J)$ by the assumption that all the balls $B[S]$ are open-and-closed subset in $\Abs[S]$. Obviously, $\Abs(J)$ is a completely discontinuous compact set.
[Remark.]{} Obviously, hierarchomorphisms locally preserve hierarchy of balls on the absolute[^7]. Obviously, spheromorphisms are homeomorphisms of the absolute. But preserving of the hierarchy of balls is a very rigid condition on a homeomorphism.
[**5.2. New notation in the space $\H(J)$.**]{} Let us fix a vertex $\xi\in\Vert(J)$. Let $a,b\in\Vert(J)$. Consider the way $a_0=a, a_1,\dots, a_l=b$ connecting $a$, $b$. Assume $$\theta(a,b)=2\min\rho(\xi,a_j).$$
We emphasis that this function has sense also if $a$ or $b$ are points of the absolute, and the value $\theta(a,b)$ is finite except the case $a=b\in\Abs(J)$.
For $a\in\Vert(J)$, consider the vector $f_a\in\H(J)$ given by $$f_a=\lambda^{-\rho(\xi,a)}e_a.$$ Then $$\langle f_a,f_b,\rangle=\lambda^{-\theta(a,b)}.$$
[Remark.]{} Let $S$ be a subtree in $J$ containing $\xi$. For $c\in\Vert(J)$, consider the nearest vertex $b\in \Vert(S)$. Then the projection of $f_c$ to $\H(S)$ is $f_b$.
[**5.3. Measures on $\Abs(J)$ and compatible systems of measures on $\Vert(J)$.**]{} Let $R\subset J$ be a subtree. We say that $R$ is [*complete*]{} if any $a\in\Vert(R)$ satisfies one of two following conditions (see Picture 5).
1\. Any vertex $b$ of $J$ adjacent to $a$ is contained in $R$.
2\. Only one vertex of $J$ adjacent to $a$ is contained in $R$
Let $\partial R$ denote the boundary of $R$, i.e., the set of all vertices of the second type.
We also assume $\xi\in\Vert(R)\setminus \partial R$.
Consider a real-valued measure (charge) $\mu$ of [*finite variation*]{} on $\Abs(J)$. Recall that any measure $\mu$ of finite variation admits the canonical representation $$\mu=\mu^+-\mu^-,$$ where $\mu^\pm$ are nonnegative finite measures, and for some (noncanonical) Borel subset $U\subset\Abs$, $$\mu^-(U)=0;\qquad \mu^+(\Abs\setminus U)=0.$$ The [*variation*]{} of the measure $\mu$ is $${\rm var}(\mu)=\mu^+(U)+\mu^-(\Abs\setminus U).$$
For a complete subtree $R$, denote by $u_1$, $u_2$, … the points of $\partial R$. For any $u_k$, there exists a unique branch $S_{u_k}\subset J$ such that $u_k$ is the root of $S_{u_k}$ and $\xi\notin S_{u_k}$.
Consider the measure $\mu_R$ defined on the finite set $\partial R$ by $$\mu_R(u_j)=\mu\bigl(B[S_{u_j}]\bigr).$$ Consider also the vector $$\Psi[\mu|R]=\sum\limits_{u_j\in\partial R}
\mu\bigl(B[S_{u_j}]\bigr)\,f_{u_j}.$$
Let $R_2\supset R_1$ be complete subtrees. Then we have the obvious retraction $$\eta^{R_2}_{R_1}:\,\,\Vert(R_2)\to\Vert(R_1):$$ if $a\in \Vert(R_2)$, then $\eta^{R_2}_{R_1}(a)$ is the nearest vertex of $R_1$.
[Lemma 5.1.]{} a) [*$\mu_{R_1}$ is the image of $\mu_{R_2}$ under the retraction $\eta^{R_2}_{R_1}(a)$.*]{}
b\) [*The vector $\Psi[\mu|R_1]$ is the projection of $\Psi[\mu|R_2]$ to the subspace $\H(R_1)$. In particular,*]{} $$\|\Psi[\mu|R_1]\|\le\| \Psi[\mu|R_2]\|.$$
[Proof.]{} Assertion a) is obvious, and assertion b) follows from the last remark from 5.2. $\square$
Conversely, consider a family of complete subtrees $$R_1\subset R_2\subset R_3\subset\dots,$$ such that $\bigcup R_j=J$. Let for each $j$ we have a measure $\nu_j$ on $\partial R_j$, and $\eta^{R_{j+1}}_{R_j}\nu_{j+1}=\nu_j$ for all $j$. If $\sup {\rm var}(\nu_j)<\infty$, then there exists a unique measure $\nu$ on $\Abs$ such that $\nu_j=\nu_{R_j}$.
[**5.4. Boundary spaces $\E\subset\H$.**]{} Let $R_1\subset R_2\subset \dots$ be a sequence of complete subtrees in $J$, and $\bigcup R_k=J$ (the construction below do not depend on choice of the sequence).
Let $\mu$ be a measure of finite variation on $\Abs(J)$. We say that $\mu$ belongs to the class $\E=\E(J)$ if $$\lim_{j\to\infty}\|\Psi[\mu|R_j]\|_{\H} <\infty.$$
[Proposition 5.2.]{} [*For $\mu,\mu'\in \E$, the following statements hold.*]{}
a\) [*There exists the following limit in the space $\H(J)$*]{} $$\begin{aligned}
\Psi[\mu]&:=\lim_{j\to\infty} \Psi[\mu|R_j].\\
\text{b)}\qquad\,\,\, \qquad \qquad \qquad \|\Psi[\mu]\|_{\H}&=
\lim_{j\to\infty}\|\Psi[\mu|R_j]\|_{\H}.\\
\text{c)}\qquad\qquad \qquad \langle \Psi[\mu], \Psi[\mu'] \rangle_{\H}&=
\lim_{j\to\infty}
\langle \Psi[\mu|R_j], \Psi[\mu'|R_j] \rangle_{\H}.
\end{aligned}$$
[Proof.]{} All statements follow from Lemma 5.1. $\square$
Thus we obtain the embedding $\E(J)\mapsto\H(J)$ given by $\Psi:\mu\to\Psi[\mu]$. We define the inner product in $\E(J)$ by $$\langle \mu_1,\mu_2\rangle_{\E(J)}:=
\langle \Psi[\mu_1], \Psi[\mu_2] \rangle_{\H(J)}.$$
Denote by $\EE\subset\H$ the image of the embedding $\Psi$. Denote by $\ov\EE$ the closure of $\EE$ in $\H$, and also denote by $\ov \E$ the completion of the space $\E$ with respect to the norm (5.3).
[**5.5. More direct description of $\E$.**]{} We can write formally $$\begin{aligned}
\|\mu\|^2_{\E}=
\iint_{\Abs\times\Abs}\lambda^{-\theta(\omega_1,\omega_2)}
d\mu(\omega_1)\, d\mu(\omega_2) ;\\
\langle \mu_1,\mu_2\rangle_{\E}=
\iint_{\Abs\times\Abs}\lambda^{-\theta(\omega_1,\omega_2)}
d\mu_1(\omega_1)\, d\mu_2(\omega_2).
\end{aligned}$$ These integrals are very simple, since the integrand $\lambda^{-\theta(\omega_1,\omega_2)}$ has only countable set of values. Nevertheless, generally (even for the Bruhat–Tits tree $\T_2$) for $\mu_1,\mu_2\in\E$, these integrals diverge as Lebesgue integrals.
Our limit procedure is equivalent to the Riemann improper integration in the following sense. Consider a complete subtree $R\subset J$ such that $\xi\in R$. Then $J\setminus R$ is a union of disjoint branches $S_1,\dots, S_k$. Thus $$\Abs(J)=B[S_1]\cup\dots\cup B[S_k].$$ Let us define the Darboux sum $${\cal S}_R(\mu_1,\mu_2)=
\sum\limits_{i,j}
\Bigl\{\min\limits_{\omega_1\in B[S_i],\,\,\,\omega_2\in B[S_j]}
\lambda^{-\theta(\omega_1,\omega_2)}\Bigr\} \mu_1(B[S_i])\mu_2(B[S_j])
.$$
[Remark.]{} If $i\ne j$, then the value $\lambda^{-\theta(\omega_1,\omega_2)}$ is a constant on $B[S_i]\times B[S_j]$.
We have seen, that $$R_2\supset R_1\qquad \Rightarrow\qquad {\cal S}_{R_1}(\mu,\mu)
\le {\cal S}_{R_2}(\mu,\mu)
.$$ Now we can define the integral (5.5) as the limit of these Darboux sums under refinement of the partition. A measure $\mu$ is contained in $\E$ iff the Riemann integral (5.5) is finite.
After this, we can define the inner product in $\E$ as the Riemann improper integral (5.6).
Nevertheless, the space $\H$ was essentially used in the justification of this construction, since the convergence of Darboux sums and positivity of the integral (5.5) are not obvious.
[**5.6. Non-emptiness of $\E$.**]{}
[Theorem 5.3.]{} a) [*There exists $\sigma$, which belongs to $0\le\sigma\le1$, such that the space $\E$ is zero for $\lambda<\sigma$ and the space $\E$ is not zero for $\lambda>\sigma$.*]{}
b\) [*If lengths of edges of $J$ are bounded, then $\sigma<1$.*]{}
c\) [*Let $J$ contain a subtree $I$ that is isomorphic to the Bruhat–Tits tree $\T_p$ as a simplicial tree, and lengths of all edges of $I$ are $\le \tau$. Then $\sigma\le 1/\sqrt[2\tau]{p}$.*]{}
d\) [*Assume lengths of edges of $J$ are bounded away from zero. Let the number $s(N)$ of $a\in\Vert(J)$, satisfying $d_\s(\xi,a)\le N$, has exponential growth, i.e., $s(N)\le \exp(\alpha N)$ for some constant $\alpha$. Then $\sigma>0$.*]{}
The proof of the Theorem is contained below in 5.7–5.11
[**5.7. Expansion of $\|\Psi\|^2$ into series with positive terms.**]{} Let $R_0\subset R_1\subset R_2\subset\dots$ be a sequence of complete subtrees in $J$, and $\bigcup R_m=J$. We say that the sequence $R_j$ is [*incompressible*]{} if
$1^\circ$. $R_0$ consists of the vertex $\xi$;
$2^\circ$. for each $m$, there exists $u\in\partial R_m$ such that $\Vert(R_{m+1})\setminus\Vert( R_m)$ consists of vertices adjacent to $u$.
Fix a measure (charge) $\mu$ on $\Abs$.
Obviously, $\Psi[\mu|R_0]=\mu(\Abs)f_\xi$, and hence $$\|\Psi[\mu|R_0]\|^2=\mu(\Abs)^2.$$
Let us evaluate $$z^{(m)}(\lambda)=\|\Psi[\mu|R_{m+1}]\|^2_{\H}-
\|\Psi[\mu|R_{m}]\|^2_{\H}.$$ Let $u$ be the vertex defined in $2^\circ$. Let $v_1,\dots, v_n\in\partial R_{m+1}$ be the vertices adjacent to $u$, see Picture 6.
Let $\mu_{R_{m+1}}(v_k)=t_k$ (these numbers can be negative), respectively $\mu_{R_m}(u)=t_1+ \dots + t_n$. It is readily seen that $$\begin{gathered}
z^{(m)}(\lambda)=
\Bigl(\lambda^{-2\rho(\xi,u)}\sum\limits_{k\ne l} t_kt_l+
\lambda^{-2\rho(\xi,u)} \sum\limits_k \lambda^{-2\rho(u,v_k)}t_k^2\Bigr)
-\lambda^{-2\rho(\xi,u)} \bigl(\sum t_k\bigr)^2=\\
=\lambda^{-2\rho(\xi,u)} \sum\limits_k (\lambda^{-2\rho(u,v_k)}-1)\,t_k^2
.\end{gathered}$$ First, we observe that this expression is completely determined by the measure $\mu$ and the vertex $u$. The subtrees $R_m$, $R_{m+1}$ are nonessential. Hence it is natural to denote $z^{(m)}(\lambda)$ by $z_u(\lambda)$.
Thus, $$\begin{aligned}
\|\Psi[\mu]\|^2&=
\mu(\Abs)^2 + \sum\limits_{m=1}^\infty z^{(m)}(\lambda)= \\=
&
\mu(\Abs)^2 +
\sum\limits_{u\in\Vert(J),u\ne\xi}z_u(\lambda)
.\end{aligned}$$
We emphasis that
a\) all summands of this series are positive;
b\) all summands $z_u(\lambda)$ are decreasing functions on $\lambda$ for $0\le\lambda\le1$.
[**5.8. Existence of $\sigma$.**]{} The Statement a) of Theorem 5.4 follows from the last observation of previous subsection.
[**5.9. Existence of $\E$.**]{} It is sufficient to prove c), since b) is a corollary of c). Furthemore, it is sufficient to prove nontriviality of $\E(I)$ for the subtree $I$. Denote by $R_k$ the subtree of $I$, consisting of all vertices $a\in I$ such that the simplicial distance $d_{\s}(\xi,a)\le k$.
Consider the uniform measure $\mu_{R_k}$ on $\partial R_k$, i.e., the measure of each point is $1/(p^{k-1}(p+1))$. Obviously, the measures $\mu_k$ form a compatible system of the measures, denote by $\mu$ the inverse limit of the measures $\mu_{R_k}$.
Let us estimate $$\begin{aligned}
\Psi[\mu|R_k]\|^2&= \frac 1{(p+1)^2 p^{2(k-1)}}
\|\sum\limits_{a\in \partial R_k} f_a\|^2=\\
&= \frac 1{(p+1)^2 p^{2(k-1)}} \sum\limits_{a,b\in \partial R_k}
\lambda^{-2\rho(a,b)}\le
\\
&\le \frac 1{(p+1)^2 p^{2(k-1)}}
\sum\limits_{j=0}^k \lambda^{-2\tau j}\cdot\left\{
\begin{matrix}
\text{number of pairs $(a,b)\in\partial R_k$}\\
\text{ such that
$d_{\s}(a,b)=2(k-j)$} \end{matrix}\right\} =\\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
=\frac 1{(p+1)^2 p^{2(k-1)}}
\Bigl[ (p+1)p^{2k-1} +
\sum\limits_{j=1}^{k-1}\lambda^{-2\tau j} (p+1)p^{k-1} (p-1) p^{k-j-1}
+\lambda^{-2k\tau}(p+1)p^{k-1}\Bigr]
\le \\
&\le\sum\limits_{j=0}^k \lambda^{-2\tau j}p^{-j}
.\end{aligned}$$ If $\lambda^{2\tau} p>1$, then these sums are uniformly bounded in $k$; hence $\mu\in\E(I)\subset\E(J)$.
[**5.10. Localization.**]{}
[Lemma 5.4.]{} [*Let $\mu\in\E$, and let $B[S]\subset\Abs$ be a ball. Let $\nu$ be the restriction of $\mu$ to $B[S]$ [(]{}i.e., $\nu(A)=\mu(A\bigcap B([S])$ for any Borel subset $A\subset\Abs$[)]{}. Then $\nu\in\E$.*]{}
[Proof.]{} We can assume $\xi\notin S$. Denote by $v$ the root of the branch $S$. The quantity $\|\mu\|_{\E}^2$ is the sum of the series $\sum z_u(\lambda)$ given by (5.8), (5.10). The series for $\|\nu\|_{\E}^2$ is obtained from the series fo $\|\mu\|_{\E}^2$ by the following operations.
1\) For $u$ lying between $\xi$ and $v$, the summands $z_u(\lambda)$ are changed in a non-predictible way.
2\) For any $u\in S$, the summand $z_u(\lambda)$ does not change.
3\) All other summands become zero.
Obviously, the new series $z_u(\lambda)$ is convergent. $\square$
[Remark.]{} Consider a Borel subset $U$ in the absolute. Let $\nu$ be the resriction of $\mu\in\E$ to $U$. Generally, $\nu\notin\E$. Also, generally, $\mu^\pm\notin\E$.
[**5.11. Lower estimate of $\sigma$.**]{} By Lemma 5.4, if $\E\ne 0$, then there exists a measure $\mu\in\E$ such that $\mu(\Abs)\ne 0$. For definiteness, assume $\mu(\Abs)=1$.
Let $\sigma$ be a lower bound for lengths of edges. Consider a complete subtree $R\subset J$ defined by the condition $d_{\s}(\xi,a)\le N$. Consider the measure $\mu_R$ on $\partial R$. In notation 5.7, $$\|\Psi[\mu]\|^2=1+\sum\limits_{v\in \Vert J,\,\, v\ne\xi} z_u\ge
\sum\limits_{u\in\partial R} z_u\ge$$ (by formula (5.8)) $$\ge \lambda^{-2N\sigma} (\lambda^{-2\sigma}-1)
\sum\limits_{u\in\partial R} \mu_R(u)^2
.$$ The number of points of $\partial R$ is less than $\exp\{\alpha N\}$, where $\alpha$ is a constant. Furthemore, $\sum_{u\in\partial R} \mu_R(u)=1$, hence the last expression is larger than $$\ge \lambda^{-2N\sigma} (\lambda^{-2\sigma}-1) \exp\{-\alpha N\}.$$ For a sufficiently small $\lambda>0$, the last expression tends to $\infty$ as $N\to\infty$. and thus $\E=0$.
[**6. Action of group of hierarchomorphisms in $\E$**]{}
Let $J$ be a perfect locally finite metric tree. Let $\H(J)\supset\ov\EE(J)\simeq \ov\E(J)$ be the same spaces as above. Let a group $\Gamma$ act on $J$ by isometries. Let $\Spher^\circ(J,\Gamma)$, $\Spher(J,\Gamma)$ be the corresponding hierarchomorphisms groups. The group $\Spher^\circ(J,\Gamma)$ acts in $\H(J)$ by the operators $U(g)$ given by (4.1).
[**6.1. Action of hierarchomorphisms in $\E$.**]{}
[Proposition 6.1]{} a) [*The space $\ov\EE(J)\subset\H(J)$ is invariant with respect to $\Spher^\circ(J,\Gamma)$.*]{}
b\) [*For $g\in \Spher^\circ(J,\Gamma)$, the restriction of the operator $U(g)$ to $\ov\EE$ depends only on the corresponding element $\widetilde g\in \Spher(J,\Gamma)$.*]{}
c\) [*The action of $\Spher(J,\Gamma)$ in $\E(J)\simeq\EE(J)$ is given by $$T_\lambda(\widetilde g) \mu(\omega)=
\lambda^{n(g,\omega)}\cdot\mu(g\omega),
\qquad\text{where $g\in\Spher(J,\Gamma)$},$$ where the pseudoderivative $n(g,\omega)=n(\widetilde g,\omega)$ of a hierarchomorphism on the absolute was defined in [3.2]{}, and $\mu(g\omega)$ is the image of the measure $\mu$ under the transformation $\omega\mapsto g\omega$.*]{}
[Remark.]{} For $g\in \Gamma$, the operator $T_\lambda(g)$ is unitary.
[Proof.]{} Fix $g\in \Spher^\circ(J,\Gamma)$. Let $R_0\subset R_1\subset\dots$ be a incompressible sequence of complete subtrees as in 5.7, $\bigcup R_k=J$. Consider the sequence $g\cdot\partial R_1$, $g\cdot\partial R_2$, …. There exists $l$ such that for all $k\ge l$ $$g\cdot\partial R_k =
\partial T_k\qquad\text{where $T_k$ is a complete subtree}
.$$ Hence, $$U(g)\Psi[\mu|R_k]=\Psi[\nu|T_k].$$ where $\nu$ is some measure on $\Abs(J)$.
We must show that the numbers $\|\Psi[\nu|T_k]\|$ are bounded. Consider the expansion of $\|\Psi[\mu]\|^2$ and $\|\Psi[\nu]\|^2$ into the series $\sum z^k(\lambda)$, see (5.9), (5.10). The summands with numbers $<l$ are essentially different, but this do not influence on the convergence. Other summands are rearranged and multiplied by the factors $\lambda^{n(g,\omega)}$.
But $\lambda^{n(g,\omega)}$ has only finite number of values and hence the series $\sum z^k(\lambda)$ for the measure $\nu$ is also convergent. Thus $\nu\in\E(J)$.
The statement a) is proved, the statement b) is obvious, and the statement c) follows from the same considerations.
[**6.2. Almost orthogonality.**]{}
[Theorem 6.2.]{} [*Let $g\in\Spher(J,\Gamma)$. The operators $T_\lambda(g)$ in $\E(J)$ given by [(6.1)]{} admit the representation $T_\lambda(g)=A(1+Q)$, where $A$ is an orthogonal operator and $Q$ is a finite rank operator.*]{}
This statement follows from Theorem 4.1. This can also be proved directly from the explicit formulas (5.6), (6.1).
[cc]{}
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Address (spring 2001):
Erwin Schrödinger Institute for Mathematical Physics
Boltzmanngasse, 9, Wien 1020, Austria
Permanent address: Institute of Theoretical and Experimental Physics,
Bolshaya Cheremushkinskaya, 25
Moscow 117259
Russia
[e-mail neretin@main.mccme.rssi.ru]{}
[^1]: In [@Ner2], there was proposed the term [*’ball-morphisms’*]{}, which is difficult for pronouncement. In English translation, it was replaced by [*’spheromorphism’*]{}. I want to propose the neologism [*’hierarchomorphism’*]{}, this a map regarding hierarchy of balls on the absolute; see below Subsection 5.1.
[^2]: Another heuristic explanation can be obtain by the monstrous degeneration construction from [@Kapo], chapter 9; the Lobachevsky plane can be degenerated to the universal $\R$-tree.
[^3]: see also the recent preprint of Nekrashevich [@Nek].
[^4]: Subspaces $M_1$, $M_2$ in a Hilbert space are perpendicular iff there is an orthogonal system of vectors $u_1,u_2,\dots$, $v_1,v_2,\dots$, $w_1,w_2,\dots$ such that $M_1$ is spanned by the vectors $u_i$, $v_j$, and $M_2$ is spanned by the vectors $w_n$, $v_j$.
[^5]: An orthogonal operator is an invertible operator in a real Hilbert space preserving the inner product
[^6]: up to a minor variation of terminology
[^7]: Firstly, this hierarchy structure on $p$-adic manifolds was mentioned in Addendum in Serre’s book [@Ser].
|
---
author:
- 'A. Ignesti, M. Gitti, G. Brunetti, L. Feretti, G. Giovannini'
bibliography:
- 'bibliography.bib'
date: Accepted
title: |
New JVLA observations at 3 GHz and 5.5 GHz\
of the ‘Kite’ radio source in Abell 2626
---
[ We report on new JVLA observations performed at 3 GHz and 5.5 GHz of Abell 2626. The cluster has been the object of several studies in the recent years due to its peculiar radio emission, which shows a complex system of symmetric radio arcs characterized by a steep spectrum. The origin of these radio sources is still unclear. Due to their mirror symmetry toward the center, it has been proposed that they may be created by pairs of precessing jets powered by the inner AGN. ]{} [ The new JVLA observations were requested with the specific aim of detecting extended emission on frequencies higher than 1.4 GHz, in order to constrain the jet-precession model by analyzing the spectral index and radiative age patterns alongs the arcs.]{} [ We performed a standard data reduction of the JVLA datasets with the software CASA. By combining the new 3 GHz data with the archival 1.4 GHz VLA dataset we produced a spectral index maps of the extended emission, and then we estimated the radiative age of the arcs by assuming that the plasma was accelerated in moving hot-spots tracing the arcs.]{} [ Thanks to the high sensitivity of the JVLA, we achieve the detection of the arcs at 3 GHz and extended emission at 5.5 GHz. We measure a mean spectral index $<-2.5$ for the arcs up to 3 GHz. No clear spectral index, or radiative age, trend is detected across the arcs which may challenge the interpretation based on precession or put strong constraints on the jet-precession period. In particular, by analyzing the radiative age distribution along the arcs, we were able to provide for the first time a time-scale $< 26$ Myr of the jet-precession period. ]{}
Introduction
============
In recent years, radio observations have revealed that a fraction of galaxy clusters hosts diffuse synchrotron emission on cluster scale. The discovery of radio sources not associated with any individual galaxy proves the presence of non-thermal components, such as relativistic particles and magnetic fields, mixed with the thermal intra-cluster medium (ICM) on spatial scale that are comparable to the cluster size. According to their morphology and location in the cluster, cluster-scale radio sources are classified as radio relic, radio halos and radio mini-halos [e.g. @Feretti_2012]. Relics are polarized, elongated, arc-shaped synchrotron sources located in the peripheries of dynamically disturbed clusters. Radio relics are also unique probes of the properties of magnetic fields in the outskirts of galaxy cluster. Halos and mini-halos are instead roundish radio sources located in the cluster central regions. The two classes differ in size and in the dynamical properties of the hosting clusters. Radio halos are located in the center of dynamically disturbed clusters, which show signs of recent major merger activity, whereas mini-halos are detected only in relaxed, cool-core clusters. It is currently thought that relics and halos originate from complex acceleration mechanisms that are driven by shocks and turbulence in the ICM [e.g. @Brunetti-Jones_2014]. On the other hand, the mini-halo diffuse emission is always observed to surround the intense radio emission of the brightest cluster galaxy (BCG), which often shows non-thermal radio jets and lobes ejected by the central active galactic nucleus (AGN). The radio-lobe plasma strongly interacts with the ICM by inflating large X-ray cavities and triggering the so-called ‘radio-mode AGN feedback’ [e.g., @Gitti_2012]. The central radio-loud AGN is also likely to play a role in the initial injection of the relativistic particles emitting in the mini-halo region. Nonetheless, the diffuse radio emission of mini-halos is truly generated from the ICM on larger scales, where the thermal and non-thermal components are mixed, and can be explained in the framework of leptonic models which envision in situ particle re-acceleration by turbulence in the cool-core region [e.g., @Gitti_2002]. Turbulence in cool cores may be generated by several mechanisms, including the interplay between the outflowing relativistic plasma in AGN jets/lobes, and sloshing gas motions.
This work is a multi-frequency study of the inner part of Abell 2626 (hereafter A2626), which was included in the first sample of mini-halo clusters [@Gitti_2004]. Its radio morphology, more complex than that of the standard radio bubbles typically observed to fill X-ray cavities, represents a challenge to models for the ICM / radio source interaction in cool cores. A2626 is a low-redshift [z=0.0553 @Struble-Rood_1999], regular, poor cluster [@Mohr_1996], located at RA 23h36m30s, DEC+21d08m33s and it is part of the Perseus-Pegasus super-cluster. A2626 has an estimated mass of 1.3$\times10^{15}$ M$_{\odot}$ and a virial radius of 1.6 Mpc [@Mohr_1996]. It is a cool-core cluster with estimated X-ray luminosity of $1.9 \times
10^{44}$ erg s$^{-1}$ and mass accretion rate of 5 M$_{\odot}$ y$r^{-1}$ [@Bravi_2016]. Its core-dominant (cD) galaxy IC5338 hosts a pair of optical nuclei with a projected separation of 3.3 kpc, of which only the southern one has a counterpart also in the radio [@Gitti_2013b hereafter G13] and hard X-rays [@Wong_2008] bands [see @Wong_2008 Fig. 3]. The extended radio emission of the cluster has a peculiar morphology resembling a giant kite, with striking arc-like, symmetric features whose origin is puzzling [@Gitti_2004; @Gitti_2013b; @Kale_2017]. The arcs are collimated structures having largest extent of $\simeq$70 kpc, steep radio spectrum [$\alpha<$-2.5[^1], @Kale_2017], and no evidence of polarized emission (G13). Gitti et al. (2004) argued that the elongated features are distinct from and embedded in the diffuse, extended radio emission, which they classified as a radio mini-halo and successfully modeled as radio emission from relativistic electrons reaccelerated by MHD turbulence in the cool-core region. On the other hand, the origin and nature of the radio arcs is still unclear. Their morphology may suggest that the arcs are radio bubbles, or cluster relics. However, Chandra and XMM-Newton observations failed to detect any X-ray cavity or shock front associated to them, in general showing no clear spatial correlation between the non-thermal emission of the arcs and the thermal emission of the cluster [@Wong_2008].
@Wong_2008 argued that the peculiar radio morphology of the northern and southern arcs may be produced by jet precession triggered by the reciprocal gravitational interactions of the two cores of the cD galaxy. According to this scenario, the relativistic plasma of the arcs was accelerated in a pair of precessing hot-spots powered by the AGN located inside the southern core of IC5338. In particular, if two jets ejected towards the north and south direction are precessing about an axis which is nearly perpendicular to the line-of-sight and are stopped at approximately equal radii from the AGN (at a ‘working surface’), radio emission may be produced by particle acceleration, thus originating the elongated structures. The impressive arc-like, mirror symmetric morphology of these features highlighted by the high resolution radio images (G13) may support this interpretation. On the other hand, the recent discovery of the third and fourth radio arc to the east-west direction [@Gitti_2013b; @Kale_2017] further complicates the picture. They could represent other radio bubbles ejected in a different direction, similarly to what observed in RBS 797 [@Doria_2012; @Gitti_2013a], but again the absence of any correlation with the X-ray image disfavors this interpretation. In the model proposed by @Wong_2008, they could represent the result of particle acceleration produced at a working surface by a second pair of jets ejected to the east-west direction. This interpretation would imply the existence of radio jets emanating also from the northeast nucleus of the cD galaxy IC 5338, which however does not show a radio core.
In order to constrain the jet-precession scenario, we requested new Karl Jansky Very Large Array (JVLA) observations at 3 GHz and 5.5 GHz to estimate the spectral index distribution along the arcs and thus infer detailed information on the radiative age of the plasma in the radio arcs. According to the jet-precession model, due to the gradual formation of the structures, the plasma along the arcs should exhibit a monotonic trend from one end to the other in the radiative age distribution, and thus also in the spectral index. The differences in radiative age between the opposite ends of each arc would represent a measure of the time required to create the arcs, and therefore it would also provide an estimate of the precession time of the jets.
We adopt a $\mathrm{\Lambda CDM}$ cosmology with $\mathrm{H_{0}=70}$ km $\mathrm{s^{-1}Mpc^{-1}}$, $\Omega_{M} = 1 - \Omega_{\Lambda} =
0.3$. The cluster luminosity distance is 232 Mpc, leading to a conversion of 1 arcsec = 1.1 kpc[^2].
Observation and Data Reduction
==============================
We performed new observations of the radio source A2626 at 3 GHz and 5.5 GHz with the JVLA in C-configuration (see Table \[obs.tab\] for details regarding these observations). In all observations the source 3C 48 (J0137$+$3309) was used as the primary flux density calibrator, while the sources J0016$-$0015 and 3C 138 (J0521$+$1638) were used as secondary phase and polarization calibrators, respectively. Data reduction was done using the NRAO Common Astronomy Software Applications package (CASA), version 4.6. As a first step we carried out a careful editing of the visibilities. In particular, the 3 GHz dataset required an accurate flagging to remove every radio frequency interference (RFI), that are quite common in this radio band. Overall, we removed about 10$\%$ of the visibilities in the 5.5 GHz dataset and 20$\%$ in the 3 GHz one, by repeating manual and automatic flagging with the modes [MANUAL]{} and [RFLAG]{} of the CASA task [ FLAGDATA]{}. Due to the visibility loss, the bandwidth of the 3 GHz observation decreased from 2.0 GHz (2.0-4.0 GHz) to 1.6 GHz (2.4-4.0 GHz), thus moving the central frequency from 3.0 GHz to 3.2 GHz. We performed a standard calibration procedure[^3] for each dataset, and the target was further self-calibrated.
Due to the presence of several bright radio sources in the field of view, we carried out the imaging procedure, with the task [ CLEAN]{}, on a $32'\times32'$ region centered on the cluster, in order to remove as best as possible their secondary lobes. To this purpose, we used [gridmode=WIDEFIELD]{} to parametrize the curvature of the sky regions far from the phase center. We also used the MS-MFS algorithm [@Rau_2011] by setting a two-terms approximation of the spectral model ([nterms=2]{}) and the multi-scale clean ([multiscale=\[0,5\]]{} on the point-like and beam scales) to reconstruct as best as possible the faint extended emission.
-------------------------- ------------- -------------
PI: Dr. Myriam Gitti 5.5 GHz 3 GHz
\[C-band\] \[S-band\]
Observation Date 19-Oct-2014 14-Dec-2014
Frequency Coverage (GHz) 4.5-6.5 2.0-4.0
Array Configuration C C
On source Time 44 m 63 m
\[obs.tab\]
-------------------------- ------------- -------------
: New JVLA data analysed in this work (project code:14B-022)
Results
=======
We report here the most relevant results of our analysis. For each observing band, we produced three different types of maps by varying the relative weight between short and long baselines. The [ UNIFORM]{} maps, obtained by setting [weighting=UNIFORM, nterms=1]{}, give a uniform weight to all spatial frequencies, thus enhancing angular resolution. The [NATURAL]{} maps, obtained by setting [weighting=NATURAL, nterms=1]{}, give more weight to low spatial frequencies thus degrading angular resolution but at the same time maximizing the sensitivity to the diffuse, extended emission sampled by short baselines. Finally, the [ROBUST 0]{} maps, obtained by setting [weighting=BRIGGS, ROBUST=0, nterms=2]{}, have resolution and sensitivity half-way between [UNIFORM]{} and [NATURAL]{}. We report the maps in Fig. \[cmaps.fig\] (5.5 GHz) and Fig. \[smaps.fig\] (3 GHz), whereas the flux density values measured on each map are reported in Tab. \[flulli.tab\]. Typical amplitude calibration errors are at 3$\%$, therefore we assume this uncertainty on the flux density measurements. In the following analysis we assume the 7 $\times$ rms levels at 1.4 GHz (shown in red in Fig. 2, middle panel) as reference contours for the size and position of the arcs and of the unresolved core, as they best trace the morphology of the features seen at 1.4 GHz (G13).
5.5 GHz maps
------------
The published map at 4.8 GHz (G13) does not show diffuse radio emission neither discrete features like the radio arcs seen at 1.4 GHz. The new observations presented in this work were performed with a more compact configuration of the array and with the larger receivers band of the JVLA, thus reaching an unprecedented high sensitivity. This allowed us to detect faint extended emission around IC5338 and IC5337 for the first time at 5.5 GHz.
We report the 5.5 GHz maps in Figure \[cmaps.fig\]. The [ UNIFORM]{} (top panel) and the [ROBUST 0]{} (middle panel) maps, which have a resolution of $2''.8 \times 2''.7$ and $3''.2
\times 2''.8$, do not show extended emission around IC5338, except for the jet-like feature at south-west that was yet observed at 1.4 GHz by G13. On the other hand, in the [ROBUST 0]{} map the close source on the left, IC5337, exhibits an extended emission in addition to the radio core already detected by G13. Due to its optical properties indicating that the galaxy is leaving a trail of cold gas, IC5337 has also been classified by as a jellyfish galaxy [@Poggianti_2016]. Therefore, the disrupted radio morphology agrees with previous dynamical studies of the cluster [@Mohr_1996] indicating that IC5337 is moving toward IC5338.
In the bottom panel of Fig. \[cmaps.fig\] we show the [ NATURAL]{} map at a resolution of $4''.9 \times 4''.4$, obtained by further imposing a UV tapering to 20 k$\lambda$ to maximize the sensitivity to the short spatial frequencies. In this high-sensitivity, low-resolution image we detected extended emission around IC5338. The emission region extends up to 27$''$ ($\sim$ 30 kpc) and is located inside the radio arcs known at 1.4 GHz (G13). We measure a total flux densities of $S_{\rm 5.5,NAT} =$ 8.2 $\pm$ 0.2, of which 6.2$\pm$0.2 mJy are contributed by the core, thus leaving an estimate of 2.0$\pm$0.3 mJy related to the extended emission. The nature of this radio emission is uncertain. On the one hand, due to its position, it may be contributed by the emission of the radio arcs and of the inner AGN jets. On the other hand, it may indicate the presence of a diffuse radio mini-halo.
We note that the core flux density measured in these new 5.5 GHz maps differs from that reported by G13 in the same band, which is 9.6$\pm$0.3 mJy. We accurately checked the flux calibration procedure in both observations finding no trivial problems, therefore this $\approx 40\%$ flux difference may indicate variability, and therefore activity, of the IC5338 core. By observing hard X-rays emission from its core, @Wong_2008 suggested that IC5338 hosts an AGN. The jet-like features observed at 1.4 GHz by G13 and the likely variability of the radio emission that we observe are consistent with this hypothesis.
![\[cmaps.fig\] [*Top panel :* ]{} 5.5 GHz map ([UNIFORM]{}) at a resolution of $2''.8 \times 2''.7$ with rms noise of 16.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 5.5 GHz map ([ROBUST 0]{}) at a resolution of $3''.2
\times 2''.8$ with rms noise = 8.0 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 5.5 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $4''.9 \times 4''.4$ with rms noise = 8.6 $\mu$Jy beam$^{-1}$. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps produced with comparable resolution from G13 data. The rms levels of the red contours are 15.4, 21.9 $\mu$Jy beam$^{-1}$ in middle and bottom panel, respectively.](c-uniform.eps){width="100.00000%"}
![\[cmaps.fig\] [*Top panel :* ]{} 5.5 GHz map ([UNIFORM]{}) at a resolution of $2''.8 \times 2''.7$ with rms noise of 16.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 5.5 GHz map ([ROBUST 0]{}) at a resolution of $3''.2
\times 2''.8$ with rms noise = 8.0 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 5.5 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $4''.9 \times 4''.4$ with rms noise = 8.6 $\mu$Jy beam$^{-1}$. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps produced with comparable resolution from G13 data. The rms levels of the red contours are 15.4, 21.9 $\mu$Jy beam$^{-1}$ in middle and bottom panel, respectively.](c-r0-7-14-2.eps){width="100.00000%"}
![\[cmaps.fig\] [*Top panel :* ]{} 5.5 GHz map ([UNIFORM]{}) at a resolution of $2''.8 \times 2''.7$ with rms noise of 16.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 5.5 GHz map ([ROBUST 0]{}) at a resolution of $3''.2
\times 2''.8$ with rms noise = 8.0 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 5.5 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $4''.9 \times 4''.4$ with rms noise = 8.6 $\mu$Jy beam$^{-1}$. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps produced with comparable resolution from G13 data. The rms levels of the red contours are 15.4, 21.9 $\mu$Jy beam$^{-1}$ in middle and bottom panel, respectively.](c-nat-7-14-2.eps){width="100.00000%"}
3 GHz maps
----------
The 2.0-4.0 GHz band is the new radio window observed with the JVLA. This is also the transmission band of the communication satellites, so it is populated by radio interferences which may jeopardize the observations. The maps we present here have been produced after an accurate editing of the target visibilities. The [UNIFORM]{} map (Fig. \[smaps.fig\], top panel) exhibits only the emission of IC5338 and the extended emission of IC5337. The [ROBUST 0]{} map at a resolution of $8''.7 \times 5''.8$ (Figure \[smaps.fig\], middle panel) shows several patches of extended emission. By superposing the 1.4 GHz contours by G13 (in red), we confirm that the new features detected in this band are the radio arcs already seen at 1.4 GHz. Remarkably, we identify a feature to the east which resembles the diffuse emission detected by G13 (see their Fig. 3), but it does not coincide entirely with the eastern arc discovered by @Kale_2017. Due to the low resolution of this map, the inner part of IC5338 is unresolved.
For this band we produced also a polarization intensity map. We combined the stokes components, Q and U, of the emission to compute the vector and total intensity maps of linear polarization. We set the weights to [NATURAL]{} to improve the signal-to-noise ratio (SNR) of the arcs. The [NATURAL]{} map in Fig. \[smaps.fig\], bottom panel, shows an overlay of polarization vector and total intensity contours. We observe only a small percentage ($\sim2\%$) of polarized emission from the core, whereas we do not observe polarized emission from the arcs. By comparing the rms of the polarized emission map with the peak flux density of the arcs, we estimate an upper limit for the polarized emission fraction of the arcs of $\sim30\%$[^4].
![\[smaps.fig\] [*Top panel :* ]{} 3 GHz map ([UNIFORM]{}) at a resolution of $7''.5 \times 8''.5$ with rms noise = 40.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 3 GHz map ([ROBUST 0]{}) at a resolution of $8''.7
\times 5''.8$, and rms noise = 20.4 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 3 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $13''.1 \times 8''.5$, and rms noise = 18.5 $\mu$Jy beam$^{-1}$. In black there are the polarization vectors. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise in all maps, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps made with comparable resolution.The rms levels of the red contours are 21.9 $\mu$Jy beam$^{-1}$.](s-uniform.eps){width="100.00000%"}
![\[smaps.fig\] [*Top panel :* ]{} 3 GHz map ([UNIFORM]{}) at a resolution of $7''.5 \times 8''.5$ with rms noise = 40.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 3 GHz map ([ROBUST 0]{}) at a resolution of $8''.7
\times 5''.8$, and rms noise = 20.4 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 3 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $13''.1 \times 8''.5$, and rms noise = 18.5 $\mu$Jy beam$^{-1}$. In black there are the polarization vectors. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise in all maps, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps made with comparable resolution.The rms levels of the red contours are 21.9 $\mu$Jy beam$^{-1}$.](s-r0-7-14-label.eps){width="100.00000%"}
![\[smaps.fig\] [*Top panel :* ]{} 3 GHz map ([UNIFORM]{}) at a resolution of $7''.5 \times 8''.5$ with rms noise = 40.1 $\mu$Jy beam$^{-1}$. [*Middle panel :* ]{} 3 GHz map ([ROBUST 0]{}) at a resolution of $8''.7
\times 5''.8$, and rms noise = 20.4 $\mu$Jy beam$^{-1}$. [ *Bottom panel :* ]{} 3 GHz map ([NATURAL, UVTAPER=\[0,20\]]{}) at a resolution of $13''.1 \times 8''.5$, and rms noise = 18.5 $\mu$Jy beam$^{-1}$. In black there are the polarization vectors. In all panels the white contours levels are -3, 3, 6, 12, 24 $\times$ rms noise in all maps, the red ones are the 7, 14 $\times$ rms of 1.4 GHz maps made with comparable resolution.The rms levels of the red contours are 21.9 $\mu$Jy beam$^{-1}$.](s-band-natural-new.eps){width="100.00000%"}
---------------- --------------------------------------- --------------
Weighting Component Flux density
\[mJy\]
**5.5 GHz maps** (Fig. \[cmaps.fig\])
Total 6.7$\pm$0.2
[UNIFORM]{} Core 6.7$\pm$0.2
IC 5337 0.2$\pm$0.1
Total 7.4$\pm$0.2
[ROBUST 0]{} Core 6.0$\pm$0.2
IC 5337 0.5$\pm$0.1
Total 8.2$\pm$0.2
[NATURAL]{} Core+jets 6.2$\pm$0.2
IC 5337 0.5$\pm$0.1
**3 GHz maps** (Fig. \[smaps.fig\])
Total 12.1$\pm$0.5
[UNIFORM]{} Core+jets 11.7$\pm$0.4
IC 5337 1.6$\pm$0.3
Total 14.6$\pm$0.4
Core+jets 12.0$\pm$0.4
Arc Noth 0.4$\pm$0.1
Arc South 0.5$\pm$0.1
Arc West 0.5$\pm$0.1
IC 5337 1.5$\pm$0.1
Total 16.5$\pm$0.5
Core+jets 10.9$\pm$0.3
Arc Noth 0.7$\pm$0.1
Arc South 0.8$\pm$0.1
Arc West 0.4$\pm$0.1
IC 5337 2.2$\pm$0.1
\[flulli.tab\]
---------------- --------------------------------------- --------------
: Flux density values of the radio sources of A2626, measured on the maps of Fig. \[cmaps.fig\]-\[smaps.fig\] inside the 3 $\times$ rms contours of each map.
Spectral Index maps
-------------------
By combining the new JVLA observation at 3 GHz with the VLA ones at 1.4 GHz obtained with the VLA in A+B configuration (G13), we produced a spectral index map of the extended emission of A2626. We produced the input maps by setting [weighting=UNIFORM, UVRANGE=0-40, UVTAPER=0-20, RESTORINGBEAM=$13'' \times 10''$, nterms=1, multiscale=\[0,5\]]{}. We improved the sensitivity to the faint, extended emission of the [UNIFORM]{} maps by applying a [UVTAPER]{} and by enlarging the beam to improve the SNR. The rms of the 1.4 GHz and 3 GHz maps are 31.9 and 29.1 $\mu$Jy beam$^{-1}$.
In calculating the spectral index value in each pixel of the map, we excluded regions where the 1.4 GHz brightness is $<3 \times$ rms level. We further corrected the spectral index map for the bandwidth effect described by @Condon_2015. Due to the large receiver bandwidth of the JVLA, the flux density the we measure on the 3 GHz maps may not coincide with the flux at 3.2 GHz, which is the arithmetical center of our band, but instead it may be the flux density at an equivalent frequency, $\nu_{u}$, defined as: $$\nu_{u}=\left[\frac{1}{\alpha+1}\left( \frac{\nu_{max}^{\alpha+1}-\nu_{min}^{\alpha+1}}{\nu_{max}-\nu_{min}} \right) \right]^{1/\alpha}
\label{condon.math}$$ where $\nu_{max}$ and $\nu_{min}$ (expressed in GHz) are the ends of the bandwidth and $\alpha$ is the spectral index. According to this relation, $\nu_{u}$ is a function of $\alpha$, which is, in turn, a function of the frequencies. In order to correct the spectral index map, which was computed between 1.4 and 3.2 GHz, we derived numerically the correction for the spectral indices. According to our bandwidth (2.4-4.0 GHz), we estimated a first set of equivalent frequencies $\nu_{u}^{i}$ for a set of spectral indices $\alpha_{i}$ between -0.1 and -7.0. We then corrected the spectral indices for the equivalent frequencies, thus obtaining a new set $\alpha_{i+1}$ that allowed us to estimate again a new set of equivalent frequencies $\nu_{u}^{i+1}$. We repeated this cycle until it converges to the final values $\nu_{u}^{N}$ and $\alpha_{N}$. We then obtained by numerical fit the relation $\alpha_{N}(\alpha_{i})$ that we finally applied to the each pixel of the uncorrected spectral index map with the task [immath]{} of CASA. With the $\alpha_{N}(\alpha_{i})$ relation we re-scaled also the relative spectral index error map. We report the corrected spectral index map, and the relative error map, in Fig. \[spix.fig\].
The spectral index map disentangles the extended emission in two major components, the unresolved core with a flat spectral index[^5] and the arcs with steep spectral index $\alpha<-2.5$, that is consistent with the recent results by @Kale_2017. We measured the mean spectral index inside the 1.4 GHz reference contours (Fig. \[smaps.fig\], middle panel) for the northern, southern and western arcs, finding $\alpha=-3.2 \pm 0.6$, $-3.0 \pm 0.4$ and $-2.6 \pm 0.6$, respectively. The unresolved core has a mean spectral index $\alpha=-0.7\pm0.1$.
{width="100.00000%"}
{width="100.00000%"}
Radiative age maps
------------------
The continuous radiative loss modifies the spectrum of the emitting particles by steepening it. So, it is possible to estimate the radiative age, $t_{rad}$, of the emitting plasma from the break frequency, $\nu_{br}$, of its synchrotron spectrum and the magnetic field, $B$, that produces the radio emission [e.g., Eq. 2 of @Murgia_1999][^6]. From this relation, it is possible to demonstrate that the radiative life-span, for a given $\nu_{br}$, is maximized by a magnetic field $B_{ml}=B_{CMB}/\sqrt{3}$ $\mu$G, where $B_{CMB}=3.25(1+z)^{2}$ $\mu$G is the equivalent magnetic field of the inverse Compton emission and $z$ is the redshift of the radio source. By integrating numerically the synchrotron emission spectrum, we derived a $t_{rad}(\alpha)$ relation for a population of particles with a energy distribution function $n(E)\propto E^{-\Gamma}$ and a magnetic field $B$ that is uniform and constant over the radio source. We assumed a $B=B_{ml}=2.1$ $\mu$G as magnetic field, and $\Gamma$=2.4 for the energy distribution function, that we derived from the hypothesis of the jet-precession model. If the plasma of the arcs was initially accelerated in an hot-spot, then it had an initial spectral index $\alpha=-0.7$ [e.g. @Meisenheimer_1997], and so $\Gamma=1-2\alpha=2.4$. Therefore, the $t_{rad}$ map exhibits the upper limit of the time required to the spectrum to steepen from $-0.7$ to the values that we observe in Fig. \[spix.fig\]. Due to our assumptions, it was not possible to evaluate the $t_{rad}$ in those regions which emit with a spectral index flatter than $-0.7$.
We computed the spectral map in Fig. \[spix.fig\] with the $t_{rad}(\alpha)$ relation and we obtained the radiative age map. Moreover, we obtained the relative error map on the radiative age from the spectral index error map by estimating the radiative age associated to the upper and lower limits of the spectral index map. We report the radiative age map and the relative error map in Fig. \[age.fig\]. To obtain an indicative estimate of the mean upper limit on the radiative age, we averaged the values of the $t_{rad}$ map inside the 1.4 GHz reference contours (Fig. \[smaps.fig\], middle panel). We measured $t_{rad} =$ 157 Myr, 148 Myr and 136 Myr for the northern, southern and western arcs, respectively, which can be considered as an estimate of the maximum time elapsed since the particle acceleration. The average radiative age error, $\sigma_{t}$, on those regions is $\sim$12 Myr.
{width="100.00000%"}
{width="100.00000%"}
Discussion
==========
{width="110.00000%"}
{width="110.00000%"}
{width="110.00000%"}
{width="100.00000%"}
The new JVLA observations presented in this work add important information to the general picture of A2626, and allowed us to test for the first time the jet-precession model.
The spectral index map in Fig. \[spix.fig\] exhibits that the arcs are steep-spectrum radio sources, with $\alpha\leq-2.5$. We do not observe $\alpha\simeq-0.7$ at any of the arcs ends, which should be the initial spectral index of the plasma, so we argue that the activity has ceased. The radiative map that we provide (Fig. \[age.fig\]) does not show the real age of the arcs, but only an upper limit of it, because we do not know the real strength of the magnetic field in the arc regions. However, it is useful to constrain the time-scale of the jet-precession by considering the difference between the radiative age at the ends of each arc, $\Delta
t_{rad}$. This can be considered as a measure of the time required by the jets to cover the arc length, thus it is an estimate of the time-scale of the precession period of the jets.
According to the jet-precession scenario, the relativistic plasma of the arcs was accelerated gradually along the structures by a pair of precessing jets. Therefore, the spectral index distribution along the arcs may reflect the gradual formation of the structures, by showing a monotonic steepening trend from one end to the other, where the emission with the flattest spectral index comes from the most recently accelerated plasma. Moreover, the trends along the northern and southern arcs should be anti-symmetric due to the geometry of the jet-precession (if the northern arc is created from west to east, then the southern arc must be created from east to west), and they should be observable also in the radiative age maps.
Observationally, the spectral index map (Fig. 3) does not show any clear evolution in the spectrum along the arcs, hinting at the absence of the age progression expected by the precession model. In order to quantitatively confirm this result, we sampled both the spectral index and the radiative age maps with the sampling reported in Fig. \[trend.fig\] (bottom-right panel) and we measured the values, with the relative errors, of each sample. The samples are as large as the beam ($13'' \times 10''$) and cover the area delimited by the 1.4 GHz reference contours shown in Fig. \[smaps.fig\], middle panel. We report the result of our analysis on each arc in Fig. \[trend.fig\].
We estimated the radiative age trend also by assuming the reference equipartition magnetic field. From the formula provided by @Govoni-Feretti_2004 [ Eq. 22], by assuming that the spectral index of the arcs at $\leq$100 MHz is $\sim -0.7$[^7], that the arcs have a cylindrical volume and that the energy in electrons and protons is equally distributed ($k$=1), we estimated a equipartition field strength of $B_{eq} = 12$ $\mu$G. Depending on the magnetic field, the mean radiative age of the arcs decreases from $\sim$140 Myr to $\sim$10 Myr, whereas the mean error on it decreases from $\sim$10 Myr to $\sim$1 Myr. Due to the uncertainties in the assumptions made to derive the equipartition magnetic field, in the following discussion we consider the estimates of radiative age upper limits provided by $B_{ml}$.
The plots in Fig. \[trend.fig\] show that the radio arcs do not exhibit significant spectral index or radiative age trends. By considering the uncertainties of our measures, we derived an estimate of the maximum difference between the radiative time scale $\Delta t_{rad}$ of 22 Myr, 26 Myr and 22 Myr for the northern, southern and western arcs, respectively. This corresponds to an estimate of the upper limit on the time required by the jets to cover the arc length.
In order to test the jet-precession model, we compared our result with a theoretical estimate of the precession period. @Wong_2008 argued that the jet-precession may be triggered by the mutual gravitational interactions of the optical cores of IC5338. The precession period, $\tau_{prec}$, of the jets of a binary system in the center of a galaxy cluster can be estimated with the relation proposed by @Pizzolato_2005. We re-scaled the relation that they provide according to the parameters that we observe:
$$\tau_{prec}=1.6\cdot10^{4} \left( \frac{M}{10^{8}M_{\odot}} \right)^{-1/2}\left( \frac{a}{1\text{ kpc}} \right)^{3}\left( \frac{a_{D}}{1\text{ pc}} \right)^{-3/2}\frac{(1+q)^{1/2}}{q\text{ }cos\theta} \text{Gyr}$$
where $M$ and $q$ are, respectively, the sum and the ratio of the masses of the black holes, $a$ is their distance, $a_{D}$ is the radius of the accretion disk and $\theta$ is the angular difference between the accretion disk and the plane of the orbit of the black holes. In the case of A2626, the masses of the objects and the accretion disk radius are unknown, so we assumed a mass of $10^{8}$ M$_{\odot}$ (so $M=$2$\times 10^{8}$ M$_{\odot}$, $q$=1), that is the typical mass of the supermassive black holes (SMBH) located at the center of cD galaxies, and the typical accretion disk radius $a_{D}=1$ pc. The observed projected distance between the cores, which is a lower limit to their real distance, is $a$=3.3 kpc. Moreover, also $\theta$ is unknown, therefore we assume for simplicity that the accretion disk is on the same plane of the orbit ($\theta=0$). From these assumptions, we estimated a lower limit for the precession time $\tau_{prec}\gtrsim 10^{5}$ Gyr, that is not in agreement with the $\Delta t_{rad}$ that we measure. This estimate is also much greater than the age of the universe, thus challenging the hypothesis that the two nuclei are currently producing the jet-precession. More specifically, from the formula proposed by @Pizzolato_2005, the distance between the two cores that admits a precession period $\tau_{prec}\simeq \Delta t_{rad}$ is: $$a\leq10 \left( \frac{\tau}{20 \text{Myr}}\right)^{1/3}\left( \frac{M}{10^{8} \text{M}_{\odot}}\right)^{1/6}\left( \frac{a_{d}}{1 \text{pc}}\right)^{1/2}\left( \frac{qcos\theta}{(1+q)^{1/2}}\right)^{1/3}\text{ pc}$$ Consequently, at the epoch of particle acceleration (corresponding to $t_{rad} \approx 150$ Myr ago) the two cores had to have a separation of just $\approx$ few pc. Alternatively, the precession should have been generated by the interactions with a secondary SMBH which is yet undetected (likely not active), rather than with the northern optical core of IC5338. Therefore, we argue that the dynamical process behind the origin of the radio arcs may be more complex than the scenario initially proposed by @Wong_2008. At the same time, however, the lack of a clear optical core candidate makes the precession model less appealing, and other scenarios for the origin of the radio arcs should be investigated in the future.
We finally note that, by combining the results of our spectral analysis with the spectral maps provided by @Kale_2017, we may argue that the radio spectrum of the arcs has a steep spectral index $<-2.5$ from 610 MHz to 3 GHz, and this is not in agreement with the expected evolution of the radio spectrum of the hot-spots. The AGN-driven nature of the radio arcs through jet precession may be further challenged by the fact that we observe four arcs, which would require the presence of [*two pairs*]{} of radio jets, powered by [*two radio cores*]{} inside IC5338. However, only one radio AGN (with jets) is currently observed, whereas the other one remains undetected (although it is still possible that it was active in the past and has recently shut down).
Summary and Conclusion
======================
We presented the results of new JVLA observations of A2626 at 3 GHz and 5.5 GHz, which were requested in order to test the jet-precession model for the origin of the radio arcs. We achieved the first detection at high frequency of an extended component of radio emission at 5.5 GHz and we observed for the first time the radio arcs at 3 GHz. Moreover, we observed the extended emission of IC5337 at 3 GHz and 5.5 GHz.
By combining the archival 1.4 GHz observation (G13) with the new one at 3 GHz, we produced new spectral index maps that allowed us to disentangle the extended emission of the cluster in two main components – the inner, flat-spectrum emission ($\alpha \simeq -0.7$), and the ultra-steep spectrum ($\alpha \leq -2.5$) arcs. In order to find constraints for the jet-precession model, we converted the spectral index map into radiative age maps and we studied the trends of spectral index and radiative time along the arcs. We found that the radio arcs do not exhibit significant spectral index or radiative age evolution. By considering the uncertainties on our measures, we estimated an upper limit of the precession period $\Delta t_{rad}\leq 26$ Myr. We also estimated a mean radiative age of $t_{rad} \leq 150$ Myr, that may be considered as a measure of the time elapsed since the hot-spot ceased to accelerate the particles.
We then compared our results with a theoretical precession period estimated for the current kinematics of the IC5338 system, finding a disagreement. Therefore, we argue that the dynamics of the cores that produced the jet-precession and created the radio arcs may be more complex than the scenario proposed by @Wong_2008. On the other hand, our results put strong constraints for every future model which relates the origin of the radio arcs to the activity of the inner AGN, or indicate a different origin of the radio arcs.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the referee for the prompt response and constructive comments. AI thanks R. Paladino for helpful advices during the data reduction with CASA. We thank A. Bonafede, M. Bondi and D. Dallacasa for useful discussions. MG, GB, GG, LF acknowledge partial support from PRIN-INAF 2014.
[^1]: In this work the radio spectral index $\alpha$ is defined such as $S \propto
\nu^{\alpha}$
[^2]: [ https://ned.ipac.caltech.edu/]{}
[^3]: e.g., see the Tutorials,\
[https://casaguides.nrao.edu/index.php/]{}
[^4]: We note that the low resolution map is affected by the beam depolarization effect, so high-resolution observations are needed to confirm the lack of polarized emission from the radio arcs at 3 GHz.
[^5]: As a word of caution we note that the spectral index in the central region may be unreliable due to the core variability, because the observations are separated by five years.
[^6]: Note that we are only considering the effect of radiative losses on the evolution of the energy of electrons
[^7]: The classical equipartition formula [@Pacholczyk_1970] is obtained for a power law energy distribution of electrons. However, we detect a synchrotron spectrum whose properties are affected by radiative ageing. In order to correct for this effect, we estimated the spectrum of the arcs at lower frequencies, where ageing is not important and the assumption of power-law energy distribution is satisfied. In particular, we calculated the emission at frequency 100 MHz by re-scaling the spectrum measured at 1.4 GHz according to the synchrotron spectrum with a break frequency derived from the spectral slope between 1.4 GHz and 3.0 GHz, obtaining an estimated flux density at 100 MHz of $\sim$260 mJy.
|
---
abstract: 'We compute three-loop corrections to the Higgs-gluon form factor, incorporating the top quark mass dependence. Our method is based on the combination of expansions around the top threshold and for large top quark mass, using conformal mapping and Padé approximation to describe the form factor over the full kinematic range.'
author:
- |
Joshua Davies$^{(a)}$, Ramona Gröber$^{(b)}$, Andreas Maier$^{(c)}$,\
Thomas Rauh$^{(d)}$, Matthias Steinhauser$^{(a)}$,\
[*(a) Institut f[ü]{}r Theoretische Teilchenphysik,*]{}\
[*Karlsruhe Institute of Technology (KIT)*]{}\
[*Wolfgang-Gaede Straße 1, 76128 Karlsruhe, Germany*]{}\
[*(b) Humboldt-Universität zu Berlin, Institut für Physik,*]{}\
[*Newtonstr. 15, 12489 Berlin, Germany*]{}\
[*(c) Deutsches Elektronen-Synchrotron, DESY,*]{}\
[*Platanenallee 6, 15738 Zeuthen, Germany*]{}\
[*(d) Albert Einstein Center for Fundamental Physics,*]{}\
[*Institute for Theoretical Physics, University of Bern,*]{}\
[*Sidlerstrasse 5, CH-3012 Bern, Switzerland*]{}
title: |
-3cm
DESY 19-092\
HU-EP-19/13\
P3H-19-012\
TTP19-014\
1.5cm Top quark mass dependence of the Higgs-gluon form factor at three loops
---
Introduction
============
The precise measurement of the properties of the Higgs boson, in particular the coupling strength to other particles and to itself, will be among the main focuses in particle physics in the coming years. The success of this enterprise crucially depends on the accuracy of the predictions provided by the theory community.
A quantity which is available to high perturbative order is the total cross section for the production of a Higgs boson at the Large Hadron Collider (LHC). For a comprehensive collection of relevant works we refer to Ref. [@deFlorian:2016spz], but we remark here that QCD corrections including the exact dependence on the top quark mass, $m_t$, have been available at next-to-leading order (NLO) for about 25 years [@Spira:1995rr]. At higher orders only approximate results are available; at NNLO the infinite top quark mass results from Refs. [@Harlander:2002wh; @Anastasiou:2002yz; @Ravindran:2003um] have been complemented by power-suppressed terms in the inverse top quark mass in [@Harlander:2009mq; @Pak:2009dg; @Harlander:2009my; @Pak:2011hs]. The N$^3$LO result has been obtained in the $m_t \to \infty$ limit in [@Anastasiou:2016cez; @Mistlberger:2018etf].
In Ref. [@deFlorian:2016spz] several sources of uncertainties have been identified for the prediction of the total cross section. Among them is that of the exact top quark mass dependence of the NNLO corrections which has been estimated to be 1%. In this paper we provide results for the Higgs-gluon form factor at three-loop order which constitutes the virtual corrections to the production cross section. Thus the findings of this paper help to eliminate the aforementioned uncertainty to a large extent. The Higgs-gluon form factor is also an important ingredient for processes where the relevant energy in the fermion loops reaches values close to or above the top quark threshold and the infinite top quark mass limit cannot be applied anymore. This concerns, e.g., Higgs boson pair production via $gg\to H^\star \to HH$ or the measurement of the Higgs boson width from off-shell production of $Z$ boson pairs in gluon fusion via $gg\to H^\star \to ZZ$ [@Caola:2013yja]. The exact dependence on the fermion mass in the loop is also important for numerous theories beyond the Standard Model, which often contain additional heavier Higgs bosons.
At two-loop order exact results for the form factor are known from Refs. [@Spira:1995rr; @Harlander:2005rq; @Anastasiou:2006hc; @Aglietti:2006tp]. However, at three loops only expansions for large top quark mass [@Harlander:2009bw; @Pak:2009bx] and non-analytic terms in the expansion around the top threshold up to ${\cal O}(1-z)$ [@Grober:2017uho] are known, where $$z=\frac{\hat{s}}{4m_t^2}
\label{eq::zdefn}$$ with $\sqrt{\hat{s}}$ being the partonic center-of-mass energy. For later convenience we also introduce $\bar{z}=1-z$. In the next section we describe our method which we use to combine these expansions in order to obtain results for the form factor valid for all space- and time-like momentum transfers. In Section \[sec::res\] we discuss our results and Section \[sec::con\] contains a brief summary.
Method
======
The method we use for the construction of the top quark mass dependence of the Higgs-gluon form factor is based on the efficient combination of information from the large top quark mass expansion (LME) ($z\to 0$) and knowledge from the threshold where $\hat{s}\approx 4m_t^2$ ($z\to 1$), using conformal mapping and Padé approximation. The procedure was developed in Ref. [@Baikov:1995ui] (see also [@Broadhurst:1993mw; @Fleischer:1994ef]) in order to compute a certain class of four-loop contributions to the muon anomalous magnetic moment. In Refs. [@Chetyrkin:1995ii; @Chetyrkin:1998ix] the method was extended to QCD corrections with the aim to compute NNLO correction to the total cross section $\sigma(e^+e^-\to\mbox{hadrons})$. A further refinement of the method has been developed in Refs. [@Hoang:2008qy; @Kiyo:2009gb] where order $\alpha_s^3$ corrections to $\sigma(e^+e^-\to\mbox{hadrons})$ have been computed. In these references additional parameters were introduced which allow one to generate a larger number of Padé approximations and thus obtain more reliable uncertainty estimates. The systematic improvement of the Padé approximations when increasing the number of input terms has been studied in Ref. [@Maier:2017ypu]. In Ref. [@Grober:2017uho] the method has been used to obtain two-loop corrections for the three form factors relevant for Higgs boson pair production.
In the following we briefly describe the application of this method to the form factor entering the interaction of a Higgs boson and two gluons. We parameterize the corresponding amplitude as $$\begin{aligned}
{\cal A}^{\mu\nu}_{ab}(gg \to H)
&=& \delta_{ab} \frac{y_t}{\sqrt{2}m_t} \frac{\alpha_s}{\pi}
T_F
\left(q_1\cdot q_2 g^{\mu\nu} - q_1^\nu q_2^\mu\right)
F_\triangle(z)
\,,\end{aligned}$$ where $q_1$ and $q_2$ are the external momenta of the gluons with polarization vectors $\varepsilon^\mu(q_1)$ and $\varepsilon^\nu(q_2)$, respectively. $y_t=\sqrt{2}m_t/v$ is the top quark Yukawa coupling, $v$ is the vacuum expectation value, $a$ and $b$ are adjoint colour indices, $T_F=1/2$ and $\hat{s}=(q_1+q_2)^2=2q_1\cdot q_2$. It is convenient to define the perturbative expansion of $F_\triangle$ as $$\begin{aligned}
F_\triangle &=& F_\triangle^{(0)} + \frac{\alpha_s}{\pi} F_\triangle^{(1)}
+ \left(\frac{\alpha_s}{\pi}\right)^2 F_\triangle^{(2)} + \cdots
\,,\end{aligned}$$ where $\alpha_s \equiv \alpha_s^{(5)}(\mu)$ is the strong coupling constant with five active flavours evaluated at the renomalization scale $\mu$. Sample Feynman diagrams contributing to ${\cal A}^{\mu\nu}_{ab}(gg \to H)$ up to three loops can be found in Figure \[fig::diags\].
![\[fig::diags\]One-, two- and three-loop Feynman diagrams contributing to $F_\triangle$. Solid, curly and dashed lines represent quarks, gluons and Higgs bosons, respectively.](figs_gghFF/gghtri123.pdf){width="\textwidth"}
The one-loop result, $F^{(0)}_\triangle$, is finite. At two-loop order we renormalize the gluon wave function and the top quark mass in the on-shell scheme and the strong coupling constant in the $\overline{\rm MS}$ scheme. Note that the ultra-violet renormalized form factor still contains infra-red divergences which cancel against contributions from real radiation, in order to form finite physical quantities. The structure of the infra-red divergences is universal and has been studied in detail in the literature [@Catani:1998bh]. In our case finite form factors are obtained via $$\begin{aligned}
F^{(1),\rm fin}_\triangle &=& F_\triangle^{(1)} - \frac{1}{2} I^{(1)}_g F_\triangle^{(0)}\,,\nonumber\\
F^{(2),\rm fin}_\triangle &=& F_\triangle^{(2)} - \frac{1}{2} I^{(1)}_g F_\triangle^{(1)}
- \frac{1}{4} I^{(2)}_g F_\triangle^{(0)}\,,
\label{eq::FF_IR}\end{aligned}$$ where $I^{(1)}_g$ and $I^{(2)}_g$ can be found in Refs. [@Catani:1998bh; @deFlorian:2012za]. In order to fix the notation we provide an explicit expression only for $I^{(1)}_g$ which is given by $$\begin{aligned}
I^{(1)}_g &=&
{} - \left(\frac{\mu^2}{- \hat{s} -i\delta}\right)^\epsilon
\frac{e^{\epsilon\gamma_E}}{\Gamma(1-\epsilon)}
\frac{1}{\epsilon^2}
\Big[
C_A + 2\epsilon\beta_0
\Big]\,,\end{aligned}$$ with $\beta_0 = (11C_A - 4 T_Fn_l)/12$ where $C_A=3$, $T_F=1/2$ and $n_l$ is the number of massless quarks. We work in $d=4-2\epsilon$ dimensions and assume that $\delta$ is an infinitesimal small parameter. We apply the method described below to $F^{(1),\rm fin}_\triangle$ and $F^{(2),\rm fin}_\triangle$.
In the following we briefly discuss the input for the limits $z\to0$ and $z\to1$ used for the construction of the Padé approximants. For the renormalization scale we choose $\mu^2=-s$ since the $\mu$ dependence can easily be reconstructed from the one- and two-loop expressions, which are known exactly, see Ref. [@DavSte]. Furthermore, we set all colour factors to their numerical values and only keep $n_l$ as a parameter. The large-$m_t$ expansion of the three-loop form factor up to order $z^4$ has been computed in Ref. [@Harlander:2009bw; @Pak:2009bx] and the $z^5$ and $z^6$ terms are available from Ref. [@DavSte]. The analytic expressions read $$\begin{aligned}
F_{\triangle}^{(0)} &=&
\frac{4}{3}
+ \frac{14}{45} z
+ \frac{8}{63} z^2
+ \frac{104}{1575} z^3
+ \frac{2048}{51975} z^4
+ \frac{4864}{189189} z^5
+ \frac{512}{28665} z^6
+ {\cal O}(z^7)\,, \nonumber\\ \vphantom{\Bigg[}
F_{\triangle}^{(1),\rm fin} &=&
\frac{11}{3}
+ \frac{1237}{810} z
+ \frac{35726}{42525} z^2
+ \frac{157483}{297675} z^3
+ \frac{2546776}{7016625} z^4
+ \frac{194849538824}{737482370625} z^5
\nonumber\\&&\mbox{}
+ \frac{385088204192}{1917454163625} z^6
+ {\cal O}(z^7)\,, \nonumber\\ \vphantom{\Bigg[}
F_{\triangle}^{(2),\rm fin} &=&
-\frac{253 \zeta (3)}{24}+\frac{3941}{108}+\frac{19 \pi ^2}{12}+\frac{\pi ^4}{96}
+ \frac{19}{12} L_s
\nonumber\\ &&\mbox{}
+ n_l\Bigg(
-\frac{17 \zeta (3)}{36}-\frac{3239}{648}-\frac{47 \pi ^2}{432}
+ \frac{4}{9} L_s
\Bigg)
+ n_l^2 \frac{\pi ^2}{648}\nonumber\\ &&\mbox{}
\nonumber\\ &&\mbox{}+\Bigg[
\frac{9290881 \zeta (3)}{103680}-\frac{44326367}{466560}+\frac{623 \pi ^2}{1080}+\frac{7 \pi ^4}{2880}+\frac{28}{405} \pi ^2 \log (2)
+ \frac{8261}{3240} L_s
\nonumber\\ &&\mbox{} + n_l\Bigg(
-\frac{119 \zeta (3)}{1080}-\frac{107087}{291600}-\frac{259 \pi ^2}{4320}
-\frac{169}{1080} L_s
\Bigg)
+ n_l^2 \frac{7 \pi ^2}{19440}\Bigg]
z
\nonumber\\ &&\mbox{}+\Bigg[
\frac{7037623781 \zeta (3)}{69672960}-\frac{82500975779}{731566080}+\frac{121 \pi ^2}{378}+\frac{\pi ^4}{1008}+\frac{32}{567} \pi ^2 \log (2)
\nonumber\\ &&\mbox{}
+ \frac{253549}{170100} L_s
+ n_l\Bigg(
-\frac{17 \zeta (3)}{378}-\frac{6385481}{53581500}-\frac{25 \pi ^2}{648}
-\frac{4133}{36450} L_s
\Bigg)
\nonumber\\ &&\mbox{}
+ n_l^2 \frac{\pi ^2}{6804}\Bigg]
z^2
+\Bigg[
\frac{650760513719 \zeta
(3)}{412876800}-\frac{1740869750908152049}{921773260800000}+\frac{221
\pi ^2}{1050}
\nonumber\\ &&\mbox{}
+\frac{13 \pi ^4}{25200}
+\frac{208 \pi ^2 \log (2)}{4725}
+ \frac{804644}{826875} L_s
+ n_l\Bigg(
-\frac{221 \zeta (3)}{9450}-\frac{6383750249}{112521150000}
\nonumber\\ &&\mbox{}
-\frac{3107 \pi ^2}{113400}
-\frac{1147037}{14883750} L_s
\Bigg)
+ n_l^2 \frac{13 \pi ^2}{170100}\Bigg]
z^3
\nonumber\\ &&\mbox{}+\Bigg[
\frac{193543938976537 \zeta
(3)}{37158912000}-\frac{6978205934887756008911}{1115345645568000000}+\frac{4736
\pi ^2}{31185}+\frac{16 \pi ^4}{51975}
\nonumber\\ &&\mbox{}
+\frac{16384 \pi ^2 \log (2)}{467775}
+ \frac{33498106}{49116375} L_s
+ n_l\Bigg(
-\frac{2176 \zeta (3)}{155925}-\frac{2197298833}{72937816875}
\nonumber\\ &&\mbox{}
-\frac{3232 \pi ^2}{155925}
-\frac{20932}{382725} L_s
\Bigg)
+ n_l^2 \frac{64 \pi ^2}{1403325}\Bigg]
z^4
\nonumber\\ &&\mbox{}+\Bigg[
\frac{2460310706266276921 \zeta
(3)}{81155063808000}-\frac{159929147625953730170902566067}{4389031448658778521600000}
\nonumber\\ &&\mbox{}
+\frac{9424 \pi ^2}{81081}+\frac{38 \pi ^4}{189189}+\frac{48640 \pi ^2 \log (2)}{1702701}
+ \frac{945911804923}{1877227852500} L_s
+ n_l\Bigg(
-\frac{5168 \zeta (3)}{567567}
\nonumber\\ &&\mbox{}
-\frac{22552503119716043}{1395235522161731250}-\frac{27892 \pi ^2}{1702701}
-\frac{48324340168}{1191317675625} L_s
\Bigg)
+ n_l^2 \frac{152 \pi ^2}{5108103}\Bigg]
z^5
\nonumber\\ &&\mbox{}+\Bigg[
\frac{15128773883548934558969 \zeta
(3)}{114266329841664000}
+\frac{2656 \pi ^2}{28665}+\frac{4 \pi ^4}{28665}+\frac{2048 \pi ^2 \log (2)}{85995}
\nonumber\\ &&\mbox{} \vphantom{\Bigg[}
-\frac{13560383230749413568271118392175429}{85205730523295753699328000000}
+ \frac{339242844181}{871570074375} L_s
\nonumber\\ &&\mbox{} \vphantom{\Bigg[}
+ n_l\Bigg(
-\frac{544 \zeta (3)}{85995}
-\frac{2085146760850288}{259115168401464375}-\frac{3448 \pi ^2}{257985}
-\frac{35895528824}{1150472498175} L_s
\Bigg)
\nonumber\\ &&\mbox{}
+ n_l^2 \frac{16 \pi ^2}{773955}\Bigg]
z^6
+ {\cal O}(z^7)\,,
\label{eq::Flme}\end{aligned}$$ where $L_s = \log(-4z-i0)$ and $\zeta(n)$ is the Riemann zeta function.
The expansion of the three-loop form factor around threshold can be found in Eq. (50) of Ref. [@Grober:2017uho] where it was determined using the Coulomb resummed $P$-wave Green function [@Beneke:2013kia]. For convenience we reproduce the analytic expression together with the one- and two-loop results which are given by $$\begin{aligned}
F_\triangle^{(0)} \mathop{\asymp}\limits^{z\to1} & \,
2 \pi (1-z)^{3/2}
+ \frac{13 \pi}{3} (1-z)^{5/2}
+ \mathcal{O}\left((1-z)^{7/2}\right)\,, \nonumber\\
F_\triangle^{(1),\text{fin}} \mathop{\asymp}\limits^{z\to1} & \,
\frac{4 \pi ^2}{3}(1-z)\log(1-z)
- \frac{\pi}{36} \left(124+15 \pi ^2\right) (1-z)^{3/2}
+ \frac{8 \pi ^2}{9} (1-z)^2 \log(1-z) \nonumber\\
& + \frac{\pi}{216} \Big[2252-117 \pi ^2-2112 \log (2)-672
\log(1-z)\Big] (1-z)^{5/2} \nonumber\\
& - \frac{28\pi ^2}{45} (1-z)^3 \log(1-z)
+ \mathcal{O}\left((1-z)^{7/2}\right)\,, \nonumber\\
F_\triangle^{(2),\text{fin}} \mathop{\asymp}\limits^{z\to1} &
- \frac{8\pi^3}{27} \left(3+\pi ^2\right) \sqrt{1-z} \nonumber\\
& + \frac{\pi^2}{54}\Big[\left(458-15 \pi^2-44 n_l\right)
\log(1-z)-(99-6 n_l) \log^2(1-z)\Big](1-z) \nonumber\\
& + \mathcal{O}\left((1-z)^{3/2}\right)\,.
\label{eq::Fthr}\end{aligned}$$ where “$\mathop{\asymp}$” denotes that analytic terms in $(1-z)$ have been dropped on the right-hand side.
The information provided in Eqs. (\[eq::Flme\]) and (\[eq::Fthr\]) is used to construct approximations of the form factor. We first subtract the logarithmic contributions for $z\sim 1$ and define $$\tilde{F}_\triangle = F_\triangle - F_\triangle^{\rm sub}
\,,
\label{eq::Fthrsub}$$ where $F_\triangle^{\rm sub}$ is constructed in such a way that $\tilde{F}_\triangle$ remains an analytic function for $|z| < 1$, while the threshold expansion of $\tilde{F}_\triangle$ is free of logarithms up to $(1-z)^{3/2}$. Such a subtraction function $F_\triangle^{\rm sub}$ can be obtained using the vacuum polarization as a building block, see [@Grober:2017uho] for details of the construction. For explicit examples for $F_\triangle^{\rm sub}$ we refer to the sample Padé approximants in the ancillary file [@progdata]. Note that also in the limit $z\to0$ $F_\triangle$ develops logarithmic divergences which manifest in the linear $L_s$ term in Eq. (\[eq::Flme\]). Whereas in Ref. [@Chetyrkin:1998ix] these contributions are also subtracted, here we instead construct separate Padé approximants for the $L_s$-independent term and for the coefficient of $L_s$, as discussed in [@Grober:2017uho].
Next we apply a conformal mapping $$\begin{aligned}
z &=& \frac{4\omega}{(1+\omega)^2}
\,,\end{aligned}$$ to transform the $z$ plane into the interior of the unit circle in the $\omega$ plane; the time-like momentum regions $z\in[0,1]$ and $z\in[1,\infty]$ with $\mbox{Im}(z)>0$ are mapped to $\omega\in[0,1]$ and the upper semi-circle, respectively.
At this point we construct Padé approximants in the variable $\omega$. They have the form $$[n/m](\omega) = \frac{\sum\limits_{i=0}^n a_i \omega^i}{1 + \sum\limits_{j=1}^m b_j \omega^j}
\label{eq:Pade_ansatz}
\,,$$ where $n+m$ is fixed by the number of input terms from the large top mass and threshold expansions. In our case we have seven terms for $z\to 0$ and one for $z\to1$ which is sufficient to determine eight coefficients in Eq. (\[eq:Pade\_ansatz\]), i.e. Padé approximants for $n+m=7$. More precisely, we construct Padé approximants for the rescaled form factor $$[n/m](\omega) \simeq \left[1+a_R\,z(\omega)\right] \tilde{F}_\triangle(z(\omega)),
\label{eq:Pade_rescaling}$$ where $a_R$ is a real parameter. This removes the spurious condition $F_\triangle(z\to\infty)=0$ introduced by the definition of the form factor through $\mathcal{A}_{gg\to H}\propto zF_\triangle(z)$ and provides a means to test the stability of the solutions through variation of $a_R$. As discussed in [@Grober:2017uho] we only use the diagonal and next-to-diagonal Padé approximants which are $[5/2],[4/3],[3/4]$ and $[2/5]$ in the case that seven large top quark mass expansion terms and one term from the threshold expansion are taken into account. In Section \[sec::res\] we also show results which only incorporate LME terms up to $z^4$, for which we construct the Padé approximants $[4/1],[3/2],[2/3]$ and $[1/4]$.
By construction the Padé approximants develop poles in the complex $\omega$ plane. In the following we discuss our criteria which exclude approximants with poles too close to the physical region. For this discussion we have to distinguish space-like and time-like momentum regions. For $z>0$ we exclude all approximants which contain poles $\omega_0$ in the region $$\text{Re}(z(\omega_0)) \geq -2
\quad\&\quad
|\omega_0|\leq1.2\,,
\label{eq:pole_crit}$$ as they can cause unphysical behaviour in the approximation. We find that poles in the entire complex plane in $z$, i.e. in the unit disc $|\omega|\leq1$, cannot be excluded as this leads to the exclusion of all Padé approximants. In those cases where the approximants show obviously unphysical resonances we moderately increase the exclusion region. This concerns the coefficient of $L_s$ for $F_\triangle^{(2,0),\rm fin}(z)$ (cf. Eq. (\[eq::split\])) where we use $$\text{Re}(z(\omega_0)) \geq -2
\quad\&\quad
|\omega_0|\leq1.3\,.
\label{eq:pole_crit_2}$$ For each choice of $[n/m]$ we aim to construct 20 Padé approximants by choosing $a_R$ in Eq. (\[eq:Pade\_rescaling\]) randomly in the range $[0.1,10]$, leading to a maximum of 80 approximants. The mean and standard deviation of this set are used as the central value and uncertainty estimate, respectively. For some choices of $\{n,m\}$ Padé approximants satisfying criteria (\[eq:pole\_crit\]) and (\[eq:pole\_crit\_2\]) could not be found, however, we checked that at least 40 approximants remain in all cases. For such sets of fewer than 80 approximants we increase our uncertainty estimate by the ratio of the maximal number of Padé approximants (80) over the actual number in the set.
For space-like momenta our exclusion region is defined by $$\text{Re}(z(\omega_0)) \leq 2
\quad\&\quad
|\omega_0|\leq1.2\,,
\label{eq:pole_crit_space}$$ and negative values of $a_R$ in the range $[-10,-0.1]$ are chosen.
\[sec::res\]Results
===================
![\[fig:Ftri2LoopDiffs\]Differences between our approximations and the exact result. The input used is shown in the legend of each panel. The dashed lines correspond to the real part of the LME approximation up to order $z^2$, $z^4$ and $z^6$. The lower left panel corresponds to the lower panel of Figure \[fig:Ftri2Loop\].](figs_gghFF/Ftri2LoopDifferenceGridPlot.pdf){width="\textwidth"}
Before discussing the three-loop results we apply the method described in the previous section to the two-loop form factor, for which we can compare to the exact expressions [@Spira:1995rr; @Harlander:2005rq; @Anastasiou:2006hc; @Aglietti:2006tp].
We show in Figure \[fig:Ftri2Loop\] that the exact result for the two-loop form factor can be reproduced very well with the same amount of information that is available at three loops. The shaded region is spanned by the standard deviation w.r.t. to the mean value of a set of 20 approximants for each considered set $\{n,m\}$. These approximants are available in the ancillary file [@progdata]. Figure \[fig:Ftri2LoopDiffs\], where the difference between the exact result and the approximations is shown, demonstrates that the approximation can be systematically improved by including more expansion coefficients. We compare the results based on the input used in Figure \[fig:Ftri2Loop\] (lower left panel) to results where fewer expansion coefficients for large top quark masses are used (upper left panel). Furthermore, we also show results where additional information from threshold is incorporated in the construction of the Padé approximations (panels on the right).
Our approximation of the three-loop form factor is shown in Figure \[fig:Ftri3Loop\] and represents the main result of this paper. At three loops the LME and threshold coefficients develop terms linear in $L_s = \log(-4z-i0)$. We construct separate Padé approximants for the coefficient such that we obtain an approximation of the form $$F_\triangle^{(2),\rm fin}(z(\omega)) \simeq
\frac{[n/m]_0(\omega)
+ F_\triangle^{(2),\rm sub}}{1+a_{R,0} z(\omega)}
+ \frac{[k/l]_1(\omega)L_s}{1+a_{R,1} z(\omega)}
\,,
\label{eq:Ftri3l_Pade}$$ where the subscripts indicate the power of $L_s$. Note that the Padé approximants of the $L_s$-independent and linear-$L_s$ term are averaged independently using separate values of $a_R$. The threshold subtraction (cf. Eq. (\[eq::Fthrsub\])) is only needed for the first term in Eq. (\[eq:Ftri3l\_Pade\]). The lower panel in Figure \[fig:Ftri3Loop\] shows the differences from the central values (obtained using seven expansion terms for small $z$) both with seven and five input terms from the large top quark mass expansion as solid and dashed boundaries of the uncertainty bands, respectively. One observes over the whole range in $z$ (except for a small region for $z\approx 10$) that the solid bands lie within the dashed band. Below threshold ($z=1$) our method results in tiny uncertainties for both the real and imaginary parts of the form factor. For $1\le z\le2$ the form factor is numerically large and we thus observe small relative uncertainties. Although the absolute uncertainty becomes larger for higher values of $z$ we can provide a good approximation with an uncertainty which is sufficiently small for phenomenological applications.
In order to facilitate the comparison with a future exact calculation we split our three-loop result according to the number of light fermions and write $$F_\triangle^{(2),\rm fin}(z) =
F_\triangle^{(2,0),\rm fin}(z)
+ n_l F_\triangle^{(2,1),\rm fin}(z)
+ n_l^2F_\triangle^{(2,2),\rm fin}(z)\,,
\label{eq::split}$$ where $n_l=5$ is the number of light flavors. Note that $F_\triangle^{(2,0),\rm fin}(z)$ contains contributions with closed massive loops, which are numerically less important than the $n_l$ terms. There are no three-loop vertex diagrams which contain two closed fermion loops; $F_\triangle^{(2,2),\rm fin}(z)$ is completely determined by the infra-red subtraction terms. In fact, it is proportional to $F_\triangle^{(0)}$ and we will not discuss it further.
The results for $F_\triangle^{(2,0),\rm fin}(z)$ and $F_\triangle^{(2,1),\rm fin}(z)$ are shown in Figure \[fig:Ftri3LoopNf\], adopting the notation from Figure \[fig:Ftri3Loop\]. Both coefficients show a convergence which is very similar to $F_\triangle^{(2),\rm fin}$. Summing up the coefficients leads to good agreement with the result but with a larger uncertainty which is why should be used for numerical applications. Note that $F_\triangle^{(2,0),\rm fin}(z)$ is the only result where the exclusion criterion (\[eq:pole\_crit\_2\]) has been used whereas for all other results (\[eq:pole\_crit\]) is applied.
Finally, we present in Figure \[fig:Ftri3Loop\_spacelike\] results for the three-loop form factor for $z<0$. One observes small uncertainties for $|z|<5$ which become larger when $z$ becomes more negative. For $|z|>20$ the Padé approximation procedure does not lead to stable results which is also seen by the fact that the uncertainty becomes larger after incorporating more expansion terms (see lower panel). Note that the large top quark mass expansion shows an alternating behaviour.
Together with this paper we provide representative Padé approximants for all plots shown in this section in an ancillary file [@progdata].
\[sec::con\]Conclusion
======================
We compute three-loop corrections to the Higgs-gluon form factor including finite top quark mass effects. Our approach is based on the combination of analytic results from two kinematic regions: the expansion for large top quark mass and the top quark threshold. In addition, we incorporate the information that the form factors vanish at high energies by a rescaling (cf. Eq. (\[eq:Pade\_rescaling\])). For the rescaled form factors, we apply a conformal mapping and a subsequent Padé approximation. We first apply our method at two loops and show that we can reproduce the known results. The two-loop expression is also used to demonstrate that our estimate for the uncertainty works reliably. Our main result is shown in Figure \[fig:Ftri3Loop\] where we plot the three-loop form factor in the time-like momentum region. This plot can be reproduced using the approximation functions which are provided in the ancillary file [@progdata]. We have shown that our results can be systematically improved by incorporating more expansion terms into the analysis.
Acknowledgements {#acknowledgements .unnumbered}
================
RG is supported by the “Berliner Chancengleichheitsprogramm”. This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 — TRR 257 “Particle Physics Phenomenology after the Higgs Discovery” and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No. 764850, SAGEX.
[99]{}
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---
abstract: 'We perform a cosmological parameter analysis of the 75 square degree CTIO lensing survey in conjunction with CMB and Type Ia supernovae data. For $\Lambda$CDM cosmologies, we find that the amplitude of the power spectrum at low redshift is given by $\sigma_8 = 0.81^{+0.15}_{-0.10}\ (95\%\ {\rm c.l.})$, where the error bar includes both statistical and systematic errors. The total of all systematic errors is smaller than the statistical errors, but they do make up a significant fraction of the error budget. We find that weak lensing improves the constraints on dark energy as well. The (constant) dark energy equation of state parameter, $w$, is measured to be $-0.89^{+0.16}_{-0.21}\ (95\%\ {\rm c.l.})$. Marginalizing over a constant $w$ slightly changes the estimate of $\sigma_8$ to $0.79^{+0.17}_{-0.14}\ (95\%\ {\rm c.l.})$. We also investigate variable $w$ cosmologies, but find that the constraints weaken considerably; the next generation surveys are needed to obtain meaningful constraints on the possible time evolution of dark energy.'
author:
- 'Mike Jarvis, Bhuvnesh Jain, Gary Bernstein, Derek Dolney'
title: Dark Energy Constraints from the CTIO Lensing Survey
---
Introduction
============
Observations of weak gravitational lensing, the coherent distortion in the images of distant galaxies, have advanced rapidly in the past four years. The first detections of weak lensing in blank fields were reported only a few years ago [@Wi00; @KWL00; @vW00; @Ba00; @Rh00]. More recent lensing measurements [@Ho02; @Re02; @Ba03; @Br03; @Ha03; @Ja03; @vW04; @He05] have used larger and/or deeper surveys to reduce the statistical errors, which scale as $N^{-1/2}$ where $N$ is the number of galaxies measured. Better techniques for reducing systematic errors have also been developed, resulting in interesting cosmological constraints from lensing surveys.
Shear correlations measured by lensing surveys determine the projected power spectrum of matter fluctuations in the Universe. These fluctuations are believed to have grown due to gravitational instability from the early universe to the present. The growth of fluctations from last scattering, $z=1100$, to the present is sensitive to the densities of dark energy and matter via the Hubble expansion rate. Further, the measured lensing signal depends on angular diameter distances to the source galaxies. Thus weak lensing observables probe the dark energy density through both the growth function and the geometric distance factors. Present lensing measurements are sensitive to redshifts $0 \simlt z\simlt 1$.
As the size of weak lensing surveys increases and the statistical errors keep going down, it becomes more important to similarly reduce the systematic errors. There are a few different systematic errors which can contaminate a weak lensing signal at the level of typical cosmic shear measurements. The largest of these has typically been the corrections of the anisotropic point-spread-function (PSF). We have recently developed a principal component analysis approach to interpolating the PSF between the stars in the image using information from multiple exposures [@Ja05]. The PSF pattern is found to be a function of only a few underlying variables. Therefore, we are able to improve the fits of this pattern by using stars from many exposures. We have applied it to the CTIO survey data presented in @Ja03 and have shown that the level of systematic error is well below the statistical errors. Indeed the measured $B$-mode in shear correlations is consistent with zero on all scales measured. In this paper we present a new parameter analysis of the CTIO lensing survey. With the reduced systematic error, we are able to extract information from the measured signal over nearly two decades in length scale.
Type Ia supernovae observations led to the discovery of the accelerated expansion of the universe (Riess [[*et al.*]{}]{}1998; Perlmutter [[*et al.*]{}]{}1999). By combining information from the CMB at $z=1100$, large-scale structure and SNIa, interesting constraints on dark energy have been obtained [@Sp03; @bridle; @We03; @alam; @saini; @Te04; @Wa04; @Se05; @simon; @rapetti; @Jas05]. Whether the dark energy density is constant or evolving with cosmic time is one of the most interesting observational questions. It is often expressed using the parameterization $$w(a) = w_0 + w_a(1-a),
\label{eqn:wa}$$ where $w$ is the dark energy equation of state parameter $w=p/\rho$ and $a=1/(1+z)$ is the expansion scale factor [@Ch01; @Li03]. The cosmological constant corresponds to $w_0=-1, w_a=0$. A non-zero value of $w_a$ corresponds to a time-dependent equation of state. We investigate three priors for $w$: $\Lambda$CDM ($w_0=-1, w_a=0$), constant $w$ ($w_a=0$) with $-3 < w < 0$, and variable $w$ with $-8 < w < 8$ and $-8 < w_a < 8$.
In §\[datasection\] we briefly review our CTIO survey data and shape measurement technique, largely deferring to our previous papers [@Ja03; @Ja05] for more details. In §\[statssection\], we present the results from the new reduction. We use these results to constrain cosmological parameters in §\[analysissection\], and conclude in §\[discussionsection\].
Data {#datasection}
====
Our CTIO survey data was originally described in detail in @Ja03, and we refer the reader to that paper for most of the details about the data and the analysis. Here, we present a brief summary of the data set, and point out two significant changes in the analysis: our PSF interpolation and the dilution correction.
The data were taken at Cerro Tololo Interamerican Observatory (CTIO) in Chile from December, 1996 to July, 2000. We observed 12 fields, well separated on the sky, in low extinction but otherwise random locations. Each field is approximately 2.5 degrees on a side, giving us a total of 75 square degrees. The 50% completeness level occurs near $R = 23.5$ for each field, although it varies somewhat between the fields. We impose limits of $19 < R < 23$, which gives about 2 million galaxies to use for our lensing statistics.
The shape measurements of the galaxies follow the techniques of @BJ02. The galaxy shapes are measured using an elliptical Gaussian weight function which is matched to have the same ellipticity as the galaxy. (That is, we use a circular Gaussian in the sheared coordinate system in which the galaxy is round.) The observed ellipticity is then calculated as: $${\mbox{\bf e}}= e_1 + i e_2 =
\frac{\int I(x,y) W(x,y) (x+iy)^2 dxdy}{\int I(x,y) W(x,y) (x^2+y^2) dxdy}$$ where $I$ is the intensity, $W$ is our Gaussian weight, $x$ and $y$ are measured from the (weighted) centroid of the image, and the boldface indicates that ${\mbox{\bf e}}$ is a complex quantity.
We correct for the effects of the point spread function (PSF) in two steps. First, we correct for the effect of the shape of the PSF by reconvolving the observed images with a spatially varying convolution kernel which is designed to make the stars round. The galaxies in the convolved images are thus no longer affected by the [*shape*]{} of the point spread function (PSF), but are still affected by the [*size*]{} of the PSF.
Since the convolution kernel is only measured where we have an observed star, and the PSF is far from uniform across each image, we need to interpolate between the stars. In our previous analysis, we used a separate interpolation for every image. For this analysis, however, we use our new principal component algorithm which uses the information from all of the images at once. This new method, which gives a better fit, is described in @Ja05.
The effect of the size of the PSF is called dilution. A perfectly round PSF blurs the images of galaxies, which reduces the observed ellipticity. For a purely Gaussian PSF and Gaussian galaxies, the measured ellipticities are reduced by a factor $R$: $$R = 1 - \frac{\sigma^2_{\rm psf}}{\sigma^2_{\rm gal}}
\label{gaussian_response}$$
However, galaxies are certainly not Gaussian, and stars are only approximately Gaussian. In our previous analysis we used a formula intended to account fot the first order corrections due to the kurtosis of the galaxies and the PSF. Unfortunately, our formula was incorrect with respect to the PSF kurtosis, as pointed out by @Hi03. They give a more accurate correction scheme which accounts for both kurtoses correctly to first order – we implement this scheme in the analysis presented below. Their formula is quite complicated to write, but is easy to implement, so we defer to their Appendix B for the relevant equations.
Finally, we estimate the shear, ${\mbox{\boldmath $\gamma$}}$, from an ensemble of ellipticities using the formula: $$\hat{{\mbox{\boldmath $\gamma$}}} = \frac{1}{2{\mbox{$\cal R$}}} \frac{\sum w_i \,{\mbox{\bf e}}_i}{\sum w_i}$$ where the responsivity, ${\mbox{$\cal R$}}$, describes how our mean ellipticity changes in the presence of an applied distortion. It generalizes the formula given in equation \[gaussian\_response\] to non-Gaussian shapes. The factor of 2 in the denominator above converts the ellipticity to shear, and $w$ is our weight function. We use the “easy” weight function given in @BJ02 (Equation 5.36): $$w = [e^2 + (1.5\sigma_\eta)^2]^{-1/2}$$ where $\sigma_\eta$ is the shape uncertainty in the sheared coordinates where the galaxy is circular. The corresponding responsivity, ${\mbox{$\cal R$}}$, is also given in @BJ02 (Equation 5.33).
Weak Lensing Statistics {#statssection}
=======================
To describe the two-point statistics of our shear field, we use the aperture mass statistic [@Sch98; @Cr02; @Sch02; @Pen02].
The predictions from theory come in the form of the convergence power spectrum: $$P_\kappa(\ell)
= \frac{9}{4} \Omega_0^2 \int_0^{\chi_H} d\chi
\frac{g^2(\chi)}{a^2(\chi)}
P_{3D} \left( \frac{\ell}{r(\chi)}; \chi \right)
\label{Pkappa}$$ where $\chi$ is the radial comoving distance, $\chi_H$ is the horizon distance, $r(\chi)$ is the comoving angular distance, $a$ is the scale size of the universe, $P_{3D}$ is the three-dimensional power spectrum of the matter fluctuations, and $$g(\chi) = \int_\chi^{\chi_H} d\chi^\prime p(\chi^\prime)
\frac{r(\chi^\prime-\chi)}{r(\chi^\prime)}
\label{gchi}$$ where $p$ is the normalized (to give unit integral over $\chi$) redshift distribution of source galaxies. These predictions are not reliable for high values of $k$ ($k > 10 h{\rm Mpc}^{-1}$) [@Sm03] due to difficulties in predicting the non-linear growth.
The observed second moments are completely described using the two-point correlation functions: $$\begin{aligned}
\xi_+(\theta)
&=& \langle \gamma({\bf r}) \gamma^*({\bf r + {\mbox{\boldmath $\theta$}}}) \rangle \nonumber\\
\xi_-(\theta) + i\xi_\times(\theta)
&=& \langle\gamma({\bf r}) \gamma({\bf r +{\mbox{\boldmath $\theta$}}}) e^{-4i\alpha}\rangle\end{aligned}$$ where ${\mbox{\boldmath $\theta$}}= \theta e^{i\alpha}$ is the separation between pairs of galaxies, and treating the positions on the sky as complex values. The correlation functions are not measured at all on scales larger than the size of the survey fields (for this survey, at $\theta > 200\arcmin$), which correspond to low $k$ values.
Using the information from the correlation function to obtain the power spectrum, or vice versa, requires an extrapolation away from the $k$ values which are well measured or well predicted. The aperture mass statistic is in some sense a compromise between these two statistics, since it can be calculated from either the power spectrum or the correlation function using only the range of $k$ values which are either predicted or measured, respectively: $$\begin{aligned}
\langle M_{\rm ap}^2 \rangle (R)
\label{mapsq_p}
&=& \frac{1}{2\pi} \int \ell d\ell P_\kappa(\ell) W(\ell R)^2 \nonumber\\
\label{mapsq_xi}
&=& \frac{1}{2} \int \frac{\theta \,d\theta}{R^2}
\left[ \xi_+(\theta) T_+\left(\frac{\theta}{R}\right)
+ \xi_-(\theta) T_-\left(\frac{\theta}{R}\right)
\right] \end{aligned}$$ where we use the form suggested by @Cr02 for which we have: $$\begin{aligned}
W(\eta) &=& \frac{\eta^4}{4} e^{-\eta^2} \nonumber\\
T_+(x) &=& \frac{x^4-16x^2+32}{128} e^{-x^2/4} \nonumber\\
T_-(x) &=& \frac{x^4}{128} e^{-x^2/4}\end{aligned}$$ The aperture mass therefore has both good predictions from theory and accurate measurements from the data.
While the integral in is technically from 0 to infinity, the $T$ functions drop off very quickly, so that the effective upper limit is really around $5R$. Our fields are $200\arcmin$ along the long diagonal, so we can measure the aperture mass up to $R = 40\arcmin$. The lower limit, due to difficulties of measuring the correlation function on very small scales, is about $R = 1\arcmin$.
The function $W(\ell R)$ is a narrow function around $\ell R = \sqrt{2}$, so this corresponds to a range in $\ell$ of approximately $120 < \ell < 5000$.
The other big advantage to the aperture mass statistic is that it provides a natural check for systematics. Weak lensing should only produce a curl-free $E$-mode component, so any $B$-mode observed in the shear field represents a systematic error of some sort. The aperture mass statistic has a $B$-mode counterpart which can be likewise calculated from the correlation functions as: $$\langle M_\times^2 \rangle (R)
= \frac{1}{2} \int \frac{\theta \,d\theta}{R^2}
\left[ \xi_+(\theta) T_+\left(\frac{\theta}{R}\right)
- \xi_-(\theta) T_-\left(\frac{\theta}{R}\right)
\right]$$
Figure \[mapvarfigure\] shows the results for our reanalysis. The blue points are the $E$-mode signal, and the red points are the $B$-mode contamination. Points separated by at least one other point are very nearly uncorrelated. The black curve is the best fit flat $\Lambda$CDM model found below.
The $B$-mode is seen to be consistent with zero at all scales, which was not the case for our previous analysis. Further, in Jarvis & Jain (2004) we show that the measured ellipticity correlation function of stars, which is another measure of systematic errors, is one to two orders of magnitude smaller than the expected lensing signal at all scales. Therefore, we can now confidently use all of the aperture mass values from $1\arcmin$ to $40\arcmin$ for our constraints on cosmology.
In addition to the aperture mass, we also measure the variance of the mean shear in circular apertures: $$\begin{aligned}
\langle|\gamma|\rangle^2(R)
\label{var_p}
&=& \frac{1}{2\pi} \int \ell d\ell P_\kappa(\ell)
\frac{4 J_1(\ell R)^2}{(\ell R)^2} \nonumber\\
\label{var_xi}
&=& \int_0^{2R} \frac{\theta \,d\theta}{R^2}
\xi_+(\theta) S_+\left(\frac{\theta}{R}\right)\end{aligned}$$ where $$S_+(x) = \frac{1}{\pi} \left(4 \arccos(x/2) - x \sqrt{4-x^2}\right)$$
Figure \[mapvarfigure\] shows the results for the shear variance statistic. There is no $E/B$ decomposition for this statistic[^1]. We assume implicitly that there is no $B$-mode contamination in the shear variance measurements, which seems reasonable given the low $B$-mode seen for the aperture mass.
The usefulness of this statistic is that it is able to probe the power spectrum at somewhat smaller $\ell$ values than the aperture mass statistic. The upper limit in the integral in is only $2R$, so we can calculate the shear variance up to $R = 100\arcmin$ with our data. This probes the power spectrum down to $\ell$ of about 70. This leads to a total dynamic range for both statistics of almost 2 orders of magnitude. The shear variance below $R = 50\arcmin$ is degenerate with the aperture mass, so for our constraints, we only use the shear variance at large values of $R$ where it is providing extra information.
In Figure \[mapvarfigure\], we also show the overall best fit $\Lambda$CDM model (see §\[analysissection\]). This fit has a $\chi^2$ of 7.7, for effectively 6 degrees of freedom, yielding a reduced $\chi^2$ of 1.28.
Analysis {#analysissection}
========
Dark Energy Constraints from Weak Lensing
-----------------------------------------
Our lensing measurement constrains the shear power spectrum, which is a weighted projection of the mass power spectrum. The constraints on dark energy arise from two sources. The first is the angular diameter distances to the lens, to the source, and between the lens and the source that enter into the projection. The second is the amplitude of the power spectrum. The dark energy component alters the expansion rate of the universe at redshifts below about 2 (at least if its evolution is not too different from a cosmological constant). This in turns affects the growth of structure. Since the CMB fixes the amplitude of the power spectrum at $z=1100$, the lensing measurement of the amplitude at low redshift measures the growth function. See @Hu04 for a detailed discussion.
We assume that massive neutrinos make a negligible contribution to the matter density, that the primordial power spectrum index has no running and that the universe is spatially flat. The shape of the mass power spectrum is then specified by the baryon density ${\mbox{$\Omega_{\rm b}$}}h^{2}$, the matter density ${\mbox{$\Omega_{\rm m}$}}h^{2}$, and the primordial spectrum. Following the WMAP convention, we use the scalar amplitude ${\mbox{$A_{\rm s}$}}$ and spectral index $n$ such that the shape of the primordial power spectrum is ${\mbox{$A_{\rm s}$}}(k/k_0)^{(n-1)}$, where $k_0=0.05$ Mpc$^{-1}$ is the normalization scale. The current uncertainties in these parameters are at the 10% level or better. Thus the power spectrum as a function of $k$ in Mpc$^{-1}$ (not $h$ Mpc$^{-1}$) in the matter dominated regime can be considered as largely known.
The amplitude of the power spectrum at a given redshift depends on the initial normalization ${\mbox{$A_{\rm s}$}}$ and the “growth function” $G$ defined by $$P(k,z) = \left[ \frac{1}{1+z} \frac{G(z)}{G_0} \right]^2 P(k,0) \,,$$ where $G_0\equiv G(z=0)$ and we assume that all relevant scales are sufficiently below the maximal sound horizon of the dark energy.
The normalization of the linear power spectrum today is conventionally given at a scale of $r=8 h^{-1}$Mpc and can be approximated as [@Hu04] $$\sigma_{8} ~\approx~ \frac{{\mbox{$A_{\rm s}$}}^{1/2}}{0.97}
\left( \frac{ {\mbox{$\Omega_{\rm b}$}}h^{2}}{0.024} \right)^{-0.33}
\left( \frac{ {\mbox{$\Omega_{\rm m}$}}h^{2} }{ 0.14} \right)^{0.563} ~\times(3.123h)^{(n-1)/2} \left( \frac{ h }{ 0.72} \right)^{0.693}
\frac{G_0 }{ 0.76}\,,
\label{eqn:sigma8}
$$ Thus a measurement of $\sigma_8$, in conjunction with constraints on the other parameters in the above equation from the CMB, constrains a combination of dark energy parameters that affect $G_0$. Note that while the equation above illustrates how the dark energy parameters are linked to others, we do not actually use it in our parameter analysis. Instead we use the projection integral of which includes the full range of redshifts probed by our survey. The peak contribution to the lensing correlations is from $z\simeq 0.3$, though the maximum sensitivity to $w$ is at $z\simeq 0.4$ as discussed below. Combining our measurement with others such as Type Ia supernovae, which are sensitive to different redshifts, enables a probe of the time dependence of dark energy parameters. With photometric redshifts, this could be done with the lensing data alone using the auto and cross-spectra in redshift bins.
The dark energy modifies the expansion of the universe according to the equation (for a flat universe)[[[*e.g.*]{}]{} @Li02]: $$H^2(a) = H_0^2 \left[ {\mbox{$\Omega_{\rm m}$}}a^{-3} + {\mbox{$\Omega_{\rm de}$}}e^{
-3 \int_1^a {\frac{da}{a} (1 + w(a))} } \right]$$ where the dark energy density is ${\mbox{$\Omega_{\rm de}$}}(a)=8\pi G \rho_{\rm de}/3H(a)^2$, its equation of state is $w(a)=p_{\rm de}/\rho_{\rm de}$, and we indicate the present time values, ${\mbox{$\Omega_{\rm m}$}}(a=1)$ and ${\mbox{$\Omega_{\rm de}$}}(a=1)$ as simply and .
Distances are then given by $$\chi(a) = \int_a^1 { \frac{da^\prime c}{H(a^\prime){a^\prime}^2} }$$ and the growth function $G$ depends only on the dark energy throught the equation: $$\frac{d^2 G}{d \ln a^2} + \left[ \frac{5}{2} - \frac{3}{2} w(a) {\mbox{$\Omega_{\rm de}$}}(a) \right]
\frac{d G}{d \ln a} + \frac{3}{2}[1-w(a)]{\mbox{$\Omega_{\rm de}$}}(a) G =0\, .
$$
When we use a constant $w$ parameterization, it is equivalent to a measurement of $w$ at the redshift for which the errors in the constant and time-dependent piece (in a Taylor expansion) are uncorrelated. See @Hu04 for a discussion of this pivot redshift, which we find (§\[varwsection\]) to be about $0.4$ for our survey (when combined with the CMB and SN priors as described below).
Joint Constraints
-----------------
We use the results from WMAP [@Sp03] as priors for our analysis. Specifically, we use the Monte Carlo Markov Chain (MCMC) calculated by @Ve03[^2]. We choose to use the less restrictive prior of $w > -3$ rather than $w > -1$ in order to be as conservative as possible. However, we do have a hard prior that ${\mbox{$\Omega_{\rm k}$}}= 0$. While WMAP has constrained this to be $0.02 \pm 0.02$ for pure $\Lambda$CDM, and there is theoretical bias in believing it is exactly 0, it is worth emphasising that our dark energy constraints would be weakened if this prior were relaxed.
Each step in the Markov chain contains a value for each of the following parameters: $\omega_{\rm b} = {\mbox{$\Omega_{\rm b}$}}h^2$ is the density of baryons; $\omega_{\rm c} = {\mbox{$\Omega_{\rm c}$}}h^2$ is the density of cold dark matter; $\theta_{\rm A}$ is the angular scale of the acoustic peaks; $n$ is the spectral slope of the scalar primordial density power spectrum; $Z = \exp(-2\tau)$ is related to the optical depth ($\tau$) of the last scattering surface; ${\mbox{$A_{\rm s}$}}$ is the overal amplitude of the scalar primordial power spectrum; $h = H_0/100$ is the Hubble constant; $w$ is the equation of state parameter for the dark energy.
The parameters $\theta_A$ and $Z$ are not directly relevant to the aperture mass statistic which we have measured, so we marginalize over these two. The others define a cosmology from which we can predict the aperture mass statistic on the scales where we have measured it. We present results for selected parameters below, these are obtained by a full marginalization over all the other parameters listed above.
For the linear power spectrum, we use the transfer function of @BBKS. We then estimate the non-linear power spectrum using the halo-based model of model of @Sm03. Their fitting formulae provide a means of estimating the quasi-linear and non-linear halo contributions to the power spectrum based on the linear value and the effective spectral index. This model agrees with the results of $N$-body simulations better than the simpler formula of @PD96. The nonlinear correction affects the predicted variance in the mass aperture statistic on scales below about 4 arcminutes.
We also note that the model of @Sm03 that we use does not include $w$ directly. We correctly take it into account for the growth factor and the distances, but @Ma03 and @Kl03 show that dark energy changes the virial density contrast, $\Delta_c$, which results in changes in the power spectrum at high $k$ values ($k > 1\ h {\rm Mpc}^{-1}$). However, the effect is smaller than the expected error in the non-linear model even for a $w \simeq -0.5$ cosmology.
From the predicted power spectrum, we calculate the aperture mass and shear variance statistics using Equations \[Pkappa\], \[mapsq\_p\], and \[var\_p\]. Our data then give a likelihood value for each cosmology, which we combine with the CMB likelihood from the MCMC.
We also use the recent supernova measurements of @Ri04 to further constrain the results. For this, we use the $\chi^2$ calculation program of @To03[^3].
### $\Lambda$CDM Models
With dark energy priors of $w = -1$ and $w_a = 0$, the likelihood constraints are non-trivial contours through a five-dimensional parameter space. However, most of the gain in constraining power from the addition of the lensing data comes in the quantities ${\mbox{$\Omega_{\rm m}$}}$ and $\sigma_8$. We show the error contours projected onto the ${\mbox{$\Omega_{\rm m}$}}-\sigma_8$ plane in Figure \[lambda\_plots\]. The left plot shows the contours starting with the CMB data set, and sequentially adding the supernova and lensing data. The right plot shows the contours in the ${\mbox{$\Omega_{\rm m}$}}- \sigma_8$ plane separately for each of the three data sets to indicate the degree of their complementarity, which is why their combination leads to the tight overall constraints. In both plots, the contours correspond to 68% and 95% confidence regions ($\Delta \chi^2$ = 2.30 and 6.17). The crosses are at the peak likelihood in the projected plane, which is at: (${\mbox{$\Omega_{\rm m}$}}= 0.26$, $\sigma_8 = 0.82$).
### Constant $w$ Models
We show the uncertainty contours for the dark energy priors of $-3 < w_0 < 0$ and $w_a = 0$ in Figure \[w3\_plots\]. The left plot give the projection in the ${\mbox{$\Omega_{\rm m}$}}-\sigma_8$ plane, which indicates that allowing $w$ to be free does not significantly worsen the constraints on ${\mbox{$\Omega_{\rm m}$}}$ and $\sigma_8$ compared to the pure $\Lambda$CDM model. The ${\mbox{$\Omega_{\rm de}$}}-w$ plot (right) shows why. While none of the three data sets individually have tight constraints in this plane, the combination of all three leads to a fairly tight contour near (and consistent with) $w=-1$. The peak likelihood models in the two projections are: (${\mbox{$\Omega_{\rm m}$}}= 0.25$, $\sigma_8 = 0.79$) and (${\mbox{$\Omega_{\rm de}$}}= 0.75$, $w = -0.90$).
### Variable $w$ Models {#varwsection}
Finally, we consider dark energy priors of $-8 < w_0 < 8$ and $-8 < w_a < 8$. It turns out that some of the dark energy models in this range have $\Omega_{\rm DE}(z=1100) \approx 1$. That is, the mass-energy of the universe was essentially all dark energy at the epoch of recombination. This seems to be ruled out by WMAP data [@Ca03; @Ca04; @Wa04]. Therefore, we make the additional prior that $\Omega_{\rm DE}(z=1100) < 0.5$. In practice, all the models have $\Omega_{\rm DE}(z=1100) \approx 0$ or $1$, so this contraint is effectively $\Omega_{\rm DE}(z=1100) \approx 0$.
Given this constraint, the primary effect of dark energy on the CMB is through the distance to the last-scattering surface, $d_{\rm LSS}$. Therefore, we approximate the CMB likelihoods by using the WMap constant-$w$ Markov chain mentioned above, modifying the dark energy parameters to maintain a constant $d_{\rm LSS}$. Specifically, for each line in the Markov chain, we determine $d_{\rm LSS}$ from the values of $\Omega_m$ and $w$; we select $w_a$ from $-8 < w_a < 8$; then we determine what $w_0$ with this $w_a$ and the same $\Omega_m$ maintain the given value of $d_{\rm LSS}$, and we write these values out as a line in a new pseudo-chain.
The main approximation in this process is that we neglect the difference of the integrated Sachs-Wolfe (ISW) effect between the two models. There is some indication[^4] that for some of the models that fall within our contours (Figure \[wwa\_plots\], right), the ISW may spread the first peak in the power spectrum of the CMB enough to disfavor these models by 1 or 2 sigma. However, we defer a more detailed study of this effect to future work.
The error contours projected onto the ${\mbox{$\Omega_{\rm m}$}}-\sigma_8$ plane and the $w_0-w_a$ plane are shown in Figure \[wwa\_plots\]. With this much freedom in the models, we do finally lose much of our constraining power. Even with all three data sets, the error contours are still quite large. The ${\mbox{$\Omega_{\rm m}$}}-\sigma_8$ plot (left) shows that allowing $w$ to vary with cosmological time does significantly worsen the constraints, allowing much lower values of $\sigma_8$.
Also, while lensing does significantly reduce the allowed parameter space in the ${\mbox{$\Omega_{\rm m}$}}-\sigma_8$ plane, this reduction has only a small effect in the $w_0-w_a$ plane, so that our constraints there are not much better than those for the CMB + SN data alone. The data are consistent with a cosmological constant ($w_0=-1, w_a=0$) at the 95% confidence level, but $w_a$ may range as high as 2[^5].
We can use the direction of the contours in the $w_0-w_a$ plane to determine the redshift at which we have the strongest constraints on the dark energy. If we change variables from $\{w(z=0),w_a\}$ to $\{w(z=0.4),w_a\}$, the likelihood contour becomes roughly vertical. This indicates that our pivot redshift, or “sweet spot” [@Hut01; @We02; @Hu02; @Hu04], where the constraints of the dark energy are strongest, is at a redshift of about $0.4$. (Of course, the banana shape makes this impossible to do precisely, so $z_{\rm piv}=0.4$ is just an approximate value.) We constrain $w$ at this redshift, marginalizing over $w_a$ (and everything else) below.
Systematic Errors
-----------------
Systematic errors are harder to estimate than statistical errors, since by their nature they contaminate the data in unknown ways. There are four systematic errors which we investigate and attempt to estimate: residual anisotropic PSF as estimated by the residual $B$-mode, calibration error, redshift distribution error, and errors in the non-linear prediction. For shorthand, we refer to these as B, CAL, Z, and NL respectively.
When estimating the contributions of these systematic uncertainties to our error budget, we limit our consideration to constraints on single parameters, fully marginalized over all the other parameters. First we calculate the 95% error bars with only the statistical errors. Then, for each systematic effect, we change our handling of the effect as described more completely below for each case. When we do this, the 95% confidence intervals move around somewhat. We define the upper systematic error to be the maximum upper limit of the confidence interval allowed by the various changes minus the nominal upper limit with only the statistical errors. Likewise the lower systematic error is the lower limit with only the statistical errors minus the minimum allowed lower limit. Finally, we (conservatively) estimate the total error as the sum of the statistical errors and each of the systematic errors added linearly, not in quadrature.
For the residual PSF (B) systematic, we implement the same technique we used in @Ja03, namely running the analysis with the $B$-mode contamination added to and subtracted from the $E$-mode signal. Most types of contamination either add power (roughly) equally to the $E$ and $B$ modes, in which case the subtraction is appropriate, or mix power between the two modes while conserving total signal, in which case the addition is appropriate. We allow for both possibilities to estimate how the contamination could be affecting the cosmological fits.
The calibration (CAL) uncertainty includes errors in the dilution calculation, the responsivity formula, and possibly biases in the shape measurements. This systematic was the subject of significant discussion at the recent IAU symposium 225[^6]. There seem to be calibration differences of order 5% in shear estimators between different methods. Tests with simulated images with known shears indicate that we have calibration errors of less than 2% [@STEP], but we allow for $\pm$ 5% in our shear values as a conservative estimate of this systematic.
For the redshift calibration (Z) of our survey, there are two public redshift surveys with depths similar to our observations: the Caltech Faint Galaxy Redshift Survey [@CRS] (CRS), and the Canada-France Redshift Survey [@CFRS] (CFRS). We argue in @Ja03 that the CRS is a better choice, since it is more complete in the $R$ filter band pass used for our observations. However, switching to the CFRS distribution allows us to estimate the uncertainties due to the redshift calibration. Also, since we only have one other survey to use, we cannot run symmetric plus and minus versions of this test. So when the 95% confidence limit moved inward for a value, we take the absolute value of the change as the measure of the systematic error, since a different redshift survey [*might*]{} have moved the limit a similar amount outward.
Finally, for the non-linear predictions (NL), we used the @Sm03 model, which were an improvement over that of @PD96. Switching back to the older model should give us a rough (over-)estimate of the remaining uncertainties due to the non-linear modelling. Again, we cannot run symmetric tests, so we take the absolute value of any change as a measure of the systematic error.
Since this technique is non-standard and may be confusing, an example with the actual values might help explain it. For the CAL test with the $\Lambda$CDM prior, when we multiplied the shear data by 1.05 (for the +5% test), the upper limit of the 95% confidence interval for $\sigma_8$ moved from 0.910 to 0.929, and increase of 0.019. This is our estimate of the positive systematic error. Similarly, in the -5% CAL test, the lower limit decreased, providing the negative systematic error for $\sigma_8$.
In Table \[paramstable\], we present the statistical and systematic error estimates for ${\mbox{$\Omega_{\rm m}$}}$, $\sigma_8$, $w_0$ and $w_a$, for each of our dark energy priors. The first two columns present the estimates for each value with and without the lensing data, marginalized over all other parameters, and quoting only the statistical errors. The next four columns show the estimated systematic error due to each effect listed above. The final column includes the total uncertainty with the systematic errors added linearly with the statistical uncertainty. Note that we make no attempt to estimate the systematic errors present in the CMB or SN data.
[clllllll]{} ${\mbox{$\Omega_{\rm m}$}}$ & $0.301^{+0.07}_{-0.07}$ & $0.256^{+0.05}_{-0.05}$ & ${}^{+0.005}_{-0.001}$ & ${}^{+0.008}_{-0.003}$ & ${}^{+0.008}_{-0.005}$ & ${}^{+0.000}_{-0.000}$ & $0.256^{+0.07}_{-0.06}$\
$\sigma_8$ & $0.953^{+0.27}_{-0.18}$ & $0.812^{+0.10}_{-0.09}$ & ${}^{+0.028}_{-0.006}$ & ${}^{+0.019}_{-0.006}$ & ${}^{+0.020}_{-0.007}$ & ${}^{+0.001}_{-0.001}$ & $0.812^{+0.17}_{-0.11}$\
${\mbox{$\Omega_{\rm m}$}}$ & $0.285^{+0.08}_{-0.07}$ & $0.254^{+0.05}_{-0.04}$ & ${}^{+0.005}_{-0.001}$ & ${}^{+0.006}_{-0.005}$ & ${}^{+0.006}_{-0.001}$ & ${}^{+0.000}_{-0.000}$ & $0.254^{+0.07}_{-0.05}$\
$\sigma_8$ & $0.846^{+0.29}_{-0.17}$ & $0.790^{+0.11}_{-0.10}$ & ${}^{+0.022}_{-0.015}$ & ${}^{+0.018}_{-0.012}$ & ${}^{+0.019}_{-0.012}$ & ${}^{+0.001}_{-0.001}$ & $0.790^{+0.17}_{-0.14}$\
$w$ & $-0.935^{+0.16}_{-0.28}$ & $-0.894^{+0.14}_{-0.16}$ & ${}^{+0.006}_{-0.016}$ & ${}^{+0.005}_{-0.017}$ & ${}^{+0.005}_{-0.019}$ & ${}^{+0.001}_{-0.000}$ & $-0.894^{+0.16}_{-0.21}$\
${\mbox{$\Omega_{\rm m}$}}$ & $0.29^{+0.11}_{-0.06}$ & $0.29^{+0.11}_{-0.07}$ & ${}^{+0.009}_{-0.000}$ & ${}^{+0.001}_{-0.000}$ & ${}^{+0.009}_{-0.001}$ & ${}^{+0.000}_{-0.000}$ & $0.29^{+0.13}_{-0.07}$\
$\sigma_8$ & $0.79^{+0.32}_{-0.35}$ & $0.74^{+0.13}_{-0.17}$ & ${}^{+0.044}_{-0.029}$ & ${}^{+0.013}_{-0.016}$ & ${}^{+0.024}_{-0.000}$ & ${}^{+0.000}_{-0.000}$ & $0.74^{+0.21}_{-0.22}$\
$w_0$ & $-1.17^{+0.50}_{-0.57}$ & $-1.14^{+0.41}_{-0.53}$ & ${}^{+0.008}_{-0.044}$ & ${}^{+0.006}_{-0.020}$ & ${}^{+0.016}_{-0.004}$ & ${}^{+0.002}_{-0.000}$ & $-1.14^{+0.45}_{-0.60}$\
$w_a$ & $1.36^{+0.75}_{-1.79}$ & $1.48^{+0.53}_{-1.57}$ & ${}^{+0.101}_{-0.258}$ & ${}^{+0.021}_{-0.055}$ & ${}^{+0.038}_{-0.124}$ & ${}^{+0.013}_{-0.082}$ & $1.48^{+0.70}_{-2.09}$\
$w(z=0.4)$ & $-0.90^{+0.20}_{-0.33}$ & $-0.87^{+0.13}_{-0.24}$ & ${}^{+0.005}_{-0.047}$ & ${}^{+0.000}_{-0.028}$ & ${}^{+0.002}_{-0.007}$ & ${}^{+0.000}_{-0.001}$ & $-0.87^{+0.14}_{-0.32}$\
\[paramstable\]
It is apparent that the systematic uncertainties in our survey are smaller than the statistical uncertainties for each of the above cosmological parameters. However, there is still significant room for improvement in all of these sources of systematic errors. Future lensing surveys which expect to reduce the statistical uncertainties by a factor of order 10 will need to address these systematics so that they do not dominate the final error budget. And while the systematic errors due to the non-linear modelling are essentially completely negligible for our survey, they will be more significant for lensing surveys with smaller fields than ours.
Discussion {#discussionsection}
==========
We have used our measurement of the shear two-point correlations to constrain the clustering of mass at redshifts $z\sim 0.3$ and the density and equation of state of dark energy. This has been done by combining the lensing information with the CMB and supernovae data. The three probes are sufficiently complementary that the joint contraints are significantly better than from any one or two methods.
We find that the primary sytematic effects on our lensing data are the redshift distribution of the galaxies, the overall calibration of the shear estimates, and the systematic error due to the coherent PSF anisotropy. With our new analysis, the total effect of these three systematics is smaller than the statistical errors. The PCA technique has substantially reduced the contribution of the $B$-mode, which used to be the dominant systematic error [@Ja03]. We are continuing to work on methods to reduce this and the calibration uncertainty.
Improving the redshift distribution would require more data: either a larger redshift survey of similar depth as our data, or imaging the galaxies in three or four other filters to measure photometric redshifts. As discussed below, this second option would also allow us to bin the galaxies and make tomographic measurements.
We have necessarily made some choices of priors and datasets in our parameter analysis. The datasets we have used in addition to the lensing data are the WMAP first year extended data [@Ve03] (which also include CBI and ACBAR data) and the @Ri04 Type 1a Supernovae data. We have assumed that the universe is spatially flat; weakening this assumption significantly weakens constraints on dark energy, especially if $w$ is allowed to vary in time. We have also assumed no tensor contribution to the CMB power spectrum, that the primordial power spectrum is an exact power law (no running), and we have neglected the effects of massive neutrinos on the power spectrum. Current upper limits from cosmology [see e.g. @Se05] are below 1 electron volt for the the sum of neutrino masses. Allowing for massive neutrinos could lead to (at most) a few percent increase in our estimated $\sigma_8$, as the presence of massive neutrinos suppresses the power spectrum on scales that affect our observed shear correlations, but would not lead to interesting constraints on the neutrino mass.
Combining our data with CMB and SN data, our investigation of dark energy models show no evidence for the dark energy being different from a standard cosmological constant. Constant $w$ models are consistent with $w=-1$, and variable $w$ models are consistent with $w_a=0$. The constraints on dark energy from weak lensing come from a range of redshifts centered at $z\sim 0.3$, and extending by about $0.2$ in redshift on either side. Thus the measurements of its density and of $w$ should be interpreted with this redshift range in mind. When we combine our data with supernova data, we are using information from different redshifts, and the combined data have a pivot redshift of about $0.4$, where $w$ is best measured. At this redshift, we also find that $w$ is consistent with $-1$.
Our analysis can be compared with other recent work that combines CMB and supernova data with galaxy clustering, the abundance of galaxy clusters, the clustering of the Lyman alpha forest and other probes [@Sp03; @We03; @Te04; @Wa04; @Se05; @rapetti]. It is a powerful consistency check that these different methods appear to agree in their conclusions. It is interesting to compare the different redshift ranges probed by these methods, and explore constraints on the time dependence of the equation of state [[[*e.g.*]{}]{} @Li03; @Hut05].
The prospects for constraining dark energy with future lensing surveys are very interesting. With a well designed survey and the recent analysis techniques, it can be hoped that systematic errors would stay at levels comparable to statistical errors. The addition of tomographic information with photometric redshifts would allow for significantly better constraints on cosmological parameters from lensing alone [@Hu99]. The use of three point correlations would allow for some independent checks on systematics as well as improved constraints on cosmological parameters (see @Pen03 [@Ja04] for detections of the lensing skewness, and Takada & Jain 2004 for forecasts). Thus even with a survey of size similar to ours, significant improvements in parameter measurements are possible. With future surveys that will cover a significant fraction of the sky, weak lensing should allow for very precise measurements of the mass power spectrum, the dark energy density and its evolution.
We thank Wayne Hu, Eric Linder, Masahiro Takada and Martin White for helpful discussions. We are grateful to Licia Verde and the WMAP team for making available their Markov chains and to Adam Riess, John Tonry and the High-z Supernova team for making available their data and likelihood code. We also thank the anonymous referee for useful comments. This work is supported in part by NASA grant NAG5-10924, NSF grant AST-0236702, and a Keck foundation grant.
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[^1]: Technically, there is [@Sch02], but it requires extrapolation of the correlation functions past where they are measured.
[^2]: Available at http://www.physics.upenn.edu/ lverde/MAPCHAINS/mcmc.html.
[^3]: Available at http://www.ifa.hawaii.edu/ jt/SOFT/snchi.html. The code there was modified slightly to read the data of @Ri04.
[^4]: Wayne Hu, private communication
[^5]: It is for some of these high $w_a$ models that the ISW effect may be important, requiring a more careful analysis to determine what portion of the nominally allowed region is really ruled out.
[^6]: July 19-23, 2004, Lausanne, Switzerland
|
---
abstract: 'This paper explores the use of Propositional Dynamic Logic (PDL) as a suitable formal framework for describing Sign Language (SL), the language of deaf people, in the context of natural language processing. SLs are visual, complete, standalone languages which are just as expressive as oral languages. Signs in SL usually correspond to sequences of highly specific body postures interleaved with movements, which make reference to real world objects, characters or situations. Here we propose a formal representation of SL signs, that will help us with the analysis of automatically-collected hand tracking data from French Sign Language (FSL) video corpora. We further show how such a representation could help us with the design of computer aided SL verification tools, which in turn would bring us closer to the development of an automatic recognition system for these languages.'
author:
- |
Arturo Curiel [^1]\
Université Paul Sabatier\
118 route de Narbonne, IRIT,\
31062, Toulouse, France\
[curiel@irit.fr]{}\
Christophe Collet\
Université Paul Sabatier\
118 route de Narbonne, IRIT,\
31062, Toulouse, France\
[collet@irit.fr]{}\
bibliography:
- 'aclshort.bib'
title: Sign Language Lexical Recognition With Propositional Dynamic Logic
---
Introduction
============
Sign languages (SL), the vernaculars of deaf people, are complete, rich, standalone communication systems which have evolved in parallel with oral languages [@valli_linguistics_2000]. However, in contrast to the last ones, research in automatic SL processing has not yet managed to build a complete, formal definition oriented to their automatic recognition [@cuxac_problematique_2007]. In SL, both hands and non-manual features (NMF), [*e.g.*]{} facial muscles, can convey information with their placements, configurations and movements. These particular conditions can difficult the construction of a formal description with common natural language processing (NLP) methods, since the existing modeling techniques are mostly designed to work with one-channel sound productions inherent to oral languages, rather than with the multi-channel partially-synchronized information induced by SLs.
Our research strives to address the formalization problem by introducing a logical language that lets us represent SL from the lowest level, so as to render the recognition task more approachable. For this, we use an instance of a formal logic, specifically Propositional Dynamic Logic (PDL), as a possible description language for SL signs.
For the rest of this section, we will present a brief introduction to current research efforts in the area. Section \[signlanguagelogic\] presents a general description of our formalism, while section \[proofofconcept\] shows how our work can be used when confronted with real world data. Finally, section \[conclusions\_fw\] present our final observations and future work.
Images for the examples where taken from [@dictasign] corpus.
Current Sign Language Research
------------------------------
Extensive efforts have been made to achieve efficient automatic capture and representation of the subtle nuances commonly present in sign language discourse [@ong_automatic_2005]. Research ranges from the development of hand and body trackers [@dreuw_enhancing_2009; @gianni_robust_2009], to the design of high level SL representation models [@lejeune_analyse_2004; @lenseigne_using_2006]. Linguistic research in the area has focused on the characterization of corporal expressions into meaningful transcriptions [@dreuw_signspeak_2010; @stokoe_sign_2005] or common patterns across SL [@aronoff_paradox_2005; @meir_re-thinking_2006; @wittmann_classification_1991], so as to gain understanding of the underlying mechanisms of SL communication.
Works like [@losson_sign_1998] deal with the creation of a lexical description oriented to computer-based sign animation. Report [@filhol_zebedee:_2009] describes a lexical specification to address the same problem. Both propose a thoroughly geometrical parametric encoding of signs, thus leaving behind meaningful information necessary for recognition and introducing data beyond the scope of recognition. This complicates the reutilization of their formal descriptions. Besides, they don’t take in account the presence of partial information. Treating partiality is important for us, since it is often the case with automatic tools that incomplete or unrecognizable information arises. Finally, little to no work has been directed towards the unification of raw collected data from SL corpora with higher level descriptions [@dalle_high_2006].
Propositional Dynamic Logic for SL {#signlanguagelogic}
==================================
[*Propositional Dynamic Logic*]{} (PDL) is a multi-modal logic, first defined by [@fischer_propositional_1979]. It provides a language for describing programs, their correctness and termination, by allowing them to be modal operators. We work with our own variant of this logic, the [*Propositional Dynamic Logic for Sign Language*]{} ($\mathrm{PDL}_{\mathsf{SL}}$), which is just an instantiation of PDL where we take signers’ movements as programs.
Our sign formalization is based on the approach of [@liddell_american_1989] and [@filhol_modedescriptif_2008]. They describe signs as sequences of immutable [*key postures*]{} and movement [*transitions*]{}.
In general, each key posture will be characterized by the concurrent parametric state of each [*body articulator*]{} over a time-interval. For us, a body articulator is any relevant body part involved in signing. The parameters taken in account can vary from articulator to articulator, but most of the time they comprise their configurations, orientations and their placement within one or more [*places of articulation*]{}. Transitions will correspond to the movements executed between fixed postures.
Syntax
------
We need to define some primitive sets that will limit the domain of our logical language.
Let $\mathcal{B}_\mathsf{SL} = \{\mathbb{D, W, R, L}\}$ be the set of [*relevant body articulators for SL*]{}, where $\mathbb{D}$, $\mathbb{W}$, $\mathbb{R}$ and $\mathbb{L}$ represent the dominant, weak, right and left hands, respectively. Both $\mathbb{D}$ and $\mathbb{W}$ can be aliases for the right or left hands, but they change depending on whether the signer is right-handed or left-handed, or even depending on the context.
Let $\Psi$ be the two-dimensional projection of a human body skeleton, seen by the front. We define the set of [*places of articulation for SL*]{} as $\Lambda_\mathsf{SL} = \{ \mathtt{HEAD}, \mathtt{CHEST}, \mathtt{NEUTRAL}, \ldots \}$, such that for each $\lambda\in\Lambda_\mathsf{SL}$, $\lambda$ is a sub-plane of $\Psi$, as shown graphically in figure \[fig:neutralspace\].
Let $\mathbb{C}_\mathsf{SL}$ be the set of possible morphological configurations for a hand.
Let $\Delta = \{\uparrow,\nearrow, \rightarrow,\searrow,\downarrow,\swarrow,\leftarrow,\nwarrow\}$ be the set of [*relative directions*]{} from the signer’s point of view, where each arrow represents one of eight possible two-dimensional direction vectors that share the same origin. For vector $\delta \in \Delta$, we define vector $\overleftarrow{\delta}$ as the same as $\delta$ but with the inverted abscissa axis, such that $\overleftarrow{\delta} \in \Delta$. Let vector $\widehat{\delta}$ indicate movement with respect to the dominant or weak hand in the following manner: $$\widehat{\delta} = \left\{
\begin{array}{rl}
\delta &\mbox{ if $~\mathbb{D} \equiv \mathbb{R}~$ or $~\mathbb{W} \equiv \mathbb{L}$} \\
\overleftarrow{\delta} &\mbox{ if $~\mathbb{D} \equiv \mathbb{L}~$ or $~\mathbb{W} \equiv \mathbb{R}$}
\end{array}
\right.$$
Finally, let $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ be any two vectors with the same origin. We denote the rotation angle between the two as $\theta(\overrightarrow{v_1}, \overrightarrow{v_2})$.
![Possible places of articulation in $\mathcal{B}_\mathsf{SL}$. \[fig:neutralspace\]](figures/placesofarticulation.jpg)
Now we define the set of atomic propositions that we will use to characterize fixed states, and a set of atomic actions to describe movements.
\[atomicdefinitions\] The set of [*atomic propositions for SL articulators*]{} ($\Phi_{\mathsf{SL}}$) is defined as:
$$\Phi_{\mathsf{SL}}=\{{\beta_1}^{\delta}_{\beta_2}, \Xi^{\beta_1}_\lambda, \mathcal{T}^{\beta_1}_{\beta_2}, \mathcal{F}^{\beta_1}_c, \angle^\delta_{\beta_1} \}$$
where ${\beta_1}, {\beta_2} \in \mathcal{B}_\mathsf{SL}$, $\delta \in \Delta$, $\lambda \in \Lambda_\mathsf{SL}$ and $c \in \mathbb{C}_\mathsf{SL}$.
Intuitively, ${\beta_1}^{\delta}_{\beta_2}$ indicates that articulator ${\beta_1}$ is placed in relative direction $\delta$ with respect to articulator ${\beta_2}$. Let the current place of articulation of ${\beta_2}$ be the origin point of ${\beta_2}$’s Cartesian system ($\mathcal{C}_{\beta_2}$). Let vector $\overrightarrow{{\beta_1}}$ describe the current place of articulation of ${\beta_1}$ in $\mathcal{C}_{\beta_2}$. Proposition ${\beta_1}^{\delta}_{\beta_2}$ holds when $\forall \overrightarrow{v} \in \Delta$, $\theta(\overrightarrow{{\beta_1}}, \delta) \leq \theta(\overrightarrow{{\beta_1}}, \overrightarrow{v})$.
$\Xi^{\beta_1}_\lambda$ asserts that articulator ${\beta_1}$ is located in $\lambda$.
$\mathcal{T}^{\beta_1}_{\beta_2}$ is active whenever articulator ${\beta_1}$ physically touches articulator ${\beta_2}$.
$\mathcal{F}^{\beta_1}_c$ indicates that $c$ is the morphological configuration of articulator ${\beta_1}$.
Finally, $\angle^\delta_{\beta_1}$ means that an articulator ${\beta_1}$ is [*oriented*]{} towards direction $\delta \in \Delta$. For hands, $\angle^\delta_{\beta_1}$ will hold whenever the vector perpendicular to the plane of the palm has the smallest rotation angle with respect to $\delta$.
The [*atomic actions for SL articulators*]{} ( $\Pi_\mathsf{SL}$) are given by the following set:
$$\Pi_\mathsf{SL}=\{\delta_{\beta_1}, \leftrightsquigarrow_{\beta_1}\}$$ where $\delta \in \Delta$ and ${\beta_1} \in \mathcal{B}_\mathsf{SL}$. Let ${\beta_1}$’s position before movement be the origin of ${\beta_1}$’s Cartesian system ($\mathcal{C}_{\beta_1}$) and $\overrightarrow{{\beta_1}}$ be the position vector of ${\beta_1}$ in $\mathcal{C}_{\beta_1}$ after moving. Action $\delta_{\beta_1}$ indicates that ${\beta_1}$ moves in relative direction $\delta$ in $\mathcal{C}_{\beta_1}$ if $\forall \overrightarrow{v} \in \Delta$, $\theta(\overrightarrow{{\beta_1}},\delta) \leq \theta(\overrightarrow{{\beta_1}},\overrightarrow{v})$.
Action $\leftrightsquigarrow_{\beta_1}$ occurs when articulator ${\beta_1}$ moves rapidly and continuously ([*thrills*]{}) without changing it’s current place of articulation.
\[actionlanguage\] The [*action language for body articulators*]{} ($\mathcal{A}_\mathsf{SL}$) is given by the following rule:
$$\alpha::=\pi~|~\alpha\cap\alpha~|~\alpha\cup\alpha~|~\alpha;\alpha~|~\alpha^*$$ where $\pi \in \Pi_\mathsf{SL}$.
Intuitively, $\alpha\cap\alpha$ indicates the concurrent execution of two actions, while $\alpha\cup\alpha$ means that at least one of two actions will be non-deterministically executed. Action $\alpha;\alpha$ describes the sequential execution of two actions. Finally, action $\alpha^*$ indicates the reflexive transitive closure of $\alpha$.
\[languagedefinition\] The formulae $\varphi$ of [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}are given by the following rule:
$$\varphi ::= \top~|~p~|~\neg\varphi~|~\varphi \wedge \varphi~|~[\alpha]\varphi$$
where $p \in \Phi_{\mathsf{SL}}$, $\alpha \in \mathcal{A}_\mathsf{SL}$.
Semantics
---------
[$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{} formulas are interpreted over labeled transition systems (LTS), in the spirit of the possible worlds model introduced by [@hintikka_knowledge_1962]. Models correspond to connected graphs representing key postures and transitions: states are determined by the values of their propositions, while edges represent sets of executed movements. Here we present only a small extract of the logic semantics.
\[kripkedefinition\] A [*sign language utterance model*]{} ($\mathcal{U}_\mathsf{SL}$), is a tuple $\mathcal{U}_\mathsf{SL}=(S, R, \llbracket \cdot \rrbracket_{\Pi_{\mathsf{SL}}}, \llbracket \cdot \rrbracket_{\Phi_{\mathsf{SL}}})$ where:
- $S$ is a non-empty set of states
- $R$ is a transition relation $R \subseteq S \times S$ where, $\forall s\in~S, \exists s'\in S \text{ such that } (s,s')\in R$.
- $\llbracket \cdot \rrbracket_{\Pi_{\mathsf{SL}}}:~\Pi_{\mathsf{SL}} \rightarrow R$, denotes the function mapping actions to the set of binary relations.
- $\llbracket \cdot \rrbracket_{\Phi_{\mathsf{SL}}}:~S\rightarrow 2^{\Phi_{\mathsf{SL}}}$, maps each state to a set of atomic propositions.
We also need to define a structure over sequences of states to model internal dependencies between them, nevertheless we decided to omit the rest of our semantics, alongside satisfaction conditions, for the sake of readability.
Use Case: Semi-Automatic Sign Recognition {#proofofconcept}
=========================================
We now present an example of how we can use our formalism in a semi-automatic sign recognition system. Figure \[architecture\] shows a simple module diagram exemplifying information flow in the system’s architecture. We proceed to briefly describe each of our modules and how they work together.
= \[fill=blue!20, rectangle, minimum width=1.4cm, minimum height=1.4cm\] = \[fill=red!20, circle, minimum width=1.2cm\] = \[coordinate, node distance=1.6cm\] = \[text width=4.6em, font=, align=center\]
(corpus) ; (tracking); (segmentation); (modeliser); (checker); (model); (formules); (rules); (lexicaldata);
(corpustag) [Corpus]{}; (trackingtag)[Tracking and Segmentation Module]{}; (segmentationtag) [Key postures & transitions]{}; (modelisertag)[[$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}\
Model Extraction Module]{}; (checkertag) [[$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}\
Verification Module]{}; (modeltag)[[$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}\
Graph]{}; (formulestag)[Sign Formulæ]{}; (rulestag)[User\
Input]{}; (lexicaldatatag)[Sign Proposals]{};
(corpus) edge (tracking) (tracking) edge (segmentation) (segmentation) edge (modeliser) (modeliser) edge (model) (model) edge (checker) (formules) edge (checker) (rules) edge (checker) (checker) edge (lexicaldata);
Tracking and Segmentation Module
--------------------------------
The process starts by capturing relevant information from video corpora. We use an existing head and hand tracker expressly developed for SL research [@gonzalez_robust_2011]. This tool analyses individual video instances, and returns the frame-by-frame positions of the tracked articulators. By using this information, the module can immediately calculate speeds and directions on the fly for each hand.
The module further employs the method proposed by the authors in [@gonzalez_sign_2012] to achieve sub-lexical segmentation from the previously calculated data. Like them, we use the relative velocity between hands to identify when hands either move at the same time, independently or don’t move at all. With these, we can produce a set of possible key postures and transitions that will serve as input to the modeling module.
Model Extraction Module
-----------------------
This module calculates a propositional state for each static posture, where atomic [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}formulas codify the information tracked in the previous part. Detected movements are interpreted as [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}actions between states.
![Example of modeling over four automatically identified frames as possible key postures.[]{data-label="fig:modeling"}](figures/1.png "fig:"){width="0.23\columnwidth"} ![Example of modeling over four automatically identified frames as possible key postures.[]{data-label="fig:modeling"}](figures/2.png "fig:"){width="0.23\columnwidth"} ![Example of modeling over four automatically identified frames as possible key postures.[]{data-label="fig:modeling"}](figures/4.png "fig:"){width="0.23\columnwidth"} ![Example of modeling over four automatically identified frames as possible key postures.[]{data-label="fig:modeling"}](figures/6.png "fig:"){width="0.23\columnwidth"}
\
=\[circle, fill=blue!65, draw=none, text=white, text width=.2cm, inner sep=0pt, minimum size=10pt, node distance=1.75cm\] =\[fill=none,draw=none,text=black, node distance=.4cm\]
(s0) [$\vdots$]{}; (s0tag1) \[below=0.05cm of s0\] [$\mathbb{R}^\nearrow_{\mathbb{L}}$]{}; (s0tag2) \[below of=s0tag1\] [$\Xi^\mathbb{L}_{\mathtt{TORSE}}$]{}; (s0tag3) \[below of=s0tag2\] [$\Xi^\mathbb{R}_{\mathtt{R\_SIDEOFBODY}}$]{}; (s0tag4) \[below of=s0tag3\] [$\neg\mathcal{F}^{\mathbb{R}}_{\mathtt{L\_CONFIG}}$]{}; (s0tag5) \[below of=s0tag4\] [$\neg\mathcal{F}^{\mathbb{L}}_{\mathtt{FIST\_CONFIG}}$]{}; (s0tag6) \[below of=s0tag5\] [$\neg\mathcal{T}^{\mathbb{R}}_{\mathbb{L}}$]{}; (s0tag7) \[below of=s0tag6\] [$\vdots$]{}; (s1) \[right of=s0\] [$\vdots$]{}; (s1tag1) \[below=0.05cm of s1\] [$\mathbb{R}^\leftarrow_{\mathbb{L}}$]{}; (s1tag2) \[below of=s1tag1\] [$\Xi^\mathbb{L}_{\mathtt{L\_SIDEOFBODY}}$]{}; (s1tag3) \[below of=s1tag2\] [$\Xi^\mathbb{R}_{\mathtt{R\_SIDEOFBODY}}$]{}; (s1tag4) \[below of=s1tag3\] [$\mathcal{F}^{\mathbb{R}}_{\mathtt{KEY\_CONFIG}}$]{}; (s1tag5) \[below of=s1tag4\] [$\mathcal{F}^{\mathbb{L}}_{\mathtt{KEY\_CONFIG}}$]{}; (s1tag6) \[below of=s1tag5\] [$\neg\mathcal{T}^{\mathbb{R}}_{\mathbb{L}}$]{}; (s1tag7) \[below of=s1tag6\] [$\vdots$]{}; (s0) edge node [$\nearrow_{\mathbb{L}}$]{} (s1); (s1) edge\[loop above, distance=.5cm\] node [$\leftrightsquigarrow_\mathbb{D} \cap \leftrightsquigarrow_\mathbb{G}$]{} (s1); (s2) \[right of=s1\] [$\vdots$]{}; (s2tag1) \[below=0.05cm of s2\] [$\mathbb{R}^\leftarrow_{\mathbb{L}}$]{}; (s2tag2) \[below of=s2tag1\] [$\Xi^\mathbb{L}_{\mathtt{CENTEROFBODY}}$]{}; (s2tag3) \[below of=s2tag2\] [$\Xi^\mathbb{R}_{\mathtt{R\_SIDEOFHEAD}}$]{}; (s2tag4) \[below of=s2tag3\] [$\mathcal{F}^{\mathbb{R}}_{\mathtt{BEAK\_CONFIG}}$]{}; (s2tag5) \[below of=s2tag4\] [$\mathcal{F}^{\mathbb{L}}_{\mathtt{INDEX\_CONFIG}}$]{}; (s2tag6) \[below of=s2tag5\] [$\neg\mathcal{T}^{\mathbb{R}}_{\mathbb{L}}$]{}; (s2tag7) \[below of=s2tag6\] [$\vdots$]{}; (s1) edge node [$\swarrow_{\mathbb{L}}$]{} (s2); (s3) \[right of=s2\] [$\vdots$]{}; (s3tag1) \[below=0.05cm of s3\] [$\mathbb{R}^\leftarrow_{\mathbb{L}}$]{}; (s3tag2) \[below of=s3tag1\] [$\Xi^\mathbb{L}_{\mathtt{L\_SIDEOFBODY}}$]{}; (s3tag3) \[below of=s3tag2\] [$\Xi^\mathbb{R}_{\mathtt{R\_SIDEOFBODY}}$]{}; (s3tag4) \[below of=s3tag3\] [$\mathcal{F}^{\mathbb{R}}_{\mathtt{OPENPALM\_CONFIG}}$]{}; (s3tag5) \[below of=s3tag4\] [$\mathcal{F}^{\mathbb{L}}_{\mathtt{OPENPALM\_CONFIG}}$]{}; (s3tag6) \[below of=s3tag5\] [$\neg\mathcal{T}^{\mathbb{R}}_{\mathbb{L}}$]{}; (s3tag7) \[below of=s3tag6\] [$\vdots$]{}; (s2) edge node [$\nearrow_{\mathbb{L}}$]{} (s3);
Figure \[fig:modeling\] shows an example of the process. Here, each key posture is codified into propositions acknowledging the hand positions with respect to each other ($\mathbb{R}^\leftarrow_{\mathbb{L}}$), their place of articulation ([*e.g.*]{} “left hand floats over the torse” with $\Xi^\mathbb{L}_{\mathtt{TORSE}}$), their configuration ([*e.g.*]{} “right hand is open” with $\mathcal{F}^{\mathbb{R}}_{\mathtt{OPENPALM\_CONFIG}}$) and their movements ([*e.g.*]{} “left hand moves to the up-left direction” with $\nearrow_{\mathbb{L}}$).
This module also checks that the generated graph is correct: it will discard simple tracking errors to ensure that the resulting LTS will remain consistent.
Verification Module
-------------------
First of all, the verification module has to be loaded with a database of sign descriptions encoded as [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}formulas. These will characterize the specific sequence of key postures that morphologically describe a sign. For example, let’s take the case for sign “route” in FSL, shown in figure \[fig:routelsf\], with the following [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}formulation,
$$\begin{split}
(\Xi^\mathbb{R}_\mathtt{FACE} \wedge
\Xi^\mathbb{L}_\mathtt{FACE} \wedge
\mathbb{L}^\rightarrow_\mathbb{R} \wedge
\mathcal{F}^\mathbb{R}_\mathtt{CLAMP} \wedge
\mathcal{F}^\mathbb{L}_\mathtt{CLAMP} \wedge
\mathcal{T}^\mathbb{R}_\mathbb{L})
\rightarrow \\
[\leftarrow_\mathbb{R} \cap
\rightarrow_\mathbb{L}](
\mathbb{L}^\rightarrow_\mathbb{R} \wedge
\mathcal{F}^\mathbb{R}_\mathtt{CLAMP} \wedge
\mathcal{F}^\mathbb{L}_\mathtt{CLAMP} \wedge
\neg\mathcal{T}^\mathbb{R}_\mathbb{L})
\end{split}
\label{eq:route}$$
![$\mathtt{ROUTE}_\mathsf{FSL}$ production. \[fig:routelsf\]](./figures/route1DS.png "fig:") ![$\mathtt{ROUTE}_\mathsf{FSL}$ production. \[fig:routelsf\]](./figures/route2DS_arr.png "fig:")
Formula (\[eq:route\]) describes $\mathtt{ROUTE}_\mathsf{FSL}$ as a sign with two key postures, connected by a two-hand simultaneous movement (represented with operator $\cap$). It also indicates the position of each hand, their orientation, whether they touch and their respective configurations (in this example, both hold the same `CLAMP` configuration).
The module can then verify whether a sign formula in the lexical database holds in any sub-sequence of states of the graph generated in the previous step. Algorithm \[alg:verification\] sums up the process.
SL model $\mathcal{M}_{{\mathsf{SL}}}$ connected graph $\mathcal{G}_{{\mathsf{SL}}}$ lexical database $\mathcal{DB}_{{\mathsf{SL}}}$ Proposals\_For\[$state\_qty$\] Proposals\_For\[$s$\].append($\varphi$)\
For each state, the algorithm returns a set of possible signs. Expert users (or higher level algorithms) can further refine the process by introducing additional information previously missed by the tracker.
Conclusions and Future Work {#conclusions_fw}
===========================
We have shown how a logical language can be used to model SL signs for semi-automatic recognition, albeit with some restrictions. The traits we have chosen to represent were imposed by the limits of the tracking tools we had to our disposition, most notably working with 2D coordinates. With these in mind, we tried to design something flexible that could be easily adapted by computer scientists and linguists alike. Our primitive sets, were intentionally defined in a very general fashion due to the same reason: all of the perceived directions, articulators and places of articulation can easily change their domains, depending on the SL we are modeling or the technological constraints we have to deal with. Propositions can also be changed, or even induced, by existing written sign representation languages such as Zebedee [@filhol_modedescriptif_2008] or HamNoSys [@hanke_hamnosysrepresenting_2004], mainly for the sake of extendability.
From the application side, we still need to create an extensive sign database codified in [$\mathrm{PDL}_{{\mathsf{SL}}}$ ]{}and try recognition on other corpora, with different tracking information. For verification and model extraction, further optimizations are expected, including the handling of data inconsistencies and repairing broken queries when verifying the graph.
Regarding our theoretical issues, future work will be centered in improving our language to better comply with SL research. This includes adding new features, like incorporating probability representation to improve recognition. We also expect to finish the definition of our formal semantics, as well as proving correction and complexity of our algorithms.
[^1]: Supported by CONACYT (Mexico) scholarship program.
|
---
abstract: 'We analyze numerically the magnetorotational instability of a Taylor-Couette flow in a helical magnetic field [\[]{}helical magnetorotational instability (HMRI)\] using the inductionless approximation defined by a zero magnetic Prandtl number ($\Pm=0).$ The Chebyshev collocation method is used to calculate the eigenvalue spectrum for small amplitude perturbations. First, we carry out a detailed conventional linear stability analysis with respect to perturbations in the form of Fourier modes that corresponds to the convective instability which is not in general self-sustained. The helical magnetic field is found to extend the instability to a relatively narrow range beyond its purely hydrodynamic limit defined by the Rayleigh line. There is not only a lower critical threshold at which HMRI appears but also an upper one at which it disappears again. The latter distinguishes the HMRI from a magnetically modified Taylor vortex flow. Second, we find an absolute instability threshold as well. In the hydrodynamically unstable regime before the Rayleigh line, the threshold of absolute instability is just slightly above the convective one although the critical wavelength of the former is noticeably shorter than that of the latter. Beyond the Rayleigh line the lower threshold of absolute instability rises significantly above the corresponding convective one while the upper one descends significantly below its convective counterpart. As a result, the extension of the absolute HMRI beyond the Rayleigh line is considerably shorter than that of the convective instability. The absolute HMRI is supposed to be self-sustained and, thus, experimentally observable without any external excitation in a system of sufficiently large axial extension.'
author:
- Jānis Priede
- Gunter Gerbeth
title: 'Absolute versus convective helical magnetorotational instability in a Taylor-Couette flow'
---
Introduction
============
The magnetorotational instability (MRI) is known to be able to destabilize hydrodynamically stable flows by means of an externally imposed magnetic field as originally shown by Velikhov [@Velikhov-1959] and analyzed in more detail by Chandrasekhar [@Chandrasekhar-1960] for cylindrical Taylor-Couette flow of a perfectly conducting fluid subject to an axial magnetic field. Three decades later Balbus and Hawley [@Balbus-Hawley-1991] suggested that, in a similar way, the hydrodynamically stable Keplerian velocity distribution in accretion disks could be rendered turbulent by the MRI accounting for the formation of stars and entire galaxies proceeding much faster than it could be accomplished by the viscous angular-momentum transport alone. Meanwhile this proposition has triggered not only numerous theoretical and numerical studies [@Balbus-Hawley-1998] but also some experimental efforts as well [@Sisan-etal; @Nature-2006]. However, one of the main technical challenges to laboratory MRI is the magnetic Reynolds number $\Rm$ which is required to be $\sim10$ at least. For a liquid metal with the magnetic Prandtl number $\Pm\sim10^{-5}-10^{-6}$ this translates into a hydrodynamic Reynolds number $\RE=\Rm/\Pm\sim10^{6}-10^{7}$ [@Goodman-Ji-2002]. Thus, the base flow on which the MRI is supposed to be observable may easily be turbulent at such Reynolds numbers independently of MRI as in the experiment of Sisan *et al.* [@Sisan-etal]. A way to circumvent this problem was proposed by Hollerbach and Rüdiger [@Hollerbach-Ruediger-2005] who suggested that MRI can take place in the Taylor-Couette flow at $\RE\sim10^{3}$ when the imposed magnetic field is helical rather than purely axial as in the classical case. The theoretical prediction of this new type of helical MRI (HMRI) was soon succeeded by a confirming experimental evidence provided by the so-called PROMISE facility [@Rued-apjl; @Stefani-etal; @Stefani-NJP]. Nevertheless, these experimental observations have subsequently been questioned by Liu *et al.* [@Liu-etal2006] who find no such instability in their inviscid theoretical analysis of finite length cylinders with insulating end caps. They suspect the observed phenomenon to be a transient growth rather than a self-sustained instability [@Liu-etal2007; @Liu2008].
Indeed, such an interpretation of the HMRI is possible when the analysis is based only on the conventional linear stability analysis for separate Fourier modes as done by Hollerbach and Rüdiger [@Hollerbach-Ruediger-2005] following the classical MRI approach. However, there is a principal difference between the classical and the helical MRIs, namely, the former is stationary whereas the latter is traveling. It is important to emphasize that the conventional stability analysis for traveling waves yields the so-called convective instability threshold at which the system becomes able to amplify certain externally excited perturbations. At this threshold the perturbation grows in time only in the frame of reference moving with its group velocity while it asymptotically decays in any other frame of reference including the laboratory one [@Landau-87]. Eventually, such a growing while traveling perturbation reaches the end wall where it is absorbed unless the system is able to reflect it back. The latter supposes reflection symmetry in the system which, however, is not the case provided that the magnetic field is helical. Thus, it is indeed unclear whether the HMRI can be self-sustained in an ideal Taylor-Couette flow of large but finite axial extension.
This question is addressed in the second part of the present study where the absolute HMRI is found to exist besides the convective one which, in turn, is analyzed in detail in the first part. Note that the existence of absolute instability is nontrivial as known, for instance, for the Ponomarenko dynamo [@Pono] which has a convective but no absolute instability threshold [@Agris1]. The latter requires an additional return flow to be included in the original Ponomarenko model [@Agris2]. The distinction between convective and absolute instabilities is relevant mainly for open flows and unbounded geometries [@Huerre-Monkewitz-1990]. In finite geometries, it is important to distinguish transiently growing and noise-sustained perturbations from the self-sustained linear instabilities, which are always global with the threshold asymptotically approaching from above that of the absolute instability as the system size increases [@Tobias-etal-1998; @Proctor-etal-2000].
We consider both the convective and the absolute HMRIs in the inductionless approximation corresponding to $\Pm=0$ that was suggested in our previous work [@Priede-etal-2007]. This approximation, which leads to a significant simplification of the problem, allows us to focus exclusively on the HMRI because it does not capture the conventional MRI [@Herron-Goodman-2006]. We show that the HMRI is effective only in a relatively narrow range of the ratio of rotation rates of the inner and outer cylinders beyond the limit of purely hydrodynamic instability defined by the so-called Rayleigh line. For the convective HMRI, the range of instability is considerably larger for perfectly conducting cylinders than that for insulating ones. In addition we find that the HMRI is effective only in a limited range of Reynolds numbers. Namely, for any unstable mode, there is not only a lower critical Reynolds number by exceeding which the HMRI sets in but also an upper one by exceeding which it disappears again. It is this upper threshold that distinguishes HMRI from a magnetically modified Taylor vortex flow. Absolute HMRI exists in a significantly narrower range of parameters than the convective one. In contrast to the convective HMRI, the absolute one is much less dependent on the conductivity of the boundaries.
The paper is organized as follows. In Sec. \[sec:prob-form\] we formulate the problem using the inductionless approximation. Numerical results concerning the convective and absolute instability thresholds for both insulating and perfectly conducting cylinders are presented in Secs. \[sub:Conv\] and \[sub:Abs-inst\], respectively. Section \[sec:summ\] concludes the paper with a summary and a comparison with experimental results of Stefani *et al.* [@Rued-apjl; @Stefani-etal; @Stefani-NJP].
![\[fig:sketch\]Sketch to the formulation of the problem.](fig1){width="30.00000%"}
\[sec:prob-form\]Problem formulation
====================================
Consider an incompressible fluid of kinematic viscosity $\nu$ and electrical conductivity $\sigma$ filling the gap between two infinite concentric cylinders with inner radius $R_{i}$ and outer radius $R_{o}$ rotating with angular velocities $\Omega_{i}$ and $\Omega_{o}$, respectively, in the presence of an externally imposed steady magnetic field $\vec{B}_{0}=B_{\phi}\vec{e}_{\phi}+B_{z}\vec{e}_{z}$ with axial and azimuthal components $B_{z}=B_{0}$ and $B_{\phi}=\beta B_{0}R_{i}/r$ in cylindrical coordinates $(r,\phi,z),$ where $\beta$ is a dimensionless parameter characterizing the geometrical helicity of the field (Fig. \[fig:sketch\]). Further, we assume the magnetic field of the currents induced by the fluid flow to be negligible relative to the imposed field. This corresponds to the so-called inductionless approximation holding for most of liquid-metal magnetohydrodynamics characterized by small magnetic Reynolds numbers $\Rm=\mu_{0}\sigma v_{0}L\ll1,$ where $\mu_{0}$ is the magnetic permeability of vacuum and $v_{0}$ and $L$ are the characteristic velocity and length scale. The velocity of fluid flow $\vec{v}$ is governed by the Navier-Stokes equation with electromagnetic body force $$\frac{\partial\vec{v}}{\partial t}+(\vec{v}\cdot\vec{\nabla})\vec{v}=-\frac{1}{\rho}\vec{\nabla}p+\nu\vec{\nabla}^{2}\vec{v}+\frac{1}{\rho}\vec{j}\times\vec{B}_{0},\label{eq:N-S}$$ where the induced current follows from Ohm’s law for a moving medium $$\vec{j}=\sigma\left(\vec{E}+\vec{v}\times\vec{B}_{0}\right).\label{eq:Ohm}$$ In addition, we assume the characteristic time of velocity variation to be much longer than the magnetic diffusion time $\tau_{0}\gg\tau_{m}=\mu_{0}\sigma L^{2}$ that leads to the quasi-stationary approximation, according to which $\vec{\nabla}\times\vec{E}=0$ and $\vec{E}=-\vec{\nabla}\Phi,$ where $\Phi$ is the electrostatic potential. Mass and charge conservations imply $\vec{\nabla}\cdot\vec{v}=\vec{\nabla}\cdot\vec{j}=0.$
The problem admits a base state with a purely azimuthal velocity distribution $\vec{v}_{0}(r)=\vec{e}_{\phi}v_{0}(r),$ where $$v_{0}(r)=r\frac{\Omega_{o}R_{o}^{2}-\Omega_{i}R_{i}^{2}}{R_{o}^{2}-R_{i}^{2}}+\frac{1}{r}\frac{\Omega_{o}-\Omega_{i}}{R_{o}^{-2}-R_{i}^{-2}}.$$ Note that the magnetic field does not affect the base flow because it gives rise only to the electrostatic potential $\Phi_{0}(r)=B_{0}\int v_{0}(r)dr$ whose gradient compensates the induced electric field, so that there is no current in the base state $(\vec{j}_{0}=0)$. However, a current may appear in a perturbed state, $$\left\{ \begin{array}{c}
\vec{v},p\\
\vec{j},\Phi\end{array}\right\} (\vec{r},t)=\left\{ \begin{array}{c}
\vec{v}_{0},p_{0}\\
\vec{j}_{0},\Phi_{0}\end{array}\right\} (r)+\left\{ \begin{array}{c}
\vec{v}_{1},p_{1}\\
\vec{j}_{1},\Phi_{1}\end{array}\right\} (\vec{r},t)$$ where $\vec{v}_{1},$ $p_{1},$ $\vec{j}_{1},$ and $\Phi_{1}$ present small-amplitude perturbations for which Eqs. (\[eq:N-S\]) and (\[eq:Ohm\]) after linearization take the form $$\frac{\partial\vec{v}_{1}}{\partial t}+(\vec{v}_{1}\cdot\vec{\nabla})\vec{v}_{0}+(\vec{v}_{0}\cdot\vec{\nabla})\vec{v}_{1}=-\frac{1}{\rho}\vec{\nabla}p_{1}+\nu\vec{\nabla}^{2}\vec{v}_{1}+\frac{1}{\rho}\vec{j}_{1}\times\vec{B}_{0},\label{eq:v1}$$ $$\vec{j}_{1}=\sigma\left(-\vec{\nabla}\Phi_{1}+\vec{v}_{1}\times\vec{B}_{0}\right).\label{eq:j1}$$ In the following, we focus on axisymmetric perturbations which are typically much more unstable than nonaxisymmetric ones [@Rued-ANN]. For such perturbations the solenoidity constraints are satisfied by meridional stream functions for fluid flow and electric current as $$\vec{v}=v\vec{e}_{\phi}+\vec{\nabla}\times(\psi\vec{e}_{\phi}),\qquad\vec{j}=j\vec{e}_{\phi}+\vec{\nabla}\times(h\vec{e}_{\phi}).$$ Note that $h$ is the azimuthal component of the induced magnetic field which is used subsequently instead of $\Phi$ for the description of the induced current. Thus, we effectively retain the azimuthal component of the induction equation to describe meridional components of the induced current while the azimuthal current is explicitly related to the radial velocity. The use of the electrostatic potential $\Phi,$ which provides an alternative mathematical formulation for the induced currents in the inductionless approximation, would result in slightly more complicated governing equations. In addition, for numerical purposes, we introduce also the vorticity $\vec{\omega}=\omega\vec{e}_{\phi}+\vec{\nabla}\times(v\vec{e}_{\phi})=\vec{\nabla}\times\vec{v}$ as an auxiliary variable. The perturbation is sought in the normal mode form $$\left\{ v_{1},\omega_{1,}\psi_{1},h_{1}\right\} (\vec{r},t)=\left\{ \hat{v},\hat{\omega},\hat{\psi},\hat{h}\right\} (r)e^{\gamma t+ikz},\label{eq:pert}$$ where $\gamma$ is, in general, a complex growth rate and $k$ is the axial wave number which is real for the conventional stability analysis and complex for absolute instability. Henceforth, we proceed to dimensionless variables by using $R_{i},$ $R_{i}^{2}/\nu,$ $R_{i}\Omega_{i},$ $B_{0},$ and $\sigma B_{0}R_{i}\Omega_{i}$ as the length, time, velocity, magnetic field, and current scales, respectively. The nondimensionalized governing equations then read as $$\begin{aligned}
\gamma\hat{v} & = & D_{k}\hat{v}+\RE ik(r^{2}\Omega)'r^{-1}\hat{\psi}+\Ha^{2}ik\hat{h},\label{eq:vhat}\\
\gamma\hat{\omega} & = & D_{k}\hat{\omega}+2\RE ik\Omega\hat{v}-\Ha^{2}ik(ik\hat{\psi}+2\beta r^{-2}\hat{h}),\label{eq:omghat}\\
0 & = & D_{k}\hat{\psi}+\hat{\omega},\label{eq:psihat}\\
0 & = & D_{k}\hat{h}+ik(\hat{v}-2\beta r^{-2}\hat{\psi}),\label{eq:hhat}\end{aligned}$$ where $D_{k}f\equiv r^{-1}\left(rf'\right)'-(r^{-2}+k^{2})f$ and the prime stands for $d/dr,$ $\RE=R_{i}^{2}\Omega_{i}/\nu$ and $\Ha=R_{i}B_{0}\sqrt{\sigma/(\rho\nu)}$ are Reynolds and Hartmann numbers, respectively, and $$\Omega(r)=\frac{\lambda^{-2}-\mu+r^{-2}\left(\mu-1\right)}{\lambda^{-2}-1}$$ is the dimensionless angular velocity of the base flow defined by $\lambda=R_{o}/R_{i}$ and $\mu=\Omega_{o}/\Omega_{i}$. The boundary conditions for the flow perturbation on the inner and outer cylinders at $r=1$ and $r=\lambda,$ respectively, are $\hat{v}=\hat{\psi}=\hat{\psi}'=0.$ Boundary conditions for $\hat{h}$ on insulating and perfectly conducting cylinders, respectively, are $\hat{h}=0$ and $(r\hat{h})'=0$ at $r=1;\lambda.$
The governing equations (\[eq:vhat\])–(\[eq:hhat\]) for perturbation amplitudes were discretized using a spectral collocation method on a Chebyshev-Lobatto grid with a typical number of internal points $N=32-96$. Auxiliary Dirichlet boundary conditions for $\hat{\omega}$ were introduced and then numerically eliminated to satisfy the no-slip boundary conditions $\hat{\psi}'=0.$ The electric stream function $\hat{h}$ was expressed in terms of $\hat{v}$ and $\hat{\psi}$ by solving Eq. (\[eq:hhat\]) and then substituted in Eqs. (\[eq:vhat\]) and (\[eq:omghat\]) that eventually resulted in the $2N\times2N$ complex matrix eigenvalue problem which was solved by the LAPACK’s ZGEEV routine.
Numerical results
=================
\[sub:Conv\]Convective instability
----------------------------------
In this section, we consider the so-called convective instability threshold supplied by the conventional linear stability analysis with real wave numbers $k,$ as done in most of previous studies [@Hollerbach-Ruediger-2005; @Stefani-etal; @Priede-etal-2007]. Note that at the convective instability threshold the system becomes able to amplify certain perturbations which however might be not self-sustained and, thus, experimentally unobservable without an external excitation. The following results concern the radii ratio of outer to inner cylinder $\lambda=2$ and we start with insulating cylinders which form the side walls of the system.
### \[sub:conv-insl\]Insulating cylinders
{width="45.00000%"}{width="45.00000%"}\
{width="45.00000%"}
The critical Reynolds number, wave number, and frequency are shown in Fig. \[cap:Rec-mu\] versus the angular velocity ratio $\mu$ of outer to inner cylinder for Hartmann number $\Ha=15$ and various geometrical helicities $\beta.$ For $\beta=0$ corresponding to a purely axial magnetic field, the critical Reynolds number tends to infinity as $\mu$ approaches the Rayleigh line $\mu_{c}=\lambda^{-2}=0.25$ defined by $d\left(r^{2}\Omega\right)/dr=0.$ Thus, for $\beta=0,$ the range of instability is limited by the Rayleigh line, *i.e.*, $\mu<\mu_{c},$ as in the purely hydrodynamic case. For helical magnetic fields defined by $\beta\not=0,$ the instability extends well beyond the Rayleigh line, as originally found by Hollerbach and Rüdiger [@Hollerbach-Ruediger-2005]. Note that it is this extension of the instability beyond its purely hydrodynamic limit, that for ideal Taylor-Couette flow is defined by the Rayleigh line, which constitutes the essence of the MRI. Comparing the stability curves presented in Fig. \[cap:Rec-mu\](a) for fixed $\Ha$ to those of Hollerbach and Rüdiger [@Hollerbach-Ruediger-2005], which are presented for $\Pm\not=0$ and variable $\Ha$ yielding minimal $\RE_{c},$ there are two differences to note. First, the range of instability is limited by a certain $\mu_{\max},$ which depends on the helicity $\beta$ and the Hartmann number, as shown in Fig. \[cap:muc-bt\](a). Second, the destabilization beyond the Rayleigh line is effective only in a limited range of Reynolds numbers bounded by an upper critical value which tends to infinity as $\mu$ approaches the Rayleigh line from the right.
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The origin of the upper critical Reynolds number, by exceeding which the flow becomes linearly stable again, is illustrated in Fig. \[cap:re-wk\] showing (a) the Reynolds number and (b) the corresponding frequency of marginally stable modes versus their wave number $k$ for $\beta=5,$ $\Ha=15,$ and $\mu=0.27.$ As seen, the marginal stability curves for $\mu$ beyond the Rayleigh line $(\mu>0.25)$ form closed loops which collapse at $\mu=\mu_{\max}.$ Thus, unstable modes exist only within limited ranges of wave and Reynolds numbers. Obviously, at sufficiently large Reynolds numbers the flow becomes effectively non-magnetic as inertia starts to dominate over the electromagnetic forces suppressing the HMRI.
Note that the suppression of HMRI at high $\RE$ is related to the negligibility of the induced magnetic field as long as $\Rm\ll1.$ In this case, the electric current is induced only by the velocity perturbation crossing the imposed magnetic field. In the conventional MRI conversely to the HMRI, also the induced field is relevant, which crossed by the base flow induces an additional electric current. These two effects are easily noticeable in the induction equation. Thus, in the HMRI, the resulting electromagnetic force is not affected by the base flow which is not the case for the conventional MRI, where the electromagnetic force remains significant with respect to inertia also at high $\RE.$
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As seen in Figs. \[cap:Rec-Ha\](a) and \[cap:Rec-Ha\](b), the critical Reynolds number can vary with the Hartmann number in three different ways depending on $\mu.$ For $\mu=0,$ the critical Reynolds number is bounded at $\Ha=0$ because a purely hydrodynamic instability is possible before the Rayleigh line. Numerical results evidence that the increase in the Hartmann number results in the growth of the critical Reynolds number with asymptotics $\sim\Ha.$ For $\mu=0.25,$ which lies exactly on the Rayleigh line, the flow is hydrodynamically stable without the magnetic field. Thus, in this case, the critical Reynolds number increases as $\sim\Ha^{-2}$ as $\Ha\rightarrow0$ because there is no finite value of the critical Reynolds number without the magnetic field. The corresponding critical wave number tends to a finite value independent of $\beta.$ With increase in the Hartmann number, the critical Reynolds number attains a minimum at $\Ha\sim10$ and starts to grow at larger Hartmann numbers similarly to the previous case. The corresponding critical wave number decreases asymptotically as $\sim\Ha^{-1}$ that means a critical wavelength increasing directly with the magnetic field strength. The critical frequency plotted in Fig. \[cap:Rec-Ha\] changes from a constant value of 30 at small $\Ha$ to another nearly constant value of about $100$ slightly varying with $\beta$ at large $\Ha.$ At large helicities $(\beta=15),$ another relatively short-wave instability mode dominates up to a Hartmann number $\Ha\approx30,$ where the most unstable mode switches back to the long-wave one which is characteristic for smaller helicities. Transition to this large-$\beta$ mode is also obvious in Fig. \[cap:Rec-mu\] for $\beta=15$ at $\mu\approx0.235.$ As seen in Fig. \[cap:Rec-Ha\](b) for $\mu=0.27$, which is beyond the Rayleigh line, there is no instability as $\Ha\rightarrow0.$ Consequently, a finite minimal value of $\Ha$ depending on $\beta$ is necessary in this case. Moreover, the instability is limited by the upper branch of the critical Reynolds number discussed above which merges with the lower branch at the minimum of the Hartmann number for the given helicity $\beta.$
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The variation of the critical Reynolds number with the helicity $\beta$ shown in Fig. \[cap:Rec-bt\](a) for $\mu=0.25=\mu_{c}$ lying exactly on the Rayleigh line differs considerably from the other case with $\mu=0.27>\mu_{c}$ [\[]{}see Fig. \[cap:Rec-bt\]b\]. In the first case, the flow can be destabilized by the magnetic field of however small helicity $\beta\rightarrow0$ that results in the critical Reynolds number increasing as $\sim1/\beta.$ For $\mu>\mu_{c},$ a certain minimal helicity depending on the Hartmann number is needed. Moreover, in this case, there is also an upper critical Reynolds number. In both cases, there is some optimal $\beta\approx5-8$ at which the lower critical Reynolds attains a minimum. Further increase in $\beta$ results in the growth of the critical Reynolds number with a significantly different asymptotic behavior in both considered cases. For $\mu>\mu_{c},$ there is a maximal $\beta$ depending on the Hartmann number at which the upper and lower branches of the critical Reynolds number merge together and the instability disappears whereas there seems to be no such merging point at any finite $\beta$ when $\mu=\mu_{c}.$ The critical wave number plotted in Figs. \[cap:Rec-bt\](c) and \[cap:Rec-bt\](d) is seen to increase with $\beta$ with some jumps at larger $\Ha$ as discussed above.
### Perfectly conducting cylinders
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For perfectly conducting cylinders, the marginal stability curves shown in Fig. \[cap:re-wk-c\] differ considerably from those for insulating walls (see Fig. \[cap:re-wk\]). Although in both cases beyond the Rayleigh line large wave numbers $(k\gg1)$ are always stable, the range of instability for perfectly conducting cylinders at moderate $\beta\lesssim10$ extends to arbitrary small wave numbers $k\rightarrow0$ whereas for insulating cylinders it is limited to sufficiently large $k$. As in the insulating case, for each unstable mode there is not only the lower but also the upper marginal Reynolds number both increasing as $\sim1/k$ toward small $k.$ Thus, the increase in the Reynolds number results in the shift of instability to smaller wave numbers, *i.e.*, longer waves. As a result, there is no upper critical Reynolds number for moderate $\beta\lesssim10$ when both cylinders are perfectly conducting.
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Consequently, as seen in Figs. \[cap:Rec-mu-c\](a) and \[cap:Rec-mu-c\](b), the critical Reynolds number becomes very large while the critical wave number tends to zero as $\mu$ approaches some critical $\mu_{\max}$ which varies with $\beta.$ The critical frequency $\omega_{c}$ shown in Fig. \[cap:Rec-mu-c\](c) tends, respectively, to some finite value. This behavior changes at larger $\beta$ becoming similar to that for insulating cylinders. As seen in Fig. \[cap:Rec-mu-c\](a), for $\beta\gtrsim10,$ the curves of the critical Reynolds number start to bend back at $\mu_{\max}$ toward smaller $\mu$ rather than tend to infinity. The corresponding critical wave numbers remain finite whereas the critical frequency increases with the Reynolds number [\[]{}see Figs. \[cap:Rec-mu-c\]b and \[cap:Rec-mu-c\]c\]. At intermediate $\beta$ the limiting value of $\mu_{\max},$ up to which the instability extends beyond the Rayleigh line, is seen in Fig. \[cap:muc-bt\](a) to attain a maximum which is considerably larger than that for insulating walls. At larger $\beta$, the limiting values $\mu_{\max}$ decrease approaching those for insulating walls.
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The dependence of the critical Reynolds number on the Hartmann number plotted in Fig. \[cap:Rec-Ha-c\](a) at various helicities is similar to that for insulating cylinders. First, $\RE_{c}$ attains a minimum at $\Ha=7-10$ and a finite minimal value of the Hartmann number is required for instability when $\mu>\mu_{c}.$ For moderate $\beta\lesssim10,$ in contrast to the insulating case, $\RE_{c}$ and $k_{c}$ tend to infinity and zero, respectively, as the Hartmann number approaches this minimal value which depends on $\beta.$ For $\beta\gtrsim10,$ the critical $\RE_{c}$ has an upper branch which merges with the lower one at the minimal value of $\Ha$ as in the case of insulating cylinders. At sufficiently large Hartmann numbers, the instability is seen to switch to a long-wave mode with the critical wave numbers and Reynolds numbers varying asymptotically as $\sim\Ha^{-1}$ and $\sim\Ha,$ respectively.
\[sub:Abs-inst\]Absolute instability
------------------------------------
In this section, we turn to the absolute instability for which the wave number $k$ is in general a complex quantity with real and imaginary parts $k_{r}$ and $k_{i},$ respectively [@Lifshitz-Pitaevskii-81; @Schmid-Henningson]. It is important to realize that the convective instability threshold considered above is not sufficient for the development of a self-sustained instability unless the system is mirror symmetric along the direction of propagation which, however, is not the case when the magnetic field is helical. The convective instability just ensures the ability of the system to amplify external perturbations excited with the critical frequency. From the mathematical point of view, the problem is that in an axially bounded system the perturbation has to meet certain boundary conditions at two end walls that, however, can not be accomplished by a single Fourier mode. When the critical Fourier mode is replaced with a corresponding wave packet of a limited spatial extension, one finds such a perturbation to grow only in the frame of reference traveling with its group velocity while it decays asymptotically in any other frame of reference including the laboratory one which is at rest. The growth of a perturbation in the laboratory frame of reference is ensured by the absolute instability threshold, at which the group velocity of the wave packet becomes zero. Thus, formally, the absolute instability requires one more condition to be satisfied, *i.e.*, zero group velocity, by means of an additional free parameter—the imaginary part of the wave number. Note that although the group velocity in non-conservative media is, in general, a complex quantity, for the most dangerous perturbations satisfying $\partial\gamma_{r}/\partial k_{r}=0,$ it is real and, thus, coincides with its common definition.
Alternatively, the absolute instability may be regarded as an asymptotic case of the global instability when the axial extension of the system becomes very large [@Landau-87; @Lifshitz-Pitaevskii-81; @Kulikovskii-66]. The basic idea is that for a convectively unstable perturbation to become self-sustained a feedback mechanism is needed which could transfer a part of the growing perturbation as it leaves the system back to its origin. Such a feedback can be provided by the reflections of perturbation from the end walls or, generally, by the end regions where the base state becomes axially non-uniform. If the base state is both stationary and axially uniform, the coefficients of the linearized perturbation equations do not depend on time and on the axial coordinate, respectively. Then, as for linear differential equations with constant coefficients, the particular solution for the perturbation varies exponentially in both time and axial coordinate as supposed by Eq. (\[eq:pert\]) where both the growth rate $\gamma$ and the wave number $k$ may be in general complex. At the end walls, where the base state is no longer axially invariant, the particular solutions with different wave numbers become linearly coupled while their time variation remains unaffected as long as the base state is stationary. Thus, the reflection of a perturbation by the end wall in general couples modes with different wave numbers but with the same $\gamma.$ Sufficiently far away from the end walls the reflected perturbation is dominated by the mode with the imaginary part of the wave number $k_{i}$ corresponding to either the largest growth or lowest decay rate along the axis. Consequently, sufficiently away from the end walls a global mode is expected to consist of two such waves coupled by reflections from the opposite end walls. Taking into account that the amplitude of the reflected wave is proportional to that of the incident wave, it is easy to find that in a sufficiently extended system both waves must have the same imaginary part $k_{i}$ of the wave number whereas the real parts may be different. Additionally, for two such waves to be coupled by reflections from the end walls, they have to propagate in opposite directions.
### Insulating cylinders
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We search for such a pair of modes by considering the conventional neutral stability curves at various $k_{i}.$ As seen in Figs. \[cap:re-wki\](a) and (b), the increase in $k_{i},$ on one hand, results in the reduction in the wave number range admitting such neutrally stable modes. On the other hand, the lower branch of marginal $\RE$ and the corresponding frequency first increase with $k_{i}$ in the whole wave number range and then start to decrease at larger $k_{r}$ when $k_{i}$ becomes sufficiently large $(k_{i}\gtrsim1.8).$ However, more important information is obtained by plotting the marginal $\RE$ and frequency from the previous curves against each other as in Fig. \[cap:re-wki\](c). First, similarly to the previous curves, these ones also form closed loops that shrink as $k_{i}$ is increased. It is important to notice that at sufficiently large $k_{i}$ these loops start to intersect themselves in some point as shown in the inset at the top of Fig. \[cap:re-wki\](c). This self-intersection, which is of primary importance here, occurs only in the limited range of positive $k_{i}.$ The point of intersection means that at the given Reynolds number there are two modes with the same frequency and the same imaginary but possibly different real parts of the wave number. As discussed above, two such modes could be coupled by reflections from the end walls and thus form a neutrally stable global mode in an axially bounded system provided that they propagate in opposite directions [@Lifshitz-Pitaevskii-81; @Kulikovskii-66]. To determine the direction of propagation we use a local criterion [@Priede-Gerbeth-97] which we showed to be equivalent to the Briggs pinching criterion [@Briggs-1964] for the upper instability branch at the given $k_{i}.$ Namely, the direction of propagation of both intersecting branches can be deduced from their variation with $k_{i}.$ If upon a small variation of $k_{i}$ one branch rises to higher $\RE$ while the other descends to lower $\RE$, which is the case here, it can be shown that both intersecting branches correspond to oppositely propagating modes. The lowest possible Reynolds number admitting two such modes is attained when the loop below the intersection point collapses to a cusp as seen in the inset at the bottom of Fig. \[cap:re-wki\](c) for $k_{i}=1.8.$ The cusp is formed as both intersection points of the loop merge together. It means that at the cusp point not only the imaginary but also the real parts of both wave numbers become equal. This point corresponds to the absolute instability at which the length of the wave packet of the global mode formed by two waves with merging wave numbers tends to infinity. Further we focus on this absolute instability which, in contrast to the convective one considered above, can be self-sustained in a sufficiently extended system. Note that the approach outlined above to find the absolute instability is an extension of the well-known cusp map for the complex frequency plane to the ($\RE$-$\omega$) plane [@Schmid-Henningson; @Kupfer_87] by using the neutral stability condition $\lambda_{r}(\RE)=0$ which maps the real part of the growth rate $\lambda_{r}$ to the marginal Reynolds number.
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The critical Reynolds number, frequency, and the critical complex wave number for the absolute instability threshold is plotted in Fig. \[cap:Reg-mu\] versus $\mu$ at $\Ha=15$ and various helicities of the magnetic field. Comparison with the corresponding convective instability, the critical parameters of which are plotted in Fig. \[cap:Rec-mu\], shows that before the Rayleigh line the threshold of absolute instability is only slightly above the convective one. The difference between both thresholds becomes significant at the Rayleigh line. Although the absolute instability extends beyond the Rayleigh line when the magnetic field is helical, the range of extension is noticeably shorter than that of the convective instability (see Fig. \[cap:muc-bt\]). Moreover, the upper critical Reynolds number for the absolute instability is considerably lower than that of the convective one. Although the difference between the critical Reynolds numbers for the absolute and convective instabilities is insignificant before the Rayleigh line, the critical wave numbers for the absolute instability shown in Fig. \[cap:Reg-mu\](c) are considerably larger than those for the convective instability [\[]{}see Fig. \[cap:Rec-mu\]b\]. This difference increases with $\beta$ that results in the rise of the critical wave number for absolute instability while the increase in the corresponding quantity for the convective instability threshold is insignificant. In contrast to this, the imaginary part of the critical wave number for the lower instability branch $k_{i}\approx1.8$ is almost invariable with both $\beta$ and $\mu$ except for $\beta=1,$ where a jump of the instability to a larger wave number takes place at $\mu\approx0.235.$ Note that positive $k_{i}$ corresponds to the amplitude of the critical perturbation growing axially downward which is an additional feature predicted by the absolute instability. Beyond the Rayleigh line the absolute instability similarly to the convective one is effective only in a limited range of Reynolds numbers which is bounded from above by the upper critical branch tending to the Rayleigh line from the right as the Reynolds number increases. The critical complex wave number is seen in Figs. \[cap:Reg-mu\](c) and \[cap:Reg-mu\](d) to tend to a certain limiting value independent of $\beta.$
### Perfectly conducting cylinders
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The neutral stability curves for perfectly conducting cylinders plotted in Fig. \[cap:re-wki-c\] are seen to start forming closed loops when $k_{i}>0$ becoming similar to the corresponding curves for insulating cylinders shown in Fig. \[cap:re-wki\]. In a certain range of $k_{i}$ the curves of the critical Reynolds number plotted against the frequency in Fig. \[cap:re-wki-c\](c) intersect themselves that implies the existence of two neutrally stable modes with the same Reynolds number, frequency, and imaginary part of the wave number but different real parts of the wave number. As discussed above, two such modes can be coupled by reflections from the end walls and thus form a neutrally stable small-amplitude global mode in the system of a large but finite axial extension provided that those modes propagate in opposite directions that is implied by the variation of the marginal Reynolds number upon a small variation of $k_{i}.$
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For perfectly conducting cylinders, the critical Reynolds number, frequency, and complex wave number for the absolute instability threshold plotted in Fig. \[cap:Reg-mu-c\] versus $\mu$ for $\Ha=15$ and various helicities differ significantly from the corresponding critical parameters for the convective instability threshold (see Fig. \[cap:Rec-mu-c\]). First, the range of extension of the absolute instability beyond the Rayleigh line is much shorter than that of the convective instability. Note that in contrast to the convective instability there is no significant difference with respect to the extension of the absolute instability beyond the Rayleigh line between insulating and perfectly conducting cylinders (see Fig. \[cap:muc-bt\]). Second, beyond the Rayleigh line, similarly to insulating cylinders, for all $\beta$ the range of unstable Reynolds numbers is bounded from above by the upper critical branches which approach the Rayleigh line from the right as the upper critical Reynolds number tends to infinity. Similarly to the insulating cylinders, the corresponding critical complex wave number tends to a certain asymptotic value independent of $\beta$ [\[]{}see Figs. \[cap:Reg-mu-c\]c and \[cap:Reg-mu-c\]d\]. Third, beyond the Rayleigh line the critical wave numbers for the absolute instability are noticeably greater than those for the convective instability, especially for $\beta\lesssim10$ when the critical wave numbers for the convective instability tend to zero [\[]{}see Fig. \[cap:Rec-mu-c\]b\].
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\[sec:summ\]Conclusion
======================
![\[cap:muc-Ha\] Maximal value of $\mu$ versus the Hartmann number for absolute instability at various $\beta$.](fig14){width="45.00000%"}
In this study we have analyzed numerically the MRI of Taylor-Couette flow with a helical external magnetic field. The problem was considered in the inductionless approximation defined by a zero magnetic Prandtl number ($\Pm=0).$ First, we carried out a conventional linear stability analysis for perturbations in the form of Fourier modes specified by real wave numbers. The helical magnetic field was found to extend the original instability to a relatively narrow range beyond its purely hydrodynamic limit defined by the Rayleigh line. The range of destabilization was found to be considerably larger for perfectly conducting cylinders than that for insulating ones. For insulating cylinders, the instability beyond the Rayleigh line is effective only in a limited range of wave and Reynolds numbers. Unstable Reynolds numbers are bounded by an upper critical value which tends to infinity right beyond the Rayleigh line. For perfectly conducting cylinders and moderate helicities of the magnetic field, the range of unstable wave numbers is bounded only from the short-wave end. Although there is an upper marginal Reynolds number for each unstable wave number, no bounded upper critical Reynolds number exists in this case because the range of unstable wave numbers extends to zero, *i.e.*, infinitely long waves. Nevertheless, at sufficiently large helicities, the range of unstable wave numbers becomes bounded also from below, and an upper critical Reynolds number appears in the same way as for insulating cylinders.
It is important to note that these instabilities predicted by the conventional stability analysis in the form of single traveling waves correspond to the so-called convective instability threshold at which the system becomes able to amplify certain externally imposed perturbations that, however, are not self-sustained and thus may be experimentally unobservable without a proper external excitation. The problem is that convectively unstable perturbations grow asymptotically in time only in the frame of reference traveling with their group velocity, whereas they decay in any other frame of reference including the laboratory one. For an instability to be self-sustained and thus observable it has to grow in the laboratory frame of reference. In an extended system, this condition is satisfied by the so-called absolute instability which ensures a zero group velocity of a growing perturbation. This additional condition is satisfied by regarding the wave number as a complex quantity with a nonzero imaginary part which describes an exponential axial modulation of the wave amplitude. Using this concept, we found that there is not only a convective but also an absolute HMRI implying that this instability can be experimentally observable in a system of sufficiently large but finite axial extension. In the hydrodynamically unstable range before the Rayleigh line, the threshold of absolute instability is slightly higher than the convective one. Nevertheless, the critical wavelength for absolute instability is significantly shorter than that for the convective one that may allow us to distinguish between both. The absolute instability threshold rises significantly above the convective one beyond the Rayleigh line. As a result, the extension of the absolute instability beyond the Rayleigh line is considerably shorter than that of the convective instability without a marked difference between insulating and perfectly conducting cylinders in contrast to the convective HMRI.
The extension of HMRI beyond the Rayleigh line is of particular interest from the astrophysical point of view regarding a Keplerian velocity profile [@Liu-etal2006; @Ruediger-Hollerbach2007]. For a Couette-Taylor flow with a radius ratio $\lambda=2$ considered here, the Keplerian velocity profile approximately corresponds to a ratio of rotation rates of $\mu=\lambda^{-3/2}\approx0.35.$ As seen in Figs. \[cap:muc-bt\](b) and \[cap:muc-Ha\] for the absolute instability, no such value of $\mu$ is reached up to $\beta=30$ and $\Ha=150$. Whether or not it can be reached at higher $\beta$ and $\Ha$ is still an open question requiring a more detailed study using either higher numerical resolution or asymptotic analysis. On the other hand, HMRI in a system of large axial extension with a radius ratio of $\lambda=2$ might be of limited astrophysical relevance for accretion disks anyway.
Conversely to Liu *et al.* [@Liu-etal2006; @Liu-etal2007] we find that the HMRI can be a self-sustained instability rather than just a transient growth. This contradiction may be due to a couple of additional simplifications underlying the analysis of Liu *et al.* First, the viscosity is neglected. Second, the electromagnetic force is treated as a small perturbation which is a sensible approach within the inviscid approximation where an infinitesimal magnetic field can cause a correspondingly slow growth of an unstable Fourier mode. However, such a perturbative approach may be inadequate for the absolute instability which requires a finite temporal growth rate of the corresponding convectively unstable Fourier mode. In addition, note that our results do not support the recent findings of Liu [@Liu-2009]. According to his estimates for the cases considered here, the absolute HMRI occurs above $\RE\sim10^{5}$ only. We predict the absolute HMRI to occur at the lower value of $\RE\sim10^{3},$ and to disappear again above $\RE\sim10^{5}$ due to the mechanism discussed in Sec. \[sub:conv-insl\]. This disagreement may be due to the absolute instability analysis which is carried out by Liu using real wave numbers only. According to the conventional absolute instability analysis [@Lifshitz-Pitaevskii-81; @Schmid-Henningson], such an approach yields the long-time asymptotics only in the frame of reference traveling with the group velocity of the fastest growing perturbation. Moreover, owing to its linearity, the analysis is limited to sufficiently small perturbation amplitudes only. Namely, it is limited up to the point where the first exponentially growing perturbation appears, which is the convective instability threshold.
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Finally, let us compare the convective and absolute instability thresholds calculated for perfectly conducting cylinders with the experimental data of Stefani *et al.* [@Stefani-NJP] who reports on the observation of HMRI-like traveling waves at $\mu=0.27,$ $\RE=$1775, and a fixed rod current $6\, kA$ for the coil currents $40-100\, A$ that corresponds to $\beta\approx7.4$–3 and $\Ha\approx6.3$-15.8. Another observation was done at the same $\mu$ but different other parameters: $\RE=1479,$ $\beta=6,$ and $\Ha=9.5.$ As seen in Fig. \[fig:rec-mu-exp\], in both cases the experimental points lie well inside the range of $\mu$ for convective instability but outside that for absolute instability. This discrepancy with the experimental observations may be due to the deviation of the real base flow from the idealized one used in this study. In particular, the Ekman pumping driven by the end walls in the experiment, which is not taken into account in the present analysis, may affect the hydrodynamic stability limit of the base flow, *i.e.*, its actual Rayleigh line, which however serves as the reference point for the observation of MRI. A more detailed comparison with the experimental observations lies outside the scope of the present paper.
In conclusion, the main result of the present paper is the finding of absolute HMRI in addition to the convective one which can be self-sustained and, thus, experimentally observable without external excitation in a system of sufficiently large axial extension. A characteristic feature of HMRI is the upper critical threshold existing besides the lower one that distinguishes it from a magnetically-modified Taylor vortex flow.
This research was supported by Deutsche Forschungsgemeinschaft in frame of the Collaborative Research Centre SFB 609. We would like to thank Frank Stefani and Thomas Gundrum for stimulating discussions.
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|
---
abstract: 'It is shown that a quintic form over a $p$-adic field with at least $26$ variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds $9$.'
author:
- 'Jan H. Dumke'
bibliography:
- 'Bib.bib'
title: ' $p$-adic Zeros of Quintic Forms'
---
Introduction
============
Let $F(x_1,\dots,x_n)$ denote a form of degree $d$ over a $p$-adic field $\mathbb{K}$. It is a conjecture of E. Artin from the 1930s, that $F$ has a non-trivial zero as soon as $n>d^2$. Although this is known to be false for many $d$ (for instance, see [@MR0197450] for a $2$-adic quartic form) the conjecture has been partially verified by Ax and Kochen [@MR0184930]. They showed that for every $d$ there exists a positive integer $q_0(d)$, such that Artin’s conjecture holds whenever the cardinality $q$ of the residue class field exceeds $q_0(d)$. However, little is known about the actual values of $q_0(d)$. Brown [@MR494980] has given a huge, but explicit bound on $q_0(d)$. If we write $a\uparrow b$ for $a^b$ it can be stated as $$\begin{aligned}
q_0(d)\leq 2\uparrow (2\uparrow (2\uparrow (2\uparrow (2\uparrow (d\uparrow (11\uparrow (4d))))))). \notag\end{aligned}$$ If $d$ is neither composite nor a sum of composite numbers, better bounds are available. Besides the classical result $q_0(2)=1$ (Hasse [@Hasse]) and $q_0(3)=1$ (Lewis [@Lewis1]) this concerns in fact $d=5,7,11$ only. Leep and Yeomans [@MR1382749] have shown $q_0(5)\leq 43$ and later this has been improved by Heath-Brown [@MR2595750]. He proved that a quintic form over $\mathbb{Q}_p$ possesses a non-trivial zero if $p\geq 17$. For septic and unidecic forms bounds $q_0(7)\leq 883$ and $q_0(11)\leq 8053$ are due to Wooley [@MR2413363]. In this paper we shall establish $q_0(5)\leq 9$.
\[maintheorem\] Let $F(x_1,\dots,x_n)=F(\mathbf{x})$ be a quintic form with at least $n\geq 26$ variables over a $p$-adic field $\mathbb{K}$ with residue class field of cardinality $q>9$. Then there exists a non-zero vector $\mathbf{x}\in \mathbb{K}^n$ with $F(\mathbf{x})=0$.
The proof relies on a $p$-adic minimisation procedure applicable to forms of degree $d=2,3,5,7$ and $11$ which has been developed by Lewis [@Lewis1], Birch and Lewis [@MR0123534] and Laxton and Lewis [@MR0175884]. They showed that one may assume that $F$ is reduced, that is, the resultant of the partial derivatives does not vanish and is of minimal normalised $p$-adic valuation. It then follows from a result of Leep and Yeomans that the reduction of $F$ over the residue class field, denoted by $\theta(F)$, is a non-degenerate form with at least $6+s$ variables, where $s$ is the maximal affine dimension of a vector space on which $\theta(F)$ vanishes. If $\theta(F)$ possesses a non-singular zero, it can be lifted by Hensel’s Lemma to a non-trivial zero of $F$. We recall that a non-singular zero is one which is not a simultaneous zero of the partial derivatives.\
We shall use certain properties of quintic forms to choose a suitable subspace and show that it contains a non-singular zero. For $q=11,13,16,25,27,32$ this is accomplished with the help of computer calculations. The author was able to carry those out on his personal notebook. This, together with the previously mentioned results of Leep and Yeomans and Heath-Brown, yields Theorem \[maintheorem\].\
There is numerical evidence to suggest that the imposed constraint on $q$ can be further reduced. Given the current state of technology, it certainly seems doubtful to expect an answer for all $q$ at this stage.
Preliminaries
=============
Let $\mathbb{K}$ denote a $p$-adic field with normalised valuation $\nu$, residue class field $\mathbb{F}_q$ and ring of integers $\mathcal{O}_{\mathbb{K}}$. As we are interested in a zero, we may assume from now on that $F$ has coefficients in $\mathcal{O}_{\mathbb{K}}$ and is non-degenerate.\
We call two forms $F$ and $G$ over $\mathcal{O}_{\mathbb{K}}$ equivalent if there exists a matrix $A\in GL_n(\mathbb{K})$ and $c\in \mathbb{K}^{\times}$ such that $cF(A\mathbf{x})=G(\mathbf{x})$. In order to state the first lemma we denote by $\mathcal{I}(F)$ the resultant of the $n$ partial derivatives of $F$. Laxton and Lewis have shown that if $\mathcal{I}(F)=0$, then there exists a sequence of forms $F_i$ with $\mathcal{I}(F_i)\neq 0$ converging to $F$. This observation results in the following lemma.
\[Lemma1\] In order to prove that any form of degree $d$ over a $p$-adic field $\mathbb{K}$ in $n>d^2$ variables has a non-trivial zero it is sufficient to prove this fact for forms with $\mathcal{I}(F)\neq 0$.
We call $F$ reduced if $\mathcal{I}(F)\neq 0$ and $\nu(\mathcal{I}(F))$ is minimal among all forms equivalent to $F$. Thus we may assume by Lemma \[Lemma1\] that $F$ is reduced. This yields suitable implications on the number of variables of $\theta(F)$.
\[lemma2\] Let $F$ be a reduced quintic form in at least $26$ variables over a $p$-adic field and $s\geq 0$ be an integer such that $\theta(F)$ vanishes on an affine $s$-dimension linear plane $V$. If $s>1$ we assume in addition that $q\geq 5$. We then obtain that $\theta(F)$ is a non-degenerate form in at least $6+s$ variables.
The next lemma shows in particular that $s\geq 1$. Throughout this paper we shall denote by $Z(f)$ the set of projective zeros of a form $f$ over $\mathbb{F}_q$.
\[Warning\] Let $f$ be a form of degree $d$ over $\mathbb{F}_q$ in $n$ variables. If $n>d$ we have $$\begin{aligned}
|Z(f)|\geq \frac{{q^{n-d}-1}}{q-1}\notag.\end{aligned}$$
A proof of this classical result can be found in [@2847218848221]. Lemmas \[lemma2\] and \[Warning\] yield the following consequence.
\[Cor1\] Let $F$ be a quintic form in at least $26$ variables over $\mathcal{O}_{\mathbb{K}}$ that does not have a non-trivial zero. Let $s$ be as defined in Lemma \[lemma2\]. We then have $$\begin{aligned}
|Z(\theta(F))|\geq \frac{q^{s+1}-1}{q-1}\notag.\end{aligned}$$
A zero of $\theta(F)$ is not sufficient for a non-trivial zero of $F$, instead we require a non-singular zero. Once we have found one, we can apply the version of Hensel’s Lemma given below.
\[sdkdkwiuujdjude\] Let $F\in \mathcal{O}_{\mathbb{K}}[x_1,\dots,x_n]$. If $\theta(F)$ has a non-singular zero, then $F$ has a non-trivial zero in $\mathbb{K}^n$.
For a discussion of Hensel’s Lemma see [@MR0241358], for example.
Proof of Theorem \[maintheorem\]
================================
Let $F$ be a quintic form in at least $26$ variables over a $p$-adic field $\mathbb{K}$ with residue class field of cardinality $q>9$. Throughout this section we shall write $f$ for the reduction $\theta(F)$. We denote the linear span of vectors $\mathbf{v}_1,\dots,\mathbf{v}_l\in \mathbb{F}_q^n$ by $\langle \mathbf{v}_1,\dots,\mathbf{v}_l\rangle$.\
By Lemma \[Lemma1\] we may assume that $F$ is reduced. It then follows by Lemma \[lemma2\], that $f$ is a non-degenerate form in at least $6+s$ variables, where $s$ is the maximal affine dimension of a linear subspace of $Z(f)$.\
Suppose that $f$ does not have a non-singular zero. We show that there are at least four linearly independent zeros $$\begin{aligned}
\mathbf{z}_1, \mathbf{z}_2, \mathbf{z}_3, \mathbf{z}_4\in Z(f)\text{ such that } \langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f) \notag \end{aligned}$$ for all $1\leq i<j\leq 4$. Hence the form $$\begin{aligned}
g(x_1,x_2,x_3,x_4):=f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3+x_4\mathbf{z}_4)\notag \end{aligned}$$ must be of a certain shape. In particular, certain coefficients of $g$ do not vanish. We then prove the existence of a non-singular zero of $g$, contrary to our assumption. This is achieved by considering successively larger subspaces of $\langle \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3,\mathbf{z}_4 \rangle $ and sieving out forms possessing non-singular zeros.\
As a first step, we prove that there are five distinct non-zero vectors $$\begin{aligned}
\mathbf{z}_1, \dots, \mathbf{z}_5 \in Z(f) \notag\end{aligned}$$ such that $\mathbf{z}_1$, $\mathbf{z}_2$, $\mathbf{z}_3$ are linearly independent and $f$ does not vanish on any plane spanned by two vectors of one of the quadruples $$\begin{aligned}
\{\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3,\mathbf{z}_i\}\quad \text{where $i=4,5$.}\notag\end{aligned}$$ In order to establish this, we begin by showing that there are three distinct subspaces $V_1, V_2, V_3\subseteq Z(f)$ of maximal dimension and two zeros $\mathbf{z}_1$, $\mathbf{z}_2\in Z(f)$ such that $$\begin{aligned}
\mathbf{z}_1,\mathbf{z}_2\notin \bigcup_{i=1}^3V_i \quad \text{ and $\quad\langle \mathbf{z}_1$, $\mathbf{z}_2 \rangle \nsubseteq Z(f)$}. \notag\end{aligned}$$ Secondly, we prove the existence of a third zero $\mathbf{z}_3\in V_3\backslash(V_1\cup V_2)$ such that $\mathbf{z}_1$, $\mathbf{z}_2$, $\mathbf{z}_3$ are linearly independent. Thirdly, we show that there is a fourth zero $\mathbf{z}_4\in V_2\backslash V_1$ completing the first quadruple and finally, we will choose a fifth zero $\mathbf{z}_5 \in V_1$ completing the second quadruple.\
For convenience, we first state a basic lemma and give the details of the argument outlined afterwards.
\[Lemma5\] Let $f$ be a quintic form over $\mathbb{F}_q$ possessing two distinct non-trivial zeros $\mathbf{z}_1$ and $\mathbf{z}_2$. Then $f$ either has a non-singular zero or $$\begin{aligned}
f(x_1\mathbf{z}_1+x_2\mathbf{z}_2)=c_{12}x_1^3x_2^2+c_{21}x_2^3x_1^2\notag \end{aligned}$$ and $c_{12}c_{21}=0$. If, in addition, $|\langle \mathbf{z}_1, \mathbf{z}_2 \rangle \cap Z(f)|\geq 3$, then $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2)$ either possesses a non-singular zero or is the zero polynomial.
We write $$\begin{aligned}
f(x_1\mathbf{z}_1+x_2\mathbf{z}_2)= a_1x_1^5+b_{12}x_1^4x_2+c_{12}x_1^3x_2^2+c_{21}x_2^3x_1^2+b_{21}x_2^4x_1+a_2x_2^5. \notag \end{aligned}$$ We may assume that $\mathbf{z}_1$ and $\mathbf{z}_2$ are singular zeros and hence $$\begin{aligned}
f(x_1\mathbf{z}_1+x_2\mathbf{z}_2)=(c_{12}x_1+c_{21}x_2)x_1^2x_2^2\notag.\end{aligned}$$ If $c_{12}c_{21}\neq 0$ then $(-c_{21},c_{12})$ is a non-singular zero and otherwise $\langle \mathbf{z}_1, \mathbf{z}_2 \rangle \cap Z(f)=\{\mathbf{z}_1,\mathbf{z}_2\}$ or $\langle \mathbf{z}_1, \mathbf{z}_2 \rangle\subseteq Z(f)$.
Since $f$ has at least $6$ variables, Lemma \[Warning\] yields a non-trivial zero and thus we may assume $s\geq 1$. By Corollary \[Cor1\] we have $$\begin{aligned}
\label{equation1}|Z(f)|> \frac{4(q^{s}-1)}{q-1},\end{aligned}$$ provided $q\geq 4$. Thus we can pick four distinct subspaces $$\begin{aligned}
V_1,V_2,V_3,V_4\subseteq Z(f)\notag\end{aligned}$$ such that $V_i$ is of maximal dimension for $1\leq i \leq 4$. By equation (\[equation1\]) we can choose an additional zero $\mathbf{z}_1\in Z(f)\backslash\bigcup_{i=1}^4V_i$. We set $S_3:=\bigcup_{i=1}^3 V_i$ and show that there exists a vector $\mathbf{z}_2\in V_4\backslash S_3$ such that $\langle \mathbf{z}_1,\mathbf{z}_2 \rangle \nsubseteq Z(f)$. Suppose by the contrary that $$\begin{aligned}
\text{for all $\mathbf{z}\in V_4\backslash S_3$ we have $ \langle \mathbf{z}_1,\mathbf{z} \rangle \subseteq Z(f)$.}\label{equation2}\end{aligned}$$ If $V_4\cap S_3=\{0\}$, then (\[equation2\]) contradicts the maximality of $V_4$ and otherwise we shall argue as follows. Let $\mathbf{s}\in V_4\cap S_3$ be arbitrary. As $V_4$ is distinct from $S_3$ we can choose a non-zero vector $\mathbf{v}\in V_4\backslash S_3$ and consider the projective line $L_{\mathbf{s}}:=\langle \mathbf{v},\mathbf{s} \rangle$. Since $\mathbf{v}\notin S_3$, the projective line $L_{\mathbf{s}}$ can not contain two vectors of $V_i$ for each $1\leq i\leq 3$. Thus the intersection $L_{\mathbf{s}} \cap S_3$ contains at most three non-zero points. On the other hand, since $q\geq 5$, there are at least three points $\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3 \in L_{\mathbf{s}}$ not contained in $S_3$. It follows from our assumption (\[equation2\]) that $\langle \mathbf{z}_1,\mathbf{p}_i \rangle\subseteq Z(f)$ for all $1\leq i \leq 3$.
\[Lemma6\] Let $f$ be a quintic form over $\mathbb{F}_q$ without a non-singular zero, $L$ a projective line, $\mathbf{z}$ a non-zero point not on $L$ and $\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\in L$ three distinct non-zero points. Assume that $$\begin{aligned}
\langle \mathbf{p}_i,\mathbf{z} \rangle \subseteq Z(f) \quad \text{for all $1\leq i \leq 3$}\notag.\end{aligned}$$ Then $\langle L,\mathbf{z} \rangle \subseteq Z(f)$.
Let $\mathbf{x}\in \langle L, \mathbf{z} \rangle$ and $\mathbf{x}\notin \bigcup_{i=1}^3 \langle \mathbf{p}_i,\mathbf{z} \rangle$. There exists a projective line $H$ in $\langle L, \mathbf{z} \rangle$ through $ \mathbf{x} $ that does not contain $ \mathbf{z} $. Since we have assumed that $\mathbf{x}\notin \langle \mathbf{p}_i,\mathbf{z} \rangle$ and $\langle \mathbf{p}_i,\mathbf{z} \rangle$ has co-dimension $1$ in $\langle L,\mathbf{z} \rangle$, the line $H$ intersects $\langle \mathbf{p}_i,\mathbf{z} \rangle$ in exactly one point $\mathbf{s}_i$, say, for each $1\leq i\leq 3$. Since $\bigcap_{i=1}^3\langle \mathbf{p}_i,\mathbf{z} \rangle= \mathbf{z} $ and $\mathbf{z} \notin H$, we conclude that there are at least three distinct points, namely $\mathbf{s}_i$ for $1\leq i \leq 3$, in $H$ that are contained in $Z(f)$. By Lemma \[Lemma5\] we have $H\subseteq Z(f)$ and hence $\mathbf{x}\in Z(f)$. We conclude that $\langle L, \mathbf{z} \rangle \subseteq Z(f)$.
By applying Lemma \[Lemma6\] we have $\langle \mathbf{z}_1,V_4 \rangle\subseteq Z(f)$, contrary to the maximality of the dimension of $V_4$. We conclude that there are three non-identical subspaces $V_1,V_2,V_3\subseteq Z(f)$ of maximal dimension and two zeros $\mathbf{z}_1,\mathbf{z}_2 \notin \bigcup_{i=1}^3 V_i$ such that $$\begin{aligned}
\langle \mathbf{z}_1,\mathbf{z}_2 \rangle \cap Z(f)=\{\mathbf{z}_1,\mathbf{z}_2\}\notag. \end{aligned}$$ As mentioned above we shall proceed by proving the existence of a third vector $\mathbf{z}_3 \in V_3\backslash (V_1\cup V_2)$ with the property $\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f)$ for all $1\leq i<j\leq 3$. Suppose by the contrary that for every $\mathbf{z} \in V_3\backslash (V_1\cup V_2)$ at least one of the following holds $$\begin{aligned}
\label{equation3}\langle \mathbf{z},\mathbf{z}_1 \rangle \subseteq Z(f)\quad \text{or} \quad \langle \mathbf{z},\mathbf{z}_2 \rangle \subseteq Z(f).\end{aligned}$$ We set $S_2:=V_1\cup V_2$ for shorter notation and shall argue that we may assume $S_2\cap V_3=\{0\}$. Suppose there exists at least one non-zero vector $\mathbf{s}\in S_2\cap V_3$. We then pick a vector $\mathbf{v}\in V_3\backslash S_2$ and define for any vector $\mathbf{s}\in S_2\cap V_3$ the projective line $L_{\mathbf{s}}:=\langle \mathbf{s},\mathbf{v} \rangle$. We show that $$\begin{aligned}
\label{equation4}\langle L_{\mathbf{s}},\mathbf{z}_1 \rangle \subseteq Z(f)\quad \text{or} \quad \langle L_{\mathbf{s}},\mathbf{z}_2 \rangle \subseteq Z(f). \end{aligned}$$ Since $\mathbf{v}\notin S_2$, neither two vectors of the subspace $V_1$ nor two of the subspace $V_2$ can be contained in $L_{\mathbf{s}}$. Thus there are at least $5$ projective points in $L_{\mathbf{s}}\backslash S_2$, provided $q\geq 6$. By our assumption (\[equation3\]) there are three points $\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3$ among them such that $\langle \mathbf{p}_i,\mathbf{z}_k \rangle \subseteq Z(f)$ for all $1\leq i \leq 3$ and a certain $1\leq k\leq 2$. Equation (\[equation4\]) then follows from Lemma \[Lemma6\] and thus, we have that for every $\mathbf{z}\in V_3$ at least one of the following holds $$\begin{aligned}
\label{equation5}\langle \mathbf{z},\mathbf{z}_1 \rangle \subseteq Z(f)\quad \text{or} \quad \langle \mathbf{z},\mathbf{z}_2 \rangle \subseteq Z(f). \end{aligned}$$
\[Lemma7\] Let $f$ be a quintic form over $\mathbb{F}_q$ without a non-singular zero, $V\subseteq Z(f)$ an $m$-dimensional subspace where $m\geq 2$ and $\mathbf{z}_1,\dots, \mathbf{z}_k$ non-trivial zeros not contained in $V$. We assume $q\geq 2k$ and that there exists for any projective plane $W\subseteq V$ of co-dimension $1$ an index $i\in \{1,\dots,k\}$ such that $\langle W,\mathbf{z}_i \rangle \subseteq Z(f)$. Then there exists an index $i\in \{1,\dots,k\}$ such that $$\begin{aligned}
\langle V,\mathbf{z}_i \rangle \subseteq Z(f). \notag \end{aligned}$$
We write $[x_1:\dots:x_m]$ for a projective point in $V$. Since $m\geq 2$ we can define the following subspaces $$\begin{aligned}
W_{(a,b)}:=\{[x_1:\dots : ax_{m-1}:bx_{m-1}]\mid x_i \in \mathbb{F}_q \text{ for $1 \leq i \leq m$} \} \notag \end{aligned}$$ for $(a,b)\in (\{1\}\times \mathbb{F}_q)\cup \{(0,1)\}$.\
Since $q\geq 2k$ there are at least $2k+1$ subspaces $W_{(a,b)}$. Thus we may assume that there are at least three subspaces, $W_1$, $W_2$, $W_3$ say, among these and a zero $\mathbf{z}\in\{\mathbf{z}_1,\dots, \mathbf{z}_k\}$ such that $$\begin{aligned}
\langle W_i, \mathbf{z} \rangle \subseteq Z(f) \quad \text{ for $1\leq i \leq 3$.} \notag\end{aligned}$$ We shall complete the proof of this lemma by following Leep and Yeomans \[[@MR1382749], Lemma 5.3\]. For $W_1,W_2,W_3$ as above, we have $$\begin{aligned}
\langle W_i,\mathbf{z} \rangle \cap \langle W_j,\mathbf{z} \rangle &= \langle W_i\cap W_j, \mathbf{z}\rangle, \label{equation6}\\
\langle W_i, \mathbf{z}\rangle \cap \langle W_j,\mathbf{z} \rangle &= \bigcap_{i=1}^3 \langle W_i, \mathbf{z} \rangle \label{equation6b}\end{aligned}$$ for any $1\leq i<j\leq 3$. We notice that for equation (\[equation6\]) we have for each pair $i\neq j$ with $\langle W_i,\mathbf{z} \rangle$ and $\langle W_j,\mathbf{z} \rangle$ two non-identical $m$-dimensional planes and that $\langle W_i\cap W_j, \mathbf{z}\rangle$ is an $m-1$ dimensional plane. Equation (\[equation6b\]) follows from (\[equation6\]) and the fact that $$\begin{aligned}
W_i\cap W_j=\bigcap_{i=1}^3 W_i \quad\text{for distinct $i,j$.} \notag\end{aligned}$$\
Let $\mathbf{x}$ be a point in $\langle V,\mathbf{z} \rangle \backslash \bigcup_{i=1}^3 \langle W_i,\mathbf{z} \rangle$. We observe that $\bigcap_{i=1}^3 W_i$ has co-dimension $2$ in $ V $. Thus, we conclude by (\[equation6\]) and (\[equation6b\]) that $\bigcap_{i=1}^3 \langle W_i,\mathbf{z} \rangle$ has co-dimension $2$ in $\langle V,\mathbf{z} \rangle$. Hence we can choose a projective line $H$ through the point $\mathbf{x}$ that does not intersect with $\bigcap_{i=1}^3\langle W_i,\mathbf{z} \rangle $. Since $\mathbf{x}\notin \langle W_i,\mathbf{z} \rangle$ and $\langle W_i,\mathbf{z} \rangle$ has co-dimension $1$ in $\langle V,\mathbf{z} \rangle$, we conclude that there exists for each $i$ a point $\mathbf{p}_i\in \langle W_i,\mathbf{z} \rangle \cap H$. Since $ \langle W_i,\mathbf{z}\rangle \subseteq Z(f)$ and $H$ does not intersect $\bigcap_{i=1}^3\langle W_i,\mathbf{z} \rangle$ there are at least three distinct non-trivial zeros of $f$ on $H$. Thus we conclude by Lemma \[Lemma5\] that $\langle V,\mathbf{z} \rangle \subseteq Z(f)$.
We apply Lemma \[Lemma7\] to (\[equation5\]) and thus, we have $$\begin{aligned}
\langle V_3,\mathbf{z}_1 \rangle \subseteq Z(f)\quad \text{or} \quad \langle V_3,\mathbf{z}_2 \rangle \subseteq Z(f). \notag\end{aligned}$$ However, this contradicts the maximality of the dimension of $V_3$. Moreover, the vectors $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3$ are linearly independent, since by Lemma \[Lemma5\] there are at most two zeros on the projective line $\langle \mathbf{z}_1,\mathbf{z}_2 \rangle $. Thus we have found three linearly independent vectors $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f) \quad \text{for all $1\leq i<j\leq 3$}.\notag\end{aligned}$$ We show that there exists a fourth vector $\mathbf{z}_4 \in V_2\backslash V_1$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f) \quad \text{ for all $1\leq i<j\leq 4$.}\notag \end{aligned}$$ Suppose by the contrary that for all $\mathbf{z} \in V_2\backslash V_1$ at least one of the following holds $$\begin{aligned}
\label{equation7}\langle \mathbf{z},\mathbf{z}_1 \rangle \subseteq Z(f),\quad \langle \mathbf{z},\mathbf{z}_2 \rangle \subseteq Z(f)\quad \text{or} \quad \langle \mathbf{z},\mathbf{z}_3 \rangle \subseteq Z(f).\end{aligned}$$ We shall argue that there is no loss of generality if we assume $V_1\cap V_2=\{0\}$. As there exists a point $\mathbf{v}\in V_2\backslash V_1$ we consider for any vector $\mathbf{s}\in V_2\cap V_1$ the plane $L_{\mathbf{s}}:=\langle \mathbf{s},\mathbf{v} \rangle $. We show that $$\begin{aligned}
\langle L_{\mathbf{s}},\mathbf{z}_1 \rangle \subseteq Z(f),\quad \langle L_{\mathbf{s}},\mathbf{z}_2 \rangle \subseteq Z(f)\quad \text{or} \quad \langle L_{\mathbf{s}},\mathbf{z}_3 \rangle \subseteq Z(f). \notag \end{aligned}$$ Since $q\geq 7$ there are at least $7$ projective points in $L_{\mathbf{s}}$ not contained in $V_1$. Thus, by (\[equation7\]) there are three points $\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3$ among them such that $\langle \mathbf{p}_i,\mathbf{z}_k \rangle \subseteq Z(f) $ for all $1\leq i \leq 3$ and a certain $1\leq k\leq 3$. By Lemma \[Lemma6\], we have that for every $\mathbf{z}\in V_2$ at least one of the following holds $$\begin{aligned}
\label{equation8}\langle \mathbf{z},\mathbf{z}_1 \rangle \subseteq Z(f), \quad \langle \mathbf{z},\mathbf{z}_2 \rangle \subseteq Z(f) \quad \text{or} \quad \langle \mathbf{z},\mathbf{z}_3 \rangle \subseteq Z(f). \end{aligned}$$ It then follows in conjunction with Lemma \[Lemma7\] that $$\begin{aligned}
\langle V_2,\mathbf{z}_1 \rangle \subseteq Z(f),\quad \langle V_2,\mathbf{z}_2 \rangle \subseteq Z(f)\quad \text{or} \quad \langle V_2,\mathbf{z}_3 \rangle\subseteq Z(f). \notag\end{aligned}$$ However, any of those contradicts the maximality of the dimension of $V_2$ and hence we may assume the existence of a vector $\mathbf{z}_4 \in V_2\backslash V_1$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f)\quad \text{for all $1\leq i<j\leq 4$}.\notag\end{aligned}$$ We show that there exists a fifth vector $\mathbf{z}_5 \in V_1$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_5 \rangle \nsubseteq Z(f)\notag \quad \text{for all $1\leq i\leq 3$.}\end{aligned}$$ Suppose by the contrary that for all $\mathbf{z} \in V_1$ at least one of the conditions in equation (\[equation7\]) holds. By Lemma \[Lemma7\] this implies $$\begin{aligned}
\langle V_1,\mathbf{z}_1 \rangle \subseteq Z(f),\quad \langle V_1,\mathbf{z}_2 \rangle \subseteq Z(f) \quad \text{or} \quad \langle V_1,\mathbf{z}_3 \rangle \subseteq Z(f).\notag\end{aligned}$$ However, any of these contradicts the maximality of the dimension of $V_1$ and thus we conclude that there is a vector $\mathbf{z}_5 \in V_1$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_5 \rangle \nsubseteq Z(f)\quad \text{for all $1\leq i\leq 3$.}\notag\end{aligned}$$ In summary, we have shown that there are two quadruples of zeros, $$\begin{aligned}
\text{$\mathbf{z}_1$, $\mathbf{z}_2$, $\mathbf{z}_3$, $\mathbf{z}_4\quad$ and $\quad\mathbf{z}_1$, $\mathbf{z}_2$, $\mathbf{z}_3$, $\mathbf{z}_5$,}\notag\end{aligned}$$ such that $f$ does not vanish on any two-dimensional plane spanned by two zeros of one quadruple. Moreover, we know that $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3$ are linearly independent. We will now estimate the number of zeros of $f$ in $\langle \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3 \rangle$.
\[Lemma8\] Let $f$ be a quintic form over $\mathbb{F}_q$ with three linearly independent zeros $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3\in Z(f)$ such that $\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f) $ for all $1\leq i<j\leq 3$. Then the following holds.\
If $q\geq 17$, then $f$ has a non-singular zero. If $ 11\leq q<17$, it possesses a non-singular zero or $|\langle \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3 \rangle\cap Z(f)|= 3$ holds. If $q<11$ it has a non-singular zero or $|\langle \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3 \rangle\cap Z(f)|\leq 4$ holds.
The last inequality is sharp. For instance, $$\begin{aligned}
2x_1^3x_2^2+2x_1^3x_3^2+4x_2^3x_3^2+x_1x_2x_3(5x_1^2+6x_2^2+2x_3^2+x_1x_2+x_1x_3+x_2x_3) \notag \end{aligned}$$ is a form over $\mathbb{F}_7$ possessing exactly four singular zeros, namely $$\begin{aligned}
\langle (1,0,0) \rangle, \langle (0,1,0) \rangle, \langle (0,0,1) \rangle, \langle (1,6,2) \rangle.\notag\end{aligned}$$
Suppose that $f$ does not have a non-singular zero. Thus we can write $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3)$ as $$\begin{aligned}
x_1x_2x_3Q(x_1,x_2,x_3)+\sum_{\substack{1\leq i< j \leq 3}}c_{ ij }x_i^3x_j^2+c_{ ji }x_j^3x_i^2 \notag
\notag \end{aligned}$$ where $Q(x_1,x_2,x_3)$ is a quadratic form. By applying Lemma \[Lemma5\] to any two variables of $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3)$ we have $c_{ij}c_{ji}=0$ for all $1\leq i<j\leq 3$. Since $f$ does not vanish on any of the projective lines $\langle \mathbf{z}_i,\mathbf{z}_j \rangle$ with $1\leq i<j\leq 3$, we have either $$\begin{aligned}
\text{$c_{ij}\neq 0\quad$ or $\quad c_{ji}\neq 0\quad$ for all $1\leq i<j\leq 3$.}\notag\end{aligned}$$ Hence, we see after permuting the variables that $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3)$ takes one of the following shapes $$\begin{aligned}
&t_1(x_1,x_2,x_3)=c_{12}x_1^3x_2^2+c_{13}x_1^3x_3^2+c_{23}x_2^3x_3^2+x_1x_2x_3Q(x_1,x_2,x_3), \notag\\
&t_2(x_1,x_2,x_3)=c_{12}x_1^3x_2^2+c_{31}x_3^3x_1^2+c_{23}x_2^3x_3^2+x_1x_2x_3Q(x_1,x_2,x_3),\notag\end{aligned}$$ where $Q(x_1,x_2,x_3)$ is a quadratic form and $c_{12},c_{13},c_{23}$ and $c_{31}$ are all non-zero coefficients.\
It has been proved by Leep and Yeomans [@MR1382749] using the Lang-Weil Bound that $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3)$ has always a non-singular zero, provided $q\geq 43$. Heath-Brown [@MR2595750] has extended this to prime values of $q\geq 17$.\
Similarly, we show by computer calculations that $f$ has a non-singular zero for $q=25,27,32$. In each case there are, after an appropriate rescaling of both, the forms $t_1$, $t_2$ and the variables, just $6$ degrees of freedom. A computer program can verify the existence of a non-singular zero for each form $t_1$, respectively each form $t_2$, by successively testing points in $\mathbb{F}_q^3$.\
If $q< 17$ it can be checked by an analogous computer calculation that $t_1$ and $t_2$ either possess a non-singular zero or that the bound on $|\langle \mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3 \rangle\cap Z(f)|$ holds.
Lemma \[Lemma8\] establishes Theorem \[maintheorem\], provided $q\geq 17$. Moreover, it shows that not both quadruples $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3,\mathbf{z}_4$ and $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3,\mathbf{z}_5$ can consist of linearly dependent vectors. Thus we may assume, after renaming, that we have linearly independent vectors $\mathbf{z}_1,\mathbf{z}_2,\mathbf{z}_3,\mathbf{z}_4$ such that $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_j \rangle \nsubseteq Z(f)\quad \text{for all $1\leq i<j\leq 4$}.\notag\end{aligned}$$ We write $f(x_1\mathbf{z}_1+x_2\mathbf{z}_2+x_3\mathbf{z}_3+x_4\mathbf{z}_4)$ as $$\begin{aligned}
\label{equation10}
\sum_{\substack{i\neq j}}a_{ij}x_i^3x_j^2 + \sum_{\substack{ k\neq i,j\\ i<j}}b_{ijk}x_ix_jx_k^3+ \sum_{\substack{i \neq j,k\\ j<k}}c_{ijk}x_ix_j^2x_k^2 + \sum_{\substack{l \neq i,j,k\\i<j<k}}d_{ijkl}x_ix_jx_kx_l^2,\end{aligned}$$ where $1 \leq i,j,k \leq 4$. By applying Lemma \[Lemma5\] and since $f$ does not vanish on any of the projective lines $\langle \mathbf{z}_i,\mathbf{z}_j \rangle$, we conclude that for each pair $(i,j)$ with $i\neq j$ exactly one of $a_{ij}$ and $a_{ji}$ is zero. It then follows that, after a permutation of the variables, the form (\[equation10\]) can take only four different shapes. If we write $h$ for $$\begin{aligned}
a_{23}x_2^3x_3^2&+a_{24}x_2^3x_4^2+a_{34}x_3^3x_4^2+\notag \\ & \sum_{\substack{ k\neq i,j\\ i<j}}b_{ijk}x_ix_jx_k^3+ \sum_{\substack{i \neq j,k\\ j<k}}c_{ijk}x_ix_j^2x_k^2 + \sum_{\substack{l \neq i,j,k\\ i<j<k}}d_{ijkl}x_ix_jx_kx_l^2 \notag\end{aligned}$$ those are $$\begin{aligned}
%(x_1,x_2,x_3,x_4)
&g_1:=a_{12}x_1^3x_2^2+a_{13}x_1^3x_3^2+a_{14}x_1^3x_4^2+h, \notag \\
&g_2:=a_{12}x_1^3x_2^2+a_{31}x_3^3x_1^2+a_{14}x_1^3x_4^2+h, \notag \\
&g_3:=a_{12}x_1^3x_2^2+a_{13}x_1^3x_3^2+a_{41}x_4^3x_1^2+h, \notag\\
&g_4:=a_{21}x_2^3x_1^2+a_{13}x_1^3x_3^2+a_{41}x_4^3x_1^2+h. \notag \end{aligned}$$ As indicated it has been checked on a computer that each of those forms has a non-singular zero, provided $9< q\leq 16$. We briefly describe the assembling process.\
Along the way, we have already excluded, via Lemma \[Lemma5\], all forms that have a non-singular zero on one of the projective lines $\langle \mathbf{z}_i,\mathbf{z}_j \rangle$ for some $1\leq i<j\leq 4$. Furthermore, we know from the proof of Lemma \[Lemma8\] all forms which do not have a non-singular zero in one of the subspaces $$\begin{aligned}
\langle \mathbf{z}_i,\mathbf{z}_j,\mathbf{z}_k \rangle \quad \text{for some $1\leq i<j<k\leq 4$.}\notag \end{aligned}$$ Note that $g_1,g_2,g_3$ and $g_4$ restricted to such a subspace are, after permuting the variables, equal to $t_1$ or $t_2$ as stated in the proof of Lemma \[Lemma8\]. The computer programs for $g_1,g_2,g_3$ and $g_4$ are analogous. Suppose $g_s$ for some $1\leq s \leq 4$ is one of these cases. We save the rearranged coefficients of those forms of shape $t_1$, respectively $t_2$, without a non-singular zero in four multidimensional arrays $$\begin{aligned}
A_{ijk}[\star,\star] \quad \text{where $1\leq i<j<k\leq 4$}\notag\end{aligned}$$ such that they represent the coefficients of $g_s$ restricted to the subspace $\langle \mathbf{z}_i,\mathbf{z}_j,\mathbf{z}_k \rangle$. Thus, every set of coefficients of the form $g_s|_{\langle \mathbf{z}_i,\mathbf{z}_j,\mathbf{z}_k \rangle}$ without a non-singular zero corresponds to a line $A_{ijk}[r,\star]$.\
We use these data to construct all remaining forms by combining data in these arrays and four additional degrees of freedom. Let $r_{ijk}$ denote the $r_{ijk}$-th line of $A_{ijk}[\star,\star]$ for $1\leq i<j<k \leq 4$. The non-negative integers $r_{123},r_{124},r_{134},r_{234}$, provided the corresponding lines are compatible with respect to the coefficients they share, determine a form $$\begin{aligned}
C(r_{123},r_{124},r_{134},r_{234})\notag\end{aligned}$$ in four variables, $x_1,x_2,x_3,x_4$ say, with each monomial in at most three variables. Thus any relevant form of shape $g_s$ can be written as $$\begin{aligned}
C(r_{123}&,r_{124},r_{134},r_{234};a,b,c,d)\notag \\&=C(r_{123},r_{124},r_{134},r_{234})+x_1x_2x_3x_4(ax_1+bx_2+cx_3+dx_4).\notag \end{aligned}$$ For all admissible $r_{123},r_{124},r_{134},r_{234}$ and for all $a,b,c,d\in \mathbb{F}_q$ we then search for a non-singular zero $(x_1,x_2,x_3,x_4) \in \mathbb{F}_q^4$ of $$\begin{aligned}
C(r_{123},r_{124},r_{134},r_{234};a,b,c,d)\notag \end{aligned}$$ by trying points successively. To do this efficiently, one can rescale both the forms and variables. For instance, rescale $g_1,g_2,g_3$ such that $$\begin{aligned}
a_{12}=1,\quad a_{23}=1,\quad a_{34}=1\notag\end{aligned}$$ and $g_4$ such that $$\begin{aligned}
a_{21}=1,\quad a_{23}=1,\quad a_{34}=1.\notag\end{aligned}$$ It is easier to choose a rescaling that is compatible with the one used in Lemma \[Lemma8\] (and hence with the data in the arrays $A_{ijk}[\star,\star]$). Besides these considerations, we put a general effort on implementing the algorithm efficiently.\
The full C++ program and the data used in the assembling process are available at [@2013arXiv1308.0999D]. This completes the proof of Theorem \[maintheorem\].\
Note that apart from the computer checks we have not used any assumption other than $q>5$. For $q=8,9$ it is likely that one can also find by a computer search a non-singular zero of every form of the shapes $g_1,g_2,g_3$ and $g_4$. Whereas the case $q=7$ seems more doubtful than $q=8,9$, one can easily find counterexamples, for instance of shape $g_1$, for $q=5$ using the same algorithm.
<span style="font-variant:small-caps;">Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen,</span>\
<span style="font-variant:small-caps;">Germany</span>\
*E-mail address:* `jdumke@uni-math.gwdg.de`
|
---
abstract: |
The spiral structure of the low surface brightness galaxies F568-6 (Malin 2) and UGC 6614 is large scale, with arms that wrap more than half a revolution, and extend out to 50 and 80 kpc in UGC 6614 and F568-6 respectively. The density contrasts observed in the maps are high, with arm/interarm contrasts of $\sim$2:1, whereas the velocity perturbations due to spiral structure are low, in the range 10–20 km/s and 10–30 km/s in UGC 6614 and F568-6 respectively.
Upper limits for the disk mass-to-light ratios are estimated by considering the minimum velocity perturbations in the velocity field that should result from the spiral structure observed in the $R$ band images. The weak observed response in the $\phi$ velocity component limits the mass-to-light ratios of the disk inside a scale length to $M/L \lta 3$ and 6 for UGC 6614 for F568-6 respectively (in solar units) based upon azimuthal variations observed in the $R$ band images. These limits are sufficiently strong to require a significant dark matter component even in the central regions of these galaxies. Our limits furthermore imply that this dark matter component cannot be in the form of a cold disk since a cold disk would necessarily be involved in the spiral structure. However, a more massive disk could be consistent with the observations because of a non-linear gas response or if the gas is driven by bar-like distortions instead of spiral structure.
To produce the large observed arm/interarm density variations it is likely that the spiral arm potential perturbation is sufficiently strong to produce shocks in the gas. For a forcing that is greater than $2\%$ of the axisymmetric force, $M/L \gta 1$ is required in both galaxies in the outer regions. This is equivalent to a disk surface density between $r =$ 60–120 in UGC 6614 of 2.6–1.0 $M_\odot/{\rm pc}^2$ and between $r =$ 40–90 in F568-6 of 6.6–1.0 $M_\odot/{\rm pc}^2$ assuming that the amplitude of the variations in the disk mass is the same as that observed in the $R$ band. These lower limits imply that the stellar surface density is at least of the same order as the gas surface density. This is consistent with the large scale morphology of the spiral structure, and the stability of the gas disk, both which suggest that a moderate stellar component is required to produce the observed spiral structure.
author:
- 'A. C. Quillen$^,$ & T. E. Pickering$^,$'
title: |
Spiral Structure Based Limits on the Disk Mass\
of the Low Surface Brightness Galaxies UGC 6614 and F568-6
---
\#1[to 0pt[\#1]{}]{}
Introduction
============
The observed rotation curve shapes of both giant, so-called Malin-type, low surface brightness galaxies (e.g. [@deb96a], [@pic97]) and dwarf low surface brightness galaxies (e.g. [@deb96a], [@vz97]) do not match the rotation curves shapes inferred from their light distributions assuming constant mass-to-light ratios. In both cases, the observed rotation curves rise more slowly and continue rising out to radii where the rotation curves inferred from the light distributions are declining. In the giant low surface brightness galaxies, very high disk mass-to-light ratios of about 20–30 (in $R$ band) would be required to match the high observed rotation speeds ([@pic97]). As a result, to fit the rotation curves of these galaxies a substantial dark matter component is required, even at small radii, and they are said to be dark matter dominated. This is in contrast to normal high surface brightness galaxies where maximal disk solutions with lower mass-to-light ratios yield good fits to the rotation curves (e.g. [@ken87a] & [@ken87b]) in the central few disk scale lengths. Comparison of low and high surface brightness galaxies with equal total luminosity suggest that low surface brightness galaxies have low mass surface density disks ([@deb96b]). This is consistent with the fact that these galaxies appear at near normal locations on the Tully Fisher relation ([@zwa95]).
In spite of their low mass surface densities, some low surface brightness disk galaxies do show spiral structure including, of course, the two cases discussed here as well as all of the giant low surface brightness galaxies presented in [@spray95] and some of the dwarfs in the samples of [@vz97] and [@deb96b] (e.g. UGC 11820, UGC 5716, and F568-1). Since spiral structure requires non-axisymmetric mass perturbations in the disk of the galaxy itself, its properties can be used to limit the [*mass*]{} of the [*disk*]{} involved in the spiral density waves. If velocity perturbations caused by the spiral structure are small, a limit can be placed on the total mass in the spiral structure, yielding an upper limit on the mass-to-light ratio of the luminous stellar disk. If evidence for strong spiral shocks is seen in the gas response, then a sufficiently strong spiral gravitational force is required to produce this gas response. This critical forcing yields a lower limit for the mass of the disk.
Indeed the concept of using spiral arm patterns to limit the disk mass-to-light ratio was considered by [@vis80] who compared the predictions of spiral density wave theory to observations of M81. He stated “The second test is whether the amplitude of the wave as measured” (by photometry) “converted into the amplitude of the spiral arm potential perturbation, is consistent with the observed amplitude of the density and velocity perturbations of the gas.” He treated the forcing spiral wave amplitude as a free parameter and found that large amplitudes produced velocity residuals too large to be consistent with the observations, whereas weak forcing amplitudes did not produce shocks leading to the large density contrasts which are observed in emission in M81. Using a different approach, [@ath87] placed limits on the mass-to-light ratios of some normal disk galaxies by assuming that the disks should amplify $m=2$ or bisymmetric spiral modes but inhibit asymmetric $m=1$ modes.
Low surface brightness galaxies are an important setting to place limits on the mass in the disk for two reasons: 1) They have low density disks and 2) They appear to be dark matter dominated. Normal high surface brightness galaxies with strong spiral structure have bright disks which are adequately massive to produce a strong gravitational spiral force even when the spiral structure is relatively weak. However, in the low surface brightness galaxies the disk surface brightness is 1–3 magnitudes fainter even though rotational velocities are nearly equivalent so it is quite surprising that spiral structure exists in any of these galaxies. Many normal galaxies can be well fit by maximal disk or near maximal disk rotation curve models (e.g. [@ken87a], [@ken87b]) and models of spiral gas response require gravitational forcing which is consistent with the amplitude variations observed from images of these disks (e.g. [@low94]). By placing limits on the mass density of the disk from the existence of spiral structure we can test the degree to which these galaxies are dark matter dominated and the possibility that these galaxies might have a very massive disk component.
In this paper we use the spiral structure observed in optical and images of the low surface brightness galaxies UGC 6614 and F568-6 (Malin 2) to place both upper and lower limits on the mass-to-light ratio of the optical disk. In §2 we review the and $R$ band images which show spiral structure in these two galaxies. In §3 we place upper limits on the mass-to-light ratio of this spiral structure based on the low level of spiral arm induced velocity perturbations. In §4 we consider the mass density in spiral structure required to produce shocks in the gas that would be consistent with the large density contrasts observed in the column density maps. A discussion follows in §5.
Spiral Structure in UGC 6614 and F568-6
=======================================
UGC 6614 and F568-6 were observed in $R$ band and in by [@pic97] to investigate their neutral hydrogen and kinematic properties. These data are displayed as overlays in Figures 1 and 2 and are described in detail in [@pic97]. Table 1 lists some observation parameters and basic properties from [@pic97]. UGC 6614 and F568-6 are both low surface brightness galaxies with central surface brightnesses several magnitudes fainter than sky level ($\mu_{R}(0) = 22.9$ and 22.1 for UGC 6614 and F568-6 respectively). Their disk exponential scale lengths are large (14 and 18 kpc for UGC 6614 and F568-6 respectively) and they both contain copious amounts of ($2.5$ and $3.6 \times 10^{10} M_\odot$ for the two galaxies respectively, [@pic97]). We assume here the distances of $D=85$ and $184$ Mpc to UGC 6614 and F568-6 respectively (following [@pic97]; derived from a Hubble constant of $75$ km s$^{-1}$ Mpc$^{-1}$).
Both galaxies show clear evidence of spiral structure (see Figures 1 and 2). Spiral arms are visible in the optical images, the column density maps, and as kinks in the velocity field at the level of 10–30 km/s. This spiral arm structure is coincident in the $R$ band and images as well as in the velocity field. The spiral structure for the two galaxies is large scale extending to a radius of more than 50 and 90 kpc in UGC 6614 and F568-6 respectively. This is in contrast to normal galaxies such as M81 where the entire spiral arm structure lies within the optical disk or within $\sim 20$ kpc. We note however that in terms of disk scale lengths the extent of spiral structure in UGC 6614 and F568-6 is not so large, extending only to 3.6 and 5 scale lengths respectively, which is small compared to 8 scale lengths in M81 (the disk exponential scale length in M81 $\sim 2.5$ kpc, [@ken87b]).
Spiral Arm Morphology
---------------------
For both galaxies the morphology of the spiral structure consists of coherent spiral arms each of which wrap around the galaxy more than half a revolution. In this respect the spiral structure does not resemble that of flocculent late-type galaxies where pieces of arms exist only locally. Both spiral structures are strongly asymmetric, particularly in their inner regions. The outer arms in UGC 6614, however, are close to being a bi-symmetric or two arm spiral pattern which is centered about a position located to the east of the nucleus. In flocculent galaxies, the spiral structure is thought to be primarily propagating in the gas disk without strong coupling to the stellar disk (e.g. [@bra93]). Also, simulations have shown that when the disk mass is low, the spiral structure is more likely to be flocculent ([@sel84], [@car85]). The more coherent nature of the spiral structure observed in these low surface brightness galaxies suggests that the stellar disk is coupled to the gas disk and actively involved in the spiral structure.
The column density map shows large arm/interarm contrasts of $\sim$ 2:1 and velocity variations seen as kinks in the velocity field detectable at the level of 10–20 and 10–30 km/s in UGC 6614 and F568-6 respectively. We note that these values for the arm/interarm contrast and velocity residuals are similar to those observed in M81 which has radial velocity perturbations of $\sim 10$ km/s observable in ([@vis80]). High surface brightness galaxies can also have significantly stronger velocity perturbations. For example, M51 has radial velocity perturbations of 60–90 km/s seen in the CO velocity field ([@vog88]), and UGC 2885 has perturbations of 50–70 km/s observed in H$\alpha$ ([@can93a]).
Along the kinematic minor axis of the galaxy velocity residuals are expected to be radial. The pattern of the velocity residual pattern depends on whether the spiral arm pattern lies within the corotation radius ([@can93]). Kinks in the velocity field alternate sign only a few times as a function of radius along the minor axes of the two galaxies. This residual pattern which is consistent with a single residual alternating sign pair associated with each arm is similar to the velocity residual field of M81 and suggests that the entire spiral pattern lies within the corotation resonance ([@can93]).
Low surface brightness galaxies can have gas densities ([@vdh93], [@mcg92]) which fall below the threshold $\Sigma_{crit}$ which [@ken89] found was required for massive star formation in normal spiral galaxies. As proposed by [@ken89] this critical density is directly related to the local stability of the disk where the Toomre stability parameter $Q$, defined as $$Q \equiv {\kappa \sigma \over 3.36 G \Sigma }$$ (see [@B+T]) can be written in terms of the critical gas density as $$Q = {\Sigma_{crit} \over \Sigma \alpha}.$$ Here $\Sigma$ is the local gas surface density, $\kappa$ is the epicyclic frequency, $\sigma$ is the velocity dispersion, and $\alpha$ was emperically determined by [@ken89] to be $\sim 0.7$. When $Q > 2$ amplification processes such as the swing amplifier are inefficient and the disk is unresponsive to tidal perturbations which could excite spiral density waves in a more unstable disk (see [@B+T] and references theirin). It is therefore unlikely for a disk with $Q>2$ to show spiral structure. This implies that a disk with gas density below the critical gas density of [@ken89] is also unlikely to show spiral structure assuming there is no other disk mass component. (Note that $Q\sim 1.4 $ for $\Sigma_{crit} /\Sigma = 1$.) Consequently if a gas disk is well below the critical gas density and yet shows spiral structure, a natural explanation is that there is another massive component in the disk (see [@jog84] for instability in a two fluid disk).
Low surface brightness galaxies can show spiral structure despite the fact that their gas densities fall below the critical density ([@vdh93], [@deb96b]). Unfortunately the data of [@pic97] is not of sufficiently high velocity resolution to measure the gas velocity dispersion $\sigma$ required to calculate the $\Sigma_{crit}$ predicted by [@ken89]. However, [@pic97] found that even if $\sigma$ had the low value of 6km/s, UGC 6614 had gas densities below the critical density for $r <50''$, $r>120''$ and that Malin 2 had gas densities below this threshold everywhere. This suggests that a gas plus stellar disk may be required for these disks to be sufficiently unstable to support the observed spiral structure.
Spiral Arm Amplitudes
---------------------
To estimate the amplitude of the spiral arm structure observed in and in the $R$ band images, we must first correct for the inclination of the galaxy. This is relatively straightforward in the case of UGC 6614 since the $R$ band isophote, isophote and velocity field derived position and inclination angles all agree (PA = $116^\circ$, $i=35^\circ\pm 3 ^\circ$, [@pic97]). For F568-6 the $R$ band isophote and velocity field derived position and inclination angles agree (PA = $75 ^\circ$, $i=38^\circ\pm 3 ^\circ$), though the distribution is more asymmetric than the $R$ band light ([@pic97]). We have adopted these orientations to correct for the galaxy inclinations.
In Figures 3 and 4 we show azimuthal profiles in the inclination corrected $R$ band images and maps for the two galaxies. The amplitudes as a function of radius of the $m=1$ and $m=2$ azimuthal Fourier components expressed as a percentage of the azimuthal average for the two galaxies are shown in Figure 5 for the $R$ band images and in Figure 6 for the images. UGC 6614 has an oval distortion in its central region ($r < 40''$) which is particularly noticeable after correction for inclination. (By oval distortion we mean an elongation in the isophotes that does not vary in position angle over a range of radius.) This oval distortion is seen as a large azimuthal density variation (evident as high $m=2$ components in Figure 5) which is not an actual intensity variation in the spiral arms. In the outer regions of both galaxies the faint spiral arms are far brighter than the underlying disk which is only marginally detected. Although peaks in the and $R$ band surface brightness are correlated (see Figures 1 and 2) they do not correspond to maximum densities or surface brightnesses in the azimuthal cuts except at large radii in UGC 6614 (see Figures 3 and 4). Better correlation might be observed with higher angular resolution observations.
Since the amplitudes shown in Figures 3 and 4 depend on the assumed galaxy orientations we recomputed them for moderate variations in the inclination angles. For a $5^\circ$ lower inclination of $30^\circ$ in UGC 6614, the change in amplitudes were largest in the region of the oval distortion ($r<40''$) and were $10-15\%$ smaller than at an inclination of $35^\circ$. Variations in the amplitudes elsewhere in UGC 6614 and in F568-6 for a corresponding difference in inclination were smaller, $\lta 5\%$. These amplitude uncertaintites are not sufficiently large to significantly change the limits for the mass-to-light ratio we estimate below.
Placing an Upper Limit on the Mass-to-Light Ratio
=================================================
Spiral density waves are a resonant wave phenomenon. Because resonances can exist a small gravitational perturbation can give a large response, however the opposite is not true. Given a particular spiral gravitational perturbation there is a minimum possible response. This makes it possible to place an upper limit upon the strength of the gravitational perturbation if a measure of the response, such as kinks in the velocity field, are small. To place an upper limit on the mass-to-light ratio of the spiral structure, we consider that the mass involved in the spiral density wave is insufficient to drive a strong response. We therefore estimate the smallest possible gas response to a non-axisymmetric or spiral gravitational perturbation. For such a perturbation, an expansion to first order should give an appropriate description for the gas flow in regions not directly affected by resonances. In this section we follow the notation and derivation given in chapter 6 of [@B+T],
We assume that there is a non-axisymmetric perturbation to the gravitational potential in the plane of the galaxy of the form $$\Phi_1( r,\phi) = Re[\Phi_{a}(r) \exp{i(m\phi - \omega t)}]$$ where the pattern speed of the perturbation is $\Omega_p \equiv
\omega/m$. We consider perturbations in the gas velocity field in response to this perturbation. The response in velocity to a tightly wound spiral density wave to first order for the $\phi$ (non-radial) component is of the same form as that of the potential and in the tight winding or WKB approximation is ([@B+T], equation 6-37) $$%v_{R a}(R) = {(m\Omega - \omega) ik(\Phi_a + h_a) \over \Delta} ~~~~~~
v_{\phi a}(r) = {-2Bik(\Phi_a + h_a) \over
\kappa^2 - (m \Omega - \omega)^2}$$ where $\kappa$ is the epicyclic frequency, $\Omega$ and $v_c =
\Omega/r $ are the angular rotation rate and circular velocity of the unperturbed system at radius $r$, and $B \equiv -\kappa^2/4\Omega$. The wavenumber of the spiral arm perturbation, $k$, is given by $k \equiv df(r)/dr$ where $\Phi_a(r) \propto e^{ i f(r)}$. $h_a$ corresponds to variations in the specific enthalpy of the same form as (equation 1) and can be neglected in the limit $c_s << v_c$ which is appropriate for the rotation curves of the low surface brightness galaxies considered here.
Although these velocity perturbations become large near resonances, a limit on the [*minimum*]{} possible velocity perturbation can be placed from the observations. We can rewrite the above equation as $$v_{\phi a} = {ik \Phi_a \over
(1 - (m \Omega - \omega)^2 \kappa^{-2}) 2 \Omega}.$$ Because the magnitude of the denominator of the above equation reaches a maximum near corotation we can place the approximate limit $$\vert v_{\phi a} \vert \gta
\left\vert {k \Phi_a \over 2 \Omega }\right\vert.$$ A density perturbation of the same form as Eq. 3 causes the spiral potential perturbation given above where $$\Phi_a = {-2 \pi G \Sigma_a \over \vert k \vert}$$ ([@B+T], equation 6-17) for an infinitely thin disk. Substituting this into the above equation gives us the limit $$\vert v_{\phi a} \vert \gta
\left\vert { \pi G \Sigma_a \over \Omega } \right\vert.
%= \left\vert { \pi G \Sigma_a R \over v_c } \right\vert$$ If we consider the possibility that the stellar component is more massive than the gas component the above expression can be inverted to yield an upper limit on the mass-to-light ratio, $M/L$, of the disk given an upper limit on the $\phi$ component velocity perturbations across the spiral arm: $$M/L \lta {v_{\phi a} v_c \over \pi G S_a r}$$ where $S_a$ is the surface brightness variation about radius $r$ (of the same form as Eq. 3), and $v_{\phi a}$ is the maximum velocity perturbation detected (or detectable in the case of no detection).
The $\phi$ component velocity perturbations are particularly noticeable along the kinematic major axis where the line of sight component of the velocity is in the azimuthal direction. From Figures 1 and 2 we can see that for UGC 6614 these velocity perturbations are smaller than $\sim 15$ km/s, whereas for F568-6 larger perturbations are detected but are $\lta 25$ km/s. Using these velocities as upper limits and the $m=1$ and $m=2$ components of the surface brightness variations from the azimuthal profiles (shown in Figure 5) upper limits to the mass-to-light ratios as a function of radius for the two galaxies are shown in Figure 7.
The $m=2$ component oval distortion for $r \sim 40''$ in UGC 6614 is sufficiently strong to produce larger velocity perturbations than detected if the mass-to-light ratio is greater than $\sim 3$ (for $R$ band in solar units) in this region. From this we derive a limit $M/L
\lta 3$ in the central region of the galaxy. Alternatively we can think of this limit in terms of mass and say that if the disk has perturbations of the same amplitude as observed in the $R$ band image, the surface density of the disk must be less than $30 M_\odot/{\rm
pc}^2$ (for $M/L = 3$ and a surface brightness of 23.5 mag/arcsec$^2$ in the $R$ band at $r =$ 40 or 16 kpc). For F568-6 the spiral structure in the central regions is not as strong and only gives a limit of $M/L \lta 6 $ at $r \sim$ 30 which is equivalent to a surface density which must be less than $60 M_\odot/{\rm pc}^2$ (for $M/L = 6$ and a surface brightness of 23.5 mag/arcsec$^2$ in the $R$ band at $r =$ 30 or 24 kpc).
The fits to the rotation curves of [@pic97] for both galaxies have $M/L \lta 1$ for the bulge and disk and require a substantial dark matter component even at small radii. For these fits the peak of the velocity contribution from the disk is only $\sim 40$ and 90 km/s for UGC 6614 and F568-6 respectively. For our upper limit of $M/L = 3 $ and 6 for the disks in UGC 6614 and F568-6 a substantial dark matter component is still required to reach the observed maximum rotational velocity. Our upper limits reinforce the findings of [@pic97] and others based on fits to the rotation curve that a substantial dark matter component is required even in the central regions of these low surface brightness galaxies. This limit further requires that the dark component cannot be in the form of a cold stellar disk (which would have to be involved in the spiral structure).
The above high stellar surface densities are consistent with our neglect of the gas density in Eq. 9 since they are significantly higher than the gas density (see Figure 5). However, even though strong density variations variations exist in the for these galaxies we have estimated the minimum possible velocity response using linear perturbation theory. We note here that it is possible that the non-linear response of the gas could cause the velocity perturbations to be somewhat smaller than inferred from the above limit. In this case a higher mass-to-light ratio could be consistent with the small size of the observed velocity perturbations. If future higher angular resolution observations reveal larger velocity perturbations along the spiral arms, then the upper limit for the mass-to-light ratio would also be higher than stated here. An underlying more massive disk at these radii could exist if it had smaller azimuthal density variations than observed in the $R$ band. For example if near-infrared observations (which might more accurately trace surface density varitations) show smaller amplitude variations in the spiral structure, $S_a$ in the above equation would be smaller and a more massive disk could be consistent with the observations. The spiral arms observed in the $R$ band images show fine structure which can only exist when the stellar disk is thin. It is therefore reasonable to use the approximation of a thin disk for the potential (Eq. 7). We also note that our lowest value for the mass-to-light ratio in UGC 6614 coincided with density perturbations that were part of an oval distortion in the galaxy with position angle that varied only slowly with radius. While it is not unreasonable that this oval distortion could be driving a strong gas response, the WKB or tight winding approximation is no longer valid. Although the minimum velocity response is of the same order as given in Eq. 8, this equation could be inaccurate by a factor of a few.
Critical Forcing Required to Produce Shocks – A Lower Limit for $M/L$
=====================================================================
The large density variations in the spiral structure of UGC 6614 and F568-6 are evidence for shocks in the ISM induced by a spiral gravitational potential. Indeed the high arm/interarm density contrast of observed in galaxies such as M81 and M51 is one of the major predictions of the spiral density wave theory. In this section we consider how much mass is required in the form of spiral structure to drive shocks in the gas that would be consistent with the $\sim$2:1 arm/interarm density contrasts observed in the of these two low surface brightness galaxies.
The response of the gas in a spiral density wave is primarily dependent on the forcing gravitational field and the effective ISM sound speed and only weakly dependent on the cloud dependent properties such as the cloud mean-free path and the cloud number density ([@rob84], [@hau84], and [@rob69]). A critical forcing parameter to produce shocks or large density contrasts in the ISM was explored by [@rob69] and [@shu73]. These authors considered the role of $F$, the spiral gravitational force expressed as percentage of the axisymmetric force. Originally [@rob69] found that a forcing of $F > 2\%$ was required to produce a density contrast of greater than 2 and that for $F< 1\%$ no shock was produced. Subsequent work by [@shu73] found that the critical forcing parameter depended on the effective sound speed of the gas, $c_g$, and on the speed of the imposed spiral force $c_k\equiv
m\Omega_p/k$. For a wide variety of pattern speeds, [@shu73] found that a forcing amplitude of at least $3-4\%$ was required to produce shocks in the gas, although when $c_k/c_g$ is moderately greater than 1, a forcing of only $1\%$ could produce shocks. This differed from the critical forcing parameter introduced in [@too77] (see also [@B+T]), valid in the limit of low effective sound speed, which requires forcing amplitudes of a few percent for moderate values of $c_k/c_g > 1$ ([@too77]).
Subsequent models and simulations of the gas response find that forcing of at least a few percent is required to produce the observed gas density contrasts. For example using a forcing spiral gravitational field based on photometry of M81 [@vis80] found that when $F \lta 4\%$ no shocks were produced in his model. Although more recent modeling of M81 does not vary the forcing amplitude, [@low94] produce the observed density contrast with simulations driven by a spiral forcing amplitude of $5-10\%$. By exploring the properties of cloud-particle simulations [@rob84] found that density contrasts of a few resulted where $F = 5-10\%$ using a spiral structure based on M81 even for the low effective gas sound speed of 8 km/s.
To summarize, the critical forcing value of a few percent ($\gta$ 3–4%) seems to be required to produce shocks in the ISM that result in significant gas density contrasts, although this value is dependent on $c_g$ and $c_k$ so that an ISM with a low effective sound speed forced by a fast spiral pattern could produce shocks with the weak forcing amplitude of only $\sim 1\%$.
Forcing Amplitudes in F568-6 and UGC 6614
-----------------------------------------
For the potential given in Eq. 3, the forcing amplitude expressed as a ratio of the unperturbed axisymmetric gravitational force can be written $$F \equiv \left\vert{\Phi_a k \over r \Omega^2}\right\vert.$$ Using Eq. 7 we can rewrite this in terms of the density variation as $$F = {2 \pi G \Sigma_a r \over v_c^2}.$$
Using the above expression we have computed the forcing amplitudes using $\Sigma_a$ estimated from the $m=1$ and $m=2$ gas and $R$ band azimuthal components with $M/L =1$ in the $R$ band. These forcing amplitudes are shown in Figure 8. We note that the forcing amplitudes are typically quite small. The distortions in the central regions of both galaxies are of sufficient strength to drive shocks in the gas assuming a mass-to-light ratio of 1. In these regions the gas contribution to the spiral gravitational field are negligible. At larger radii the gas and stellar spiral gravitational forcing in both galaxies are quite small, $\lta 2\%$ of the axisymmetric force.
If we assume that a minimum particular forcing strength is required to produce the observed density contrasts, then we derive a lower limit for the mass-to-light ratio of the disk. We have chosen a critical value of $2\%$ because even though a forcing of only $1\%$ might be able to cause shocks, is unlikely to cause the observed gas density contrast. Larger values typical of normal galaxies such as M81 might be inconsistent with the low rate of star formation observed in these galaxies ([@pic97]). Assuming that forcing of at least $2 \%$ is required to produce the observed density contrasts then in both galaxies we derive the lower limit $M/L \gta 1$ in the outer parts of the disks ($r>60''$ or 24 kpc and $r>40''$ or 32 kpc in UGC 6614 and F568-6 respectively). This can be expressed as a mass density (if the underlying stellar disk has the same amplitude azimuthal variations as observed in $R$ band). In UGC 6614 for $r =$ 60–120 the $R$ band surface brightness varies from 25–26 mag/arcsec$^2$ which for $M/L=1$ is equivalent to a disk stellar surface density of 2.6–1.0 $M_\odot/{\rm pc}^2$. In F568-6 for $r =$ 40–90 the $R$ band surface brightness varies from 24–26 mag/arcsec$^2$ which for $M/L=1$ is equivalent to a disk stellar surface density of 6.6–1.0 $M_\odot/{\rm pc}^2$. These lower limits imply that the stellar surface density is of the same order as the gas surface density. This is consistent with the large scale morphology of the spiral structure and the stability of the gas disk, both of which suggest that a moderate stellar component is required to produce the observed spiral structure.
We note that the forcing assuming $M/L=1$ given in Figure 8 (expressed as a ratio of the unperturbed axisymmetric gravitational force) is smaller than derived for normal high surface brightness galaxies but not 10 times smaller as expected from the 1-3 lower magnitude surface brightness disks of the low surface brightness galaxies, and the equivalent size of their rotational velocites. This is due to the fact that at the large radii where spiral structure exists in these low surface brightness galaxies, the axisymmetric force ($v_c^2/r$) is lower than that at the smaller radii where the spiral structure exists in normal galaxies.
One possible concern is that a moderate external tidal field could be driving the gas response instead of a spiral disk component. In this case the disks of these galaxies could have very low mass-to-light ratios. However, UGC 6614 appears to be isolated; no gas near its redshift was evident in the velocity channel maps of [@pic97]. On the other hand, in F568-6 there is evidence for interacting material; there is a high velocity clump of gas to the south of the nucleus which seems to be due to a superimposed, possibly interacting dwarf ([@pic97]). This scenario is unlikely, though, because an external tidal field should cause an oval (non-spiral) perturbation to the gravitational potential which would drive spiral shocks in the gas that are more open than the tightly wound spiral arms observed. It is also difficult for an external tidal field to drive shocks in the gas over a large range of radius. An external tidal field could however be ultimately responsible for exciting the spiral density waves in the gas plus stellar disk (as is likely in the case of M51), but then the resulting stellar and gas derived spiral gravitational field must be sufficiently strong to cause shocks in the gas as assumed here.
Summary and Discussion
======================
The spiral structure of the low surface brightness galaxies F568-6 and UGC 6614 is large scale, with arms that wrap more than half a revolution, and extend out to 50 and 80 kpc in UGC 6614 and F568-6 respectively. These spiral arms are visible in the $R$ band images, the column density maps and as kinks in the velocity fields. The density contrasts observed in the maps are high, with arm/interarm contrasts of $\sim 2:1$, whereas the velocity perturbations due to spiral structure are low, in the range 10-20 km/s and 10-30 km/s in UGC 6614 and F568-6 respectively.
We use the small velocity response to place upper limits on the mass-to-light ratio of the stellar disk. The strongest limits occur at small radii ($r \sim 40''$ or 16 kpc and $r \sim 30''$ or 24 kpc in UGC 6614 and F568-6 respectively) where there are strong distortions observed in the $R$ band images. The weak observed response in the $\phi$ velocity component limits the mass-to-light ratios of the disk in these regions to $M/L < 3$ and 6 for UGC 6614 for F568-6 respectively. This is equivalent to requiring the densities of the disks (if they have azimuthal variations of the same size as that observed in the $R$ band images) to be less than 30 and 60 $M_\odot/{\rm pc}^2$ at a radius of 16 and 24 kpc for UGC 6614 and F568-6 respectively. An underlying more massive disk at these radii could exist if it had smaller azimuthal density variations than observed in the $R$ band. These limits are sufficiently strong to require a significant dark matter component even in the central regions of this galaxy, confirming the findings of previous studies. Our limits furthermore imply that this dark matter component cannot be in the form of a cold disk since a cold disk would necessarily be involved in the spiral structure, though a hot disk cannot be excluded. We note that this upper limit was derived assuming a linear gas response in the tight winding or WKB approximation, and that a non-linear gas response driven by bar like or oval distortions could cause small velocity perturbations to be present even in the case of stronger potential perturbations (or higher mass-to-light ratios).
To produce the large arm/interarm density variations it is likely that the spiral arm potential perturbation is sufficiently strong to produce shocks in the gas. For a forcing that is greater than $2\%$ of the axisymmetric force, $M/L \gta 1$ is required in both galaxies in the outer regions. This is equivalent to a disk surface density between $r =$ 60–120 in UGC 6614 of 2.6–1.0 $M_\odot/{\rm pc}^2$ and between $r =$ 40–90 in F568-6 of 6.6–1.0 $M_\odot/{\rm pc}^2$ assuming that the amplitude of the variations in the disk mass is the same as that observed in the $R$ band. These lower limits imply that the stellar surface density is of the same order as the gas surface density. This is consistent with the large scale morphology of the spiral structure, and the stability of the gas disk, both of which suggest that a moderate stellar component is required to produce the observed spiral structure. The gas disks alone probably fall below the critical gas density ([@pic97]) emperically found by [@ken89] for the onset of masive star formation and so are not likely to be unstable enough to support spiral density waves. However it is likely that the combined stellar and gas disks are (see [@jog84] for instability in a two fluid disk). The coupled gas and stellar disk would be consistent with the large scale spiral arm morphologies which do not resemble that of gas dominated flocculent galaxies.
The limits for the mass-to-light ratio of the disk derived here suggest that the disks of these two low surface brightness galaxies lie in the range $1 < M/L < 6$ (in the $R$ band). This range is identical to that found for normal higher surface brightness disk galaxies derived from maximal disk fits to the observed rotation curves ([@ken87a], [@ken87b]). In other words the disk mass-to-light ratio limits placed here are not abnormal compared to those of normal galaxies. Our limits on the disk surface densities remain consistent with previous studies which find that low surface brightness galaxies have substantially lower mass surface densities than normal galaxies ([@deb96b], [@spray95b]).
In low surface brightness galaxies since the disk contributions to the rotation curves are small compared to the halo, good fits to the rotation curves can be acheived with a range of mass-to-light ratios. Fits to the rotation curves using halo profiles of the form proposed by [@nav95] yeilded the best fit mass-to-light ratios of $\sim 0.8$ and $0.5$ for UGC 6614 and F568-6 respectively ([@pic97]), whereas [@imp97] using an isothermal halo found a good fit to the rotation curve of F568-6 with a much higher $M/L_B = 8$. The disk mass-to-light ratios derived from these fits probably depend upon the halo profile assumed and whether the bulge and disk are allowed to have different mass-to-light ratios and so are not tightly constrained.
We note that multi-wavelength observations (particularly those in the infrared) find that amplitude variations across spiral arms can be a strong function of wavelength (e. g. [@rix93]). While the low column depths in UGC 6614 and F568-6 suggest that extinction from dust is not a large effect in these low surface brightness galaxies, it would not be surprising if an older population of stars (which would be more apparent in the near infrared wavelengths) might have smaller spiral arm amplitudes. In this case a lower limit for the mass-to-light derived as we have done here but from a near infrared image might yield even stronger limits requiring an even more massive stellar disk. If high mass-to-light ratios are required, then a mass-to-light ratio variation across the disk could be required to yield a good fit to the rotation curve assuming a smooth halo component, and a large dark matter component might not necessarily be required in the central regions. Multi-wavelength observations coupled with metallicity measurements should also provide useful information about what kinds of stellar populations would be consistent with the range of mass-to-light ratios given here.
It might prove to be interesting to place similar mass-to-light ratio limits in low surface brightness dwarf galaxies which also can have spiral structure (for example UGC 11820, UGC 5716, and F568-1, from [@vz97] and [@deb96b]). These galaxies can have regular, symmetrical velocity fields and smooth column density maps, even when the optical components lack symmetry or contain strong spiral structure. The optical components therefore don’t strongly influence the velocity field (although some of these effects might be visible in observations at higher angular resolution). This suggests that the upper limits discussed here might be particularly revealing.
Strong spiral structure detected in in the outer regions of higher surface brightness galaxies could be used to estimate the mass density of a low surface brightness stellar disk. Some interesting candidates for such a study might be the dark blue compact dwarf NGC 2915 which may contain spiral structure well outside its optical disk ([@meu96]), and the polar ring galaxy NGC 4650A which shows evidence for spiral arms in its ring ([@arn97]).
We acknowledge helpful discussions and correspondence with R. Kennicutt, L. van Zee, A. Nelson, G. Rieke, J. Navarro and H-W. Rix. We acknowledge support from NSF grant AST-9529190 to M. and G. Rieke.
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[lrrrrr]{}
Total absolute R band magnitude & $-23.6\pm0.1$ & $-22.3\pm 0.1$ Distance assumed (Mpc) & $184$ & $85$ Total HI line flux (Jy km/s) & $4.4\pm 0.3$ & $15.0\pm 0.8$ Total HI mass ($10^{10} M_\odot$) & $3.6\pm 0.4$ & $2.5\pm 0.2$ FWHM of synthesized beam ($'' \times ''$) & $19.5 \times 18.4$ & $21.4 \times 19.8$ P.A. of synthesized beam (degrees) & $-73.0$ & $52.3$ Channel spacing (km/s) & $22.6$ & $10.7$ RMS noise in channel maps (mJy/beam) & $0.25$ & $0.7$ Limiting column density, $N_H$ ($10^{19} {\rm cm}^{-2}$) & $4.4$ & $5.8$
|
---
abstract: 'The relativistic mean field theory with the Green’s function method is taken to study the single-particle resonant states. Different from our previous work \[\], the resonant states are identified by searching for the poles of Green’s function or the extremes of the density of states. This new approach is very effective for all kinds of resonant states, no matter it is broad or narrow. The dependence on the space size for the resonant energies, widths, and the density distributions in the coordinate space has been checked and it is found very stable. Taking $^{120}$Sn as an example, four new broad resonant states $2g_{7/2}$, $2g_{9/2}$, $2h_{11/2}$ and $1j_{13/2}$ are observed, and also the accuracy for the width of the very narrow resonant state $1h_{9/2}$ is highly improved to be $1\times 10^{-8}$ MeV. Besides, our results are very close to those by the complex momentum representation method and the complex scaling method.'
author:
- 'C. Chen'
- 'Z. P. Li'
- 'Y.-X. Li'
- 'T.-T. Sun'
title: 'Study of single-particle resonant states with Green’s function method'
---
Introduction {#sec:intr}
============
The single-particle resonant states in the continuum are playing crucial roles in the formation of halos in exotic nuclei [@PRC2003Sandulescu_68_054323]. For example, studies by the relativistic continuum Hartree-Bogoliubov theory suggested that giant halos can be formed in the neutron-rich Zr and Ca isotopes if more than two valence neutrons occupy the resonant states with low angular momentums [@PRL1998MengJ_80_460; @PRC2002MengJ_65_041302], and the existence of a possible deformed halo in $^{40,42}$Mg and $^{22}$C is mainly decided by the single-particle states around the Fermi surface including the resonant states in the continuum [@PRC2010Zhou_82_011301; @PRC2012Li_85_024312; @CPL2012LiLL_29_042101; @PLB2018SunXX_785_530]. As a result, the exploration of resonant states are becoming significantly important and attracting more and more attentions.
During the past years, a series of approaches have been taken or developed in the exploration of the single-particle resonant states. Some approaches are based on the conventional scattering theories, such as $R$-matrix theory [@PRL1987Hale_59_763; @PR1947Wigner_72_29], $K$-matrix theory [@PRC1991Humblet_44_2530], $S$-matrix theory [@Book1972Taylor-ScatteringTheor; @PRC2002CaoLG_66_024311], Jost function approach [@PRL2012BNLu_109_072501; @PRC2013BNLv_88_024323], and the scattering phase shift (SPS) method [@Book1972Taylor-ScatteringTheor; @PRC2010LiZP_81_034311; @SCP2010Li_53_773]. Besides, some techniques which are used for bound states have also been extended to study the single-particle resonant states, such as the complex momentum representation (CMR) method [@PRC2006Hagen_73_034321; @PRL2016Li_117_062502; @PLB2020ShiXX_801_135174], the complex scaling method (CSM) [@PRC1986Gyarmati_34_95; @PRC1988Kruppa_37_383; @PRL1997Kruppa_79_2217; @PRC2006Arai_74_064311; @PR1983Ho_99_1; @PRC2010JYGuo_82_034318; @PRC2014ZLZhou_89_034307; @PRC2012QLiu_86_054312; @PRC2014Shi_90_034319], the complex-scaled Green’s function (CGF) method [@PLB1998Kruppa_431_237; @PTP2005Suzuki_113_1273; @EPJA2017Shi_53_40], the real stabilization method (RSM) [@PRC2008ZhangL_77_014312], and the analytical continuation of the coupling constant (ACCC) method [@PRC2005Guo_72_054319; @Book1989Kukulin-TheorResonance; @PRC1997Tanaka_56_562; @PRC1999Tanaka_59_1391; @PRC2000Cattapan_61_067301; @CPL2001SCYang_18_196; @PRC2004ZhangSS_70_034308; @EPJA2007SSZhang_32_43; @PRC2012SSZhang_86_032802; @EPJA2012SSZhang_48_40; @EPJA2013SSZhang_49_77; @PLB2014SSZhang_730_30; @PRC2015Xu_92_024324].
The Green’s function (GF) method [@SJNP1987Belyaev_45_783; @PRB1992Tamura_45_3271; @PRA2004Foulis_70_022706; @Book2006Eleftherios-GF] is also a successful candidate for studying resonances. It can treat the continuum exactly. With this method, the single-particle spectrum covering the bound states and the continuum are treated on the same footing, exact energies and widths can be obtained for resonant states of all kinds, and correct asymptotic behaviors are kept well for the density distributions. Besides, it is very convenient to be combined with nuclear models. As a result, Green’s function method has been used extensively in the study of the nuclear structure and excitations. For example, by applying the GF method to the Hartree-Fock-Bogoliubov (HFB) theory in the coordinate representation, halos in both spherical and deformed nuclei are described very well [@PRC2009Oba_80_024301; @PRC2011ZhangY_83_054301; @PRC2012YZhang_86_054318; @PRC2019SunTT_99_054316]. Besides, the continuum quasiparticle random-phase approximation (QRPA) formulated with Green’s function method [@NPA2001Matsuo_696_371] is developed to describe many interesting phenomena, such as the collective excitations [@PTPS2002Matsuo_146_110; @PRC2005Matsuo_71_064326; @NPA2007Matsuo_788_307; @PTP2009Serizawa_121_97; @PRC2009Mizuyama_79_024313; @PRC2010Matsuo_82_024318; @PRC2011Shimoyama_84_044317], monopole pair vibrational modes and associated two-neutron transfer amplitudes [@PRC2013Shimoyama_88_054308], and neutron capture reactions [@PRC2015Matsuo_91_034604].
The covariant density functional theory (CDFT) [@ANP1984SerotBD_16_1; @RPP1989Reinhard_52_439; @PPNP1996Ring_37_193; @JPG2015JMeng_42_093101] has got remarkable achievements in describing many systems and interesting phenomena, such as the stable and exotic nuclei [@PPNP2006MengJ_57_470; @PRC2017ZhangW_96_054308; @CPC2017Zhang_41_094102; @PRC2018ZhangW_97_054302], hypernuclei [@PRC2011BNLu_84_014328; @PRC2014BNLu_89_044307; @PRC2016TTSun_94_064319; @PRC2018LiuZX_98_024316], neutron stars [@CPC2018SunTT_42_025101; @PRD2019SunTT_99_023004], pseudospin symmetries [@PhysRep2015HZLiang_570_1; @JPG2017Lu_44_125104; @PRC2017Sun_96_044312; @PRC2019Sun_99_034310], and $r$-process simulations [@PRC2008SunB_78_025806; @PRC2009NiuZM_80_065806; @PLB2013NiuZM_723_172]. Thus, in recent years, we applied the Green’s function method to the framework of the covariant density functional theory. In 2014, the relativistic mean field theory formulated with the Green’s function method (RMF-GF) is developed, and as the first time, it is successfully applied to study the single-neutron resonant states [@PRC2014TTSun_90_054321]. It is also confirmed effective for the proton and $\Lambda$-hyperon single-particle resonant states [@JPG2016TTSun_43_045107; @PRC2017Ren_95_054318]. In 2016, the relativistic continuum Hartree-Bogoliubov theory combining the Green’s function method (RCHB-GF) is developed by containing the pairing correlation, which can describe the halos very well [@Sci2016Sun_46_12006]. Very recently, Green’s function method is further extended to study the resonances in deformed nuclei by solving a coupled-channel Dirac equation with quadrupole-deformed Woods-Saxon potential [@PRC2020Sun_101_014321].
In our previous works [@PRC2014TTSun_90_054321; @JPG2016TTSun_43_045107; @PRC2017Ren_95_054318], the single-particle resonances are identified by comparing the density of states (DOS) displayed by nucleons moving in the mean-field potentials and those by free particles. According to the DOS difference between the nucleons and free particles, the energy and width of resonant state are given by reading the position and the full-width at half-maximum (FWHM) of the resonant peak, respectively. In this way, energies and widths can be obtained easily for narrow resonances with good accuracy. However, the accuracy decreases for the wide resonances due to the irregular shapes of the resonant peaks. In our recent work [@PRC2020Sun_101_014321], a direct but very effective approach is proposed to study the resonant states by searching for the extremes of Green’s function in terms of the fact that the resonant states are poles locating in the fourth quadrant of the complex energy plane. In this work, we applied this new approach with Green’s function method to study the single-particle resonances based on the RMF theory.
The paper is organized as follows. In Sec. \[sec:Theory\], the Green’s function method is given briefly. In Sec. \[sec:Numer\], numerical details are presented. After the results and discussions in Sec. \[sec:resu\], a brief summary is drawn in Sec. \[sec:Sum\].
THEORETICAL FRAMEWORK {#sec:Theory}
=====================
In the RMF-GF theory [@PRC2014TTSun_90_054321], the Green’s function is applied in the coordinate space to calculate the densities for nucleons and the single-particle spectrum of the Dirac equation. The Dirac equation for nucleons in the RMF theory [@ANP1984SerotBD_16_1; @RPP1989Reinhard_52_439; @PPNP1996Ring_37_193] is, $$[\bm{\alpha}\cdot\bm{p}+V(\bm{r})+\beta(M+S(\bm{r}))]\psi_n(\bm{r})=\varepsilon_n\psi_n(\bm{r}),
\label{EQ:Dirac}$$ with the nucleon mass $M$, the Dirac matrices $\bm{\alpha}$ and $\beta$, and the scalar and vector potentials $S(\bm{r})$ and $V(\bm{r})$, respectively.
A relativistic single-particle Green’s function $\mathcal{G}(\bm{r},\bm{r'};\varepsilon)$ satisfying the following definition is needed to be constructed, $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}'),
\label{Eq:GF_define}$$ with $\hat{h}(\bm{r})$ being the Hamiltonian of the Dirac equation (\[EQ:Dirac\]). Starting from Eq. (\[Eq:GF\_define\]) and taking a complete set of eigenstates $\psi_{n}(\bm{r})$ and eigenvalues $\varepsilon_{n}$, the Green’s function can be represented as $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n\frac{\psi_{n}(\bm{r})\psi_{n}^{\dag}(\bm{r}')}{\varepsilon-\varepsilon_{n}},
\label{EQ:GF}$$ which has a form of a $2\times2$ matrix due to the two components of the Dirac spinor $\psi_{n}(\bm{r})$, $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=
\left(
\begin{array}{cc}
\mathcal{G}^{(11)}(\bm{r},\bm{r}';\varepsilon) & \mathcal{G}^{(12)}(\bm{r},\bm{r}';\varepsilon) \\
\mathcal{G}^{(21)}(\bm{r},\bm{r}';\varepsilon) & \mathcal{G}^{(22)}(\bm{r},\bm{r}';\varepsilon)
\end{array}
\right).$$
It is noted that the eigenvalues $\varepsilon_{n}$ of Dirac equation are poles of the Green’s function in Eq. (\[EQ:GF\]). As a result, one can obtain the eigenvalues $\varepsilon_{n}$ by searching for the poles of the Green’s function. In practice, following Ref. [@Book2006Eleftherios-GF], it can be done with the help of the density of states (DOS) $n(\varepsilon)$, $$n(\varepsilon)=\sum_n\delta(\varepsilon-\varepsilon_{n}),
\label{EQ:dos}$$ which displays like discrete $\delta$-function peaks for bound states at the eigenvalues $\varepsilon=\varepsilon_{n}$ but distributes continuously in the continuum with peaks for resonances. DOSs $n(\varepsilon)$ in a wide energy range will be calculated by scanning single-particle energy $\varepsilon$. One notes that for the continuum, energies $\varepsilon$ are complex $\varepsilon=\varepsilon_r+i\varepsilon_i$ and the energies for the resonant states can be written as $\varepsilon_n = E-i\Gamma/2$ with the resonance energy $E$ and width $\Gamma$.
Taking the imaginary part of the Green’s function, the DOSs can be calculated by the integrals in the coordinate ${\bm r}$ space [@PRC2014TTSun_90_054321]. For the bound states, it is $$\begin{aligned}
\label{EQ:Sdos}
&&n(\varepsilon)\\
&=&-\frac{1}{\pi}\int d\bm{r}\mathrm{Im}[\mathcal{G}^{(11)}(\bm{r},\bm{r};\varepsilon+i\epsilon)+\mathcal{G}^{(22)}(\bm{r},\bm{r};\varepsilon+i\epsilon)],\nonumber\end{aligned}$$ where $``i\epsilon"$ is the introduced positive infinitesimal imaginary part to the single-particle energy $\varepsilon$, with which the $\delta$-function shaped DOSs for bound states are simulated by Lorentzian functions with the FWHM of $2\epsilon$. For the resonant states, one does not need to introduce the infinitesimal imaginary part$``i\epsilon"$ since the single-particle energy $\varepsilon$ is complex. Besides, when scanning the imaginary part of complex energy $\varepsilon_i$, before and after it crossing the resonant states, the DOSs $n(\varepsilon)$ differ by a minus sign. The DOSs for the resonant states can be written as, $$\begin{aligned}
\label{Eq:DOSres}
&&n(\varepsilon)=\delta(\varepsilon_r-E)\\
&=&\left\{
\begin{array}{ll}
{\displaystyle -\frac{1}{\pi}\int d\bm{r}\mathrm{Im}[\mathcal{G}^{(11)}(\bm{r},\bm{r};\varepsilon)+\mathcal{G}^{(22)}(\bm{r},\bm{r};\varepsilon)]} , &\\ \hbox{ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~if $\varepsilon_i>-\Gamma/2$;} \\
{\displaystyle ~~~\frac{1}{\pi}\int d\bm{r}\mathrm{Im}[\mathcal{G}^{(11)}(\bm{r},\bm{r};\varepsilon)+\mathcal{G}^{(22)}(\bm{r},\bm{r};\varepsilon)]}, &\\
\hbox{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~if $\varepsilon_i<-\Gamma/2$.}
\end{array}
\right.\nonumber\end{aligned}$$ In practice, we calculate DOSs for resonances by scanning the whole complex energy range taking the first equation in Eq. (\[Eq:DOSres\]) and they will reverse and become negative when $\varepsilon_i$ over the resonant states. According to those changes, the widths of resonant states $\Gamma/2$ can be determined.
In the spherical case, the Green’s function can be expanded as $$\mathcal{G}({\bm r},{\bm r'};\varepsilon)=\sum_{\kappa m}Y_{\kappa m}(\theta,\phi)\frac{\mathcal{G}_{\kappa}(r,r';\varepsilon)}{rr'}Y_{\kappa m}^{*}(\theta',\phi'),$$ where $Y_{\kappa m}(\theta,\phi)$ is the spin spherical harmonic, $\mathcal{G}_{\kappa}(r,r';\varepsilon)$ denotes the radial Green’s function, and the quantum number $\kappa$ labeling different partial waves, which can give the values of the angular momentums $l$ and $j$, $$\left\{
\begin{array}{ll}
l=\kappa,j=\kappa-\frac{1}{2}, & \hbox{if $\kappa>0$;} \\
l=-\kappa-1,j=-\kappa-\frac{1}{2}, & \hbox{if $\kappa<0$.}
\end{array}
\right.$$ Then the DOS for each partial wave $\kappa$ is $$\begin{aligned}
n_{\kappa}(\varepsilon)&=&-\frac{2j+1}{\pi}\int dr\mathrm{Im}\left[\mathcal{G}_{\kappa}^{(11)}(r,r;\varepsilon)\right.\nonumber\\
&&\left.+\mathcal{G}_{\kappa}^{(22)}(r,r;\varepsilon)\right].
\label{Eq:DOS_kappa}\end{aligned}$$ Practically, we do the integrals in Eq. (\[Eq:DOS\_kappa\]) in a finite box and obtain an approximate DOS $n_{\kappa}^{R}(\varepsilon)$ for a fixed $R_{\rm box}$.
Finally, a Green’s function $\mathcal{G}_{\kappa}(r,r';\varepsilon)$ with angular momentum $\kappa$ and complex single-particle energy $\varepsilon$ is constructed as [@PRB1992Tamura_45_3271]
\_(r,r’;)= &, \[EQ:RGF\]
where $\theta(r-r')$ is the step function, $\phi^{(1)}_{\kappa}(r,\varepsilon)$ and $\phi^{(2)}_{\kappa}(r,\varepsilon)$ are two linearly independent Dirac spinors
\^[(1)]{}\_(r,)=(
[c]{} g\^[(1)]{}\_(r,)\
f\^[(1)]{}\_(r,)
),\
\^[(2)]{}\_(r,)=(
[c]{} g\^[(2)]{}\_(r,)\
f\^[(2)]{}\_(r,)
),
obtained by the Runge-Kutta integrals in the whole $r$ space from the asymptotic behaviors of the Dirac spinors at $r\rightarrow0$ and $r\rightarrow\infty$, respectively, and $W_{\kappa}(\varepsilon)$ is the $r$-independent Wronskian funciton defined by $$W_{\kappa}(\varepsilon)=g^{(1)}_{\kappa}(r,\varepsilon)f^{(2)}_{\kappa}(r,\varepsilon)-g^{(2)}_{\kappa}(r,\varepsilon)f^{(1)}_{\kappa}(r,\varepsilon).$$
Exact asymptotic behaviors in the origin and in the infinity are taken for the Dirac spinor. In particular, it is regular at $r\rightarrow 0$ and satisfies $$\begin{aligned}
\phi^{(1)}_{\kappa}(r,\varepsilon)
&\longrightarrow& r\left(
\begin{array}{c}
j_l(k r) \\
\frac{\kappa}{|\kappa|}\frac{\varepsilon-V-S}{k}j_{\tilde{l}}(kr)\\
\end{array}
\right),
\label{Eq:behavior_r0}
\end{aligned}$$ where $\tilde{l}=l+(-1)^{j+l-1/2}$ is the angular momentum of the small component of the Dirac spinor, $k=\sqrt{(\varepsilon-V-S)(\varepsilon-V+S+2M)}$ is the single-particle momentum for all states, and the spherical Bessel function of the first kind $j_l(k r)$ satisfies $$j_l(k r)\longrightarrow \frac{(kr)^l}{(2l+1)!!},~~~\text{ when}~~r\rightarrow 0.$$
The Dirac spinor at $r\rightarrow\infty$ behaves exponentially decaying for the bound states while oscillating outgoing for the continuum, which can be written uniformly as, $$\begin{aligned}
\phi^{(2)}_{\kappa}(r,\varepsilon)
&\longrightarrow&\left(
\begin{array}{c}
rk h^{(1)}_l(k r) \\
\frac{\kappa}{|\kappa|}\frac{rk^2}{\varepsilon+2M}h^{(1)}_{\tilde{l}}(k r) \\
\end{array}
\right),
\label{Eq:behavior_rinf}\end{aligned}$$ with the single-particle momentum $k=\sqrt{\varepsilon(\varepsilon+2M)}$ and the spherical Hankel function of the first kind $h^{(1)}_l(k r)$.
Numerical details {#sec:Numer}
=================
In this work, to compare the results with those obtained by the previous GF calculations [@PRC2014TTSun_90_054321] and also those by CMR [@PRL2016Li_117_062502], CSM [@PRC2010JYGuo_82_034318], RSM [@PRC2008ZhangL_77_014312], and ACCC [@PRC2004ZhangSS_70_034308] methods, we take the same nucleus $^{120}$Sn as an example, and investigate the single-particle resonant states for neutrons by taking the GF method based on the RMF theory. The energies, widths, and the density distributions in coordinate space for resonant states are given and compared with other methods. Both PK1 [@PRC2004WHLong_69_034319] and NL3 [@NL3] parameters are taken in these RMF calculations.
The equations in the RMF-GF theory are solved in the coordinate space, with different space sizes $R_{\rm{box}}$ and a step of $dr=0.1$ fm. In Eq. (\[EQ:Sdos\]), the infinitesimal imaginary parameter $\epsilon$ is taken as $1\times 10^{-6}$ MeV when calculating DOSs for bound states. In calculating the DOSs $n_{\kappa}^{R}(\varepsilon)$ by scanning energies $\varepsilon$ in the fourth quadrant of the complex energy plane, the energy steps $d\varepsilon$ is taken as $1\times10^{-4}$ MeV for both the real energy part and the imaginary energy part in searching for most of resonances. As a result, the predicted energies and widths of the resonant states by the GF method own an accuracy of $0.1$ keV. Besides, much higher accuracy can be easily achieved by taking smaller energy steps $d\varepsilon$.
Results and discussion {#sec:resu}
======================
![(Color online) (a) Single-particle complex energy plane $\varepsilon=\varepsilon_r+i\varepsilon_i$ and the single-neutron resonant state $2f_{5/2}$ in $^{120}$Sn located in the fourth quadrant. (b) DOSs $n_{\kappa}^{R}(\varepsilon)$ as functions of the complex energy $\varepsilon$ including the real part $\varepsilon_r$ and the imaginary part $\varepsilon_i$, obtained by the RMF-GF method by taking the PK1 effective interaction and space size $R_{\rm box}=20$ fm.[]{data-label="Fig1"}](Fig1.eps){width="45.00000%"}
The resonant states are well known as poles locating in the fourth quadrant of the single-particle complex energy plane. Therefore, in this work, we take a direct way to explore for these poles which are also the extremes of GF according to Eq. (\[EQ:GF\]). In practice, the definition of density of states in Eq. (\[Eq:DOSres\]) is applied and a series of DOSs $n_{\kappa}^{R}(\varepsilon)$ will be calculated by scanning the complex energy $\varepsilon$ in the fourth quadrant, both in the directions of the real energy $\varepsilon_{r}$ axis and the imaginary energy $\varepsilon_i$ axis.
![(Color online) Comparison of the DOSs $n_{\kappa}^{R}(\varepsilon)$ for the resonant state $2f_{5/2}$ obtained in different space sizes $R_{\mathrm{box}}=20$ fm (a), 25 fm (b), and 30 fm (c), respectively.[]{data-label="Fig2"}](Fig2.eps){width="45.00000%"}
As an example, in Fig. \[Fig1\], we give the details in determining the single-neutron resonant state $2f_{5/2}$ in $^{120}$Sn. To explore for the pole corresponding to the resonant state, as shown in Fig. \[Fig1\](a), the complex energy $\varepsilon$ in a wide range containing both the real part $\varepsilon_{r}$ and the imaginary part $\varepsilon_{i}$ are covered to calculate the DOSs. The PK1 effective interaction and the coordinate space of $R_{\rm box}=20~$fm are taken the RMF-GF calculations. In Fig. \[Fig1\](b), the calculated DOSs are plotted as functions of $\varepsilon_{r}$ for different $\varepsilon_i$. In particular, it is noted that with the imaginary energy $\varepsilon_i$ varying from $-0.0175~$MeV to $-0.0475~$MeV, the DOSs alter significantly in the energy range from $\varepsilon_r=0.75~$MeV to $1.00~$MeV. With the imaginary energy $\varepsilon_i$ approaching to $-0.0325$ MeV, the peaks of DOS evolves sharper and sharper and finally reaches the extreme. A peak in the shape of $\delta$-function locates at $\varepsilon_r=0.8705~$MeV. Besides, just after $\varepsilon_i$ crossing the energy $-0.0325$ MeV, the peak of DOS reverses sharply. After that, the peak of DOS evolves in an apposite way and becomes lower and lower with $\varepsilon_i$ going farther. This indicates a pole locating at $\varepsilon=0.8705-i0.0325~$MeV.
![(Color online) The same as Fig. \[Fig2\], but for the single-neutron resonant state $2g_{9/2}$ in $^{120}$Sn.[]{data-label="Fig3"}](Fig3.eps){width="45.00000%"}
In the following, we check the dependence of the obtained resonance energy and width $E-i\Gamma/2$ on the space size, as we know that they should be constant against the changes of the coordinate space size $R_{\rm box}$. In Fig. \[Fig2\], DOSs calculated by taking different coordinate space sizes $R_{\mathrm{box}}=20$ fm (a), $25$ fm (b), and $30$ fm (c) are plotted for the single-neutron resonant state $2f_{5/2}$ in $^{120}$Sn. It is noted that the shapes of DOSs for $2f_{5/2}$ in different $R_{\rm box}$ are quite similar and all of them reach the extreme at $\varepsilon_i=-0.0325$ MeV and reverse immediately at the next energy point $-0.0326$ MeV. Besides, the peak of DOS in each case also locates at the same energy $\varepsilon_r=0.8705~$MeV. Accordingly, we can conclude that the energy and width of the resonant state $2f_{5/2}$ obtained by the RMF-GF method is independent on the coordinate space size.
The same check plotted in Fig. \[Fig2\] is also performed for a wide resonant state. In Fig. \[Fig3\], the DOSs in different $R_{\mathrm{box}}$ are plotted for the resonant state $2g_{9/2}$ with a width around $3$ MeV. In general, for a wide resonant state, the DOS is more sensitive to the changes of imaginary part of complex energy $\varepsilon_i$. In Fig. \[Fig3\], although the DOSs do not have exactly the same shapes with the changes of the space size $R_{\rm box}$, extremes at the same energy $\varepsilon=5.4428-i1.6948$ MeV are observed, demonstrating that the same resonant state with energy $E=5.4428$ MeV and width $\Gamma/2=1.6948$ MeV is obtained in different space sizes. Combing the checks in Figs. \[Fig2\] and \[Fig3\], it is proved that the descriptions of resonate states by the new approach with the GF method keeps very stable with the changes of the space size, even for a resonant state with broad width.
![(Color online) Density distributions $\rho_{\kappa}(r,\varepsilon)$ for the single-neutron resonant state $2f_{5/2}$ at resonant energy $\varepsilon= 0.8705$ MeV plotted in the coordinate space. Calculations are done with different space sizes $R_{\mathrm{box}}=20$ fm, 25 fm, and 30 fm.[]{data-label="Fig4"}](Fig4.eps){width="45.00000%"}
Another advantage of GF method for resonant states is that it can also describe the density distributions in the coordinate space. Here, following Refs. [@PRC2009Oba_80_024301; @PRC2011ZhangY_83_054301; @PRC2012YZhang_86_054318], we use the density $\rho_{\kappa}(r,\varepsilon)$ defined at resonance energy $\varepsilon=E$ to describe the distribution for a resonant state in the coordinate space, which is calculated by $$\begin{aligned}
&&\rho_{\kappa}(r,\varepsilon)\\
&=&-\frac{(2j+1)}{4\pi r^2}\frac{1}{\pi}{\mathrm{Im}}\left[\mathcal{G}_{\kappa}^{(11)}(r,r;E)+\mathcal{G}_{\kappa}^{(22)}(r,r;E)\right].\nonumber\end{aligned}$$ In Fig. \[Fig4\], the density distribution $\rho_{\kappa}(r,\varepsilon)$ at the resonance energy $\varepsilon=0.8705~$MeV for the state $2f_{5/2}$ in $^{120}$Sn is shown. The space dependence is also checked by doing calculations in different box sizes $R_{\mathrm{box}}=20~$fm, $25~$fm, and $30~$fm. Exactly the same density distribution in the whole coordinate space is obtained with different space sizes, demonstrating again the advantage of GF method. Besides, we can see that the density distribution for the narrow $2f_{5/2}$ is very localized, behaving as a bound state.
![(Color online) Single-neutron resonant states in $^{120}$Sn obtained by the RMF-GF method with PK1 effective interaction.[]{data-label="Fig5"}](Fig5.eps){width="45.00000%"}
positive parity $2g_{7/2}$ $2g_{9/2}$ $1i_{11/2}$ $1i_{13/2}$
----------------- ------------------ ------------------ --------------------------- ------------------- ------------------- -------------------
this work $6.3585-i3.1052$ $5.4428-i1.6948$ $9.8544-i0.6413$ $3.4786-i0.0024$
previous work $9.700-i0.636 $ $3.469-i0.002 $
negative parity $3p_{1/2}$ $2f_{5/2}$ $1h_{9/2}$ $2h_{11/2}$ $1j_{13/2}$ $1j_{15/2}$
this work $0.0504-i0.0164$ $0.8705-i0.0325$ $0.2508-i4\times 10^{-8}$ $10.5130-i6.7681$ $18.1846-i3.1531$ $12.8929-i0.5322$
previous work $0.031-i0.043$ $0.887-i0.032$ $0.251-i0.0001$ $12.956-i0.688$
\[Tab1\]
According to the above studies, GF method is effective and reliable in describing resonant states, no matter it is narrow or broad. Resonance energies $E-i\Gamma/2$ can be easily obtained by searching for the poles of Green’s function or extremes of DOS. In Fig. \[Fig5\], we plot all the obtained single-neutron resonant states in $^{120}$Sn, identified by scanning the complex energy $\varepsilon$ in a wide range for different $\kappa$ blocks and searching for resonant states by observing extremes. Compared with the results in our previous work (see Fig.6 in Ref. [@PRC2014TTSun_90_054321]), in which the resonant states were identified by comparing the DOSs for nucleons moving in the mean field potentials with those for free particles, new resonant states $2g_{7/2}$, $2g_{9/2}$, $2h_{11/2}$, and $1j_{13/2}$ with very broad widths ranging from $3$ MeV to $13$ MeV are also observed. In Table \[Tab1\], we list the energies $E-i\Gamma/2$ of the single-neutron resonant states obtained by RMF-GF method, in comparison with the results in the previous GF calculations [@PRC2014TTSun_90_054321]. It is found that the accuracy is highly improved with the new approach by GF method, both for the narrow resonant states and broad ones. For example, the uncertainty is well constrained within $1.0\times 10^{-8}$ MeV for the extremely narrow resonant state $h_{9/2}$. It is noted that for the very narrow resonant states, the scanning energy step $d\varepsilon$ for the imaginary part in calculating DOSs $n(\varepsilon)$ should be much smaller. It is $1\times 10^{-8}$ MeV for the resonance $h_{9/2}$, and only with such a small imaginary energy step, the reverse of DOSs extremes can be observed.
$nl_{j}$ GF CMR [@PRL2016Li_117_062502] CSM [@PRC2010JYGuo_82_034318] RSM [@PRC2008ZhangL_77_014312] ACCC [@PRC2004ZhangSS_70_034308]
------------- ----------------- ----------------------------- ------------------------------- -------------------------------- ----------------------------------
$2f_{5/2}$ $ 0.674-i0.015$ $ 0.678-i0.015$ $ 0.670-i0.010$ $ 0.674-i0.015$ $ 0.685-i0.012$
$1i_{13/2}$ $ 3.263-i0.002$ $ 3.267-i0.002$ $ 3.266-i0.002$ $ 3.266-i0.002$ $ 3.262-i0.002$
$1i_{11/2}$ $ 9.601-i0.607$ $ 9.607-i0.608$ $ 9.597-i0.606$ $ 9.559-i0.602$ $ 9.600-i0.555$
$1j_{15/2}$ $12.579-i0.496$ $12.584-i0.496$ $12.577-i0.496$ $12.564-i0.486$ $12.600-i0.450$
\[Tab2\]
Finally, to compare our results with those by CMR [@PRL2016Li_117_062502], CSM [@PRC2010JYGuo_82_034318], RSM [@PRC2008ZhangL_77_014312], and ACCC [@PRC2004ZhangSS_70_034308] methods, we also calculate the resonant states with the RMF-GF method by taking NL3 [@NL3] effective interaction. The energies $E-i\Gamma/2$ for the single-neutron resonant states $2f_{5/2}$, $1i_{11/2}$, $1i_{13/2}$, and $1j_{15/2}$ by those methods are listed in Table \[Tab2\]. We can find that the results by the GF method are all consistent with those of other four methods, especially the CMR and CSM methods. In fact, according to our previous study for the resonances in deformed nuclei [@PRC2020Sun_101_014321], it was found that GF method and CMR can obtain exactly the same energies for most of the resonant states. One possible reason for the slight difference in the present results may come from the mean-field potential obtained in the iteration calculations of RMF theory. Besides, the GF and CMR methods are performed very differently. The GF method is worked in the coordinate space, in which the resonant states are obtained by searching for its poles corresponding to the eigenvalues of the Dirac equation. However, the CMR method is implemented in the momentum space by diagonalizing the Dirac Hamiltonian.
Summary {#sec:Sum}
=======
It is well known that the single-particle resonances are playing crucial roles in the structures of exotic nuclei. Many methods such as CMR, CSM, RSM, and ACCC has been proposed to study resonant states. In this work, we apply the Green’s function method to study the single-particle resonances based on the RMF theory. Instead of searching for resonant states by comparing the density of states for nucleons in the mean field potentials with those for free particles, a direct and effective approach, that is to search for the extremes of the density of states or the poles of Green’s function, is implemented for all kinds of resonant states, either narrow or broad.
Taking $^{120}$Sn as an example, the resonant states are studied with RMF-GF method by taking PK1 effective interaction. The obtained energies and widths are very stable with the change of coordinate space size. The density distributions for resonant states can also be plotted. Compared with our previous work [@PRC2014TTSun_90_054321], new resonant states $2g_{7/2}$, $2g_{9/2}$, $2h_{11/2}$, and $1j_{13/2}$ with broad widths are identified. Furthermore, the accuracy for the very narrow resonant state $h_{9/2}$ is improved to be $1\times 10^{-8}$MeV. Besides, to compare our results with those by CMR, CSM, RSM and ACCC methods, calculations for $^{120}$Sn by taking NL3 parameter are also done. It is found that results by Green’s function method are very close to those by CMR and CSM although they are very different methods.
This work was partly supported by the Physics Research and Development Program of Zhengzhou University (Grant No. 32410217) and the National Natural Science Foundation of China (Grant Nos. 11505157 and 11875225).
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|
-1.5cm +0.2cm
\
Jin He\
Department of Physics, Huazhong University of Science and Technology,\
Wuhan, Hubei 430074, China\
E-mail:mathnob@yahoo.com\
Gravity whose nature is fundamental to the understanding of solar system, galaxies and the structure and evolution of the Universe, is theorized by the assumption of curved spacetime, according to Einstein$^,$s general theory of relativity (EGR). Particles which experience gravity only, move on curved spacetime along straight lines (geodesics). The geodesics are determined by curved-spacetime metric. In the last year, I proposed the mirrored version of EGR, the flat-spacetime general relativity (FGR), in which particles move along curved lines on flat spacetime. This puts gravitational study back to the traditional Lagrangian formulation. The Lagragian on flat spacetime is simply taken to be the curved spacetime metric of EGR. In fact, all acclaimed accurate verification of general relativity is the verification of FGR, because relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in Schwarzschild solution or in post Newtonian formulation. For example, two famous tests of general relativity are about angles. All mainstream textbooks and papers calculate the angles by directly using the coordinate $ \phi $. However, only when spacetime is flat does there exist one coordinate system which has direct meaning of time, distance, angle, and [*vice verse*]{}. This is the famous Riemann theorem when he pioneered the concept of curved space. According to the theorem, all coordinates on a curved space are merely parameters. Real angles and distances have to be calculated by employing the coefficients of spatial metric. If we do follow the geometry of curved spacetime (EGR) then the deflection of light at the limb of the sun is 1.65 arcseconds (Crothers, 2005). The publicly cited value (1.75 arcseconds) which best fits observational data is predicted by FGR. Therefore, the more claims are made that classical tests of general relativity fit data with great accuracy, the more falsified is the curved-spacetime assumption. That is, the claim is specious to EGR. Relativists made three specious claims as collected in the present paper. However, FGR predicts observationally verified results for solar system, galaxies, and the universe on the whole. Because FGR uses the single consistent Lagrangian principle, it is straightforward to show that the possibility of curved spacetime, black holes, and big bang is less than one in billion. An experiment is proposed whose results will completely decide the fate of curved spacetime assumption. Dare to make public the recent result of new Brillet and Hall experiment with one vertical light beam? Note that the original article “Einstein‘s Geometrization vs. Holonomic Cancellation of Gravity via Spatial Coordinate-rescale and Nonholonomic Cancellation via Spacetime Boost“ is attached.\
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Special Relativity — General Relativity -–- Classical Tests\
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[**Flat-spacetime Lagrangian vs. Curved spacetime** ]{}\
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People found four physical interactions (i.e., forces): gravitational, electromagnetic, weak, and strong ones. Gravity, the weakest one, is fundamental to the understanding of solar system, galaxies and the structure and evolution of the Universe. By now, people have suggested two fundamental principles which are used to construct physical theories. One is flat-spacetime Lagrangian principle and the other is curved-spacetime assumption. Mainstream gravitational theory falls into the latter category (Einstein$^,$s general relativity, EGR) while all other interactions are unified by the former principle. A flat spacetime means a global inertial frame. Therefore, Einstein denied the existence of global inertial frames and denied the possibility that our universe might provide the unique inertial frame. If Einstein is wrong then all theories of black holes and big bang are wrong, and the theory of gravity simply returns to normal Lagrangian formulation and all four interactions are hopeful to be unified.
Any theory must be testified. The curved-spacetime assumption, however, is not verified. All hailed accurate verification of general relativity is in fact the verification of flat spacetime, because relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in Schwarzschild metric or in post Newtonian formulation. For example, two famous tests of general relativity are about angles. All mainstream textbooks and papers calculate the angles by directly using the coordinate $ \phi $. However, Riemann proved the famous theorem: only when spacetime is flat does there exist one coordinate system which has direct meaning of time and distance, and [*vice verse*]{}. That is, all coordinates on a curved spacetime are merely parameters. Real angles and distances have to be calculated by employing the coefficients of spatial metric. If we do follow the geometry of curved spacetime (EGR) then the deflection of light at the limb of the sun is 1.65 arcseconds (Crothers, 2005). The publicly cited value (1.75 arcseconds) which best fits observational data is based on direct coordinate calculation. Therefore, the more claims are made that direct coordinate calculation fits data very well, the more falsified is the curved-spacetime assumption. That is, the claim is specious to EGR. Relativists made three specious claims as collected in the present paper.
Now, I review the flat-spacetime interpretation of general relativity (FGR) which I proposed in the last year. Firstly, I introduce the concept of Newton$^,$s as well as Einstein$^,$s inertial frames. In Section 2, Einsten$^,$s equivalence principle is demonstrated to be false, the principle being the second specious claim made by relativists. The principle is the only excuse for Einstein to suggest curved spacetime. [**In the final part of the Section, I show that the possibility is less than one in billion that the assumptions of curved spacetime, black holes, and big bang are true.**]{} Section 3 presents the third specious claim made by relativists and proposes an experiment whose result will completely decide the fate of curved-spacetime assumption. The final Section is conclusion.
[**(i) Newton$^,$s Inertial Frames and his First Law of Motion.** ]{} An inertial frame (flat spacetime) is the one in which the particles which do not suffer any net force are either static or moving in straight lines at constant speeds relative to the frame. This is also called Newton$^,$s first law of motion which can be formulated by an optimization principle (Lagrangian principle). The required Lagrangian per unit mass for the particles is the following $$L\left(\frac{dx ^i}{dt}\right)
=\frac{1}{2} \left(\left(\frac{dx }{dt}\right )^2 + \left(\frac{dy }{dt}\right )^2 + \left(\frac{dz }{dt}\right )^2 \right)$$ $$(1)$$ where $(x^1=x, x^2=y, x^3=z)$ is the Cartesian rectangular coordinate in the inertial frame. The Lagrangian is the kinetic energy per unit mass of particles.
Now we discuss in inertial frame the motion of particles which do suffer gravity. Firstly, we pay attention to the gravity due to single large point mass $M$ which sits at coordinate origin and is static relative to the inertial frame. Particles no longer move in straight lines at constant speeds and the simple Lagrangian (1) can not describe the motion under gravity. According to Newton theory, the required Lagrangian is the following, $$L\left(x^i, \frac{dx ^i}{dt}\right)
=\frac{1}{2} \left( \left(\frac{dx }{dt}\right )^2 + \left(\frac{dy }{dt}\right )^2 + \left(\frac{dz }{dt}\right )^2 \right) + \frac{GM}{r}$$ $$(2)$$ where $G$ is the gravitational constant and $r^2=x^2+y^2+z^2$. Note that spacetime is flat and particles$^,$ motion are curved lines on the flat Euclidean spacetime. Newton theory explains the solar system very well with very little error.
[**(ii) The Difficulty in Newton$^,$s Concept of Inertial Frames.** ]{} The coordinates $t,x,y,z$ are mathematical numbers. How are these numbers achieved? Newton did not give much discussion. He required that the number of time were the same for all inertial frames, the universal (absolute) time. In later nineteen century, this assumption led to difficulty in the explanation of light speed. Let us consider two inertial frames, one moving at constant speed $v_0$ with respect to the other. The universal time assumption leads to the conclusion that a light beam at a speed $c_0$ observed by the first frame will have a speed $c_0\pm v_0$ observed by the second one if both the light beam and the frame move in parallel directions. For example, earth$^,$s orbital speed around the sun is $v_0 \approx $ 30km/s. Therefore, if we assume that the speed of light observed by solar frame is $c_0$ in the same orbital direction, then the light speed observed by earth frame is $c_0 \pm v_0$. The difference of the two speeds is $$\Delta c = v_0 \approx 30\,{\rm km/s}.$$ $$(3)$$ Such large difference of light speeds is never observed. Therefore, Newton$^,$s concept of inertial frames must be corrected. Einstein assumes that light speed is the same for all inertial frames.
[**(iii) Einstein$^,$s Special Relativity (SR) — The Innovative Concept of Inertial Frames.** ]{} Therefore, the assumption of single universal time must be abandoned. Time is given by clocks which themselves are physical processes and the physical processes (the clocks) at all places of the frame are static relative to the frame itself. Similarly the rulers which are used to measure spatial distances are static with respect to the frame too. Therefore, the clocks belonging to one inertial frame have relative motion with respect to the clocks belonging to the other one. Therefore, one will find out that the timing of one$^,$s clocks is different from the timing observed by oneself of the clocks in other frame. Timing and spatial length of a physical process are not universal. If we talk about a time, we need to say by which inertial frame it is given. Therefore, according to Einstein$^,$s special relativity, we have different universal time given by different inertial-frames, instead of single universal time.
Einstein initiated the special theory of relativity (SR), the new concept of inertial frames which assumed universal value of light speed, instead of universal time. That is, light speed is the same for all inertial frames. Its universal value is $$c_0= \frac{dx}{dt}= 299,792,458\; {\rm m/s}.$$ $$(4)$$ This is a theoretical value. Modern technique can measure light speed to the accuracy of decimeters per second directly. However, modern technique can measure the difference of light-speeds of two light beams to the accuracy of $10^{-6}$ meters per second. The formula (4) indicates that $$-c^2_0 dt^2+dx^2=0$$ $$(5)$$ which suggests a different “Pythagoras theorem$^{,,}$ $$ds^2=-c_0^2 dt^2+dx^2+dy^2+dz^2.$$ $$(6)$$ This is called Minkowski indefinite metric on flat Minkowski spacetime, which is the basis of SR. Now we need to determine the Lagrangian which describes the motion of particles which do not suffer any force (interaction) in Einstein$^,$s inertial frame.
The Lagrangian and the light speed are both invariant quantities. Therefore, they must be connected. This is suggested by the above formula (6), $$\begin{array}{l}
L\left(x^0, x^i, c_0\frac{dt}{dp}, \frac{dx ^i}{dp}\right) \\
=\frac{1}{2} \left( -c_0^2\left(\frac{dt}{dp}\right )^2+ \left(\frac{dx }{dp}\right )^2 + \left(\frac{dy }{dp}\right )^2 + \left(\frac{dz }{dp}\right )^2 \right)
\end{array}$$ $$(7)$$ where $x^0=c_0t$, and $p$ is the curve parameter of particle$^,$s motion. The Lagrangian is called the Lagrangian per unit mass because it does not involve the quantity of mass.
Because light sets an upper limit for all particles$^,$ speeds, the values of the Lagrangian are not arbitrary. Because we always deal with causal motion, we have $ds^2 \le 0 $, i.e., $$L\le 0.$$ $$(8)$$
Now we derive the Hamiltonian per unit mass for the Lagrangian (7). The momentums per unit mass canonical to $x^\alpha , \alpha =0,1,2,3$ are the following, $$\begin{array}{l}
P_0 =\frac { \partial }{\partial (dx^0/dp)} L=-\frac { c_0d t }{dp } \\
P_i =\frac { \partial }{\partial (dx^i/dp)} L=\frac { dx^i }{dp },
\mbox{ }i=1,2,3.
\end{array}$$ $$(9)$$ Because the Lagrangian does not depend on time and position coordinates, the momentums are constants, which indicates that the motion of particles governed by the Lagrangian is the one in straight lines at constant speeds, $$t=a_0p, \; x=a_1p, \;y=a_2p, \;z=a_3p$$ $$(10)$$ where $a_\alpha , \alpha =0,1,2,3$ are constants.
The Hamiltonian of test particles is $$\begin{array}{ll}
H &= \frac { dx^0}{dp} P_0 +\sum^3_{i=1}\frac { dx^i}{dp} P_i –- L \\
&= -\frac{1}{2} c_0^2\left (\frac {dt }{dp}\right )^2 +\frac{1}{2} \sum ^3_{i=1}\left (\frac {dx^i }{dp}\right )^2\\
&\equiv L.
\end{array}$$ $$(11)$$ If we choose $a_0 =1$ in (10) then the Hamiltonian (total energy) is $$H =E=L=-\frac{1}{2} c_0^2+ \frac{1}{2} \sum ^3_{i=1}\left (\frac {dx^i }{dt}\right )^2.$$ $$(12)$$ Because $E=0$ corresponds to photon$^,$s motion, the formula indicates that light speed is $c_0$ as we expect. We see that the spatial part of the Hamiltonian corresponds to kinetic energy while the temporal part corresponds to potential energy. Both energies are constants. The potential energy is $-c_0^2/2$ which is chosen to be zero in Newtonian theory.
Einstein$^,$s SR further requires that coordinate transformations between inertial frames are the Lorentz ones and the formulation of all physical laws must be covariant with respect to the transformations. Einstein$^,$s SR is verified by many experiments and is the basis of my FGR.
[**(iv) Under which Condition is Einstein$^,$s Special Relativity (SR) True?** ]{} Einstein$^,$s special relativity is not true in real condition of local universe. SR is actually the concept of global inertial frames and describes the property of spatially and temporally homogeneous world. It is very important to know that SR would be perfectly and globally true if the matter content of the universe were both spatially and temporally homogeneous. However, it is a fact that the universe is evolving (temporally in-homogeneous). Fortunately, large-scale spatial homogeneity of the universe is observationally proved. Therefore, SR is globally true for any short period of time of the large-scale universe.
The local in-homogeneous distribution of matter of the universe introduces local spatial in-homogeneity, which is the subject of Newtonian gravity and Einstein$^,$s general relativity. In this case, special relativity must break down. Especially, light speed is anisotropic (not constant). Because gravity is the weakest interaction, the anisotropy of light speed is hard to detect.
[**(v) Einstein$^,$s General Theory of Relativity (EGR).** ]{} Einstein$^,$s SR (7) (or (6)) is the innovated version of Newton$^,$s concept of inertial frames (1). The Lagrangian (2) generalizes (1) to deal with particles$^,$ motion under gravity. Einstein$^,$s general relativity (EGR) which deals with gravity too, does not generalize his SR. SR describes the full property of homogeneous spacetime while gravity introduces inhomogeneity on spacetime. Therefore, SR must more or less break down in any theory of gravity. Einstein chose to stake at the assumption that SR is perfectly true in any infinitesimal area of spacetime. Accordingly SR can not be perfectly true in any finite area of spacetime. Otherwise, the corresponding Lagrangian would be (7) and no gravity would be present in the area. The unavoidable consequence of Einstein$^,$s choice is that spacetime must curve. Therefore, Einstein gave up the global flat Minkowski spacetime by introducing the curved spacetime whose curvature is gravity, and he abandoned Lorentz coordinate transformations by considering all curvilinear coordinate transformations on the curved spacetime.
Einstein$^,$s assumption of curved-spacetime brings more complexity than truth. Firstly, curved spacetime is embodied by non-trivial topology. Because topology is a very complicated mathematical subject, most relativists never take a look at it. Secondly, the concept of curved spacetime is nothing but temporal and spatial in-homogeneity. Therefore, all coordinates on a curved space are merely parameters. Real time and distance have to be calculated by employing coefficients of the spacetime metric. The calculation of time and distance by employing metric is very complicated too. Therefore, all relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in Schwarzschild solution or in post Newtonian formulation. However, there is the famous Riemann theorem: only when spacetime is flat does there exist one coordinate system which has direct meaning of time and distance, and [*vice verse*]{}. Therefore, the hailed accurate tests of GR verified the flat-spacetime interpretation of GR (my FGR). The more claims are made that classical tests of general relativity fit data with great accuracy, the more falsified is the curved-spacetime assumption. That is, the claim is specious to EGR.
[**(vi) Flat-spacetime Interpretation of Schwarzschild Metric.** ]{} In the last year, I proposed the mirrored version of EGR, the flat-spacetime general relativity (FGR) in which particles move along curved lines on flat spacetime. This puts gravitational study back to traditional Lagrangian formulation. For example, Schwarzschild metric of single point mass is, $$\frac {1}{2}\left( \frac{d s}{dp}\right )^2 = L\left(x^0, x^i, \frac{dx ^0}{dp} , \frac{dx ^i}{dp}\right) = -\frac{1}{2} B(r ) \left(\frac{c_0dt}{dp}\right)^2
+ \frac{1}{2}A(r ) \left(\frac{dr}{dp}\right)^2 +\frac{1}{2}r^2 \left(\frac{d\phi}{dp}\right)^2 ,$$ $$(13)$$ where $$B(r )=1-\frac{2r_g}{r}, \;\; A(r )=\frac{1}{B(r )}=\frac{ 1}{1-2r_g/r }$$ $$(14)$$ and the constant $$r_g=GM/c_0^2$$ $$(15)$$ is the Schwarzschild radius. The Schwarzschild metric (13) on curved spacetime is simply taken to be the Lagragian on flat spacetime. Although the background spacetime is flat and Cartesian coordinates have direct meaning of time and distance, one of the fundamental assumptions of SR breaks down globally. That is, light speed varies with spatial position and spatial direction as indicated in the following. However, it is still the maximum one at each position and in each direction. The Lagrangian (13) is the generalization to the one of no-interaction (7). I call this type of Lagrangian by homogeneous Lagrangian because it is a homogeneous order-two form of the components of the generalized particle velocity. The value of the Lagrangian is negative so that it describes causal motions of material particles. It can be zero and describes the motion of light.
According to the optimization principle, test particle$^,$s motion follows the corresponding Lagrange$^,$s equations. The solution of the Lagrange$^,$s equations is $$\frac {dt }{dp}=\frac{1}{ B( r)},$$ $$(16)$$ $$r^2\frac {d\phi }{dp}=J \:({\rm constant}),$$ $$(17)$$ $$\frac{1}{2}\left (\frac{A( r)}{B^2(r )} \left (\frac {dr }{d t}\right )^2+\frac {J^2 }{ r^2}-\frac {c_0^2}{B( r)}\right )=\tilde E \:({\rm constant}).$$ $$(18)$$ where $J$ and $\tilde E$ are particle$^,$s angular momentum and energy per unit mass respectively. The Newtonian approximation of the formula (18) is $$\frac{1}{2}\left (\frac {dr }{d t}\right )^2+\frac {J^2 }{2 r^2} - \frac {GM}{r} –- \frac{c_0^2}{2}=E- \frac{c_0^2}{2} \approx \tilde E.$$ $$(19)$$ Therefore, $\tilde E$ differs from $E$ in Newtonian gravity by a constant $c_0^2/2$.
The formulas (16), (17) and (18) are exactly the geodesic equations of EGR. According to EGR, spacetime is curved and all the coordinates $t, r, \phi $ in (13) do not have the direct meaning of time, distance, angle. In FGR, however, they do have, because spacetime is flat. Ironically, relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in (13). Therefore, all hailed accurate verification of general relativity is in fact the verification of FGR.
Because $L\equiv H$, the upper limit of $\tilde E$, the energy per unit mass, is $$\tilde E_{{\rm max}} =0.$$ $$(20)$$ For simplicity, I consider only the radial motion of particles with respect to the central mass. That is, they do not have angular momentum $J$, $$J=0$$ $$(21)$$ Choosing $J=0$ and $dr/d t=0$ in (18), we have the lower limit of energy $\tilde E$, $$\tilde E_{ {\rm min }}=-\frac{c_0^2}{2(1-2r_g/r)}.$$ $$(22)$$ Choosing $J=0$ and $\tilde E=0$ in (18) we have varying speed of radial light beam, $$c(r )=c_0\left (1-\frac{2r_g}{r}\right ),$$ $$(23)$$ Choosing $J=0$ and taking derivatives with $ t$ on the two sides of (18), we have the formula of radial acceleration for both light and material particles, $$a(r )=\frac {d^2r}{dt^2}= \frac{2r_g}{r^2} \left (1-\frac{2r_g}{r}\right )
\left (c_0^2+3\left (1-\frac{2r_g}{r}\right ) \tilde E \right ) .$$ $$(24)$$ Zero energy ($ \tilde E =0$) corresponds to the motion of light. Taking $\tilde E =0$ in (24), we have the radial acceleration of light, $$a_{ {\rm max }}=\frac {d^2r}{dt^2}= \frac{2r_g}{r^2} \left (1-\frac{2r_g}{r}\right )
c_0^2 >0.$$ $$(25)$$ Because the radial acceleration is positive, light decelerates toward the central mass. Therefore, light suffers repulsive force from the mass, contrary to people$^,$s imagination. This result of light deceleration is verified by the radar-echo-delay experiments (Shapiro, 1968 and 1971) and other similar experiments.
Substituting the lower limit of energy (22) into (24), we have the lower acceleration limit for material particles $$a_{ {\rm min }}=\frac {d^2r}{dt^2}=- \frac{r_g}{r^2} \left (1-\frac{2r_g}{r}\right )c_0^2 <0$$ $$(26)$$ which is negative and corresponds to positive acceleration to the mass center. Therefore, low energy bodies do suffer attracting force (gravity!).
For the earth, $r_g=4.43 \times 10^{-3} $m. Near the earth surface, the formula of radial acceleration (24) can be approximated as $$a(r_e )= \frac {d^2r}{dt^2}=2g+\frac{6g}{c_0^2}\tilde E$$ $$(27)$$ where $r_e$ is the earth radius and $g$ is the absolute value of the familiar acceleration $$g\approx 9.8 \;{\rm m\,s}^{-2}.$$ $$(28)$$ From the formula (27) we know that material particles of radial motion suffer no gravity if their energy per unit mass reaches $-c_0^2/3$. That is, particles$^,$ speed approaches light speed: $v\approx c/\sqrt{3}$. From the formula (27), the minimum acceleration near earth surface is $$a_{ {\rm min }} \approx -\frac {r_g}{r^2}c^2= -g$$ $$(29)$$ which is what Einstein thought to be the constant acceleration for all test particles of whatever energy near the earth surface. Einstein generalized this false thought as the equivalence principle and suggested the geometrization of gravity, the curved spacetime assumption. I return to the discussion in the next Section.
[**(vii) Flat-spacetime General Relativity (FGR).** ]{} Special relativity describes the properties of global inertial frames in which the distribution of matter is both spatially and temporally homogeneous. Light speed is constant in all inertial frames and the formulation of physical laws is covariant with Lorentz transformations between the inertial frames. Both EGR and FGR introduce in-homogeneity into spacetime. EGR assumes curved spacetime and does not have global inertial frames. Its formulation is covariant with all possible curvilinear coordinate transformations. My FGR is based on flat spacetime. Is it true that the formulation of FGR is only covariant with all Lorentz transformations? The answer is no, which is explained in the following.
FGR maintains what is successfully testified in EGR. Einstein field equation which connects curvature tensor to matter, is a tensor equation and fits solar observation with great accuracy. The important example is the post Newtonian formulation of the equation. However, relativists when fitting the formulation to data, calculate time, distance, or angle by directly using the coordinates. Therefore, relativists verified flat-spacetime according to Riemann theorem and the curvature tensor does not describe the curved spacetime at all. Therefore, FGR maintains all formal tensor calculus and keeps Einstein field equation. Because spacetime is flat, all tensors including the curvature one do not describe curved spacetime at all. For example, the covariant derivative $$\frac {\partial V^\mu}{\partial x^\lambda }+ \sum_\nu \Gamma ^\mu_{\lambda \nu}V^\nu$$ $$(30)$$ has no geometric meaning. This idea of flat-spacetime tensor calculus is not my invention. It is employed in the fluctuation theory of thermal physics many years ago. Therefore, the answer to the above question is no and all physical law must be covariant with all coordinate transformations between real reference frames (generally they are freely falling frames). If $x^\alpha , \;\alpha =0,1,2,3$ are rectangular coordinates on the flat spacetime then the following absolute derivatives $$\frac {\partial }{\partial x^\alpha }, \;\;\alpha =0,1,2,3$$ $$(31)$$ are not a covariant vector in FGR.
The covariance with all coordinate transformations between real reference frames provides the dynamical calculation of gravity in FGR. For example, all familiar global inertial frames are actually approximate ones. The static frame on earth which is considered to be inertial frame in civil building design is an approximate one with respect to the solar frame. Therefore, the rectangular coordinate in the earth frame is actually a curvilinear one in the solar frame. Therefore, covariant transformation of the curvilinear coordinates into the solar rectangular coordinates gives more accurate calculation of earth problem.
[**(viii) Freely-Falling Frames in Flat-spacetime Gravity (FGR).** ]{} The next Section shows that there is no local common acceleration for all test particles which cancels gravity as suggested by Einstein. In my flat-spacetime theory of gravity (FGR), the freely-falling frames with their coordinate axes being parallel transported according to the non-geometric connection, cancel gravity incompletely but mostly and globally. And the degree of cancellation differs with different freely-falling frames. Therefore, we people on earth frame can not feel the gravity from sun mostly. We can feel the gravity from the earth completely because we are not in the freely-falling frame with respect to earth$^,$s gravity.
The next Section proves the existence of the unique inertial frame of the universe. Stars are in freely-falling frames with respect to galactic gravity, and the reference frames of stars can not detect much gravity from galaxies. Sun is such a star, and all our astronomic observation is based on the sun frame. [**That is why we see that all electro-magnetic waves from the universe demonstrate approximately the same physics!**]{}\
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[**FGR Provides a Consistent Explanation to Solar System and Galaxies and the Universe** ]{}\
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Gravity is fundamental to the understanding of solar system, galaxies and the structure and evolution of the Universe. The curved-spacetime theory of gravity, EGR, encounters many difficulties in the explanation of large-scale systems (galaxies and the universe) and always resorts to dark matter and/or dark energy. Relativists claimed accurate tests of general relativity (GR) in the solar system. For example, the experiment of Gravity Probe B (GPB) claims the unprecedented accurate measurement and its result will be released within half a year. However, if GPB claims almost null error of GR prediction then the curved-spacetime assumption (EGR) is wrong and FGR is confirmed. This is because relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in Schwarzschild solution or in post Newtonian formulation or in the calculation of gravitational waves, and relativists are actually assuming flat spacetime according to Riemann$^,$s theorem. Therefore, these claims are specious to EGR. In the following, Einstein$^,$s equivalence principle is shown to be completely false and serves as the second specious claim to EGR. Then I review the simple and consistent FGR explanation to galaxies and the universe, and finally I show that the possibility is less than one in billion that the assumptions of curved spacetime, black holes, and the big bang are true.
[**(i) Einstein$^,$s Equivalence Principle is False.** ]{} Einstein$^,$s equivalence principle is that, over any small region of space and time, all test particles move at approximately the same acceleration. Therefore, the observational frame which moves at the very acceleration will see each particle being either static or moving in straight lines at constant speeds, within the small region in question. That is, the local frame sees no gravity at all and we see a cancellation of gravity by choosing local frames, which are generally called the local freely-falling frames. Einstein thought that the local frames were the local tangent 4-dimensional planes to the curved spacetime. This mistake led to the assumption of curved spacetime and resulted in ninety years$^,$ dogmatic study of gravity and cosmology: black holes, big bang, inflation, etc. I make two points to prove Einstein$^,$s mistake.
Firstly, a tangent plane is an inertial frame in which particles move in straight lines at constant speeds. Different particles may have different speeds but their acceleration must be zero. Speed is the first derivative of distance coordinate with time coordinate while acceleration is the second derivative. Tangent plane to curved space is determined only by the first derivatives not the second derivatives. How did Einstein make such simple mistake?
Secondly, Einstein made further mistake and assumed that all local particles shared the same acceleration independent of their individual properties, that is, independent of their energy per unit mass and their angular momentum per unit mass. Energy and angular momentum have totally four degrees of freedom and Einstein required that locally particles have zero degree of freedom: sharing the same acceleration. Based on FGR (see part (vi) of Section 1), however, I proved that particles must have different local accelerations corresponding to their angular momentum and energy. If angular momentum is zero (radial motion), the formula of acceleration is (24) which depends on energy. Only when their angular momentum is zero and their energy is the minimum will the test particles share the same acceleration. In the case of earth surface, the shared acceleration is $9.8$ m/s$^2$. If the energy of test particles were high enough then the leaning tower experiment of Galileo Galilei would have demonstrated opposite result. Einstein thought the result of Galileo Galilei were universal truth and generalized this false result as his equivalence principle.
Therefore, there is no local common acceleration which cancel gravity locally, and Einstein$^,$s equivalence principle is completely false and serves as the second specious claim to EGR. However, a great many arguments of dogmatic gravitational theory and its applications and the theory of big bang are based on the non-existent freely-falling frames which cancel local gravity.
In my flat-spacetime theory of gravity (FGR), the freely-falling frames with their coordinate axes being parallel transported according to the non-geometric connection, cancel gravity incompletely but mostly and globally. And the degree of cancellation differs with different freely-falling frames. Therefore, we people on earth frame can not feel the gravity from sun mostly. We can feel the gravity from the earth completely because we are not in the freely-falling frame with respect to earth$^,$s gravity.
[**(ii) EGR can not Explain Galactic Phenomena while FGR can.** ]{} Solar system has the overwhelming mass point, the sun, and is generally considered to be a two-body problem. Einstein field equation can reduce to Newtonian gravity and proved to be useful in the two-body problem. However, Einstein field equation encounters many difficulties when applied to many-body systems like star clusters, galaxies, and the universe. Relativists try hard in such applications because they believe that spacetime is really curved and has the curvature tensor contained in Einstein field equation. Therefore, EGR assumes that all many-body systems of whatever shapes and scales, demonstrate the same attractive two-body phenomenon described by Newtonian gravity. For example, EGR requires that galactic rotational curves be falling in radial direction from spiral galaxy centers. However, real observational data shows the opposite rising curves. In addition, EGR has no idea about why spiral galaxies have 2-dimensional disks.
Because of modern powerful observational technique, galaxies have many established facts. The facts can be explained by my FGR. For example, the radial surface-brightness profile of spiral galaxy disks obeys exponential law. The spiral arms of spiral galaxies are curved waves in logarithmic curvature. FGR consistently explained the laws as well as the rising rotational curves (He, 2005a and 2005b). Based on FGR, stars in the 3-dimensional elliptical galaxies and in the 3-dimensional central bulges of spiral galaxies suffer attractive force towards their centers while the stars in the 2-dimensional disks of spiral galaxies suffer radial repulsive force.
[**(iii) Flat-spacetime Model of the Universe Resolves the Difficulties Encountered by Big Bang Theory.** ]{} We know that earth, sun, or a galaxy are all approximate inertial frames. Does the whole universe provide the unique accurate inertial frame? This frame is meaningful only if all galaxies slow down their motion and try to reach the final static spatial positions on the frame. The existence of the unique inertial frame (e1: event 1) is proved based on my flat-spacetime model of the universe (He, 2006b). The model employs just one simple cosmological principle. The model not only explains galactic redshifts (e2) and Hubble redshift-distance law (e3) but also predicts a decreasing speed of light with time (e4) and the accelerating universe (e5). People have shown that decreasing speed of light resolves the difficulties encountered by big bang theory (BBT) (see Magueijo (2003)). The difficulties are horizon problem, flatness problem, etc. My model does not need dark matter and dark energy. Now I present the simple principle and my model (He, 2006b).
[*Homogeneous yet evolving universe on flat spacetime (the cosmological principle).*]{} In the first half of last century, our knowledge of the universe was very limited and all models of the universe were mainly based on assumptions. Among the models was the big bang theory (BBT) which was based on curved spacetime assumption and became dogmatic. Now, cosmological study becomes an observational science and astronomical data does indicate that the large-scale universe is spatially homogeneous. That is, the universe is isotropic so that each observer sees the same in all directions. This is very strongly suggested by the observation of cosmic microwave background radiation (CBR): the temperature of CBR is independent of direction to one part in a thousand, according to a variety of experiments on various scales of angular resolution down to $1^\prime $ (Berry, 1989). However, BBT made many assumptions besides homogeneity which can not be observationally proved. The assumptions include bang-from-nothing, expansion, inflation, etc. To be fitted to data, more and more parameters were needed. When no more parameter can fix data, the un-observable stuff, dark matter and dark energy, was introduced.
My flat-spacetime model is based on the single principle of spatial homogeneity which is observationally proved, and all above-said difficulties are gone. The Lagrangian which describes the motion of particles (galaxies, photons) in spatially homogeneous universe is unique as follows, $$L\left(\tilde t, x^i, \frac{d\tilde t}{dp} , \frac{dx ^i}{dp}\right) = -\frac{1}{2} B(\tilde t ) \left(\frac{d\tilde t}{dp}\right)^2
+ \frac{1}{2}A(\tilde t ) \left(\left(\frac{dx}{dp}\right)^2 + \left(\frac{dy}{dp}\right)^2 + \left(\frac{dz}{dp}\right)^2\right)$$ $$(32)$$ where $\tilde t \equiv x^0 \equiv c_0t$, both $A(\tilde t) (>0)$ and $B(\tilde t)(>0)$ depend on time $\tilde t$ only. If both $A(\tilde t)$ and $B(\tilde t)$ are constants then the distribution of matter in the universe is also temporally homogeneous, no gravitational interaction is present on the cosmologic scale, and the Lagrangian simply returns to Einstein$^,$s special relativity. We assume that $A$ and $B$ vary with time and we have a spatially homogeneous yet evolving universe. This temporal inhomogeneity brings “gravitational interaction$^{,,}$ into the components (galaxies and photons) of the universe. Because the universe is spatially homogeneous, the “gravitational force$^{,,}$, at each spatial position, exerts in all spatial directions and the magnitude of the force is the same for all directions. Therefore, the “gravity$^{,,}$ is called pressure gravity because it has the similar property to the one of pressure. Einstein$^,$s equivalence principle definitely fails to the pressure gravity.
The motion of particles (galaxies and photons) is the solution of the corresponding Lagrange$^,$s equation, $$\frac {dx^i}{d\tilde t}= -P_i \sqrt{ \frac{ B(\tilde t)}{(P^2-2\tilde EA(\tilde t)) A(\tilde t)}}.$$ $$(33)$$ where $P_i, i=1,2,3$ are the conservative spatial momentum vector with $P$ being its amplitude. However, this is not the full story. Since we deal with causal motion only, we always have $\tilde E \leq 0$.
[*Varying light speed in the gravitational field of the universe.* ]{} Because light has the maximum speed, we have $\tilde E = 0$ for the motion of light. In its propagation direction we have $$\frac {dx}{d\tilde t}= \sqrt{ \frac{ B(\tilde t)}{A(\tilde t)}}.$$ $$(34)$$ Currently the universe is at the time of $$\tilde t=\tilde t_1 =ct_1.$$ $$(35)$$ The current light speed is $c \simeq 3\times 10^8 {\rm m\,s}^{-1}$ which is used in the definition of $\tilde t $: $\tilde t = ct$. It is not wrong that we choose other light speed for the definition.
[*Galactic redshift and Hubble law.* ]{} Galactic redshift is the formula (48) in He (2006b), $$z=\frac{\nu _1 }{\nu _2} -1= \frac{ \sqrt{g _{00}( \tilde t _1)} }{\sqrt{g _{00}( \tilde t _2)}}-1
=\frac{ \sqrt{ B(\tilde t_1 ) } }{\sqrt{B(\tilde t _2) }} -1.$$ We see that $B(\tilde t )$ must be a monotonously increasing function with time for us to have galactic redshifts rather than blueshifts, $$B(\tilde t) \uparrow .$$ The distance $D$ between the two galaxies 1 (Milky Way) and 2 is given by the integral of the light travel formula (34) $$D= \int ^{\tilde t_1} _{\tilde t _2}\frac {dx}{d\tilde t}d\tilde t= \int ^{\tilde t_1} _{\tilde t _2} \sqrt{ \frac{ B(\tilde t)}{A(\tilde t)}} d\tilde t.$$ The distance formula must have a redshift factor to give the Hubble law. This indicates that $A(\tilde t )$ depends on $B(\tilde t )$. A simple and general model of the dependence is $$A(\tilde t )= \frac {B^{m+1}(\tilde t)}{N^2B^{\prime 2} (\tilde t) }$$ $$(36)$$ where $m$ is a constant and $N(>0)$ is another constant whose unit is length. Finally we have Hubble law, $$\begin{array}{ll}
D&= \frac{2N}{m-2}\left (\frac{1}{\sqrt{ B(\tilde t_2) }^{m-2} } -\frac{1}{\sqrt{ B(\tilde t_1) }^{m-2} }\right )\\
&= \frac{2N}{m-2}\left (\frac{1}{\sqrt{ B(\tilde t_2) } } -\frac{1}{\sqrt{ B(\tilde t_1) } }\right )\left(\frac{1}{\sqrt{ B(\tilde t_2) }^{m-3} } +\dots \right ) \\
&=\frac{2Nz}{(m-2)\sqrt{ B(\tilde t_1) }}\left(\frac{1}{\sqrt{ B(\tilde t_2) }^{m-3} } +\dots \right ) \\
&=\frac{cz}{H_0(\tilde t_2, \tilde t_1)}
\end{array}$$ $$(37)$$ where the Hubble constant $H_0$ is $$H_0=\frac{c(m-2)\sqrt{ B(\tilde t_1) }}{2N}/\left(\frac{1}{\sqrt{ B(\tilde t_2) }^{m-3} } +\dots \right ).$$ $$(38)$$ As a summary, I note that the redshift requires $B(\tilde t )$ be a monotonously increasing function of time and Hubble law requires $A$ be determined by the function $B$ (see (51)). Therefore, the only one degree of freedom left is the function form of $B(\tilde t )$.
[*‘Accelerated Expanding$^,$ Universe.*]{} If $H_0$ depended only on $\tilde t_1$, the current time, then Hubble law would be perfectly true. However, it depends on the past time of the galaxy we observe, $$H_0= H_0(\tilde t_2, \tilde t_1).$$ $$(39)$$ If we assume $$m>3$$ then Hubble constant $H_0$ is not constant and increases with the time $\tilde t_2$, of which the galaxy is observed. This increase with time of $H_0$ is explained as the ‘accelerating expansion$^,$ of the universe. However, in my model, spacetime is flat (no expansion of curved spacetime) and the redshift is gravitational one which results from the evolution of the universe (mass density varies with time). Because redshift requires increasing $B(\tilde t)$, we see that ‘accelerating expansion$^,$ is consistent to galactic redshift.
[*Infinite Light Speed and the Birth of the Universe.*]{} Positive and increasing quantity $B(\tilde t)$ indicates a time $\tilde t _0$, when $B(\tilde t _0)=0$. This is the starting time of the universe. We can choose $\tilde t _0= 0$ to be the time of birth. Currently we do not know the exact physics at the hot birth. One thing is sure that light speed at the time must be infinite. Only infinite speed of communication could result in a later spatially homogeneous mass distribution in the infinite flat universe. This resolves the horizon and flatness problems due to birth. Infinite initial light speed indicates a decrease of light speed with time. Observation during the last decade does support the result of decreasing light-speed with time. The formula of light speed is (34). Therefore, decreasing light speed imposes further condition on the evolving factor $B(\tilde t)$, $$2BB^{\prime \prime} \le m{B^\prime}^2.$$ $$(40)$$
[*Light Speed Constancy and the Death of the Universe.*]{} However, there is strong evidence that light speed is approximately constant during mature stage of the universe. Constant light speed with time means that $A(\tilde t) $ and $B(\tilde t)$ are the same $$A(\tilde t) \equiv B(\tilde t) .$$ They serve as the scaling factor. Perfect Hubble redshift-distance linear law completely determines the scaling factor, $$\frac {1} {B(\tilde t) } \equiv \frac {1} {A(\tilde t) } = \frac {1} {B_0 } –- M (\tilde t - \tilde t_0)$$ $$(41)$$ where $M$ is a constant and $ B_0 = B(\tilde t_0) $. This formula indicates a finite time $ \tilde t_1$ when $ M(\tilde t_1-\tilde t_0 )= 1/B_0 $. This is the ending time of the universe because the scaling factor reaches infinity. The possibility of a rebirth needs further investigation.
[*The Absolute Inertial Frame of the Universe.*]{} Our calculation and results are reference-frames depended. For example, photon frequency is dependent on reference frames. Our results are meaningful only when single preferred inertial frame of the universe exists and the results are calculated with respect to the frame. The absolute frame is meaningful only when all components (e.g., galaxies) of the universe have convergent motion with respect to the frame. That is, all components slow down their speed of motion with respect to the frame. Since the nineteenth century, scientific report on the evidences of absolute inertial frame has never been stopped. Because of light speed constancy we have $ A(\tilde t) \equiv B(\tilde t) $ in the formula (33). We can see that the absolute speed of material particles (galaxies) does decrease with time, slowing-down motion with respect to the absolute inertial frame (note that $\tilde E < 0$ for material particles). Here we see that the existence of absolute inertial frame is once again the direct result of galactic redshift.
[*The Variance with Time of Matter Distribution in the Universe.*]{}
Our Lagrangian is defined on flat spacetime and can be quantized according to the classical and covariant quantization procedure (He, 2006a). Because the spatial distribution of matter in the universe is homogeneous, the resulting amplitude of the wave function must be proportional to the density of the distribution. Astronomical observation suggests that the density decreases with time especially during early universe. We can see that the amplitude does decrease with time if $B(\tilde t) $ increases with time. That is, the astronomic observation is once again consistent to the result of galactic redshift.
[**(iv) The Possibility of Curved Spacetime, Black Holes, and Big Bang is Less than One in Billion.** ]{} You have probably noticed that my FGR is based on very simple principles. Now I calculate the probability that FGR is wrong. FGR generalizes special relativity (e6, event 6). Einstein$^,$s general relativity does not generalize SR (special relativity). Because SR is well verified in high energy physics, the probability is less than one in hundred ($10^{-2}$) that the requirement of a gravitational theory which generalizes SR is false. My FGR explains the phenomena of galaxies (e7), which is false with the possibility of less than a hundredth ($10^{-2}$). My FGR quantizes gravity (e8), which is false with the possibility of less than a hundredth ($10^{-2}$). My model of the universe predicts many observational facts. Its single principle is that the universe is evolving (e9). The probability is less than one in hundred ($10^{-2}$) that the universe is not evolving (e9 is false). My model of the universe involves the single function of time: $B(\tilde t)$. The function is arbitrary except satisfying some conditions. Redshifts require increasing $B(\tilde t)$ with time. Decreasing speed of light requires that $B(\tilde t)$ satisfies the condition (40). The simple conditions guarantee the existence of the unique inertial frame of the universe (e1), the redshifts (not blueshifts) of galaxies (e2), the Hubble redshift-distance law (e3), the decreasing light-speed which resolves big bang difficulties (e4), and the ‘accelerating expansion‘ universe (e5). These predictions of independent observational cosmological facts based on the two conditions of single arbitrary function are certainly not an accident. Therefore, the probability that my model of the universe is not scientific truth is less than one in million ($10^{-6}$). Because these observational facts and the principles are independent events, the probability that FGR is false is less than one in billion ($10^{-9}$): $$10^{-6} \times 10^{-2} \times 10^{-2} \times \cdots < 10^{-9}.$$ That is, the possibility that the assumptions of curved spacetime, black holes, and big bang are true, is less than one in billion.
[**(v) Why we See that All Electro-magnetic Waves from the Universe Demonstrate Approximately the Same Physics.** ]{} I have shown the existence of the unique inertial frame of the universe. Stars are in freely-falling frames with respect to galactic gravity, and the reference frames of stars can not detect much gravity from galaxies. Sun is such a star, and all our astronomic observation is based on the sun frame. [**That is why we see that all electro-magnetic waves from the universe demonstrate approximately the same physics!**]{}\
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[**Suggesting an Experiment to Test Curved-Spacetime Assumption** ]{}\
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[**(i) “No Anysotropy of Light Speed is Observed$^{,,}$: the Third Specious Claim to EGR.** ]{} Experimentalists of relativity claim that no anisotropy of light speed is observed. That is, no evidence of different light speeds is found. That means that light in vacuum demonstrates the unique value independent of its origin and reference frame. If their claim were correct then EGR must be wrong. This is because EGR is the theory of gravity and gravity introduces spatial or temporal in-homogeneity. Only if the distribution of matter were both spatially and temporally homogeneous could we have a global inertial frame where Einstein$^,$s special relativity would be perfectly true and light speed would be constant for all inertial frames. If such homogeneity does happen and SR is perfectly true in the inertial frame, light speed is definitely anisotropic in any non-inertial frame.
Ironically, all experiments measuring light-speed anisotropy were performed on earth. The rotating earth is neither an inertial frame in FGR nor a freely falling frame in EGR. According to EGR, light speed is constant only in the local inertial frames (the local tangent “planes$^{,,}$ to curved spacetime). There is no such stuff as local freely-falling frames which cancel gravity (see part (vi) of Section 1). Therefore, light speed in rotating earth frame is definitely anisotropic according to EGR (I look forward to some relativist who will derive the anisotropy formula of light-speed in non-inertial frames as predicted by EGR). Light-speed anisotropy in rotational frames was proved by Sagnac experiment and relativists admitted that light speed is not constant in non-inertial frames. Because experimental relativists claimed no measurement of light-speed anisotropy on earth frame which is against theoretical relativists$^,$ expectation, EGR is wrong. Therefore, relativists made the third specious claim to EGR.
[**(ii) Anisotropy Formula of Light-Speed Based on FGR.** ]{} According to FGR which is based on flat spacetime, gravity results in varying light speed in inertial frames. If the pattern of varying light-speed is measured to conform to the formula of FGR then the curved-spacetime assumption is dead. Now I derive the anisotropy formula of light-speed based on FGR.
In Section 1, I calculated the speed of radial light beam which is (23). Choosing $\tilde E=0$ and $dr/d t=0$ in (18), we have the speed of light beam in the perpendicular direction to the radial one, $$c(\pi/2)=r\frac{d\phi }{dt}=c_0\sqrt{1-2r_g/r }.$$ $$(42)$$ The corresponding angular momentum $J$ is the maximum, $$J_{ {\rm max}}=\frac{rc_0 }{\sqrt{1-2r_g/r }}.$$ $$(43)$$ If light beam makes the angle $\theta $ with respect to the radial direction then its angular momentum is between zero and the maximum $$J=f\frac{rc_0 }{\sqrt{1-2r_g/r }}, \;\; 0\le f\le 1.$$ $$(44)$$ It is straightforward to show that $$f\approx \sin \theta .$$ $$(45)$$ The formula of light-speed anisotropy is $$c^2(\theta ) = c_0^2( 1-2r_g/r)^2\left(1+f^2 \frac {2r_g/r}{1-2r_g/r}\right).$$ $$(46)$$ Therefore, $$c(\theta ) \approx c_0\left(1+\frac{r_g}{r}\sin^2\theta -\frac {2r_g}{r} \right)$$ $$(47)$$ which is the required anisotropy formula of light-speed in FGR.
The formula (47) is the speed of light beam staring at the radial position $r$ and running in the direction which makes a angle $\theta $ to the radial direction. No modern technique can measure the speed of single light beam to the accuracy of about kilometers per second. However, modern technique can measure the difference of light speeds of two light beams to the accuracy of about $10^{-6}$ meters per second. Therefore, the experiments for measuring anisotropy generally have two light beams starting at the same position $r$. One-way experiment lets them travel a small distance in their different directions and then measure their light-speed difference. Two-way experiment requires them return to their starting position and then measure their light-speed difference. From the formula (47) we know that the light-speed difference is the maximum ($\Delta c \approx r_g/r$) only when one beam runs in the radial direction with respect to the mass center and the other runs in the perpendicular direction.
Now let us study such experiment on earth surface. We have already known that the maximum magnitude of anisotropy is $\Delta c \approx r_g/r$ where $r$ is the radial distance to the mass center. The experiment on earth surface involves two mass centers. One is the sun and the other is the earth. It is interesting that the magnitude of anisotropy due to sun is $\Delta c \approx 3$ m/s which is about ten times larger than the one due to earth, $\Delta c \approx 2\times 10^{-1}$ m/s. However, earth is the freely-falling frame with respect to the sun whose gravitational effect can not, mostly, be detected from the experiment on earth. Therefore, if FGR is true then the measured anisotropy on earth surface must be due to earth. However, rotating earth is non-inertial frame. Because people believe that the anisotropy of light-speed due to earth$^,$s rotation is $\Delta c \approx c_0(v_e/c_0)^2$ ($v_e$ is the linear velocity of earth rotation at equator) which is even smaller than the anisotropy due to the mass of earth (see Klauber, 2006), the effect of earth rotation is neglected. Therefore, our goal is to test anisotropy of light-speed on earth due to the gravity of earth. According to the analysis given in the last paragraph, to achieve the maximum magnitude of light-speed difference we need to direct one light beam to the mass center of earth and the other beam in the perpendicular direction, i.e., the parallel direction to the earth surface. I forgot the final condition: the experiment has the ability to measure light-speed difference to an accuracy better than $ 0.2$ m/s.
[**(iii) Suggesting an Experiment to Test Curved-Spacetime Assumption.** ]{} The only experiments which claimed the above accuracy were performed by Brillet and Hall (1979), Hils and Hall (1990), and Muller [*et al.*]{} (2003). However, all the experiments aimed at testing Einstein$^,$s special relativity. That is, they test the anisotropy of light-speed due to absolute motion with respect to the absolute reference frame, the aether. I call it aether-frame anisotropy. My FGR is based on the assumption that special relativity is correct and provides the anisotropy formula of light speed (47) due to central gravity of mass point.
The formula of aether-frame anisotropy is dogmatically derived and is given as follows. If at $t=0$ a beam of light is emitted in $\Sigma $ and if $S$ (non-preferential frame) moves with the speed $v$ with respect to $\Sigma $ and if $v$ makes the angle $\theta $ with respect to the direction of the light beam then $S$ measures for light the speed $c$, where $$c(\theta , v) = c_0\left(1 + \frac{ Av^2}{c^2} \sin^2\theta + \frac{ Bv^2}{c^2}\right)$$ $$(48)$$ Parameter $A$ is a measure of light speed isotropy and is generally measured through a Michelson-Morley class of experiments. These experiments verify light speed isotropy. Parameter $A$ has been tested by to be less than $3\times 10^{-15}$. Parameter $B$ is a measure of light speed invariance relative to the speed of the emitter/receiver and it is generally measured through Kennedy-Thorndike experiments. These experiments verify light speed invariance with the movement of the emitter/observer. Parameter $B$ has been tested by to be less than $3\times 10^{-13}$. SR predicts $A=B=0$ and the experimental asymptotical limits for both $A$ and $B$ under SR are indeed zero.
However, all experiments were designed to test aether-frame anisotropy. Therefore, they do not satisfy the conditions required to testify the anisotropy due to central gravity. For example, we consider the Brillet and Hall experiment. Firstly, the two light beams were both parallel to earth surface. Therefore, it is not surprised that it gives null result of light-speed anisotropy. Secondly, they were forced to subtract out a “spurious$^{,,}$ and persistent signal of approximate amplitude $2\times10^{-13}$ at twice the rotation frequency of their apparatus (Klauber, 1999). Thirdly, for the purpose of fitting data to the formula of aether-frame anisotropy (48), they averaged out some daily periodic variation and subtract away some seasoned pattern. The formula (47) actually predicts just such an effect due to the central mass of sun and the earth rotation.
Therefore, I suggest to repeat these experiments with one light beam in gravity direction so as to test the anisotropic light-speed due to earth gravity. Dare to make public the recent result of new Brillet and Hall experiment with one vertical light beam?\
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[**Conclusion** ]{}\
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[**(i) The First Specious Claim Made for EGR.**]{} Einstein$^,$s general relativity (EGR) is the theory of curved spacetime. However, his assumption brings more complexity than truth. Firstly, curved spacetime is embodied by non-trivial topology. Because topology is a very complicated mathematical subject, most relativists never take a look at it. Secondly, the concept of curved spacetime is nothing but temporal and spatial in-homogeneity. Therefore, all coordinates on a curved space are merely parameters. Real time and distance have to be calculated by employing coefficients of the spacetime metric. The calculation of time and distance by employing metric is very complicated too. Therefore, all relativists when confronting GR to observational data, calculate time, distance, or angle by directly using the coordinates in Schwarzschild solution or in post Newtonian formulation. However, there is the famous Riemann theorem: only when spacetime is flat does there exist one coordinate system which has direct meaning of time and distance, and [*vice verse*]{}. Therefore, the hailed accurate tests of GR verified the flat-spacetime interpretation of GR (my FGR). The more claims are made that classical tests of general relativity fits data with great accuracy, the more falsified is the curved-spacetime assumption. That is, the claim is specious to EGR.
[**(ii) The Second Specious Claim Made for EGR.**]{} Einstein$^,$s equivalence principle is that, over any small region of space and time, all test particles move at approximately the same acceleration. Therefore, the observational frame which moves at the very acceleration will see each particle being either static or moving in straight lines at constant speeds, within the small region in question. That is, the local frame sees no gravity at all and we see a cancellation of gravity by choosing local frames, which are generally called the local freely-falling frames. Einstein thought that the local frames were the local tangent 4-dimensional planes of curved spacetime. This mistake led to the assumption of curved spacetime and resulted in ninety years$^,$ dogmatic study of gravity and cosmology: black holes, big bang, inflation, etc. I have made two points to prove Einstein$^,$s mistake.
Firstly, a tangent plane is an inertial frame in which particles move in straight lines at constant speeds. Different particles may have different speeds but their acceleration must be zero. Speed is the first derivative with particles$^,$ coordinates while acceleration is the second derivative. Tangent plane to curved space is determined only by the first derivatives not the second derivatives. How did Einstein make such simple mistake?
Secondly, Einstein made further mistake and assumed that all local particles shared the same acceleration independent of their individual properties, that is, independent of their energy per unit mass and their angular momentum per unit mass. Energy and angular momentum have totally four degrees of freedom and Einstein required that locally particles have zero degree of freedom: sharing the same acceleration. Based on FGR (see part (vi) of Section 1), however, I proved that particles must have different local accelerations corresponding to their angular momentum and energy. If angular momentum is zero (radial motion), the formula of acceleration is (24) which depends on energy. Only when their angular momentum is zero and their energy is the minimum will the test particles share the same acceleration. In the case of earth surface, the shared acceleration is $9.8$ m/s$^2$. If the energy of test particles were high enough then the leaning tower experiment of Galileo Galilei would have demonstrated opposite result. Einstein thought the result of Galileo Galilei were universal truth and generalized this false result as his equivalence principle.
Therefore, there is no such stuff as freely-falling frames which cancel local gravity, and Einstein$^,$s equivalence principle is completely false and serves as the second specious claim to EGR. However, a great many arguments of dogmatic gravitational theory and its applications and the theory of big bang are based on the non-existent freely-falling frames which cancel local gravity.
[**(iii) The Third Specious Claim Made for EGR.**]{} Experimentalists of relativity claim that no anisotropy of light speed is observed. That is, no evidence of different light speeds is found. That means that light in vacuum demonstrates the unique value independent of its origin and reference frame. If their claim were correct then EGR must be wrong. This is because EGR is the theory of gravity and gravity introduces spatial or temporal in-homogeneity. Only if the distribution of matter were both spatially and temporally homogeneous could we have a global inertial frame where Einstein$^,$s special relativity would be perfectly true and light speed would be constant for all inertial frames. If such homogeneity does happen and SR is perfectly true in the inertial frame then light speed is definitely anisotropic in any non-inertial frame.
Ironically, all experiments measuring light-speed anisotropy were performed on earth. The rotating earth is neither an inertial frame in FGR nor a freely falling frame in EGR. According to EGR, light speed is constant only in the local inertial frames (the local tangent “planes$^{,,}$ to curved spacetime). There is no such stuff as local freely-falling frames (see part (vi) of Section 1). Therefore, light speed in rotating earth frame is definitely anisotropic according to EGR (I look forward to some relativist who will derive the anisotropy formula of light-speed in non-inertial frames as predicted by EGR). Light-speed anisotropy in rotational frames was proved by Sagnac experiment and relativists admitted that light speed is not constant in non-inertial frames. Because experimental relativists claimed no measurement of light-speed anisotropy on earth frame which is against theoretical relativists$^,$ expectation, EGR is wrong. Therefore, relativists made the third specious claim to EGR.
However, my FGR has no such contradictory claims. When confronted to solar observation, to future GPB data, and even to the gravitational radiation damping data in a binary pulsar system (e.g., PSR 1913+16), it is directly verified without the panic of directly using coordinates as time, distance, or angle. EGR has no idea about galaxies while my FGR solves galaxy pattern and dynamics completely (He, 2005a, 2005b, 2005c, 2007). Consistent to FGR, my model of the universe proved the existence of the unique global inertial frame. What is more important, it is very simple and gives simple explanation to all available laws of cosmological observation. It is more consistent than Big Bang Theory (BBT). Because I have traditional flat spacetime, gravity is easily quantized (He, 2006a).
EGR and FGR are the mirrored versions of each other. If they are the only choice towards the truth of gravity then one must be real and the other is its illusory, tortuous, specious image. However, I have shown that the possibility of curved spacetime, black holes, and big bang, is less than one in billion. An experiment is proposed whose results will completely decide the fate of curved spacetime assumption. Dare to make public the recent result of new Brillet and Hall experiment with one vertical light beam?\
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Particle‘s acceleration in static homogeneous gravitational field is cancelled by any reference frame of the same accelerating direction and the same accelerating rate. The frame is commonly called the freely-falling one. The present paper shows that the acceleration is also cancelled by a spatial curvilinear coordinate system. The coordinate system is simply a spatial square-root coordinate rescale in the field direction, no relative motion being involved. This suggests a new equivalence principle. Spacetime is flat which has inertial frame of Minkowski metric $\eta _{ij}$. Gravity is a tensor $g _{\alpha \beta }$ on the spacetime, which is called effective metric. The effective metric emerges from the coordinate transformation. The gravitational field of an isolated point mass requires a nonholonomic spacetime boost transformation. This generalization of Newtonian gravity shares the properties of Lorentz transformation, which should help quantize gravity. The corresponding effective metric is different from that of Schwarzschild. To first order, its prediction on the deflection of light and the precession of the perihelia of planetary orbits is the same as the one of general relativity (GR). Its further implication is left for future work.\
Relativity – Gravitational Theory – Galaxies : Structure\
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Introduction
==============
[**(i) Minkowski metric description of vanishing gravity.**]{} The present paper deals with gravitational interaction only, no other interaction being involved. Newton‘s first law of motion that a particle experiencing no net force (i.e., vanishing gravitational field) must move in straight direction with a constant (or zero) velocity with respect to inertial frame $\tau \xi \eta \zeta $, can be proved geometrically by introducing Minkowski metric $\eta _{\alpha \beta }$ to the frame, $$\begin{array}{ll}
ds^2 &=d\tilde \tau ^2-d\xi ^2 -d\eta ^2 -d\zeta ^2 \\
& =-\eta _{\alpha \beta }d\xi ^\alpha d\xi ^\beta
\end{array}$$ where $\xi ^0=c\tau =\tilde \tau, \xi ^1=\xi ,\xi ^2 =\eta , \xi ^3 =\zeta $, $c$ is light speed, and $\eta _{00}=-1, \eta _{11}= \eta _{22} = \eta _{33} =1,
\eta _{\alpha \beta } = 0 (\alpha \not= \beta) $. The metric is the basis of special relativity. I call the distance $s$ along the curves of spacetime by real distance because I will introduce a new term, effective distance $\bar s$. The real distance is generally called proper distance which can be negative because the matrix $\eta _{\alpha \beta }$ is indefinite. The indefinite quadratic form (1) is the generalization of Pythagoras theorem to Minkowski spactime. It is straightforward to show that the first Newton law of motion (vanishing gravity) is equivalent to the following geodesic equation, $$\frac {d^2\xi ^\alpha }{dp^2}+\Gamma ^\alpha_{\beta \gamma }
\frac {d\xi ^\beta }{dp}\frac {d\xi ^\gamma }{dp}=0$$ where $p$ is the geodesic-curve parameter and $\Gamma ^\alpha_{\beta \gamma }$ is the affine connection. The affine connection involves the first order derivatives to $\eta _{\alpha \beta }$ and must be zero. Therefore, the first Newton law of motion is equivalent to (2). People try to generalize the equation to describe gravitational interaction.
[**(ii) Part-one assumption of general relativity.**]{} It is more important to consider test particle‘s motion in an inertial frame in which the particle does experience gravitational force. In the frame, the particle no longer moves in straight direction with a constant (or zero) velocity. The motion is described in good approximation by the Newton‘s universal law of gravitation which is, however, a non-relativistic theory and needs to be generalized to give account for the solar observations which deviate from Newton laws‘ calculation. Einstein‘s general relativity (GR) is the most important try toward the generalization. The basic assumption of GR can break into two parts. The part-one assumption of GR is the simple replacement of the above matrix $\eta _{\alpha \beta }$ by a tensor field $g _{\alpha \beta }$ whose components are, instead of the constants $\pm 1$, position functions on spacetime. Similar to the above part (i) description, particles‘ motion follows the solution of the geodesic equation $$\frac {d^2x ^\alpha }{dp^2}+\Gamma ^\alpha_{\beta \gamma }
\frac {dx ^\beta }{dp}\frac {dx ^\gamma }{dp}=0$$ where $x ^0=ct =\tilde t, x ^1=x ,x ^2 =y , x ^3 =z$ and the affine connection $\Gamma ^\alpha_{\beta \gamma }$ involves the first order derivatives to the tensor, $$\Gamma ^\alpha _{\beta \gamma }=\frac{1}{2}g^{\alpha \rho }\left (
\frac {\partial g_{\rho \beta }}{\partial x^\gamma } +
\frac {\partial g_{\rho \gamma }}{\partial x^\beta } -
\frac {\partial g_{\beta \gamma }}{\partial x^\rho } \right ).$$ The equation (3) does not involve the inertial mass of the test particles. This is an appropriate description because inertial mass equals gravitational mass in the case of gravitational interaction.
[**(iii) Part-two assumption of GR (geometrization).**]{} The present paper questions the part-two assumption of GR. The assumption is that spacetime is curved when gravity is present and $g_{\alpha \beta }$ in (4) is exactly the metric of the curved spacetime $$ds^2 =-g _{\alpha \beta }dx ^\alpha dx ^\beta .$$ The assumption is called the geometrization of gravity whose whole meaning is that $s$ must be the real distance along the curves of spacetime. Therefore, the solutions of the geodesic equation (3) extremize the following functional variation $$\delta s=\delta \int ^{p_B}_{p_A}\frac {ds}{dp}dp .$$
The geometrization is claimed to be based on the following simple fact on static homogenous gravity. Particle‘s acceleration in static homogeneous gravitational field is cancelled by any reference frame of the same accelerating direction and the same accelerating rate. This is straightforward because any test particle in the field, with fixed directions of its attached axes, sees other test particles moving on straight lines with constant speeds. The frames are commonly called the freely-falling ones, which are the exclusive property of homogeneous gravity. However, the simple fact is not the full story of homogeneous gravity. In the following part (v) I will show that the acceleration is also cancelled by a spatial curvilinear coordinate system. The coordinate system is simply a spatial square-root coordinate rescale, no relative motion being involved. Firstly, in the part (iv) which deals with freely-falling frames, I will show that geometrization does not apply to homogeneous gravity.
[**(iv) Failure of the geometrization of homogeneous gravity.** ]{}Static homogeneous gravitational field, $\vec g$, in the positive direction of $x$-axis can be canceled by a global space-time coordinate transformation, $$\begin{array}{l}
\xi=x- \frac {1}{2} g t ^2 , \\
\tau =t.
\end{array}$$ where $g (> 0)$ is constant. That is, in the $\tau \xi $ coordinate system, particles experience no gravity and follow the equation (2) in part (i) where $\Gamma ^\alpha_{\beta \gamma }$ involves the first order derivatives to $\eta _{\alpha \beta }$ and must be zero. The real distance $\bar s$ in the freely-falling $\tau \xi$ frame is the formula (1), $$d\bar s^2 =d\tilde \tau ^2-d\xi ^2$$ where I introduced a new symbol $\bar s$ instead of $s$. Its explanation will be given in the following. For simplicity, I drop off the coordinates $\eta , \zeta , y, z$ when dealing with homogeneous gravity. Substitution of the formula (7) into (8) leads to a quadratic form in the coordinates $t, x$, $$\begin{array}{ll}
d\bar s^2 &=\left (1- \frac{g^2\tilde t ^2}{c^4}\right )d\tilde t^2
+\frac{2g\tilde t}{c^2}d\tilde t dx - dx ^2 \\
&=-g _{\alpha \beta }dx ^\alpha dx ^\beta
\end{array}$$ where $$g_{00}=\frac{g^2\tilde t ^2}{c^4}-1, g_{01}=g_{10}=-\frac{g\tilde t}{c^2}, g_{11}=1.$$ Now I apply the part-one assumption of GR, i.e. the method in part (ii), to the above quantity $g _{\alpha \beta }$. It is not surprising that the solution of the corresponding geodesic equation turns out to be $x=(1/2)gt^2 +x_1$ where $x_1$ is a constant, i.e., $$\frac {d^2x}{dt^2}=g.$$ This indicates that particles in $tx$ coordinate system experience static homogeneous gravitational field $\vec g$ and the coordinate transformation (7) does cancel gravity. However, the geometric explanation of $\bar s$ to be real distance on the spacetime fails, as demonstrated in the following. In fact, Einstein‘s geometrization of gravity refuses any cancellation of gravitational field by a global spacetime coordinate transformation, because of a mathematical theorem. The theorem is that if the spacetime $txyz$ is curved then there is no global coordinate transformation $$\begin{array}{ll}
t=t(\tau ,\xi ,\eta ,\zeta ),
& x=x(\tau ,\xi ,\eta ,\zeta ), \\
y=y(\tau ,\xi ,\eta ,\zeta ),
& z=z(\tau ,\xi ,\eta ,\zeta )
\end{array}$$ which transforms the quadratic form (5) into (1), and, if there is such coordinate transformation then the spacetime must be flat. The theorem is easily understood. For simplicity, consider the case of space not the case of spacetime. For better imagination, consider two dimensional space (surface) not three dimensional space. The simplest surfaces are the flat plane and the curved sphere surface. The quadratic form for plane $\xi \eta $ is $ds^2 = d\xi ^2 +d\eta ^2$, which is exactly the Pythagoras theorem of right triangle. The quadratic form for sphere surface has a similar but definitely positive form to (5). However, it can never be transformed into the Pythagoras formula by whatever coordinate transformation. In the case of homogeneous gravity, such coordinate transformation does exist which is the formula (7). I have another coordinate transformation in part (v) which cancels the homogeneous gravity too. Therefore, the spacetime $txyz$ which presents homogeneous gravity must be flat. Because $g_{\alpha \beta }$ in (9) is not the Minkowski metric $\eta _{\alpha \beta }$, the quantity $\bar s$ in (9) is not real distance along the curves of the flat spacetime $txyz$. Therefore, Einstein‘s geometrization fails to the description of homogeneous gravity. Because the quadratic forms (9) and (8) describe homogeneous gravity successfully, they initiate a method on gravitational study. The method abandons the geometrization of gravity and requires that spacetime be flat with its only geometric quantity being the Minkowski metric $\eta _{\alpha \beta }$. The quantity $\bar s$ is called effective distance and $g _{\alpha \beta }$ is called effective metric. Both have no geometric meaning.
[**(v) Square-root rescale which cancels homogeneous gravitational field.**]{} The only observable quantity in the static homogeneous gravitational field is the quadratic motion, $x=(1/2)gt^2 +x_1$. If we can find other coordinate transformation and the application of the above procedure leads to the same “Pythagoras formula999 in the curvilinear coordinate system $\tau \xi $ and the same quadratic motion $x=(1/2)gt^2 +x_1$ in the rectangular coordinate system $tx$ then we can say that the new coordinate transformation cancels homogeneous gravity too. We try the following coordinate transformation, $$\begin{array}{l}
\sigma =\sigma _0\sqrt{x/ x _0} , \\
\tau =t
\end{array}$$ where I introduce two constants $\sigma _0, x _0$ to fulfill the requirement that both $x$ and $\sigma $ have the same length unit. In the coordinate transformations provided in the following sections, if we do not see such constants then they are understood to have values of 1 and are not presented in the formulas for simplicity. However, coordinate transformations are always understood to have homogeneous forms which are similar to the following $$x =x _0 f(\xi /\xi _0).$$ Substitution of the coordinate transformation (13) into the following “Pythagoras formula999 $$d\bar s^2 =d\tilde \tau ^2-d\sigma ^2,$$ we have the following effective metric in $tx$ coordinate system, $$\begin{array}{ll}
d\bar s^2 &=d\tilde t^2-\frac{\sigma _0^2}{4x _0 x} dx^2 \\
&=-g _{\alpha \beta }dx ^\alpha dx ^\beta
\end{array}$$ where $$g_{00}=-1, g_{01}=g_{10}=0, g_{11}= \frac{\sigma _0^2}{4x _0 x}.$$ The solution of the resulting geodesic equation (3) is any quadratic motion $x=(1/2)ht^2 +x_1 $ where $h$ is an arbitrary constant. That is, the coordinate transformation (13) cancels homogeneous gravity and particles experience no gravity in the curvilinear coordinate system $\tau \sigma $. However, $\tau \sigma $ is just a curvilinear coordinate space. It is not a reference frame because the relation between $\sigma $ and $x$ is not linear. However, the coordinate system $\tau \xi $ (see (7)) is a global freely-falling frame by which people can make measurement.
The coordinate system $\tau \sigma $ is a special curvilinear coordinate system. The main feature of the coordinate transformation (13) is that the space coordinate $\sigma $ is transformed to space coordinate $x$ independent of the time coordinate transformation. From now on, our coordinate-rescales deal with spatial coordinates only. We consider $\sigma $ to be the curvilinear coordinate relative to the rectangular space $x$ and the transformation $\sigma =\sigma _0\sqrt{x/ x _0} $ is called an uneven rescale on the coordinate $x$. The coordinate space $ \sigma $ is called a shadow of the real space $x$ and $\sigma =\sigma _0\sqrt{x/ x _0} $ is called the shadow coordinate transformation. For any real point $x$, the radial line section from the origin to the point $\sigma _0\sqrt{x/ x _0}$ is called the shadow of the real section which is from the origin to the real point $x$. The former point is called the shadow point of the latter. In the following sections, the same definitions hold except that the coordinate $x$ is replaced by the radial coordinate $r$. However, I will not repeat the definitions. The distance between a point and its shadow can be large in the case of homogeneous gravitational field. This is understandable because there must exist infinite areas of mass distribution to maintain a mathematically homogeneous gravitational field. In section 4 we will see that the distance is small ($\simeq$ 3.0 km) for the gravitational field generated by solar mass.
As argued in part (iv), if there is a global coordinate transformation which cancels gravity (i.e., the form (5) is transformed into (1)) then the spacetime must be flat (zero curvature). Its rectangular coordinate system must be $txyz$, and $\tau \xi \eta \zeta $ must be a curvilinear coordinate system of the spacetime. Both $\bar s$ and $g _{\alpha \beta }$ have no geometric meaning because otherwise the rectangular coordinate system would be $\tau \xi \eta \zeta $ (flat spacetime with Minkowski metric) and a curvilinear coordinate system would be $txyz$. [*Therefore, the gravitational theory which shares the properties of homogeneous gravity must be non-geometric.*]{} Initiation of such theory is the goal of the present paper and a new equivalence principle is proposed as follows.
[**(vi) New equivalence principle (NEP).**]{} Spacetime is always flat with Minkowski metric $\eta _{ij}$. A tensor $g _{\alpha \beta } $ with Lorentz covariant symmetry is defined on the flat spacetime, which describes gravity and has no geometric meaning $$d\bar s^2 =-g _{\alpha \beta }dx ^\alpha dx ^\beta .$$ The tensor $g _{\alpha \beta } $ is called effective metric. The effective distance $\bar s$ is not real distance on the spacetime $txyz$. Test particles follow the solution of the corresponding effective geodesic equation (3). The test particle‘s motion extremizes the following functional variation $$\delta \bar s=\delta \int ^{p_B}_{p_A}\frac {d\bar s}{dp}dp .$$ For a galaxy, the gravitational redshift due to its mass distribution is not significant to be observed, i.e., $t \simeq \tau $. Furthermore, galactic gravitational fields are shown to be cancelled by spatial coordinate rescales. Therefore, a global spacetime coordinate transformation (12) can be found which transforms the quadratic form (18) into the following $$d\bar s^2 =d\tilde \tau ^2 -d\xi ^2 -d\eta ^2-d\zeta ^2.$$ That is, the corresponding effective curvature is zero. This kind of effective metric (18) is called holonomic because the relation between $dx^{\alpha }$ and $d\xi ^{\alpha }$ (i.e., the equality of (18) to (20)), can be integrated into a global spacetime coordinate transformation (12). However, the gravitational field of an isolated point mass (e.g., a star, which is the basic component of galaxies), is non-holonomic because solar gravitational redshift is observed to be significant. That is, the corresponding effective curvature is non-zero. The present paper shows that a nonholonomic spacetime boost transformation cancels the gravity.
Now we understand that the method of NEP is similar to the one of GR except that the spacetime of the former is flat while the one of the latter is curved. In NEP, therefore, the coordinate system $\tau \xi \nu \zeta $ is curvilinear and the inertial frame $txyz$ is the real rectangular coordinate system of the flat spacetime. The effective metric $g_{\alpha \beta }$ in NEP is a tensor field on the flat spacetime and measures the gravitational 999medium999 which is generated by the corresponding mass distribution. The 999medium999 curves the motion of (test) particles (i.e., extremizing effective distance $\bar s$) in the similar way the dielectric medium curves the propagation of light waves (extremizing refractive index $n$). GR which attributes gravity to spacetime curvature, is actually based on the assumption that locally at each spacetime point there is a tangent flat Minkowski spacetime instead of a freely-falling one.
[**(vii) Square-root, logarithmic, reciprocal, and translational rescales.**]{} We have already shown that homogeneous gravity is governed by the above NEP principle and the rescale is square-root. He (2005a) found that the gravitational field of 2-dimensional mass distributions of spiral galaxy disks can be derived by the principle too and the spatial radial coordinate rescale is logarithmic. A simple explanation of galactic rotation curves is given by the corresponding new stellar dynamics. The gravitational field of elliptical galaxies is possibly described by radial reciprocal rescale. In section 4, I will show that the gravitational field of an isolated point mass is also governed by the principle and is canceled by nonholonomic spacetime boost transformation. The radial translational rescale may contribute to the cancellation too. The test particles follow the geodesic motion determined by the effective metrics and we have corresponding gravitational dynamics to all the cases. The corresponding effective curvature-tensor may be zero, i.e., holonomic (in the cases of homogeneous gravity and the gravitational fields of galaxies) or may be non-zero, i.e., nonholonomic (in the case of the gravitational field of an isolated point mass). However, the spacetime is always flat and $g_{\alpha \beta }$ has no connection to it.
[**(viii) Weakness of GR.** ]{} Any inhomogeneous gravitational field can be considered to be static and homogeneous within small zone of spacetime. On the other hand, the geometrization of gravity (GR) is claimed to be based on the cancellation of static homogeneous gravity by freely-falling frames. However, I have shown in part (iv) that geometrization fails to the description of homogeneous gravity. This is the main weakness of GR. Further more, the metric of the geometrization has to be determined by spacetime curvature and the corresponding Einstein equation is highly nonlinear and complicated. The common spatial parameters like distances and angles can not be computed directly from any coordinate system. They are determined by the metric because spacetime is curved.
GR encounters many difficulties. Theoretically, the total gravitational energy is not well defined and the gravitational field can not be quantized because it is connected to the space-time background itself. Realistically, Einstein equation permits very few metric solutions. Anisotropic and non-vacuum metric solutions which deal with 2-dimentional mass distributions like spiral galaxy disks do not exist in literature, to my knowledge. Astronomic observations reveal many problems which can not be resolved by GR and people resort to dark matter. Zhytnikov and Nester (1994)‘s study indicates that the possibility for any geometrized gravity theory to explain the behavior of galaxies without dark matter is rather improbable. Therefore, looking for a non-geometrized yet relativistic gravitational theory of galaxies is of great interest. The present paper, following He (2005a), provides a preliminary theory of the kind. In the theory, static gravity can be cancelled by spatial coordinate rescales (holonomic transformation) or by nonholonomic boost. Four types of rescales are found. The corresponding dynamical equations are ready for tests.
GR is widely accepted because some of its calculation are testified by solar measurements. However, [*the curvature of space-time was never measured and it is never proved that there exists no other dynamical equation similar to (3) whose solution gives the same or similar first-order predictions for solar system as GR.*]{} The present paper shows the existence.
Section 2 discusses general diagonal effective metric and the solution of its effective geodesic equation, taking spatial radial coordinate rescale as example. Section 3 discusses the spatial logarithmic and reciprocal coordinate rescales which cancel the gravitational fields of spiral galaxy disks and elliptical galaxies respectively. Section 4 discusses nonholonomic boost transformation which cancels the gravitational field of isolated point mass. The metric is different from the Schwarzschild one. To first order, its prediction on the deflection of light and the precession of the perihelia of the planetary orbits is the same as the one of GR. Section 5 is conclusion.
General Discussion of Diagonal Effective Metric
=================================================
He (2005a) indicates that the logarithmic arms of ordinary spiral galaxies are the evidence that the gravitational field generated by the mass distribution of spiral galaxy disks can be canceled by the $t \xi \phi \theta $ coordinate system, $$\begin{array}{l}
t=t, \\
\xi = \xi _0 \ln (r /r _0), \\
\phi =\phi , \\
\theta = \theta
\end{array}$$ where $ r \phi \theta \,(0\leq \phi <2\pi , 0\leq \theta <\pi )
$ is the spherical polar coordinate system in the real rectangular $xyz$ space and $\xi _0 $, $r _0$ are constants. The $\xi \phi \theta $ coordinate system is simply the uneven rescale on the spatial radial lines in the $xyz$ space. Because $\xi =p(r ) = \xi _0\ln (r /r _0)$, the rescale is called a logarithmic one. The rescale and all others discussed in the following are the ones on the spatial radial lines in $xyz$ space. Therefore, we give the general result on spatial radial rescale $\xi =p (r)$ in the present section.
[**(i) Effective metric and geodesic equation.** ]{} Let $$\xi =p(r )$$ be spatial radial rescale. Because the rescale cancels gravity, we have $$\begin{array}{ll}
d\bar l^2&=d\xi ^2 +\xi ^2 (d\theta ^2 +\sin ^2\theta d\phi ^2 ) \\
&=p^{\prime 2}(r ) dr ^2 + p^{ 2}(r ) ( d\theta ^2 +\sin ^2\theta d\phi ^2 )
\end{array}$$ where $\bar l$ is the spatial effective distance in the rectangular space $xyz$. If the curvilinear coordinate $\xi \eta \zeta $ is imagined to be the rectangular one of an independent space then $\bar l$ is its real spatial distance and $(\xi , \theta ,\phi )$ its polar coordinates.
The diagonal effective metric of the flat spacetime $txyz$ which describes the static radial gravitational fields must be the following $$\begin{array}{ll}
d\bar s ^2 &=B(r )d\tilde t^2-(p^{\prime 2}(r ) dr ^2 + p^{ 2}(r ) ( d\theta ^2 +\sin ^2\theta d\phi ^2 ))\\
&=B(r) d\tilde t^2-(A(r) dr ^2 + C( r) ( d\theta ^2 +\sin ^2\theta d\phi ^2 ))\\
&=-g_{\alpha \beta }dx^\alpha dx^\beta
\end{array}$$ where $dx^\alpha =(ct, r, \phi , \theta ) $ and $$\begin{array}{l}
g_{00}(\equiv g_{tt})=-B( r),\\
g_{11}(\equiv g_{rr})= p^{\prime 2}(r ) =A( r), \\
g_{22}(\equiv g_{\theta \theta })= p^{2}(r ) =C( r), \\
g_{33}(\equiv g_{\phi \phi })= p^{2}(r )\sin ^2\theta , \\
g_{\alpha \beta } \equiv 0, \: \alpha \not= \beta.
\end{array}$$ The coefficient $B(r )$ describes the gravitational redshift as suggested by GR. All other coefficients are determined by the spatial radial rescale $\xi =p (r)$. If $B(r )\equiv 1$ then the above quadratic form of effective metric can be transformed into the following 999Pythagoras formula999, $$d\bar s ^2 =d\tilde \tau^2-(d\xi ^2 +\xi ^2(d\theta ^2 +\sin ^2\theta d\phi ^2))$$ by a global spacetime coordinate transformation $$t=\tau, r=r _0f(\xi /\xi _0 ), \phi =\phi ,\theta =\theta$$ where $ r= r _0f(\xi /\xi _0 )$ is the inverse of the radial coordinate rescale $\xi =p(r )$. This is true for homogeneous gravitational field and approximately true for the gravitational fields generated by the mass distributions of spiral galaxy disks and elliptical galaxies. However, it is not true for the gravitational field generated by a point mass $M$ where $B(r )=1-2MG/(c^2r)$.
In spiral galaxy disks, stars are approximately planar motion. This is also true for any individual star of elliptical galaxies and any individual test particle in the gravitational field of an isolated point mass because the gravitational fields are described by the above effective metric. Therefore, we take $\theta =$ constant $= \pi/2$ and our formulas involve three variables $t,r,\phi$ only, $$\begin{array}{ll}
d\bar s ^2 &= B(r) d\tilde t^2-(p^{\prime 2}(r ) dr ^2 + p^{ 2}(r )d\phi ^2 ) \\
&=B(r) d\tilde t^2-(A(r) dr ^2 + C( r) d\phi ^2 )\\
&=-g_{\alpha \beta }dx^\alpha dx^\beta
\end{array}$$ where $dx^\alpha =(ct, r, \phi )$ and $$\begin{array}{l}
g_{00}(\equiv g_{tt})=-B( r),\\
g_{11}(\equiv g_{rr})= p^{\prime 2}(r ) =A( r), \\
g_{22}(\equiv g_{\phi \phi })= p^{2}(r )=C(r ) , \\
g_{\alpha \beta } \equiv 0, \: \alpha \not= \beta.
\end{array}$$ Test particles move on curved orbits due to the effective metric $g_{\alpha \beta }$. Their motion follows the geodesic equation (3). The only non-vanishing components of its affine connection are $$\begin{array}{ll}
\Gamma ^r_{rr} = \frac {A^\prime (r)}{2A(r)}, &
\Gamma ^r_{\phi \phi }=-\frac {C^\prime (r)}{2A(r)},\\
\Gamma ^r_{\tilde t \tilde t }= \frac {B^\prime (r)}{2A(r)},&
\Gamma ^\phi _{r \phi }=\Gamma ^\phi _{ \phi r}
=\frac {C^\prime (r)}{2C(r)},\\
\Gamma ^{\tilde t}_{r \tilde t }=\Gamma ^{\tilde t}_{ \tilde t r}=\frac {B^\prime (r)}{2B(r)}&
\end{array}$$ where $ A^\prime (r) =dA (r)/dr$, etc..
[**(ii) Constants of the motion.**]{} [*Note that the above formula (30) and the solution of its corresponding effective geodesic equation in the following (i.e., the formulas (32), (33), and (35)) hold to arbitrary diagonal effective metric (24) or (28) which is not necessarily a coordinate rescale.* ]{} The geodesic equation is solved by looking for constants of the motion. In fact, the following solutions (32), (33) and (35) are standardized ones which can be found in, e.g., Weinberg (1972). The only difference is about $A(r ), B(r ), C(r )$. For example, $A(r )=1/B(r ), B(r )=1-2MG/(c^2r), C(r )=r^2$ is the Schwarzschild solution of Einstein‘s geometrodynamics. I repeat Weinberg (1972)‘s argument in deriving the solutions.
The geodesic equations which involve $d^2\tilde t /dp^2$ and $d^2\phi /dp^2$ are called time component equation and polar-angle component equation respectively. They can be rewritten as the following, $$\begin{array}{l}
\frac {d }{dp}(\ln \frac {dt }{dp}+\ln B( r) )=0, \\
\frac {d }{dp}(\ln \frac {d\phi }{dp}+\ln C( r) )=0.
\end{array}$$ These yield two constants of the motion. The first one is absorbed into the definition of $p$. I choose to normalize $p$ so that the solution of the time component equation is $$\frac {dt }{dp}=1/ B( r).$$ The other constant is obtained from the polar-angle component equation, $$C( r)\frac {d\phi }{dp}=J.$$ The formula is used to study spiral galaxy rotation curves in the next section. If $C( r)$ is $r^2$ as suggested by the Schwarzschild solution then $J$ is the conservative angular momentum per unit mass and the rotation speed is $rd\phi /dt=JB( r)/r$. The Schwarzschild solution further suggests $B( r) \approx 1$ at large distance $r$ from the galaxy center and we expect a decreasing rotation curve, $V( r)= rd\phi /dt = J/r$. Real rotation curves are often constant over a large range of radius and rise outwards in some way. In our proposition (the formula (21)), however, $C( r)=\xi _0^2\ln ^2(r/r _0), B( r)=1$ and we have a non-decreasing rotation curve.
Note that the common angular momentum is no longer conserved ($C(r ) \not= r^2$) in NEP theory, because of the radial coordinate-rescale cancellation of gravity. Instead, the shadow angular momentum is conserved whose direct result is that galactic rotation curves no longer decrease outward and no dark matter is required for their explanation (He, 2005a).
Furthermore, we have a gravitational dynamic equation which is the third component of the geodesic equation (the polar-distance component equation), $$\frac {d^2r }{dp^2}+ \frac {A^\prime }{2A}\left (\frac {dr }{dp}\right )^2-
\frac {C^\prime }{2A}\left (\frac {d\phi }{dp}\right )^2+
\frac {c^2B^\prime }{2A}\left (\frac {dt }{dp}\right )^2=0.$$ With the help of the other solutions, we have the last constant of the motion, $$A( r)\left (\frac {dr }{dp}\right )^2+\frac {J^2 }{C( r)}-\frac {c^2}{B( r)}=-E\:({\rm constant}).$$
New Stellar Dynamics of Galaxies
=================================
[**(i) New Stellar dynamics of spiral galaxy disks.**]{} For the logarithmic rescale $\xi=\xi _0 \ln (r /r _0)$ (see (21)), we have $$\begin{array}{l}
p(r )=\xi _0 \ln (r/r _0), \\
A(r )= p^{\prime 2}(r )= \xi _0^2/r^2 , \\
C(r )= p^{2 }(r )= \xi _0^2\ln ^2(r/r _0).
\end{array}$$ Because astronomic observation does not show any significant gravitational redshift due to galaxy mass distributions, it is good approximation to choose $B(r )=1$. Therefore, the effective metric for spiral galaxy disks is $$d\bar s ^2 =d\tilde t^2-\left (\frac {\xi _0^2}{r^2} dr ^2 + \xi _0^2\ln ^2
\left (\frac {r}{r _0}\right ) d\phi ^2 \right ).$$ The stellar dynamics is the solutions (32), (33), and (35) with the corresponding $A(r ), B(r ), C(r )$ being substituted. It is ready for test on galaxy observations.
[**(ii) Galaxy patterns (the origin of the coordinate rescale).** ]{} The coordinate rescale origins from the study of galaxy patterns (i.e., the light distributions $\rho (x, y)$). I proposed to use curvilinear coordinate systems to study galaxy light distribution patterns with the help of a symmetry principle (He, 2003). The light pattern of spiral galaxy disks is associated with an orthogonal curvilinear coordinate system $(\lambda , \mu )$ on the disk plane and the symmetry principle is that the components of the gradient vector $\nabla f(x, y)$ associated with the local reference system of the curvilinear coordinate lines depend on single curvilinear coordinate variables $\lambda $ and $\mu $ respectively, where $f(x,y)= \ln \rho (x,y)$. The curvilinear coordinate system turns out to be the symmetrized one of the spatial part of the coordinate system (21). The coordinate system together with the symmetry principle determines the light distributions of spiral galaxy disks uniquely. This method determines all regular galaxy patterns (He, 2005a and b). The light distributions of arms can not be obtained in this manner. Arms are density waves in the coordinate space (21) and destroy the above symmetry principle.
[**(iii) Curved waves.**]{} People generally consider harmonic plane waves respective to the real Cartesian coordinate system $(x,y)$ itself, $\cos (ax+by+ct)$, where $a,b,c$ are constants. The lines of wave crests (i.e., the lines parallel to $ax+by =$ constant at fixed time $t$) cross any line of propagating direction at uniformly distributed points in the real space. Some people consider the harmonic waves respective to the polar coordinate systems $(r,\phi )$, $\cos (ar+b\phi +ct)$ whose lines of crests are curved on the real spiral galaxy disk plane and cross any line of propagating direction on the plane at uniformly distributed points too. In fact, the lines of crests are $r \propto \phi $ which express the unreal linear arms in spiral galaxies. We know that “free999 light waves (i.e., light propagation in vacuum) are straight while inhomogeneous dielectric medium curves the waves. Similarly, the density waves (arms) in spiral galaxies experience inhomogeneous gravitational fields and they have logarithmic curvatures: $\ln r \propto \phi $, that is, the crest lines of the density waves cross any line of propagating direction at unevenly distributed points in the real space. Therefore, we need to rescale the radial lines from the galaxy centers, $\xi = \xi _0 \ln (r/r _0)$, to obtain a new coordinate systems $(u=\xi \cos \phi ,v=\xi \sin \phi )$. The harmonic plane waves $$\cos (a\xi +b\phi +ct),$$ respective to the new polar coordinate system $(\xi , \phi )$ present logarithmic curvatures in real spaces of spiral galaxy disks, $\xi \propto \ln r \propto \phi $. The crest lines (the arms) on the real spiral disk plane cross any line of traveling direction at unevenly distributed points. Therefore, the waves which experience no gravity in the “free-fall999 curvilinear coordinate system $\xi \phi $ are the physical waves which experience gravity in the real rectangular Cartesian space.
[**(iv) A model of galactic rotation curves.** ]{} The solution (33) of the polar-angle component equation suggests a model of galactic rotation curves: $$V(r)=r\frac {d\phi }{dt}=rJB(r)/C(r)$$ with $B(r)\equiv 1$. I review the model presented in He (2005a) in the following. Except the constants of the motion, all other parameters from the effective metric of a specific galaxy disk are determined by the gravitational field of the background disk and are identical for all stars from the disk. The constant of motion $J$ has different values for different stars. But its averaged value $\bar J$ by all stars is a constant of the galaxy and does not depend on the radial distance $r$ of the galaxy. Finally we have a rotation curve model for spiral galaxy disks: $$V(r)=r\frac {d\theta }{dt}=r\bar J/(\xi _0^2\ln ^2(r/r _0)).$$ This rotation curve of pure spiral galaxy disks never decreases outwards. Instead, the model predicts a final rise of the curves at large distances from the galaxy centers. This is consistent to the astronomic observations of some galaxies. For the other galaxies, we need the rotation speed observations of large distances from the galaxy centers and compare the data with the prediction.
For the model (40), we see a singularity near the galaxy centers, $V(r_0)=+\infty $. The calculation of Newtonian theory for pure galactic disks suggests a peaked but smooth rotation speed (Binney and Tremaine 1987; Courteau 1997). My theory suggests a singularly peaked rotation curves for pure disk galaxies. This indicates that my theory is a correction to Newtonian theory on spiral galaxies if we assume that there is no significant dark matter. Real spiral galaxies, however, are always accompanied by 3-dimensional bulges near their centers. We expect that the mass distributions of bulges pare off the singular peaks. Therefore, we multiply a function $b(r)$ to the formula (40) to give account for the contribution of spiral galaxy bulges. Because the bulges have zero contribution at far distances from the galaxy centers, we require $b(r)\rightarrow 1$ for $r \rightarrow \infty $. One of the simplest choices is the following $$b(r)=(r-r_0)^2/(r\sqrt{(r-r_0)^2+c_0^2})$$ where the numerator helps remove the singularity and the factor $r$ in the denominator results in a steep rise of the rotation curve near the galaxy center (a suggestion from the shapes of real rotation curves). The parameter $c_0$ determines the degree in which the singular peak is pared off. A subscript $b$ in the formula of the phenomenological model is used to indicate the bulge contribution besides the disk one, $$\begin{array}{l}
V_b(r)=b(r)V(r)\\
=\bar J(r-r_0)^2/(\xi _0^2\ln ^2(r/r _0)\sqrt{(r-r_0)^2+c_0^2}).
\end{array}$$ The curve fits real rotation data (He, 2005a).
[**(v) Stellar dynamics of elliptical galaxies.**]{} As shown in the following, the radial gravitational field in 3-dimensional elliptical galaxies has the reciprocal rescale $\xi=\xi_ 0r _0/r $. Therefore, $$\begin{array}{l}
p(r )= \xi _0 r _0/r, \\
A(r )= p^{\prime 2}(r )= \xi _0^2 r _0^2/r^4, \\
C(r )= p^{2 }(r )= \xi _0^2 r _0^2/r^2 .
\end{array}$$ Because astronomic observation does not show any significant gravitational redshift due to galaxy mass distributions, it is good approximation to choose $B(r )=1$. Therefore, the effective metric for elliptical galaxies is $$d\bar s ^2 =d\tilde t^2-\left (\frac {\xi _0^2r^2 _0}{r^4} dr ^2 + \frac {\xi _0^2r _0^2} {r^2} d\phi ^2 \right ).$$ The stellar dynamics is the solutions (32), (33), and (35) with the corresponding $A(r ), B(r ), C(r )$ being substituted.
[**(vi) The origin of the reciprocal rescale.**]{} He (2005b) studied elliptical galaxy patterns by employing a curvilinear coordinate system $(\lambda , \mu ,\nu )$. The corresponding 3-dimensional spatial coordinate transformation is $$\left \{
\begin{array}{l}
\lambda = x /(x ^2 +y^2+z ^2) , \\
\mu = y /(x ^2 +y^2+z ^2) , \\
\nu = z /(x ^2 +y^2+z ^2) , \\
-\infty <x, y, z <+\infty .
\end{array}
\right .$$ With the help of the above-said symmetry principle and a cut-off method, 3-dimensional light patterns are achieved and their projected light distributions on the sky plane fit real elliptical galaxy images very well. Therefore, it is good assumption that the above coordinate transformation cancels the gravitational field of elliptical galaxy mass distributions. That is, the effective metric of the coordinate space $(\lambda ,\mu ,\nu )$ is Pythagoras theorem: $$\begin{array}{l}
d\lambda ^2 +d\mu ^2 +d\nu ^2 \\
= \frac {1}{r^4}dr^2 + \frac {1}{r^2}( d\theta ^2 +\sin ^2\theta d\phi ^2 )
\end{array}$$ where $(r, \theta ,\phi )$ is the common spherical polar coordinates in the rectangular coordinate space $(x,y,z)$. The formula is the spatial part of (44) with the constants ignored, which indicates a radial reciprocal rescale. The corresponding stellar dynamics is ready for test.
Because any 2-dimensional spiral galaxy disk is always accompanied by 3-dimensional elliptical bulge, the disk dynamics (37) must be combined with the elliptical dynamics (46) to study stellar motion in spiral galaxies.
[**(v) Gravitational dynamics and pattern dynamics.**]{} By now we have studied the “gravitational999 dynamics of galaxies. Note that I added quotation marks to the word “gravitational999 because I deduced the galactic dynamics without any consultation to Newtonian theory of gravity. What I employed is the result about galaxy patterns and the principle of interactions of finite speed $(\leq c)$, a principle of special relativity. In solar gravitational dynamics, we are never worried about the speed because the sun is the dominated cause and light $c$ can be considered to be infinite. In galactic scale systems, however, the speed is infinitesimal when compared with the spatial scales. All applications of Newtonian dynamics to galactic systems encounter difficulties and resort to dark matters. Rejecting the assumption of dark matters means giving up Newtonian dynamics in galaxy study. Therefore, I would call the galactic dynamics developed in the present paper the pattern dynamics rather than gravitational dynamics. This bears a resemblance to statistical mechanics. Galaxies are the large-scale ensemble of stars in the similar way that gas is the macroscopic ensemble of atoms which demonstrates totally different thermal properties from the ones of single isolated “cold999 atom.
Nonholonomic Boost Transformation and the Gravitational Field Generated by an Isolated Point Mass
===================================================================================================
[**(i) Nonholonomic boost transformation and the effective metric of an isolated point mass $M$.**]{} The main difference of the gravitational field of an isolated point mass from the one of galaxies is that the gravitational redshift due to galaxy mass distribution is not significant while the solar gravitational redshift is significant to be observed. Because the time coefficient $g_{00}$ of the diagonal effective metric describes the redshift and is the function of spatial radial position, we expect that the corresponding effective curvature is nonzero. That is, there is no global spacetime coordinate transformation (12) which transforms the effective metric into Minkowski form (i.e., nonholonomic).
We know that Lorentz boost transformation $$\left \{
\begin{array}{ll}
x&=\frac {\xi - \beta \tilde \tau }{\sqrt{1-\beta ^2}}, \\
\tilde t&=\frac {\tilde \tau - \beta \xi}{\sqrt{1-\beta ^2}}, \\
\beta &=v/c
\end{array}
\right .$$ leads to the equations between coordinate differentials $$\left \{
\begin{array}{ll}
d\tilde t &=\frac {d \tilde \tau }{\sqrt{1-\beta ^2}}, \\
d l&=d \bar l \sqrt{1-\beta ^2 }
\end{array}
\right .$$ which are the time dilation and length contraction respectively. In the formulas, $\tilde \tau \bar l$ is the frame in which clock and length are at rest. This equation system of differentials, (48), is holonomic because it can be integrated and the resulting coordinate transformation is (47). The holonomic boost (48) (or (47)) has a boost direction which is the $x$-axis. Now I propose a boost which is nonholonomic and boosts to all spatial directions $$\left \{
\begin{array}{ll}
d \tilde t &=\frac {d \tilde \tau }{\sqrt{1-2r_g/r}}, \\
d l&=d \bar l \sqrt{1-2r_g/r}
\end{array}
\right .$$ where $r_g =GM/c^2$ and $$d l^2= dx^2+dy^2+dz^2 = dr^2+ r^2(d\theta ^2+\sin ^2\theta d\phi ^2)$$ is the spatial square distance of arbitrary direction (not necessarily the radial direction) in the real inertial spacetime $txyz$ while $d\bar l$ is the spatial distance in the curvilinear spacetime $\tau \xi \eta \zeta $ which does not present gravity. The formula (49), which resemble (48) with $2r_g/r$ playing the role of $\beta =v/c$, is considered to be a generalized boost transformation. However, it is different from the Lorentz boost. Firstly, Lorentz boost (48) has a boost direction ($x$-axis) and is holonomic (the resulting Lorentz transformation (47)) while the generalized one is nonholonomic. That is, (49) can not be integrated to give a global coordinate transformation (12). Secondly, Lorentz boost is the relationship between two inertial frames and is symmetric about the two frames while the generalized boost is the relationship between the inertial frame $txyz$ and the curvilinear coordinate system $\tau \xi \eta \zeta $. However, the generalized boost cancels the gravity of isolated point-mass. The effective metric of the isolated point mass $M$ is $$\begin{array}{ll}
d\bar s^2 &=d\tilde \tau ^2 -–
d\bar l^2 = (1-2r_g/r) d \tilde t^2 –- \frac {dl^2}{1-2r_g/r} \\
&= (1-2r_g/r) d \tilde t^2
–-
\left (\frac {dr^2}{1-2r_g/r}+\frac {r^2}{1-2r_g/r}( d\theta ^2+\sin ^2\theta d\phi ^2)\right ) \\
&=B(r )d \tilde t^2 -–
(A(r )dr^2+ C(r )(d\theta ^2+\sin ^2\theta d\phi ^2))
\end{array}$$ where $(r, \theta ,\phi )$ is the common spherical polar coordinates in the rectangular coordinate space $(x,y,z)$ and $$\begin{array}{l}
B(r )= 1-2r_g/r, \\
A(r )= \frac{1}{1-2r_g/r } , \\
C(r )= \frac{r^2}{1-2r_g/r } .
\end{array}$$ Note that $B(r )$ and $A(r )$ are exactly the ones from Schwarzschild metric in GR. However, $C(r )(=r^2/(1-2r_g/r)\simeq (r+r_g)^2 )$ is different, which means that the common angular momentum is not conserved. Instead, the shadow angular momentum is conserved. Because the distance between a point and its shadow is $2r_g= 2GM/c^2$ ($\approx $ 3.0km for the mass of sun (GR suggests a distance of $r_g$)), the phenomenon can be resolved only by high precision solar observation.
[*Note that all solar GR tests were made on its first order (in $r_g/r$) predictions. Therefore, if $B(r ), A(r )$, and $C(r )$ in (51) all have the same first order (in $r_g/r$) approximations as GR then the corresponding effective metric gives the same predictions as GR, i.e., the same predictions of the deflection of light by the sun, the precession of perihelia, and the excess delay of radar echo near the sun.* ]{} In the remaining parts of the present section I verify the assertion and provide new results.
[**(ii) Effective metrics which generalize Newtonian theory and fulfill GR‘s first-order prediction.**]{} We have shown that geometrization of gravity is not needed. While geometrization requires spacetime curvature to determine gravitational metric (Einstein equation), the effective metric of NEP has no such constraint. Without the constraint, we are free to see which kinds of effective metrics give exactly the same first order predictions as GR. Here I present examples of such effective metrics. Firstly, $B(r )$ in (52) and (51) can be any one of the following $$B_1:\: 1-2r_g/r;\: B_2:\: (1-r_g/r)^2;\: B_3:\: \frac{1}{1+2r_g/r};\: B_4:\:\exp(-2r_g/r);\:\cdots$$ where $r_g=GM/c^2,\;G$ is gravitational constant, $M$ the isolated point mass, and $c$ light speed. Secondly, $A(r )$ can be any one of the following $$A_1:\: \frac{1}{1-2r_g/r};\: A_2:\: (1+r_g/r)^2;\: A_3:\: 1+2r_g/r;\: A_4:\: \exp(2r_g/r);\:\cdots .$$ The $A(r )$ and $B(r )$ work because they have the same first order (in $r_g/r$) approximations as the Schwarschild ones in GR. I have further result. I will show that arbitrary function $C(r )$ together with the above $A(r )$ and $B(r )$ give exactly the same predictions of the deflection of light by the sun and the precession of perihelia. But the prediction on the excess radar echo delay depends on the choice of $C(r )$, i.e., depends on its first order approximation in $r_g/r$. Therefore, in the following examples of $C(r )$ $$C_1:\: r^2;\: C_2:\: \frac{r^2}{1-2r_g/r};\: C_3:\: r^2(1+jr_g/r)^2;\: C_4:\: r^2\exp(jr_g/r);\:\cdots$$ where $j (\not= 0)$ is arbitrary constant, only the first one gives the same first order prediction on excess radar echo delay as GR. The nonholonomic boost metric corresponds to $j=1$ ($C_2$).
For verification of the assertion, following the formulas of Weinberg (1972) saves much time. Therefore, the following notes need to be read with the book and any three number set with parentheses in the following, e.g., (8.5.6), refers to the formulas in the book. The sole difference of NEP calculation from GR is to change $r^2$ to $C(r )$ in the formulas. For simplicity, I choose $C(r )$ to be $C_3$ in (55) $$C(r )=r^2(1+jr_g/r)^2$$ to verify the assertion. To begin, I note that the roughest approximation to the dynamical equations (32), (33), and (35) depends only on $B(r )$ because the only term which involves the large number $c$ is the third term in (35). Because all $B_i$ in (53) have the same first-order approximation $$1-2r_g/r,$$ this approximation of the dynamical equations gives $$\begin{array}{c}
r^2 \frac{ d\phi }{dt} \simeq J \\
\frac{1}{2} (\frac{ dr }{dt} )^2 +\frac {J^2}{2r^2}-\frac{GM}{r} \simeq \frac{c^2-E}{2}
\end{array}$$ which are the formulas between (8.4.20) and (8.4.21) in Weinberg (1972). These are the same equations of the Newtonian theory with $(c^2-E)/2$ playing the role of energy per unit mass. Therefore, $B(r )$ alone decides if the dynamics contains Newtonian theory as limiting case.
[**(iii) First-order (in $r_g/r_0$) prediction in the deflection of light by the sun.**]{} Inserting the formulas (33), (53), (54), and (56) into (35) gives test particle‘s orbital motion around an isolated point mass $M$ (the formula (8.4.30) in Weinberg (1972)), $$\phi (r)=\pm\int \frac {A^{1/2}(r ) dr}{C(r ) \left ( \frac {1}{J^2B(r )}-
\frac {E}{J^2}-\frac{1}{C(r )}\right )^{1/2}}.$$ Formula (8.5.6) is the following with $E=0$ in the case of light, $$\phi (r)-\phi (\infty )=\int ^\infty _rA^{1/2}(r )\cdot \left ( \frac {C(r )}{C(r_ 0)}
\frac {B(r_0 )}{B(r)}-1\right )^{-1/2}\cdot \frac{1}{\sqrt{C(r )}}dr .$$ The argument of the second square root, in the first order approximation of $r_g/r_0$ where $r_0$ is the closest approach to the central isolated point-mass (the sun), is $$\frac {C(r )}{C(r_ 0)}\frac {B(r_0 )}{B(r)}-1=
\left (\frac {r^2}{r_ 0^2}-1\right )\left (1- \frac {2 (1+j)r_g r}{r_0(r+r_0)}+\cdots \right ).$$ Therefore, $$\begin{array}{ll}
\phi (r)-\phi (\infty )&=\int ^\infty _r \frac {dr}{r\sqrt{r^2/r_ 0^2-1} }
\left (1+\frac{(1-j)r_g }{r}+ \frac { (1+j)r_g r}{r_0(r+r_0)} +\cdots \right )\\
&=\sin^{-1}\left ( \frac {r_ 0}{r}\right )+\frac {r_ g}{r_0}\left ((1-j)+(1+j)-(1-j) \sqrt{1-\frac{r_0^2}{r^2}}-(1+j) \sqrt{\frac{r-r_ 0}{r+r_ 0}}\right )+\cdots
\end{array}$$ We see a cancellation of $j$ on the first two terms after $r_g/r_0$ in the last result (comparing (8.5.7)). Therefore, the deflection of the orbit from a straight line $$\Delta \phi =2|\phi (r_ 0)-\phi _\infty |-\pi =\frac{4r_ g}{r_ 0}$$ remains unchanged. I have proved the first part of my assertion.
[**(iv) First-order (in $r_g/r_0$) prediction in the precession of perihelia.**]{} The formula (8.6.3) is $$\begin{array}{l}
\phi (r)-\phi (r_- )=\int ^r_{r_-}dr A^{1/2}(r )C^{-1}(r ) \\
\cdot \left ( \frac { C(r_-)(B^{-1}(r )-B^{-1}(r_- ))- C(r_+)(B^{-1}(r )-B^{-1}(r_+ ))
}
{ C(r_+)C(r_-)(B^{-1}(r_+ )-B^{-1}(r_- )) } -\frac{1}{C(r )}\right )^{-1/2}.
\end{array}$$ We consider the first-order approximation of the argument of the second square root. Note that $$B^{-1}(r ) = \bar B^{-1}(u ) =1+\frac{2r_g}{u}+\frac{2(2+j)r^2_g}{u^2} +\cdots$$ where $u=r+jr_g $. Because the first terms $1$ in (65) are completely canceled in (64), the second-order approximation of $\bar B^{-1}(u )$ is needed to achieve a first order approximation of the argument. Note that $C(r )=u^2$. Therefore, the first order approximation of the argument is an algebraic polynomial of $1/u$ of order two. Furthermore, it vanishes at $u_{\pm }(=r_{\pm }+jr_g )$, so $$\begin{array}{l}
\frac { u_-^2(\bar B^{-1}(u )- \bar B^{-1}(u_- ))- u_+^2(\bar B^{-1}(u )- \tilde B^{-1}(u_+ ))}
{ u_+^2u_-^2(\bar B^{-1}(u_+ )- \bar B^{-1}(u_- )) } -\frac{1}{u^2} \\
= D\left ( \frac {1}{u_-}-\frac {1}{u} \right )\left (\frac {1}{u}- \frac {1}{u_+} \right )
\end{array}$$ The constant $D$ can be determined by letting $u \rightarrow \infty $: $$D= \frac { u_+^2(1-\bar B^{-1}(u_+ ))- u_-^2(1-\bar B^{-1}(u_- ))}
{ u_+u_-(\bar B^{-1}(u_+ )- \bar B^{-1}(u_- )) }.$$ Because $1/u=1/r -jr_g/r^2$ and $jr_g/r^2 \ll r_g/r_0$, we have, in first order approximation of $r_g/r$, $$D=1-(2+2j)r_g\left ( \frac{1}{r_+}+ \frac{1}{r_-}\right ).$$ Similarly, the factor on the right hand side of (66) can be approximated by
$$\left ( \frac {1}{u_-}-\frac {1}{u} \right )\left (\frac {1}{u}- \frac {1}{u_+} \right )\simeq
\left ( \frac {1}{r_-}-\frac {1}{r} \right )\left (\frac {1}{r}- \frac {1}{r_+} \right ).$$
By introducing a new variable $\psi $, we have finally $$\begin{array}{ll}
\phi(r )-\phi(r_-)&\simeq \left [1+\frac{1}{2}(2+2j)r_g \left ( \frac{1}{r_+}+ \frac{1}{r_-} \right ) \right ]\int^r_{r_-}\frac {(1+(1-2j)r_g /r)dr }{
r^2 [( \frac {1}{r_-}-\frac {1}{r} )(\frac {1}{r}- \frac {1}{r_+} )]^{1/2} }\\
&=[1+\frac{1}{2}(2+2j +1-2j)r_g ( \frac{1}{r_+}+ \frac{1}{r_-} )][\psi +\frac{\pi }{2} ]\\
&-\frac{1}{2}(1-2j)r_g ( \frac{1}{r_+}- \frac{1}{r_-} )\cos \psi .
\end{array}$$ Again we see a cancellation of $j$ in the final result and the precession in one revolution remains unchanged $$\Delta \phi =\frac {6\pi GM}{L} \; {\rm radians/revolution }$$ where $$\frac{1}{L}=\frac{1}{2}\left ( \frac{1}{r_+}+ \frac{1}{r_-} \right ) .$$
[**(v) First-order (in $r_g/r_0$) prediction in the radar echo excess delay.**]{} However, $j$ is not cancelled in the formula of excess radar echo delay and the first-order approximation in $r_g/r_0$ is different from that of GR. The formula of the time required for light to go from $r_0$ to $r$ or from $r$ to $r_0$ is (8.7.2) $$t(r, r_0)= \frac {1}{c}\int ^r_{r_0} \left (\frac{A(r )/B(r ) }{
1- \frac {B(r )}{B(r_0)} \frac {C(r_0 )}{C(r)} } \right )^{1/2} dr .$$ In first-order approximation, the denominator is $$1- \frac {B(r )}{B(r_0)} \frac {C(r_0 )}{C(r)}
\simeq \left (1- \frac {r_0^2}{r^2}\right )\left (1-\frac {2(1+j)r_g r_0}{r(r+r_0)}\right ).$$ Therefore, $$\begin{array}{ll}
t(r, r_0)&\simeq \frac {1}{c}\int ^r_{r_0}\left (1-\frac {r_0^2}{r^2}\right )^{-1/2}\left (1+\frac{2r_g}{r} + \frac{(1+j)r_gr_0}{r(r+r_0)} \right)dr\\
&= \left (\sqrt{r^2-r^2_0}+ 2r_g\ln\frac{r+\sqrt{r^2-r^2_0} }{r_0}
+(1+j)r_g (\frac{r-r_0}{r+r_0})^{1/2}\right )/c.
\end{array}$$ The leading term $\sqrt{r^2-r_0^2}/c$ in the formulas is what we should expect if light traveled in straight lines at its speed $c$. The remaining terms in the formulas produce NEP delay and GR $(j=0)$ delay respectively in the time it takes a radar signal to travel to Mercury and back. These excess delays are maximums when Mercury is at superior conjunction and the radar signal just grazes the sun. Detailed calculation (Weinberg, 1972) shows that the maximum round-trip excess delay is $$(\Delta t)_{{\rm max }}\simeq 19.7 (1+j+11.2) \; \mu \,{\rm sec }.$$ The nonholonomic boost metric corresponds to $j=1$ and its prediction is $(\Delta t)_{{\rm max }}\simeq 260 \mu \,{\rm sec } $ while the corresponding result of GR $(j=0)$ is 240 $\mu$sec. The difference is less than 8 percents.
However, there is difficulty in their tests. We can transmit radar signals to Mercury at its series of orbital positions around the event of superior conjunction. The time for single round-trip is many minutes and an accuracy of the order of 0.1 $\mu $sec can be achieved (Anderson [*et al.*]{}, 1975). In order to compute an excess time delay, we have to know the time $t_0$ that the radar signal would have taken in the absence of the sun‘s gravitation to that accuracy. This accuracy of time corresponds to an accuracy of 15 meters in distance. This presents the fundamental difficulty in the above test. In order to have a theoretical value of $t_0$, Shapiro‘s group proposed to use GR itself to calculate the orbits of Mercury as well as the earth (Shapiro, 1964; Shapiro [*et al.*]{}, 1968; Shapiro [*et al.*]{}, 1971). The data of time for the above series of real round-trips minus the corresponding theoretical values of $t_0$ presents a pattern of excess time delay against observational date and was fitted to the excess delay calculated by the formula (75) with a fitting parameter $\gamma $. The group and the following researchers found that GR, among other similar theories of gravity represented by $\gamma $, fits the pattern best.
Now I have a NEP gravitational theory of nonholonomic boost. Because angular momentum is not conserved, the orbits and orbital motion are much different. It would be very interesting to use the new theory to test the same radar echo data. That is, we use the new gravitational dynamics to calculate the orbits of Mercury and the earth to achieve the above-mentioned theoretical values $t_0$. Similarly we add the same parameter $\gamma $ to the nonholonomic boost metric (52) and obtain a corresponding formula to (75). If the NEP metric fits the above-said pattern best then we can say that it is a competing theory to GR. However, the actual values of excess time delays are as difficult to resolve as the shadow angular momentum conservation of solar mass.
[**(vi) Recovery of Schwarzschild metric.**]{} We can recover Schwarzschild metric by combining a spatial radial translational coordinate rescale to the above nonholonomic boost transformation (49). That is, $txyz$ in (49) is considered to be curvilinear coordinate system. The real rectangular coordinate system in an inertial frame is $TXYZ$ and the spatial radial translational coordinate rescale is $r=p(R )=R+mr_g$ where $m$ is another arbitrary constant. Therefore, $ B_2(R )=1,\, A_2(R )= p^{\prime 2}(R )= 1,\, C_2(R )= p^{2 }(R )= (R+mr _g)^2$. Combined with the nonholonomic boost, $ B_1(r )=1-2r_g/r,\, A_1(r )=1/(1-2r_g/r),\, C_1(r )=r^2/(1-2r_g/r)$, finally we have $$\begin{array}{ll}
d\tilde \tau ^2&=B_1(r )d\tilde t^2 \simeq (1-2 r_g/R)d\tilde T^2, \\
d\bar l^2 &=A_1(r )dl^2\simeq \frac {dl^2 }{1-2 r_g/R} \\
&=\frac {dR^2 }{1-2 r_g/R}+\frac {(R+mr_g)^2 }{1-2 r_g/R}( d\theta ^2+\sin ^2\theta d\phi ^2) .
\end{array}$$ Therefore, $$\begin{array}{l}
B(R )\simeq 1-2 r_g/R, \\
A(R )\simeq \frac {1 }{1-2 r_g/R}, \\
C(R )\simeq (R+2(m+1)r_g )^2.
\end{array}$$ It can be seen that Schwarzschild metric is recovered when choosing $m=-1$.
Conclusion
============
It is shown that Einsten‘s geometrization fails to describe homogeneous gravity. A new equivalence principle (NEP) is proposed. Spacetime is always flat with Minkowski metric. A tensor $g _{\alpha \beta } $ which is covariant with respect to all curvilinear coordinate transformations (including non-curvilinear Lorentz transformations) is defined on the flat spacetime, which describes gravity and has no geometric meaning. Test particles follow the solution of the corresponding effective geodesic equation. Static gravity can be canceled by holonomic or nonholonomic coordinate transformations.
I proposed galaxy pattern dynamics which explains galaxy rotation curves successfully. That is, I deduced the stellar dynamics without any consultation to Newtonian gravitational dynamics. Instead, I employed the results about galaxy patterns and the principle of interactions of finite speed $(\leq c)$, a principle of special relativity. In solar gravitational dynamics, we are never worried about the speed $c$ because it can be considered to be infinite. In galactic scale systems, however, the speed is infinitesimal compared with the large spatial scales. All applications of Newtonian dynamics to galactic systems encounter difficulties and resort to dark matters. Rejecting dark matters means the abandonment of Newtonian dynamics in galaxy study. Therefore, I would call the galactic dynamics developed in the present paper the pattern dynamics rather than gravitational dynamics. This bears a resemblance to statistical mechanics. Galaxies are the large-scale ensemble of stars in the similar way that gas is the macroscopic ensemble of atoms which demonstrates totally different thermal properties from the ones of single isolated “cold999 atom.
Also I proposed to use nonholonomic boost to generalize Newtonian dynamics. This generalization shares the properties of Lorentz transformation, which should help quantize gravity. I explored its classical solar application. The corresponding effective metric is different from that of Schwarzschild. To first order, its prediction on the deflection of light and the precession of the perihelia of planetary orbits is the same as the one of general relativity (GR). Its further implication is left for future work.
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|
---
abstract: 'As modern architectures introduce additional heterogeneity and parallelism, we look for ways to deal with this that do not involve specialising software to every platform. In this paper, we take the Join Calculus, an elegant model for concurrent computation, and show how it can be mapped to an architecture by a Cartesian-product-style construction, thereby making use of the calculus’ inherent non-determinism to encode placement choices. This unifies the concepts of placement and scheduling into a single task.'
author:
- Peter Calvert
- Alan Mycroft
title: Mapping the Join Calculus to Heterogeneous Hardware
---
Introduction {#sec:introduction}
============
The [*Join Calculus*]{} was introduced as a model of concurrent and distributed computation [@Fournet1996]. Its elegant primitives have since formed the basis of many concurrency extensions to existing languages—both functional [@Conchon1999; @Odersky2000] and imperative [@Benton2004; @vonItzstein2005]—and also of libraries [@Russo2007]. More recently, there has also been work showing that a careful implementation can match, and even exceed, the performance of more conventional primitives [@Turon2011].
However, most of the later work has considered the model in the context of shared-memory multi-core systems. We argue that the original Join Calculus assumption of [*distributed computation*]{} with disjoint local memories lends itself better to an automatic approach. This paper adapts our previous work on Petri-nets [@Calvert2011] to show how the non-determinism of the Join Calculus can express placement choices when mapping programs to heterogeneous systems; both data movement between cores and local computation are seen as scheduling choices. Thus we argue that a JVM-style bytecode with Join Calculus primitives can form a universal intermediate representation, that not only models existing concurrency primitives, but also adapts to different architectures at load-time.
This paper introduces our construction by considering the following Join Calculus program that sorts an array of integers using a [*merge-sort*]{}-like algorithm. There is clearly scope for parallelising both the `split` and `merge` steps—although this may require moving data to another memory.
\
` def sort(numbers, k) =`\
` let N = numbers.length in`\
` def split(a) =`\
` let n = a.length`\
` in if n == 1 then merge(a)`\
` else split(a[0..(n/2)-1]) & split(a[(n/2)..(n-1)])`\
` merge(a) & merge(b) =`\
` if a.length + b.length == N then do_merge(a, b, k)`\
` else do_merge(a, b, merge)`\
` in split(numbers)`
where `do_merge` is a functional-style procedure that merges the sorted arrays `a` and `b` into a new sorted array that is passed to its continuation (`k` or `merge`). We assume moderate familiarity with Join Calculus primitives. In particular, we will:
- Restrict the Join Calculus to make all data usage explicit, showing that existing programs can be desugared into this form (Section \[sec:non-nested\]).
- Briefly show how our existing work manifests itself in the Join Calculus (Section \[sec:construction\]).
- Introduce [*workers*]{} to the Join Calculus semantics as a substitute for the [*resource constraints*]{} in the Petri-net version (Section \[sec:workers\]).
We offer a discussion of the scheduling issues and how we believe these to be tractable in Section \[sec:scheduling\], before concluding in Section \[sec:conclusion\].
The Non-Nested Join Calculus {#sec:non-nested}
============================
As in our previous work, our construction introduces explicit data transfer transitions. For these to cover all required data transfers, we disallow references to free variables which may be located on other processors—i.e. values that are not received as part of the transition’s left-hand-side join pattern. Unfortunately, nested Join Calculus definitions capture values in this way. In our running example, observe that both `N` and `k` are used implicitly by `merge`.
Our formulation of the Join Calculus therefore forbids the nesting of definitions. Instead, programs consist of a list of definitions. This necessitates a special type of signal, [*constructors*]{}, that are used to create and initialise a join definition. A new version of our program is shown in box “A” of Figure \[fig:code\].
\
`definition {`\
` `[`(tlA)` `;`]{}` .ctor sort_x(numbers, k) { `[`(tlB)` `;`]{}` .ctor sort_y(numbers, k) {`\
` split_x(numbers); split_y(numbers);`\
` info_x(numbers.length, k); info_y(numbers.length, k);`\
` } }`\
` `\
` info_x(N, k) { info_x(N,k); info_x(N,k); } info_y(N, k) { info_y(N,k); info_y(N,k); }`\
` `\
` split_x(a) { split_y(a) {`\
` let n = length(a); let n = length(a);`\
` `\
` if(n == 1) { merge_x(a); } if(n == 1) { merge_y(a); }`\
` else { split_x(a[0..(n/2)-1]); else { split_y(a[0..(n/2)-1]);`\
` split_x(a[(n/2)..(n-1)]); } split_y(a[(n/2)..(n-1)]); }`\
` } }`\
` `\
` merge_x(a) & merge_x(b) & info_x(N, k) { merge_y(a) & merge_y(b) & info_y(N, k) {`\
` info_x(N, k); info_y(N, k);`\
` if(a.length + b.length == N) if(a.length + b.length == N)`\
` do_merge(a, b, k); do_merge(a, b, k);`\
` else do_merge(a, b, merge_x); else do_merge(a, b, merge_y);`\
` } }`\
` `[`(brA)` `;`]{}` `[`(brB)` `;`]{}\
` `[`(tlC)` `;`]{}` split_x(a) { split_y(a); } split_y(a) { split_x(a); }`\
` merge_x(a) { merge_y(a); } merge_y(a) { merge_x(a); }`\
` info_x(N, k) { info_y(N, `“`k on y`”`); } info_y(N, k) { info_x(N, `“`k on x`”`); }`\
`} `[`(brC)` `;`]{}[`(tlA)` `rectangle` `(brA);` `at` `(brA)` `[fill=black,` `text=white,` `above` `left]` `;`]{}[`(tlB)` `rectangle` `(brB);` `at` `(brB)` `[fill=black,` `text=white,` `above` `left]` `;`]{}[`(tlC)` `rectangle` `(brC);` `at` `(brC)` `[fill=black,` `text=white,` `above` `left]` `;`]{}
However, despite this restriction, nested definitions can easily be encoded by a process similar to both [*lambda-lifting*]{} and Java’s [*inner-classes*]{}. In particular, any program similar to:
\
` a(x,k) {`\
` definition { .ctor Nested() { k(f); }`\
` f(m) { m(x 2); } }`\
` construct Nested();`\
` }`
can be rewritten in a similar-style to:
\
` definition { .ctor UnNested(x,k) { temp(x); k(f); }`\
` temp(x) { temp(x); temp(x); }`\
` f(m) & temp(x) { temp(x); m(x 2); } }`\
` a(x,k) { construct UnNested(x,k); }`
Unfortunately, the extra signal would cause serialisation of many transitions within the definition. This is resolved by the [*duplication*]{} transition that allows us to create as many copies of the `x` message as we require. We rely on the scheduler not to perform excessive duplications. We might also be able to optimise this ‘peek’ behaviour in our compiler.
As we will later build on our previous work involving Petri-nets [@Calvert2011], it is worth highlighting Odersky’s discussion [@Odersky2000] on the correspondence between the Join Calculus and (coloured) Petri-nets. Just as a Petri-net transition has a fixed multi-set of [*pre-places*]{}, each transition in the Join Calculus has a fixed [*join pattern*]{} defining its input signals. The key difference is that the Join Calculus is higher-order, allowing signals to be passed as values, and for the output signals to depend on its inputs—unlike Petri-nets where the [*post-places*]{} of a transition are fixed. This simple modification allows use of continuations to support functions. Moreover, while nets are static at runtime, a Join Calculus program can create new instances of definitions (containing signals and transition rules) at runtime, and although these cannot match on existing signals, existing transitions can send messages to the new signals.
Mapping Programs to Heterogeneous Hardware {#sec:construction}
==========================================
We will use the same simple hardware model as in our previous work. This considers each processor to be closely tied to a local memory. It then defines interconnects between these. The construction itself will be concerned with a finite set of [*processors*]{} $P$, a set of directed [*interconnects*]{} between these $I \subseteq P \times P$, and a [*computability*]{} relation $C
\subseteq P \times R$ (where $R$ is the set of transition rules in the program), such that $(p,r) \in C$ implies that the rule $r$ can be executed on the processor $p$. In our example, we take $P = \{\mathtt{x}, \mathtt{y}\}$, $I =
\{(\mathtt{x}, \mathtt{y}), (\mathtt{y}, \mathtt{x})\}$ and a computability relation equal to $P \times R$. However, it is easy to imagine more complex scenarios—for instance, if one processor lacked floating point support, $C$ would not relate it to any transitions using floating point operations.
A scheduler will also need a cost model, however this is not needed for this work. We would expect an affine (i.e. latency plus bandwidth) cost for the interconnect. In practice, this and the approximate cost of each transition on each processor would be given by profiling information.
There are two parts to our construction. Firstly, we produce a copy of the program for each $p \in P$, omitting any transition rules $r$ for which $(p,r)
\not\in C$, giving box “B” of Figure \[fig:code\].
Secondly, we add transitions that correspond to possible data transfers (box “C”). This requires one rule per signal and interconnect pair. However, the higher-order nature of the Join Calculus means these need more careful definition than in our Petri-net work to preserve locality. Specifically, when a signal value such as `k` is transferred it needs to be modified so that it becomes local to the destination processor. This maintains the invariant that the ‘computation transitions’ introduced by the first part of the construction can only call signals on the same processor.
Workers in place of Resource Constraints {#sec:workers}
========================================
In the Petri-net version of this work, there was a third part to the construction. We introduced [*resource constraint places*]{} to ensure that each processor or interconnect only performed one transition at once. Equivalent signals would be illegal in the Join Calculus, as they would need to be matched on by transitions from multiple definition instances (since processor time is shared between these). Changing the calculus to allow this would make it harder to generate an efficient implementation. Instead, we introduce the notion of [*workers*]{} to the semantics.
Rather than allowing any number of transition firings to be mid-execution at a given time, we restrict each worker to performing zero or one firing at a time. We also tag each transition with the worker that may fire it. In our example, we would have four workers: `x`, `y`, `(x,y)` and `(y,x)`. The `_x` and `_y` copies of the original program are tagged with the `x` and `y` CPU workers respectively, while the data transfer transitions are tagged with the relevant interconnect worker.
To accommodate vector processors such as GPUs, we augment $n$ copies of an existing transition with a single merged transition. The new transition will take significantly less time than performing the $n$ transitions individually. Obviously, a real implementation will not enumerate these merged transitions, but we can view it this way in the abstract. A similar argument also applies to data transfers, where we can benefit from doing bulk operations.
This gives the formal semantics for our calculus as defined in Figure \[fig:semantics\]. We give this for an abstract machine, however just as Java trivially compiles to the JVM, our non-nested language can trivially be compiled to this JCAM. We also use a small step semantics rather than the ChAM [@Fournet1996] or rewriting [@Odersky2000] style used previously, as this is more appropriate for our ongoing work on analysis and optimisation. Each of the workers can be either processing a transition rule, or `IDLE`. The initial state is for all workers to be `IDLE`, and some messages corresponding to program arguments to be available in $\Gamma$.
**Domains**: $$\begin{aligned}
(f,t), (f, \theta) \in \mathrm{SignalValue} &= \mathrm{Signal} \times \mathrm{Time}\\
\Gamma \in \mathrm{Environment} &= \textbf{m}(\mathrm{SignalValue} \times \mathrm{Value}^{*}) & \text{(messages available)}\\
t \in \mathrm{Time} &= \mathbb{N}_0\\
\Sigma \in \mathrm{GlobalState} &= \mathrm{Worker} \rightarrow (\mathrm{LocalState} \cup \{\mathtt{IDLE}\})\\
(l, \theta, \sigma) \in \mathrm{LocalState} &= \mathrm{Label} \times \mathrm{Time} \times \mathrm{Value}^{*} & \text{(program counter, context, local stack)}\\
v \in \mathrm{Value} &= \mathrm{SignalValue} \cup \mathrm{Primitive}
\end{aligned}$$
**Rules** (judgement form of $\Gamma, t, \Sigma \rightarrow \Gamma', t', \Sigma'$): $$\begin{aligned}
\Gamma + \Delta,\; t,\; \Sigma + \{w \mapsto \mathtt{IDLE}\} &\rightarrow \Gamma,\; t,\; \Sigma + \{w \mapsto (l_0, \theta, \vec{v_1} \cdot \ldots \cdot \vec{v_n})\} & \text{(fire)}\\
\shortintertext{where $\Delta = \{((f_1, \theta), \vec{v_1}), \ldots, ((f_n, \theta), \vec{v_n})\}$ and $r^w = f_1(\ldots) \& \ldots \& f_n(\ldots) \{ l_0, \ldots \}$}
\Gamma,\; t,\; \Sigma + \{w \mapsto (\texttt{EMIT}^l, \theta, \vec{v} \cdot s \cdot \sigma)\} &\rightarrow \Gamma + (s, \vec{v}),\; t,\; \Sigma + \{w \mapsto (\mathrm{next}(l), \theta, \sigma)\} & \text{(emit)}\\
\Gamma,\; t,\; \Sigma + \{w \mapsto (\texttt{CONSTRUCT<}f\texttt{>}^l, \theta, \vec{v} \cdot \sigma)\} &\rightarrow \Gamma + ((f,t), \vec{v}),\; t+1,\; \Sigma+\{w \mapsto (\mathrm{next}(l), \theta, \sigma)\} & \text{(construct)}\\
\Gamma,\; t,\; \Sigma + \{w \mapsto (\texttt{LOAD.SIGNAL<}f\texttt{>}^l, \theta, \sigma)\} & \rightarrow \Gamma,\; t,\; \Sigma + \{w \mapsto (\mathrm{next}(l), \theta, (f,\theta) \cdot \sigma)\} & \text{(load)}\\
\Gamma,\; t,\; \Sigma + \{w \mapsto (\texttt{FINISH}^l, \theta, \sigma)\} &\rightarrow \Gamma,\; t,\; \Sigma+\{w \mapsto \mathtt{IDLE}\} & \text{(finish)}
\end{aligned}$$
Unmapped programs can be considered to have just a single worker.
Future Work on Scheduling {#sec:scheduling}
=========================
As before, we rely on a scheduler to be able to make non-deterministic choices corresponding to the fastest execution—and clearly these need to be made quickly. In this section, we briefly discuss our thoughts on this problem.
It is clear that to optimise the expected execution time, we need transition costs, and also a probability distribution for the output signals of a transition. We believe that these could be effectively provided by profiling. This is already commonly used in auto-tuning (i.e. transition costs) and branch predictors (i.e. signal probabilities). It could also be used for [*ahead-of-time scheduling*]{} or just for determining baselines.
For practical scheduling, it is most likely that a form of machine learning will be used to adapt to new architectures. This has been used successfully for streaming applications [@Wang2010], which are not dissimilar to a very restricted Join Calculus. Existing implementations of the Join Calculus have not considered the scheduling problem, and simply pick the first transition found to match. In order to maintain this simplicity, we would consider whether the output of such a learning algorithm could be a priority list of transitions, that evolves over time to offer some load balancing.
Prior work has shown it is best to check for transition firings each time a message is emitted, rather than having a separate firing process [@Turon2011]. This will result in a queue of transitions (perhaps picked by a first-found approach). For load balancing, idle workers could then [*steal*]{} from these transition queues. However, unlike in standard [*work stealing*]{}, there are two possible forms of stealing—as well as taking a matched transition, individual messages could be taken by first decomposing some of the existing matches.
Conclusions {#sec:conclusion}
===========
In this paper, we have adapted our existing work on mapping Petri-net programs to heterogeneous architectures to the Join Calculus. In doing so, we showed how to remove the problematic environments introduced by nested definitions, and also avoid global matching on resource signals by modifying the semantics slightly to incorporate [*workers*]{}. This allows programs to be agnostic of the architecture they will run on, with any placement and scheduling choices that depend on the architecture being left in the program. We have also listed several challenges for building a scheduler that can optimise over these choices, and initial ideas to solve them. Such an implementation is our current research goal.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank the Schiff Foundation, University of Cambridge, for funding this research through a PhD studentship; the anonymous reviewers for useful comments and corrections; and also Tomas Petricek for useful discussions about these ideas.
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|
---
abstract: 'The aim of this paper is to analyze a mixed discontinuous Galerkin discretization of the time-harmonic elasticity problem. The symmetry of the Cauchy stress tensor is imposed weakly, as in the traditional dual-mixed setting. We show that the discontinuous Galerkin scheme is well-posed and uniformly stable with respect to the mesh parameter $h$ and the Lamé coefficient $\lambda$. We also derive optimal a-priori error bounds in the energy norm. Several numerical tests are presented in order to illustrate the performance of the method and confirm the theoretical results.'
author:
- '[Antonio Márquez]{}[^1], $\,\,$ [Salim Meddahi]{}[^2] and [Thanh Tran]{}[^3]'
title: 'A mixed discontinuous Galerkin method for the time harmonic elasticity problem with reduced symmetry [^4]'
---
**Mathematics Subject Classification.** 65N30, 65N12, 65N15, 74B10
**Keywords.** Mixed elasticity equations, finite elements, discontinuous Galerkin methods, error estimates.
Introduction {#section:1}
============
In this paper we are interested in the dual-mixed formulation of the elasticity problem with weakly imposed symmetry. We introduce and analyze a mixed interior penalty discontinuous Galerkin (DG) method for the elasticity system in time-harmonic regime. The interior penalty DG method can be traced back to [@arnoldIP; @douglas] and its application for elliptic problems is now well understood; see [@DiPietroErn] and the references cited therein for more details. The mixed interior penalty method introduced here can be viewed as a discontinuous version of the Arnold-Falk-Winther div-conforming finite element space [@afw-2007]. It approximates the unknowns of the mixed formulation, given by the Cauchy stress tensor and the rotation, by discontinuous finite element spaces of degree $k$ and $k-1$ respectively. This permits one to enjoy the well-known flexibility properties of DG methods for $hp$-adaptivity and to implement high-order elements by using standard shape functions. Moreover, our scheme is immune to the locking phenomenon that arises in the nearly incompressible case.
The first step in our study of the mixed DG scheme consists in providing a convergence analysis for the corresponding div-conforming Galerkin method based on the Arnold-Falk-Winther element. We point out that there are many finite element methods for the mixed formulation of the elasticity problem with reduced symmetry [@afw-2007; @BoffiBrezziFortin; @bernardo1; @gopala; @st]. All of them have been analyzed in the static case, i.e., in the case $\omega=0$ in problem below. In time harmonic regime, the operator underlying the mixed formulation is not Fredholm of index zero as in the classical displacement-based formulation. The same challenge is encountered when analyzing the curl-conforming variational formulation of the Maxwell system [@buffa; @gatica2]. Actually, the abstract theory given in [@buffa] can also be applied to the dual-mixed variational formulation of linear elasticity as shown (implicitly) in the analysis given in [@gatica1] for a fluid-solid interaction problem. Instead of using this approach, we take here advantage of the recent spectral analysis obtained in [@MMR] to directly deduce the stability of the Arnold-Falk-Winther finite element approximation of the indefinite elasticity problem.
An interior penalty discontinuous Galerkin method has also been introduced in [@dominik1] for the Maxwell system. The DG formulation we are considering here is, in a certain sense, its counterpart in the $\textrm{H}({\mathop{\mathrm{div}}\nolimits})$-setting. Notice that, in contrast to [@dominik1], our approach does not rely on a duality technique. We prove the convergence of the DG scheme by exploiting the stability of the corresponding div-conforming method and without requiring further regularity assumption than the one needed to write properly the right-hand side of below. Moreover, if the analytic Cauchy stress tensor, its divergence and rotation belong to a Sobolev space with regularity exponent $s>1/2$, then it is shown that the error in the DG-energy norm converges with the optimal order $O(h^{\min(s, k)} )$ with respect to the mesh size $h$ and the polynomial degree $k$.
The paper is organized as follows. In Section \[section:2\], we recall the dual formulation of the linear elasticity problem with reduced symmetry and prove its well-posedness when the wave number is different from a countable set of singular values. In Section \[section:3\] we prove the convergence of the conforming Galerkin scheme based on the Arnold-Falk-Winther element. In Section \[section:4\], we introduce the mixed interior penalty discontinuous Galerkin method and its convergence analysis is carried out in Section \[section:5\]. Finally, in Section \[section:6\] we present numerical results that confirm the theoretical convergence estimates.
We end this section with some of the notations that we will use below. Given any Hilbert space $V$, let $V^3$ and $V^{{3\times 3}}$ denote, respectively, the space of vectors and tensors of order $3$ with entries in $V$. In particular, ${\boldsymbol{I}}$ is the identity matrix of ${\mathbb{R}}^{{3\times 3}}$ and $\mathbf{0}$ denotes a generic null vector or tensor. Given ${\boldsymbol{\tau}}:=(\tau_{ij})$ and ${\boldsymbol{\sigma}}:=(\sigma_{ij})\in{\mathbb{R}}^{{3\times 3}}$, we define as usual the transpose tensor ${\boldsymbol{\tau}}^{{\mathtt{t}}}:=(\tau_{ji})$, the trace ${\mathop{\mathrm{tr}}\nolimits}{\boldsymbol{\tau}}:=\sum_{i=1}^3\tau_{ii}$, the deviatoric tensor ${\boldsymbol{\tau}}^{{\mathtt{D}}}:={\boldsymbol{\tau}}-\frac{1}{3}\left({\mathop{\mathrm{tr}}\nolimits}{\boldsymbol{\tau}}\right){\boldsymbol{I}}$, and the tensor inner product ${\boldsymbol{\tau}}:{\boldsymbol{\sigma}}:=\sum_{i,j=1}^3\tau_{ij}\sigma_{ij}$.
Let ${\Omega}$ be a polyhedral Lipschitz bounded domain of ${\mathbb{R}}^3$ with boundary ${\partial{\Omega}}$. For $s\geq 0$, ${\lVert\cdot\rVert}_{s,{\Omega}}$ stands indistinctly for the norm of the Hilbertian Sobolev spaces ${{\mathrm{H}}^s({\Omega})}$, ${{\mathrm{H}}^s({\Omega})}^3$ or ${{\mathrm{H}}^s({\Omega})}^{{3\times 3}}$, with the convention ${\mathrm{H}}^0({\Omega}):={{\mathrm{L}}^2({\Omega})}$. We also define for $s\geq 0$ the Hilbert space ${{{\mathrm{H}}^s(\mathbf{div},{\Omega})}}:={\lbrace {\boldsymbol{\tau}}\in{{\mathrm{H}}^s({\Omega})}^{{3\times 3}}:\ {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}\in{{\mathrm{H}}^s({\Omega})}^3 \rbrace}$, whose norm is given by ${\lVert{\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}^s(\mathbf{div},{\Omega})}}}
:={\lVert{\boldsymbol{\tau}}\rVert}_{s,{\Omega}}^2+{\lVert{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}\rVert}^2_{s,{\Omega}}$ and denote ${{{\mathrm{H}}(\mathbf{div},{\Omega})}}:={{\mathrm{H}}^0(\mathbf{div};{\Omega})}$.
Henceforth, we denote by $C$ generic constants independent of the discretization parameter, which may take different values at different places.
The model problem {#section:2}
=================
Let ${\Omega}\subset {\mathbb{R}}^3$ be an open bounded Lipschitz polyhedron representing a solid domain. We denote by ${\boldsymbol{n}}$ the outward unit normal vector to ${\partial{\Omega}}$ and assume that ${\partial{\Omega}}={\Gamma}\cup{\Sigma}$ with $\textrm{int}({\Gamma})\cap \mathrm{int}({\Sigma}) = \emptyset$. The solid is supposed to be isotropic and linearly elastic with mass density $\rho$ and Lamé constants $\mu$ and $\lambda$. Under the hypothesis of small oscillations, the time-harmonic elastodynamic equations with angular frequency $\omega>0$ and body force ${\boldsymbol{f}}:{\Omega}\to {\mathbb{R}}^3$ are given by
\[model\] $$\begin{aligned}
{\boldsymbol{\sigma}}={\mathcal{C}}{\boldsymbol{\varepsilon}}({\boldsymbol{u}}) &{\qquad\hbox{in }}{\Omega},\label{model-a}
\\
{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}+\rho \omega^2\,{\boldsymbol{u}}={\boldsymbol{f}}&{\qquad\hbox{in }}{\Omega},\label{model-b}
\\
{\boldsymbol{\sigma}}{\boldsymbol{n}}=\mathbf{0} &{\qquad\hbox{on }}{\Sigma},\label{model-c}
\\
{\boldsymbol{u}}=\mathbf{0} &{\qquad\hbox{on }}{\Gamma},\label{model-d}\end{aligned}$$
where ${\boldsymbol{u}}:{\Omega}\to {\mathbb{R}}^3$ is the displacement field, ${\boldsymbol{\varepsilon}}({\boldsymbol{u}}):=\frac{1}{2}\left[\nabla{\boldsymbol{u}}+(\nabla{\boldsymbol{u}})^{{\mathtt{t}}}\right]$ is the linearized strain tensor and ${\mathcal{C}}$ is the elasticity operator defined by $${\mathcal{C}}{\boldsymbol{\tau}}:=\lambda\left({\mathop{\mathrm{tr}}\nolimits}{\boldsymbol{\tau}}\right){\boldsymbol{I}}+2\mu{\boldsymbol{\tau}}.$$
Our aim is to introduce the Cauchy stress tensor ${\boldsymbol{\sigma}}:{\Omega}\to {\mathbb{R}}^{{3\times 3}}$ as a primary variable in the variational formulation of . To this end, we consider the closed subspace of ${{{\mathrm{H}}(\mathbf{div},{\Omega})}}$ given by $${\boldsymbol{\mathcal{W}}}:= {\lbrace {\boldsymbol{\tau}}\in {{{\mathrm{H}}(\mathbf{div},{\Omega})}};\quad \text{${\boldsymbol{\tau}}{\boldsymbol{n}}= {\mathbf{0}}$ on ${\Sigma}$} \rbrace}$$ and the space of skew symmetric tensors $${\boldsymbol{\mathcal{Q}}}:= {\lbrace {\boldsymbol{s}}\in {{\mathrm{L}}^2({\Omega})}^{{3\times 3}}; \quad {\boldsymbol{s}}= - {\boldsymbol{s}}^{{\mathtt{t}}} \rbrace}.$$ Introducing the rotation ${\boldsymbol{r}}:= \frac{1}{2}\left[\nabla{\boldsymbol{u}}- (\nabla{\boldsymbol{u}})^{{\mathtt{t}}}\right]$, the constitutive equation can be rewritten as, $${\mathcal{C}}^{-1}{\boldsymbol{\sigma}}= \nabla {\boldsymbol{u}}- {\boldsymbol{r}}.$$ Testing the last identity with ${\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}$, integrating by parts and using the momentum equation to eliminate the displacement ${\boldsymbol{u}}$, we end up with the following mixed variational formulation of problem :
\[varForm\] $$\begin{aligned}
\int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}- \kappa^2 \left( \int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\sigma}}:{\boldsymbol{\tau}}+ \int_{{\Omega}}{\boldsymbol{r}}:{\boldsymbol{\tau}}\right)
= \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}& \quad \forall {\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}\label{varForm-a}
\\
\int_{{\Omega}} {\boldsymbol{\sigma}}:{\boldsymbol{s}}= {\mathbf{0}}& \quad \forall {\boldsymbol{s}}\in {\boldsymbol{\mathcal{Q}}}\label{varForm-b},\end{aligned}$$
where the wave number $\kappa$ is given by $\sqrt{\rho}\,\omega$. We notice that equation is a restriction that imposes weakly the symmetry of ${\boldsymbol{\sigma}}$, and ${\boldsymbol{r}}$ is the corresponding Lagrange multiplier. We also point out that the dual formulation degenerates as $\omega\to 0$. The static case $\omega=0$ is then not covered by our analysis.
We introduce the symmetric bilinear forms $${B\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}:= \int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\sigma}}:{\boldsymbol{\tau}}+ \int_{{\Omega}}{\boldsymbol{r}}:{\boldsymbol{\tau}}+ \int_{{\Omega}}{\boldsymbol{s}}:{\boldsymbol{\sigma}}$$ and $${A\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} := \int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}+ {B\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}$$ and denote the product norm on ${{{\mathrm{H}}(\mathbf{div},{\Omega})}}\times {\mathrm{L}}^2({\Omega})$ by $${\lVert{({\boldsymbol{\tau}}, {\boldsymbol{s}})}\rVert}^2 := {\lVert{\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} + {\lVert{\boldsymbol{s}}\rVert}^2_{0,{\Omega}}.$$
\[infsupA-cont\] There exists a constant $\alpha_A>0$, depending on $\mu$ and ${\Omega}$ (but not on $\lambda$), such that $$\label{infsupa}
\sup_{{({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}} \frac{{A\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}}{{\lVert{({\boldsymbol{\tau}}, {\boldsymbol{s}})}\rVert}} \geq \alpha_A {\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\rVert}\quad \forall {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\in
{\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}.$$
It is important to notice that the bilinear form $$\label{invcCop}
\int_{{\Omega}}{\mathcal{C}}^{-1}{\boldsymbol{\sigma}}:{\boldsymbol{\tau}}= \frac{1}{2\mu}\int_{{\Omega}}{\boldsymbol{\sigma}}^{{\mathtt{D}}}:{\boldsymbol{\tau}}^{{\mathtt{D}}} + \frac{1}{3(3\lambda + 2\mu)} \int_{{\Omega}}({\mathop{\mathrm{tr}}\nolimits}{\boldsymbol{\sigma}})({\mathop{\mathrm{tr}}\nolimits}{\boldsymbol{\tau}})$$ is bounded by a constant independent of $\lambda$ when $\lambda$ is too large in comparison with $\mu$. Moreover, it is shown in [@MMR Lemma 2.1] that there exists a constant $\alpha_0>0$, depending on $\mu$ and ${\Omega}$ (but not on $\lambda$), such that $$\label{elip0}
\int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\tau}}:{\boldsymbol{\tau}}+ \int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}\geq\alpha_0{\lVert{\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}
\qquad\forall{\boldsymbol{\tau}}\in{\boldsymbol{\mathcal{W}}}.$$
On the other hand, there exists a constant $\beta>0$ depending only on ${\Omega}$ (see, for instance, [@BoffiBrezziFortin]) such that $$\sup_{{\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}} \frac{\int_{{\Omega}}{\boldsymbol{s}}:{\boldsymbol{\tau}}}{{\lVert{\boldsymbol{\tau}}\rVert}_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}} \geq \beta {\lVert{\boldsymbol{s}}\rVert}_{0,{\Omega}}, \qquad
\forall {\boldsymbol{s}}\in {\boldsymbol{\mathcal{Q}}}.$$
The Babuška-Brezzi theory shows that, for any bounded linear form $L\in \mathcal{L}({\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}})$, the problem: find ${({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ such that $${A\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} = L\big({({\boldsymbol{\tau}}, {\boldsymbol{s}})}\big)\qquad \forall {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}},$$ is well-posed, which proves .
We deduce from Proposition \[infsupA-cont\] and the symmetry of $A(\cdot, \cdot)$ that the operator ${\boldsymbol{T}}: {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}\to {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ characterized by $${A\Bigl( {\boldsymbol{T}}{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} = {B\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} \quad \forall {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$$ is well-defined and bounded. It is clear that, for a given wave number $\kappa>0$, ${({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\neq 0$ is a solution to the homogeneous version of problem if and only if $\left( \eta=\frac{1}{1+\kappa^2}, {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\right)$ is an eigenpair for ${\boldsymbol{T}}$. The following characterization of the spectrum of ${\boldsymbol{T}}$ will be useful for our analysis.
\[specT\] The spectrum ${\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$ of ${\boldsymbol{T}}$ decomposes as follows $${\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}}) = {\lbrace 0, 1 \rbrace} \cup {\lbrace \eta_k \rbrace}_{k\in \mathbb{N}}$$ where ${\lbrace \eta_k \rbrace}_k$ is a real sequence of finite-multiplicity eigenvalues of ${\boldsymbol{T}}$ which converges to 0. Moreover, $\eta=1$ is an infinite-multiplicity eigenvalue of ${\boldsymbol{T}}$ while $\eta=0$ is not an eigenvalue.
See [@MMR Theorem 3.7].
\[wellposed\] If $\frac{1}{1+\kappa^2}\notin {\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$, then is well-posed. Moreover, there exists a constant $C>0$ independent of $\lambda$ such that, for any ${\boldsymbol{f}}\in {{\mathrm{L}}^2({\Omega})}^3$, the solution $({\boldsymbol{\sigma}}, {\boldsymbol{r}})\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ of satisfies $$\label{estimsigr}
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\rVert} \leq C {\lVert{\boldsymbol{f}}\rVert}_{0,{\Omega}}.$$
Let us first recall that, given $z\in{\mathbb{C}}\setminus{\lbrace {\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}}) \rbrace}$, the resolvent $$(z{\boldsymbol{I}}-{\boldsymbol{T}})^{-1} :{\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}\longrightarrow{\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$$ is bounded in $\mathcal{L}({\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}})$ by a constant $C$ only depending on ${\Omega}$ and ${\lvertz\rvert}$.
We deduce from and the symmetry of $A(\cdot, \cdot)$ that the problem: find $(\bar{\boldsymbol{\sigma}}, \bar{\boldsymbol{r}})\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ solution of $${A\Bigl( {(\bar{\boldsymbol{\sigma}}, \bar{\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} =
\int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}, \quad \forall {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}},$$ is well-posed. The solution of problem is then given by $${({\boldsymbol{\sigma}}, {\boldsymbol{r}})}= \frac{1}{1+\kappa^2} \Big( \frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}\Big)^{-1} {(\bar{\boldsymbol{\sigma}}, \bar{\boldsymbol{r}})},$$ and follows from the boundedness of $\Big( \frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}\Big)^{-1}$.
A continuous Galerkin discretization {#section:3}
====================================
We consider shape regular affine meshes $\mathcal{T}_h$ that subdivide the domain $\bar \Omega$ into tetrahedra $K$ of diameter $h_K$. The parameter $h:= \max_{K\in {\mathcal{T}}_h} \{h_K\}$ represents the mesh size of ${\mathcal{T}}_h$. Hereafter, given an integer $m\geq 0$ and a domain $D\subset \mathbb{R}^3$, ${\mathcal{P}}_m(D)$ denotes the space of polynomials of degree at most $m$ on $D$. The space of piecewise polynomial functions of degree at most $m$ relatively to ${\mathcal{T}}_h$ is denoted by $${\mathcal{P}}_m({\mathcal{T}}_h) :={\lbrace v\in L^2({\Omega}); \quad v|_K \in {\mathcal{P}}_m(K),\quad \forall K\in {\mathcal{T}}_h \rbrace}.$$
For any $k\geq 1$, we consider the finite element spaces $${\boldsymbol{\mathcal{W}}}_h^c := {\mathcal{P}}_k({\mathcal{T}}_h)^{{3\times 3}} \cap {\boldsymbol{\mathcal{W}}}\qquad \text{and} \qquad {\boldsymbol{\mathcal{Q}}}_h := {\mathcal{P}}_{k-1}({\mathcal{T}}_h)^{{3\times 3}} \cap {\boldsymbol{\mathcal{Q}}}.$$
Let us now recall some well-known properties of the Brezzi-Douglas-Marini (BDM) mixed finite element [@BDM]. Let $\hat K$ be a fixed reference tetrahedron. Given $K\in {\mathcal{T}}_h$, there exists an affine and bijective map ${\boldsymbol{F}}_K:\, \hat K \to {\mathbb{R}}^3$ such that ${\boldsymbol{F}}_K(\hat K) = K$. We consider ${\boldsymbol{B}}_K:= \nabla {\boldsymbol{F}}_K$ and define $${\boldsymbol{\mathcal{N}}}_{k}(K):= {\lbrace {\boldsymbol{w}}:\Omega \to \mathbb{R}^3;\quad {\boldsymbol{w}}\circ {\boldsymbol{F}}_K = {\boldsymbol{B}}_K^{-{\mathtt{t}}} \check{{\boldsymbol{w}}}; \quad \check {\boldsymbol{w}}\in {\mathcal{P}}_{k-1}(\hat K)^3 \oplus
{\boldsymbol{\mathcal{S}}}_k(\hat K) \rbrace}$$ where $${\boldsymbol{\mathcal{S}}}_k(\hat K) := {\lbrace \check {\boldsymbol{w}}\in \tilde{{\mathcal{P}}}_k(\hat K)^3; \quad \check {\boldsymbol{w}}\cdot \hat{{\boldsymbol{x}}} = 0 \rbrace}$$ with $\tilde{{\mathcal{P}}}_k(\hat K)$ representing the space of homogeneous polynomials of total degree exactly $k$ in $\hat {\boldsymbol{x}}\in
\hat K$.
A polynomial ${\boldsymbol{v}}\in {\mathcal{P}}_k(K)^3$ is uniquely determined by the set of BDM degrees of freedom $$\begin{aligned}
m_\phi({\boldsymbol{v}}) &:= \int_F{\boldsymbol{v}}\cdot {\boldsymbol{n}}_K \phi \qquad\text{for all $\phi\in {\mathcal{P}}_k(F)$, for all $F\in {\mathcal{F}}(K)$}\label{DFa}
\\
m_{{\boldsymbol{w}}}({\boldsymbol{v}}) &:= \int_K {\boldsymbol{v}}\cdot {\boldsymbol{w}}\qquad\text{for all ${\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(K)$}\label{DFb},\end{aligned}$$ where ${\boldsymbol{n}}_K$ is the outward unit normal vector to $\partial K$. Conditions are avoided in the case $k=1$.
Let us consider an arbitrary, but fixed, orientation of all internal faces $F$ of ${\mathcal{T}}_h$ by normal vectors ${\boldsymbol{n}}_F$. On the faces $F$ lying on $\partial {\Omega}$ we take ${\boldsymbol{n}}_F = {\boldsymbol{n}}|_{F}$. We can introduce the global BDM-interpolation operator $\Pi_h: {\mathrm{H}}(\textrm{div}, {\Omega})\cap {\mathrm{H}}^s({\Omega})^3 \to {\boldsymbol{\mathcal{W}}}_h^c$, characterized, for any ${\boldsymbol{v}}\in {\mathrm{H}}(\textrm{div}, {\Omega})\cap {\mathrm{H}}^s({\Omega})^3$ with $s>1/2$, by the conditions $$\begin{aligned}
\label{Pih}
\int_F \Pi_h{\boldsymbol{v}}\cdot {\boldsymbol{n}}_F \phi & = \int_F {\boldsymbol{v}}\cdot {\boldsymbol{n}}_F \phi
\quad&\text{for all $\phi\in {\mathcal{P}}_k(F)$, for all $F\in {\mathcal{F}}_h$},
\\
\int_K \Pi_h{\boldsymbol{v}}\cdot {\boldsymbol{w}}&= \int_K {\boldsymbol{v}}\cdot {\boldsymbol{w}}\quad&\text{for all ${\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(K)$, for all $K\in {\mathcal{T}}_h$}\label{DFGc}.\end{aligned}$$ We have the following classical error estimate, see [@BoffiBrezziFortin], $$\label{asymp}
{\lVert{\boldsymbol{v}}- \Pi_h {\boldsymbol{v}}\rVert}_{0,{\Omega}} \leq C h^{\min(s, k+1)} {\lVert{\boldsymbol{v}}\rVert}_{s,{\Omega}} \qquad \forall {\boldsymbol{v}}\in {{\mathrm{H}}^s({\Omega})}^{3}, \text{with $s>1/2$}.$$ Moreover, thanks to the commutativity property, if ${\mathop{\mathrm{div}}\nolimits}{\boldsymbol{v}}\in {{\mathrm{H}}^s({\Omega})}$, then $$\label{asympDiv}
{\lVert{\mathop{\mathrm{div}}\nolimits}({\boldsymbol{v}}- \Pi_h {\boldsymbol{v}}) \rVert}_{0,{\Omega}} = {\lVert{\mathop{\mathrm{div}}\nolimits}{\boldsymbol{v}}- \mathcal R_h {\mathop{\mathrm{div}}\nolimits}{\boldsymbol{v}}) \rVert}_{0,{\Omega}}
\leq C h^{\min(s, k)} {\lVert{\mathop{\mathrm{div}}\nolimits}{\boldsymbol{v}}\rVert}_{s,{\Omega}},$$ where $\mathcal R_h$ is the ${{\mathrm{L}}^2({\Omega})}$-orthogonal projection onto ${\mathcal{P}}_{k-1}({\mathcal{T}}_h)$. Finally, we denote by $\mathcal S_h:\ {\boldsymbol{\mathcal{Q}}}\to{\boldsymbol{\mathcal{Q}}}_h$ the orthogonal projector with respect to the ${{\mathrm{L}}^2({\Omega})}^{{3\times 3}}$-norm. It is well-known that, for any $s\in(0,1]$, we have $$\label{asymQ}
{\lVert{\boldsymbol{r}}-\mathcal S_h{\boldsymbol{r}}\rVert}_{0,{\Omega}}
\leq C h^{\min(s, k)} {\lVert{\boldsymbol{r}}\rVert}_{s,{\Omega}}
\qquad\forall{\boldsymbol{r}}\in{{\mathrm{H}}^s({\Omega})}^{{3\times 3}}\cap{\boldsymbol{\mathcal{Q}}}.$$
We propose the following continuous Galerkin (CG) discretization of problem : find ${\boldsymbol{\sigma}}_h\in {\boldsymbol{\mathcal{W}}}^c_h$ and ${\boldsymbol{r}}_h\in {\boldsymbol{\mathcal{Q}}}_h$ such that
\[CGForm\] $$\begin{aligned}
\int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}_h \cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}- \kappa^2 \left( \int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\sigma}}_h:{\boldsymbol{\tau}}+ \int_{{\Omega}}{\boldsymbol{r}}_h:{\boldsymbol{\tau}}\right)
= \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}& \quad \forall {\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}_h^c\label{CGForm-a}
\\
\int_{{\Omega}} {\boldsymbol{\sigma}}_h:{\boldsymbol{s}}= {\mathbf{0}}& \quad \forall {\boldsymbol{s}}\in {\boldsymbol{\mathcal{Q}}}_h\label{CGForm-b}.\end{aligned}$$
\[infsupA-disc\] There exists a constant $\alpha_A^{c}>0$ independent of $h$ and $\lambda$ such that $$\label{infsupA-disceq}
\sup_{{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h} \frac{{A\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}}{{\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}}
\geq \alpha_A^{c} {\lVert{({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}\quad \forall {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in
{\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h.$$
We prove this result by following the same steps given in Proposition \[infsupA-cont\]. We deduce from that the bilinear form $
\int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}+ \int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\sigma}}:{\boldsymbol{\tau}}$ is elliptic on ${\boldsymbol{\mathcal{W}}}_h^c$. Moreover, the following discrete inf-sup condition is proved in [@afw-acta-2006; @BoffiBrezziFortin]: There exists $\beta^c>0$, independent of $h$, such that $$\sup_{{\boldsymbol{\tau}}_h\in {\boldsymbol{\mathcal{W}}}_h^c} \frac{{\displaystyle}\int_{{\Omega}}{\boldsymbol{s}}_h:{\boldsymbol{\tau}}_h}{{\lVert{\boldsymbol{\tau}}_h\rVert}_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}} \geq \beta^c {\lVert{\boldsymbol{s}}_h\rVert}_{0,{\Omega}}.$$ Therefore, we can use the Babuška-Brezzi theory to ensure that, for any bounded linear form $L\in \mathcal{L}( {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}})$, the problem: find ${({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$ such that $${A\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = L\big({({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\big)\qquad \forall {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$$ admits a unique solution and there exists a constant $C>0$ independent of $h$ and $\lambda$ such that $${\lVert{({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert} \leq C {\lVertL\rVert}_{\mathcal{L}( {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}})},$$ which gives .
We can now consider the discrete counterpart ${\boldsymbol{T}}_h:\, {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h \to {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h$ of ${\boldsymbol{T}}$ characterized, for any ${({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h$, by $${A\Bigl( {\boldsymbol{T}}_h{({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = {B\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}\quad \forall {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h.$$ As a consequence of Proposition \[infsupA-disc\], ${\boldsymbol{T}}_h$ is well-defined and uniformly bounded with respect to $h$ and $\lambda$. Moreover, we deduce from [@MMR Theorem 5.2] that, if $\frac{1}{1+\kappa^2}\notin{\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$, there exists a mesh size $h_0>0$ such that, for $h\leq h_0$, $$\label{boundTh}
{\lVert\big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h\big){({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert} \geq C_0 {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert} \quad
\forall {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h,$$ with a constant $C_0>0$ independent of $h$ and $\lambda$.
We introduce the bilinear form $${D\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}:={A\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} - (1+\kappa^2) {B\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}$$ and notice that there exists a constant $M^c_D>0$ independent of $h$ and $\lambda$ such that $$\label{boundD}
\Big| {D\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} \Big| \leq M^c_D \,
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\rVert}\, {\lVert{({\boldsymbol{\tau}}, {\boldsymbol{s}})}\rVert} \qquad \forall {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, \, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}.$$
\[infsupD\] Assume that $\frac{1}{1+\kappa^2}\notin{\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$ and let $h_0>0$ be the parameter for which holds true for all $h\leq h_0$. Then, for $h\leq h_0$, $$\label{infsupABh0}
\inf_{{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h} \frac{{D\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}}{{\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}}
\geq \alpha_D^c {\lVert{({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert},\qquad \forall {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in
{\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h,$$ with $\alpha_D^c>0$ independent of the mesh size $h$ and $\lambda$.
We deduce from Proposition \[infsupA-disc\] that there exists an operator $\Theta_h:\, {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h \to {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h$ satisfying $$\label{cota1}
{A\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Theta_h {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}^2 \quad \text{and}\quad
{\lVert\Theta_h {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert} \leq \frac{1}{\alpha_A^c} {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}$$ for all ${({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}^c_h\times {\boldsymbol{\mathcal{Q}}}_h$. It follows from and that $$\begin{gathered}
{D\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Theta_h \big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h \big) {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} \\= (1+\kappa^2)
{A\Bigl( \big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h \big) {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Theta_h \big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h \big) {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}\\
= (1+\kappa^2) {\lVert\big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h \big) {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}^2 \ge (1+\kappa^2)C_0 {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}
\, {\lVert\big(\frac{{\boldsymbol{I}}}{1+\kappa^2} - {\boldsymbol{T}}_h \big) {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}\end{gathered}$$ for all ${({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in
{\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$, with the constant $C_0>0$ from . The result follows now with $\alpha_D^c = C_0 (1+\kappa^2)$.
\[ConvCG\] Assume that $\frac{1}{1+\kappa^2}\notin{\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$ and let $h_0>0$ be the parameter for which holds true for all $h\leq h_0$. Then, for $h\leq h_0$, we have the following Céa estimate, $$\label{Cea}
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert} \leq \left( 1 + \frac{M^c_D}{\alpha_D^c} \right) \inf_{{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h}
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}, \quad \forall h\leq h_0.$$ Moreover, if the exact solution ${\boldsymbol{u}}$ of belongs to ${\mathrm{H}}^{1+s}({\Omega})^3$ and ${\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\in {\mathrm{H}}^{s}({\Omega})^3$ for some $s>1/2$ then, $${\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert} \leq \, C\, h^{\min(s, k)}\, ( {\lVert{\boldsymbol{u}}\rVert}_{1+s,{\Omega}} +
{\lVert{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\rVert}_{s,{\Omega}}), \quad \forall h\leq h_0,$$ with $C>0$ independent of $h$ and $\lambda$.
The Céa estimate is a direct consequence of and . The asymptotic error estimate follows from , and .
A discontinuous Galerkin discretization {#section:4}
=======================================
From now on we assume that there exists $s_0>1/2$ such that ${\boldsymbol{f}}|_{{\Omega}_j}\in {\mathrm{H}}^{s_0}({\Omega}_j)$ for $j= 1,\cdots, J$, where ${\lbrace {\Omega}_j,\quad j=1\cdots, J \rbrace}$ is a set of polyhedral subdomains forming a disjoint partition of $\bar {\Omega}$, i.e., $${\Omega}_j\cap{\Omega}_i = \emptyset \quad \text{for all $1\leq i\neq j\leq J$ \quad and \quad $\bar{\Omega}= \cup_{j= 1}^J \bar{{\Omega}}_j$}.$$ We deduce from this additional hypothesis on ${\boldsymbol{f}}$ and the momentum equation that $({\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}})|_{{\Omega}_j}$ belongs to ${\mathrm{H}}^{\min(s_0, 1)}({\Omega}_j)$ for any $j= 1,\cdots, J$.
In what follows, we assume that $\mathcal{T}_h$ is compatible with the partition $\bar{\Omega}= \cup_{j= 1}^J \bar{{\Omega}}_j$, i.e., $${\displaystyle}\cup_{K\in {\mathcal{T}}_h({\Omega}_j)} K= \bar{{\Omega}}_j \quad \forall j=1,\cdots, J,$$ where ${\mathcal{T}}_h({\Omega}_j):= {\lbrace K\in{\mathcal{T}}_h,\quad K\subset \bar{{\Omega}}_j \rbrace}$.
We say that a closed subset $F\subset \overline{\Omega}$ is an interior face if $F$ has a positive 2-dimensional measure and if there are distinct elements $K$ and $K'$ such that $F = K\cap K'$. A closed subset $F\subset \overline{\Omega}$ is a boundary face if there exists $K\in {\mathcal{T}}_h$ such that $F$ is a face of $K$ and $F = K\cap \Gamma$. We consider the set ${\mathcal{F}}_h^0$ of interior faces and the set ${\mathcal{F}}_h^\partial$ of boundary faces. We assume that the boundary mesh ${\mathcal{F}}_h^\partial$ is compatible with the partition ${\partial{\Omega}}= {\Gamma}\cup {\Sigma}$, i.e., $$\cup_{F\in {\mathcal{F}}_h^{\Gamma}} F = {\Gamma}\qquad \text{and} \qquad \cup_{F\in {\mathcal{F}}_h^{\Sigma}} F = {\Sigma}$$ where ${\mathcal{F}}_h^{\Gamma}:= {\lbrace F\in {\mathcal{F}}_h^\partial; \quad F\subset {\Gamma}\rbrace}$ and ${\mathcal{F}}_h^{{\Sigma}}:= {\lbrace F\in {\mathcal{F}}_h^\partial; \quad F\subset {\Sigma}\rbrace}$. We denote $${\mathcal{F}}_h := {\mathcal{F}}_h^0\cup {\mathcal{F}}_h^\partial\qquad \text{and} \qquad {\mathcal{F}}^*_h:= {\mathcal{F}}_h^{0} \cup {\mathcal{F}}_h^{{\Sigma}},$$ and for any element $K\in {\mathcal{T}}_h$, we introduce the set $${\mathcal{F}}(K):= {\lbrace F\in {\mathcal{F}}_h;\quad F\subset \partial K \rbrace}$$ of faces composing the boundary of $K$.
For any $s\geq 0$, we consider the broken Sobolev space $${\mathrm{H}}^s({\mathcal{T}}_h):=
{\lbrace {\boldsymbol{v}}\in {\mathrm{L}}^2({\Omega})^3; \quad {\boldsymbol{v}}|_K\in {\mathrm{H}}^s({\Omega})^3\quad \forall K\in {\mathcal{T}}_h \rbrace}.$$ For each ${\boldsymbol{v}}:={\lbrace {\boldsymbol{v}}_K \rbrace}\in {\mathrm{H}}^s({\mathcal{T}}_h)^3$ and ${\boldsymbol{\tau}}:= {\lbrace {\boldsymbol{\tau}}_K \rbrace}\in {\mathrm{H}}^s({\mathcal{T}}_h)^{{3\times 3}}$ the components ${\boldsymbol{v}}_K$ and ${\boldsymbol{\tau}}_K$ represent the restrictions ${\boldsymbol{v}}|_K$ and ${\boldsymbol{\tau}}|_K$. When no confusion arises, the restrictions of these functions will be written without any subscript. We will also need the space given on the skeletons of the triangulations ${\mathcal{T}}_h$ by $${\mathrm{L}}^2({\mathcal{F}}_h):= \prod_{F\in \mathcal{F}_h} {\mathrm{L}}^2(F).$$ Similarly, the components $\mu_F$ of $\mu := {\lbrace \mu_F \rbrace}\in {\mathrm{L}}^2({\mathcal{F}}_h)$ coincide with the restrictions $\mu|_F$ and we denote $$\int_{{\mathcal{F}}_h} \mu := \sum_{F\in {\mathcal{F}}_h} \int_F \mu_F\quad \text{and}\quad
{\lVert\mu\rVert}^2_{0,{\mathcal{F}}_h}:= \int_{{\mathcal{F}}_h} \mu^2,
\qquad
\forall \mu\in {\mathrm{L}}^2({\mathcal{F}}_h).$$ From now on, $h_{\mathcal{F}}\in {\mathrm{L}}^2({\mathcal{F}}_h)$ is the piecewise constant function defined by $h_{\mathcal{F}}|_F := h_F$ for all $F \in {\mathcal{F}}_h$ with $h_F$ denoting the diameter of face $F$.
Given a vector valued function ${\boldsymbol{v}}\in {\mathrm{H}}^t({\mathcal{T}}_h)^3$, with $t>1/2$, we define averages ${\{{\boldsymbol{v}}\}}\in {\mathrm{L}}^2({\mathcal{F}}_h)^3$ and jumps ${\llbracket {\boldsymbol{v}}\rrbracket}\in {\mathrm{L}}^2({\mathcal{F}}_h)$ by $${\{{\boldsymbol{v}}\}}_F := ({\boldsymbol{v}}_K + {\boldsymbol{v}}_{K'})/2 \quad \text{and} \quad {\llbracket {\boldsymbol{v}}\rrbracket}_F := {\boldsymbol{v}}_K \cdot{\boldsymbol{n}}_K + {\boldsymbol{v}}_{K'}\cdot{\boldsymbol{n}}_{K'}
\quad \forall F \in {\mathcal{F}}(K)\cap {\mathcal{F}}(K'),$$ where ${\boldsymbol{n}}_K$ is the outward unit normal vector to $\partial K$. On the boundary of ${\Omega}$ we use the following conventions for averages and jumps: $${\{{\boldsymbol{v}}\}}_F := {\boldsymbol{v}}_K \quad \text{and} \quad {\llbracket {\boldsymbol{v}}\rrbracket}_F := {\boldsymbol{v}}_K \cdot{\boldsymbol{n}}\quad \forall F \in {\mathcal{F}}(K)\cap {\partial{\Omega}}.$$
Similarly, for matrix valued functions ${\boldsymbol{\tau}}\in {\mathrm{H}}^t({\mathcal{T}}_h)^{{3\times 3}}$, we define ${\{{\boldsymbol{\tau}}\}}\in {\mathrm{L}}^2({\mathcal{F}}_h)^{{3\times 3}}$ and ${\llbracket {\boldsymbol{\tau}}\rrbracket}\in {\mathrm{L}}^2({\mathcal{F}}_h)^3$ by $${\{{\boldsymbol{\tau}}\}}_F := ({\boldsymbol{\tau}}_K + {\boldsymbol{\tau}}_{K'})/2 \quad \text{and} \quad {\llbracket {\boldsymbol{\tau}}\rrbracket}_F :=
{\boldsymbol{\tau}}_K {\boldsymbol{n}}_K + {\boldsymbol{\tau}}_{K'}{\boldsymbol{n}}_{K'}
\quad \forall F \in {\mathcal{F}}(K)\cap {\mathcal{F}}(K')$$ and on the boundary of $\Omega$ we set $${\{{\boldsymbol{\tau}}\}}_F := {\boldsymbol{\tau}}_K \quad \text{and} \quad {\llbracket {\boldsymbol{\tau}}\rrbracket}_F :=
{\boldsymbol{\tau}}_K {\boldsymbol{n}}\quad \forall F \in {\mathcal{F}}(K)\cap {\partial{\Omega}}.$$
For any $k\geq 1$ we introduce the finite dimensional space $
{\boldsymbol{\mathcal{W}}}_h := {\mathcal{P}}_k({\mathcal{T}}_h)^{{3\times 3}}
$ and consider $
{\boldsymbol{\mathcal{W}}}(h) := {\boldsymbol{\mathcal{W}}}+ {\boldsymbol{\mathcal{W}}}_h.
$ Given ${\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}_h$ we define ${\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}\in {\mathrm{L}}^2({\Omega})^3$ by $
{\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}|_{K} = {\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\tau}}|_K)$ for all $K\in {\mathcal{T}}_h
$ and endow ${\boldsymbol{\mathcal{W}}}(h)$ with the seminorm $$|{\boldsymbol{\tau}}|^2_{{\boldsymbol{\mathcal{W}}}(h)} := {\lVert{\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}\rVert}^2_{0,{\Omega}} + {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,{\mathcal{F}}^*_h}$$ and the norm $${\lVert{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} := |{\boldsymbol{\tau}}|^2_{{\boldsymbol{\mathcal{W}}}(h)} + {\lVert{\boldsymbol{\tau}}\rVert}^2_{0,{\Omega}}.$$ For the sake of simplicity, we will also use the notation $${\lVert({\boldsymbol{\tau}}, {\boldsymbol{s}})\rVert}^2_{DG} : = {\lVert{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} + {\lVert{\boldsymbol{s}}\rVert}^2_{0,{\Omega}}.$$
Given a parameter $\texttt{a}>0$, we introduce the symmetric bilinear form $$\begin{gathered}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})},{({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}:=
\int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}- \kappa^2 {B\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} + \\
\int_{{\mathcal{F}}^*_h} \texttt{a}h_{{\mathcal{F}}}^{-1}\, {\llbracket {\boldsymbol{\sigma}}\rrbracket}\cdot {\llbracket {\boldsymbol{\tau}}\rrbracket}-\int_{{\mathcal{F}}^*_h}
\left( {\{{\mathop{\mathbf{div}}\nolimits}_h{\boldsymbol{\sigma}}\}} \cdot {\llbracket {\boldsymbol{\tau}}\rrbracket}
+ {\{{\mathop{\mathbf{div}}\nolimits}_h{\boldsymbol{\tau}}\}} \cdot {\llbracket {\boldsymbol{\sigma}}\rrbracket} \right)\quad \forall {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h\end{gathered}$$ and the linear form $${L_h\Bigl( {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}:= \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}- \int_{{\mathcal{F}}^*_h} {\{{\boldsymbol{f}}\}}\cdot {\llbracket {\boldsymbol{\tau}}\rrbracket}\qquad
\forall {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h,$$ and consider the DG method: find ${({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h$ such that $$\label{DGshort}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)} = {L_h\Bigl( {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\Bigr)}\qquad \forall {({\boldsymbol{\tau}}, {\boldsymbol{s}})}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h.$$ We notice that, as it is usually the case for DG methods, the essential boundary condition is directly incorporated within the scheme. We need the following technical result to show that the bilinear form $D_h^{\texttt{a}}(\cdot, \cdot)$ is uniformly bounded on ${\boldsymbol{\mathcal{W}}}_h$.
\[card\] There exists a constant $C>0$ independent of $h$ such that $$\label{discTrace}
{\lVerth^{1/2}_{{\mathcal{F}}}{\{v\}}\rVert}_{0,{\mathcal{F}}_h}\leq C {\lVertv\rVert}_{0,{\Omega}}\quad \forall v\in {\mathcal{P}}_k({\mathcal{T}}_h).$$
It is straightforward that $${\lVerth^{1/2}_{{\mathcal{F}}}{\{v\}}\rVert}^2_{0,{\mathcal{F}}_h} = \sum_{F\in {\mathcal{F}}_h} {\lVerth_F^{1/2}{\{v\}}\rVert}^2_{0,F} \leq
\sum_{K\in {\mathcal{T}}_h} \sum_{F\in {\mathcal{F}}(K)} h_F{\lVertv\rVert}^2_{0,F} \leq
\sum_{K\in {\mathcal{T}}_h} h_K{\lVertv\rVert}^2_{0,\partial K}.$$ The result follows now from the following discrete trace inequality (cf. [@DiPietroErn]): $$\label{discreteTrace}
h_K {\lVertv\rVert}^2_{0,\partial K} \leq C_0 {\lVertv\rVert}^2_{0,K} \quad \forall v\in {\mathcal{P}}_k(K), \quad \forall K\in {\mathcal{T}}_h,$$ where $C_0>0$ is independent of $K$.
With the aid of the Cauchy-Schwarz inequality and Proposition \[card\], we can easily prove that there exists constants $M^{d}_D>0$ independent of $h$ and $\lambda$ such that $$\label{boundDh}
|{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}| \leq M^{d}_D \Big({\lVert{\boldsymbol{\sigma}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} +
{\lVerth_{{\mathcal{F}}}^{1/2} {\{{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\}}\rVert}^2_{0,{\mathcal{F}}^*_h} + {\lVert{\boldsymbol{r}}\rVert}^2_{0,{\Omega}}\Big)^{1/2} {\lVert({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)\rVert}_{DG}$$ for all ${({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\in{\boldsymbol{\mathcal{W}}}(h)\times {\boldsymbol{\mathcal{Q}}}$ with ${\mathop{\mathbf{div}}\nolimits}_h{\boldsymbol{\sigma}}\in {\mathrm{H}}^s({\mathcal{T}}_h)^3$ for a given $s>1/2$ and for all ${({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h$.
We end this section by showing that the DG scheme is consistent.
\[consistency\] Let ${\boldsymbol{u}}$ be the solution of and let ${\boldsymbol{\sigma}}:= {\mathcal{C}}{\boldsymbol{\varepsilon}}({\boldsymbol{u}})$ and ${\boldsymbol{r}}:= \frac12(\nabla {\boldsymbol{u}}- (\nabla {\boldsymbol{u}})^{\mathtt{t}})$. Then, $$\label{consistent}
D_h^{\textup{\texttt{a}}}\Big( ({\boldsymbol{\sigma}}-{\boldsymbol{\sigma}}_h, {\boldsymbol{r}}- {\boldsymbol{r}}_h), {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Big) = 0 \quad \forall {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h.$$
By definition, $$\begin{gathered}
\label{id1}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = \int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h -
\kappa^2 \left( \int_{{\Omega}} {\mathcal{C}}^{-1}{\boldsymbol{\sigma}}:{\boldsymbol{\tau}}_h + \int_{{\Omega}}{\boldsymbol{r}}:{\boldsymbol{\tau}}_h\right)\\ -
\int_{{\mathcal{F}}^*_h} {\{{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\}} \cdot {\llbracket {\boldsymbol{\tau}}_h \rrbracket}.
\end{gathered}$$ The identity ${\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}= {\boldsymbol{f}}- \kappa^2 {\boldsymbol{u}}$ and integration by parts yield $$\begin{gathered}
\int_{{\Omega}} {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h = \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h -
\kappa^2 \sum_{K\in {\mathcal{T}}_h} \int_K {\boldsymbol{u}}\cdot {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\tau}}_h = \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h +\\
\kappa^2\sum_{K\in {\mathcal{T}}_h} \int_K \nabla {\boldsymbol{u}}: {\boldsymbol{\tau}}_h - \kappa^2\sum_{K\in {\mathcal{T}}_h} \int_{\partial K} {\boldsymbol{u}}\cdot {\boldsymbol{\tau}}_h{\boldsymbol{n}}_K.\end{gathered}$$ Substituting back into by taking into account that $\nabla {\boldsymbol{u}}={\mathcal{C}}^{-1}{\boldsymbol{\sigma}}- {\boldsymbol{r}}$ and $$\sum_{K\in {\mathcal{T}}_h} \int_{\partial K} {\boldsymbol{u}}\cdot {\boldsymbol{\tau}}_h{\boldsymbol{n}}_K = \int_{{\mathcal{F}}^*_h} {\{{\boldsymbol{u}}\}}\cdot {\llbracket {\boldsymbol{\tau}}_h \rrbracket}$$ we obtain $$\begin{gathered}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}, {\boldsymbol{r}})}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = \int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h -
\int_{{\mathcal{F}}^*_h} {\{{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}+\kappa^2{\boldsymbol{u}}\}} \cdot {\llbracket {\boldsymbol{\tau}}_h \rrbracket} \\=
\int_{{\Omega}} {\boldsymbol{f}}\cdot {\mathop{\mathbf{div}}\nolimits}_h {\boldsymbol{\tau}}_h -
\int_{{\mathcal{F}}^*_h} {\{{\boldsymbol{f}}\}} \cdot {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\end{gathered}$$ and the result follows.
Well-posedness and stability of the DG method {#section:5}
=============================================
By using the transformation rules $$\label{trans}
\phi\circ {\boldsymbol{F}}_K = \hat \phi, \qquad {\boldsymbol{v}}\circ {\boldsymbol{F}}_K = {\displaystyle}\frac{{\boldsymbol{B}}_K \hat {\boldsymbol{v}}}{|\det {\boldsymbol{B}}_K|}
\qquad \text{and} \qquad {\boldsymbol{w}}\circ {\boldsymbol{F}}_K = {\boldsymbol{B}}_K^{-{\mathtt{t}}} \check{{\boldsymbol{w}}},$$ we can easily show that $$\label{ident}
\int_F{\boldsymbol{v}}\cdot {\boldsymbol{n}}_K \phi = \int_{\hat F} \hat{\boldsymbol{v}}\cdot {\boldsymbol{n}}_{\hat K} \hat \phi
\qquad \text{and} \qquad
\int_K {\boldsymbol{v}}\cdot {\boldsymbol{w}}= \int_{\hat K} \hat {\boldsymbol{v}}\cdot \check {\boldsymbol{w}},$$ where $F$ is the image of the face $\hat F$ under the affine map ${\boldsymbol{F}}_K:\, \hat K \to {\mathbb{R}}^3$ defined in Section \[section:3\].
\[equivA\] There exists a constant $C>0$ independent of $h$ such that $$\begin{gathered}
\left( {\lVert\mathrm{div}\, {\boldsymbol{v}}\rVert}^2_{0,K} + h_K^{-2} {\lVert{\boldsymbol{v}}\rVert}^2_{0,K}\right)^{1/2}
\leq C\Big( h_K^{-1} \sup_{{\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(K)} \frac{\int_K {\boldsymbol{v}}\cdot {\boldsymbol{w}}}{{\lVert{\boldsymbol{w}}\rVert}_{0,K}}\\+
\sum_{F\in {\mathcal{F}}(K)} h_F^{-1/2} \sup_{\phi\in {\mathcal{P}}_k(F)} \frac{\int_F{\boldsymbol{v}}\cdot {\boldsymbol{n}}_K \phi}{{\lVert\phi\rVert}_{0,F}}\Big) \end{gathered}$$ for all ${\boldsymbol{v}}\in {\mathcal{P}}_k(K)^3$.
We will use here the notation $a\lesssim b$ to express that there exists $C>0$ independent of $h$ such that $a\le C\,b$ for all $h$. The notation $A \simeq B$ means that $A\lesssim B$ and $B\lesssim A$ simultaneously. We first notice that, thanks to the unisolvency of conditions -, the norms $$\left( {\lVert\mathrm{div}\, \hat {\boldsymbol{v}}\rVert}^2_{0,\hat K} + {\lVert\hat {\boldsymbol{v}}\rVert}^2_{0, \hat K}\right)^{1/2}
\quad
\text{and}
\quad
\sup_{\check {\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(\hat K)} \frac{{\displaystyle}\int_{\hat K} \hat {\boldsymbol{v}}\cdot \check {\boldsymbol{w}}}{{\lVert\check {\boldsymbol{w}}\rVert}_{0,\hat K}} +
\sum_{\hat F\in {\mathcal{F}}(\hat K)} \sup_{\hat \phi\in {\mathcal{P}}_k(\hat F)} \frac{{\displaystyle}\int_{\hat F}\hat {\boldsymbol{v}}\cdot {\boldsymbol{n}}_{\hat K} \hat \phi}
{{\lVert\hat \phi\rVert}_{0,\hat F}}$$ are equivalent on the finite dimensional space ${\mathcal{P}}_k(\hat K)^3$. Standard scaling arguments show that $$h_K {\lVert{\boldsymbol{v}}\rVert}_{0,K}^2 \simeq
{\lVert\hat {\boldsymbol{v}}\rVert}^2_{0,\hat K},
\qquad
h_K^{3} {\lVert\text{div} {\boldsymbol{v}}\rVert}_{0,K}^2 \simeq {\lVert\mathrm{div}\, \hat {\boldsymbol{v}}\rVert}^2_{0, \hat K}$$ and $${\lVert\phi\rVert}^2_{0,F} \simeq h_F^2 {\lVert\hat \phi\rVert}^2_{0,\hat F}, \qquad
{\lVert{\boldsymbol{w}}\rVert}^2_{0,K} \simeq h_K {\lVert\check {\boldsymbol{w}}\rVert}^2_{0,\hat K}.$$ Hence, we deduce from that $$\begin{gathered}
\left( h_K {\lVert{\boldsymbol{v}}\rVert}^2_{0,K} + h_K^3 {\lVert\text{div} {\boldsymbol{v}}\rVert}_{0,K}^2 \right)^{1/2} \lesssim
\left( {\lVert\hat {\boldsymbol{v}}\rVert}^2_{0,\hat K} + {\lVert\text{div} \hat {\boldsymbol{v}}\rVert}^2_{0, \hat K} \right)^{1/2} \lesssim\\
\sup_{\check {\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(\hat K)} \frac{\int_{\hat K} \hat {\boldsymbol{v}}\cdot \check {\boldsymbol{w}}}{{\lVert\check {\boldsymbol{w}}\rVert}_{0,\hat K}} +
\sum_{\hat F\in {\mathcal{F}}(\hat K)} \sup_{\hat \phi\in {\mathcal{P}}_k(\hat F)} \frac{\int_{\hat F}\hat {\boldsymbol{v}}\cdot {\boldsymbol{n}}_{\hat K} \hat \phi}
{{\lVert\hat \phi\rVert}_{0,\hat F}} \lesssim\\
h_K^{1/2} \sup_{{\boldsymbol{w}}\in {\boldsymbol{\mathcal{N}}}_{k-1}(K)} \frac{\int_K {\boldsymbol{v}}\cdot {\boldsymbol{w}}}{{\lVert{\boldsymbol{w}}\rVert}_{0,K}}+
\sum_{F\in {\mathcal{F}}(K)} h_F \sup_{\phi\in {\mathcal{P}}_k(F)} \frac{\int_F{\boldsymbol{v}}\cdot {\boldsymbol{n}}_K \phi}{{\lVert\phi\rVert}_{0,F}},\end{gathered}$$ and the result follows.
We introduce the projection ${\mathcal{P}}_h:\, {\boldsymbol{\mathcal{W}}}_h \to {\boldsymbol{\mathcal{W}}}_h^c$ uniquely characterized, for any ${\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}_h$, by the conditions $$\begin{aligned}
\int_F ({\mathcal{P}}_h{\boldsymbol{\tau}}) {\boldsymbol{n}}_F \cdot {\boldsymbol{\varphi}}& = & \int_F {\boldsymbol{\tau}}{\boldsymbol{n}}_F \cdot {\boldsymbol{\varphi}}\qquad \forall {\boldsymbol{\varphi}}\in {\mathcal{P}}_k(F)^3, \quad\forall F\in {\mathcal{F}}_h^{{\Gamma}},\label{GDFb}
\\
\int_F ({\mathcal{P}}_h{\boldsymbol{\tau}}) {\boldsymbol{n}}_F \cdot {\boldsymbol{\varphi}}& = & 0
\qquad \forall{\boldsymbol{\varphi}}\in {\mathcal{P}}_k(F)^3, \quad \forall F\in {\mathcal{F}}_h^{{\Sigma}},\label{GDFa}
\\
\int_F ({\mathcal{P}}_h{\boldsymbol{\tau}}) {\boldsymbol{n}}_F \cdot {\boldsymbol{\varphi}}& = & \int_F {\{{\boldsymbol{\tau}}\}}_F {\boldsymbol{n}}_F \cdot {\boldsymbol{\varphi}}\qquad \forall {\boldsymbol{\varphi}}\in {\mathcal{P}}_k(F)^3, \quad\forall F\in {\mathcal{F}}_h^{0},\label{GDFbc}
\\
\int_K {\mathcal{P}}_h{\boldsymbol{\tau}}: {\boldsymbol{W}}&= & \int_K {\boldsymbol{\tau}}: {\boldsymbol{W}}\quad\forall \text{${\boldsymbol{W}}$ with rows in ${\boldsymbol{\mathcal{N}}}_{k-1}(K)$},
\quad \forall K\in {\mathcal{T}}_h\label{DFGd}.\end{aligned}$$ We point out that the projection ${\mathcal{P}}_h$ may be viewed as the div-conforming counterpart of the projection with curl-conforming range introduced in [@dominik1].
\[propC\] The norm equivalence $$\label{equivN}
\underbar{C}\, {\lVert{\boldsymbol{\tau}}\rVert}_{{\boldsymbol{\mathcal{W}}}(h)} \leq \Big( {\lVert{\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} +
{\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,{\mathcal{F}}^*_h}
\Big)^{1/2} \leq \bar{C} {\lVert{\boldsymbol{\tau}}\rVert}_{{\boldsymbol{\mathcal{W}}}(h)}$$ holds true on ${\boldsymbol{\mathcal{W}}}_h$ with constants $\underbar{C}>0$ and $\bar{C}>0$ independent of $h$.
Using Proposition \[equivA\] row-wise we deduce that there exists $C_0>0$ independent of $h$ such that $$\begin{gathered}
{\lVert{\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}})\rVert}^2_{0,K} + h_K^{-2} {\lVert{\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{0,K} \\
\leq C_0
\sum_{F\in {\mathcal{F}}(K)} h_F^{-1} \Big( \sup_{{\boldsymbol{\varphi}}\in {\mathcal{P}}_k(F)^3} \frac{\int_F({\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}){\boldsymbol{n}}_K\cdot {\boldsymbol{\varphi}}}{{\lVert{\boldsymbol{\varphi}}\rVert}_{0,F}}\Big)^2. \end{gathered}$$ It is easy to obtain, from the definition of ${\mathcal{P}}_h$, the identity $$\int_F({\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}){\boldsymbol{n}}_K\cdot {\boldsymbol{\varphi}}= \begin{cases}
\frac{1}{2}\int_F {\llbracket {\boldsymbol{\tau}}\rrbracket}_F\cdot {\boldsymbol{\varphi}}& \text{if $F\in {\mathcal{F}}^0_h$}\\[1ex]
\int_F {\llbracket {\boldsymbol{\tau}}\rrbracket}_F\cdot {\boldsymbol{\varphi}}& \text{if $F\in {\mathcal{F}}^{\Sigma}_h$}\\[1ex]
0 & \text{if $F\in {\mathcal{F}}^{\Gamma}_h$}.
\end{cases}$$ Hence, using the Cauchy-Schwarz inequality and summing up over $K\in {\mathcal{T}}_h$ we deduce that $$\label{proj}
{\lVert{\mathop{\mathbf{div}}\nolimits}_h ({\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}})\rVert}^2_{0,{\Omega}} + \sum_{K\in {\mathcal{T}}_h} h_K^{-2} {\lVert{\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{0,K}
\leq C_0
\sum_{F\in {\mathcal{F}}_h^*} h_F^{-1} {\lVert{\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,F},$$ which proves that $$\label{propB}
{\lVert{\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} \leq (1 + C_0) {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,{\mathcal{F}}^*_h}
\qquad \forall {\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}_h.$$ The lower bound of is then a consequence of the uniform boundedness of ${\mathcal{P}}_h$ on ${\boldsymbol{\mathcal{W}}}_h$, $$\label{PhBounded}
{\lVert{\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} = {\lVert{\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} \leq
2{\lVert{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} + 2 {\lVert{\boldsymbol{\tau}}- {\mathcal{P}}_h {\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} \leq 2(2 + C_0) {\lVert{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)}.$$ On the other hand, $${\lVert{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} \leq 2{\lVert{\mathcal{P}}_h{\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} + 2{\lVert{\boldsymbol{\tau}}- {\mathcal{P}}_h{\boldsymbol{\tau}}\rVert}^2_{{\boldsymbol{\mathcal{W}}}(h)} \leq
2(1 + C_0)\left({\lVert{\mathcal{P}}_h{\boldsymbol{\tau}}\rVert}^2_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} + {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,{\mathcal{F}}^*_h}\right)$$ for all ${\boldsymbol{\tau}}\in {\boldsymbol{\mathcal{W}}}_h$ which gives the upper bound of .
\[infsupDh\] Assume that $\frac{1}{1+\kappa^2}\notin {\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$. There exist parameters $h^*>0$ and $\textup{\texttt{a}}^*>0$ such that, for $h\leq h^*$ and $\textup{\texttt{a}}\ge \textup{\texttt{a}}^*$, $$\label{infsupABh}
\sup_{{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h} \frac{{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}, {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)}}{{\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}_{DG}}
\geq \alpha_D^{d} {\lVert{({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}_{DG}, \forall {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h$$ with $\alpha_D^{d}>0$ independent of the mesh size $h$ and $\lambda$.
We deduce from that there exists an operator $\Xi_h:\, {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h \to {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$ such that, for $h\leq h_0$, $$\label{Xi}
{D\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Xi_h {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\Bigr)} = \, {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}^2 \quad \text{and}\quad
{\lVert\Xi_h {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert} \leq \frac{1}{\alpha_D^c} {\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}$$ for all ${({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$. Given ${({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h$, the decomposition ${\boldsymbol{\tau}}_h = {\boldsymbol{\tau}}_h^c + \tilde{\boldsymbol{\tau}}_h$, with ${\boldsymbol{\tau}}_h^c := {\mathcal{P}}_h {\boldsymbol{\tau}}_h$ and $\tilde{\boldsymbol{\tau}}_h := {\boldsymbol{\tau}}_h - {\mathcal{P}}_h {\boldsymbol{\tau}}_h$, and yield $$\begin{gathered}
\label{split}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}+ {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}
= {\lVert{({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert}^2 +
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}, {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)} + \\
{D_h^{\texttt{a}}\Bigl( {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\Bigr)} +
{D_h^{\texttt{a}}\Bigl( {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}, {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}.\end{gathered}$$ Using the Cauchy-Schwarz inequality we have that $$\begin{gathered}
{D_h^{\texttt{a}}\Bigl( {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}, {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}= {\lVert{\mathop{\mathbf{div}}\nolimits}_h \tilde {\boldsymbol{\tau}}_h\rVert}_{0,{\Omega}}^2 +
\texttt{a} {\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}^*_h}^2 -
\kappa^2 \int_{{\Omega}} {\mathcal{C}}^{-1}\tilde {\boldsymbol{\tau}}_h:\tilde {\boldsymbol{\tau}}_h \\
- 2 \int_{{\mathcal{F}}^*_h} {\{{\mathop{\mathbf{div}}\nolimits}_h \tilde {\boldsymbol{\tau}}_h\}}\cdot {\llbracket \tilde {\boldsymbol{\tau}}_h \rrbracket}
\,\, \ge\,\, \texttt{a} {\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}^*_h}^2 - C_1 {\lVert \tilde {\boldsymbol{\tau}}_h \rVert}^2_{0,{\Omega}}\\
-2 {\lVerth_{\mathcal{F}}^{1/2} {\{{\mathop{\mathbf{div}}\nolimits}_h \tilde {\boldsymbol{\tau}}_h\}}\rVert}_{0,{\mathcal{F}}^*_h} {\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}^*_h}.\end{gathered}$$ It follows from and that $${D_h^{\texttt{a}}\Bigl( {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}, {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)} \geq (\texttt{a} -C_2){\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}^*_h}^2$$ with a constant $C_2>0$ independent of $h$ and $\lambda$.
The remaining two terms of are bounded from below by using , and . Indeed, it is straightforward that $$\begin{gathered}
{D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}, {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)} \geq -M^{d}_D
{\lVert{({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert} {\lVert\tilde{\boldsymbol{\tau}}_h\rVert}_{{\boldsymbol{\mathcal{W}}}(h)}\\
\ge -\frac{1}{4} {\lVert{({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert}^2 - C_3
{\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}^*_h}^2\end{gathered}$$ and $$\begin{gathered}
{D_h^{\texttt{a}}\Bigl( {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\Bigr)} \geq - M^{d}_D
{\lVert\Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert} {\lVert\tilde{\boldsymbol{\tau}}_h\rVert}_{{\boldsymbol{\mathcal{W}}}(h)} \geq - \frac{M^{d}_D}{\alpha_D^{c}}
{\lVert {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert} {\lVert\tilde{\boldsymbol{\tau}}_h\rVert}_{{\boldsymbol{\mathcal{W}}}(h)} \\
\geq -\frac{1}{4} {\lVert {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert}^2 - C_4
{\lVerth_{\mathcal{F}}^{-1/2} {\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}_h}^2,\end{gathered}$$ with $C_3>0$ and $C_4>0$ independent of $h$ and $\lambda$. Summing up, we have that, $${D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}+ {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}\ge \frac{1}{2} {\lVert {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert}^2 + \big(\texttt{a} - C^* \big)
{\lVerth_{\mathcal{F}}^{-1/2}{\llbracket {\boldsymbol{\tau}}_h \rrbracket}\rVert}_{0,{\mathcal{F}}_h}^2,$$ with $C^* := C_2+C_3+C_4$. Hence, if $\texttt{a} > C^* + 1/2$, by virtue of we have that $${D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}+ {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}\ge \frac{1}{2} \Big( {\lVert{({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}\rVert}^2 +
{\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket {\boldsymbol{\tau}}\rrbracket}\rVert}^2_{0,{\mathcal{F}}^*_h} \Big) \geq \frac{{\underbar C}^2}{2} {\lVert {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}^2_{DG}.$$ Finally, using and we deduce that there exists $\alpha^d_D>0$ such that, $${D_h^{\texttt{a}}\Bigl( {({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}, \Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}+ {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\Bigr)}
\geq \alpha^d_D
{\lVert{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\rVert}_{DG}
\Big({\lVert\Xi_h {({\boldsymbol{\tau}}_h^c, {\boldsymbol{s}}_h)}+ {(\tilde{{\boldsymbol{\tau}}}_h, \mathbf{0})}\rVert}^2_{DG}\Big),$$ provided that $h$ is sufficiently small and $\texttt{a}$ is sufficiently large, which gives .
The first consequence of the inf-sup condition is that the DG problem admits a unique solution. Moreover, we have the following Céa estimate.
Assume that $\frac{1}{1+\kappa^2}\notin{\mathop{\mathrm{sp}}\nolimits}({\boldsymbol{T}})$ and let ${({\boldsymbol{\sigma}}, {\boldsymbol{r}})}\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ be the solution of –. There exist parameters $h^*>0$ and $\textup{\texttt{a}}^*>0$ such that, for $h\leq h^*$ and $\textup{\texttt{a}}\ge \textup{\texttt{a}}^*$, $$\begin{gathered}
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}_{DG} \leq (1+ \frac{M^{d}_D}{\alpha_D^{d}}) \inf_{{({\boldsymbol{\tau}}_h, {\boldsymbol{s}}_h)}\in {\boldsymbol{\mathcal{W}}}_h \times {\boldsymbol{\mathcal{Q}}}_h}
\Big(
{\lVert{\boldsymbol{\sigma}}- {\boldsymbol{\tau}}_h\rVert}_{{\boldsymbol{\mathcal{W}}}(h)} \\+
{\lVerth_{{\mathcal{F}}}^{1/2} {\{{\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\sigma}}-{\boldsymbol{\tau}}_h)\}}\rVert}_{0,{\mathcal{F}}^*_h} + {\lVert{\boldsymbol{r}}-{\boldsymbol{s}}_h\rVert}_{0,{\Omega}}
\Big).\end{gathered}$$ Moreover, if the exact solution ${\boldsymbol{u}}$ of belongs to ${\mathrm{H}}^{1+s}({\Omega})^3$ for some $s>1/2$ and if ${\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\in {\mathrm{H}}^s({\Omega})^3$, then the error estimate $${\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}_{DG} \leq \, C\, h^{\min(s, k)}\, \Big(
{\lVert{\boldsymbol{u}}\rVert}_{1+s,K} + {\lVert{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\rVert}_{s,K}
\Big), \quad \forall h\leq h^*,$$ holds true with a constant $C>0$ independent of $h$ and $\lambda$.
The first estimate follows from , and as shown in [@DiPietroErn Theorem 1.35]. On the other hand, under the regularity hypotheses on $u$ and ${\boldsymbol{\sigma}}$, $$\begin{gathered}
{\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}_{DG} \leq (1+ \frac{M^{d}_D}{\alpha_D^{d}})
\Big(
{\lVert{\boldsymbol{\sigma}}- \Pi_h{\boldsymbol{\sigma}}\rVert}_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}} +
{\lVerth_{{\mathcal{F}}}^{1/2} {\{{\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\sigma}}-\Pi_h{\boldsymbol{\sigma}})\}}\rVert}_{0,{\mathcal{F}}^*_h}\\ + {\lVert{\boldsymbol{r}}-\mathcal{S}_h{\boldsymbol{r}}\rVert}_{0,{\Omega}}
\Big)\end{gathered}$$ and we notice that $${\lVerth_{{\mathcal{F}}}^{1/2} {\{{\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\sigma}}-\Pi_h{\boldsymbol{\sigma}})\}}\rVert}_{0,{\mathcal{F}}^*_h}
\leq
\sum_{K\in {\mathcal{T}}_h} \sum_{F\in {\mathcal{F}}(K)} h_F{\lVert {\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\sigma}}-\Pi_h{\boldsymbol{\sigma}}) \rVert}^2_{0,F}.$$ Using the commuting diagram property satisfied by $\Pi_h$, the trace theorem and standard scaling arguments we obtain that $$h_F^{1/2}{\lVert {\mathop{\mathbf{div}}\nolimits}({\boldsymbol{\sigma}}-\Pi_h{\boldsymbol{\sigma}}) \rVert}_{0,F} = h_F^{1/2}{\lVert {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}- \mathcal R_K {\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\rVert}_{0,F}
\leq C_2 h_K^{\min(k,s)} {\lVert{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\rVert}_{s,K}$$ for all $F\in {\mathcal{F}}(K)$, where the ${\mathrm{L}}^2(K)$-orthogonal projection $\mathcal R_K:= \mathcal R_h|_K$ onto ${\mathcal{P}}_{k-1}(K)$ is applied componentwise. Consequently, by virtue of the error estimates , and , $${\lVert{({\boldsymbol{\sigma}}, {\boldsymbol{r}})}- {({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)}\rVert}_{DG} \leq C_3 h^{\min(k,s)}
\Big(
{\lVert{\boldsymbol{u}}\rVert}_{1+s,K} + {\lVert{\mathop{\mathbf{div}}\nolimits}{\boldsymbol{\sigma}}\rVert}_{s,K}
\Big)$$ and the result follows.
Numerical results {#section:6}
=================
We present a series of numerical experiments confirming the good performance of the continuous Galerkin scheme and the discontinuous Galerkin scheme . For simplicity we consider our model problem in two dimensions. The corresponding theory and results from three dimensions apply with trivial modifications.
All the numerical results have been obtained by using the FEniCS Problem Solving Environment [@fenics]. We choose $\Omega=(0,1)\times (0,1)$, $\lambda=\mu=1$ and select the data ${\boldsymbol{f}}$ so that the exact solution is given by $${\boldsymbol{u}}(x_1, x_2) = \begin{pmatrix}
-x_2 \sin(\kappa \pi x_1)
\\ 0.5 \pi x_2 \cos(\kappa \pi x_1)
\end{pmatrix}.$$ We also assume that the body is fixed on the whole $\partial {\Omega}$ and the non-homogeneous Dirichlet boundary condition is imposed by adding an adequate boundary term to the right-hand side of . The numerical results obtained below for the continuous and discontinuous Galerkin schemes have been obtained by considering nested sequences of uniform triangular meshes ${\mathcal{T}}_h$ of the unit square ${\Omega}$. The individual relative errors produced by the continuous Galerkin method are given by $$\label{Ec}
\texttt{e}^\kappa_c({\boldsymbol{\sigma}}) := \frac{\| {\boldsymbol{\sigma}}- {\boldsymbol{\sigma}}_h \|_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}}{\| {\boldsymbol{\sigma}}\|_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}} \qquad \text{and} \qquad
\texttt{e}^\kappa_c({\boldsymbol{r}}) := \frac{\| {\boldsymbol{r}}- {\boldsymbol{r}}_h \|_{0,{\Omega}}}{\| {\boldsymbol{r}}\|_{0,{\Omega}}},$$ where $({\boldsymbol{\sigma}}, {\boldsymbol{r}})\in {\boldsymbol{\mathcal{W}}}\times {\boldsymbol{\mathcal{Q}}}$ and $({\boldsymbol{\sigma}}_h, {\boldsymbol{r}}_h)\in {\boldsymbol{\mathcal{W}}}_h^c\times {\boldsymbol{\mathcal{Q}}}_h$ are the solutions of and respectively. We introduce the experimental rates of convergence $$\label{EcO}
\texttt{r}_c^{\kappa}({\boldsymbol{\sigma}}) := \frac{\log (\texttt{e}_c^{\kappa}({\boldsymbol{\sigma}})/
\hat{\texttt{e}}_c^{\kappa}({\boldsymbol{\sigma}}))}{\log (h/\hat{h})},\qquad
\texttt{r}_c^{\kappa}({\boldsymbol{r}}) :=
\frac{\log (\texttt{e}_c^{\kappa}({\boldsymbol{r}})/ \hat{\texttt{e}}_c^{\kappa}({\boldsymbol{r}}))}{\log (h/\hat{h})},$$ where $\texttt{e}_c^{\kappa}$ and $\hat{\texttt{e}}_c^{\kappa}$ are the errors corresponding to two consecutive triangulations with mesh sizes $h$ and $\hat{h}$, respectively.
Similarly, we denote the individual relative errors of the discontinuous Galerkin scheme $$\label{Ed}
\texttt{e}^\kappa_{d}({\boldsymbol{\sigma}}) := \frac{\| {\boldsymbol{\sigma}}- {\boldsymbol{\sigma}}_h \|_{{\boldsymbol{\mathcal{W}}}(h)}}{\| {\boldsymbol{\sigma}}\|_{{{{\mathrm{H}}(\mathbf{div},{\Omega})}}}}, \qquad
\texttt{e}^\kappa_{d}({\boldsymbol{r}}) := \frac{\| {\boldsymbol{r}}- {\boldsymbol{r}}_h \|_{0,{\Omega}}}{\| {\boldsymbol{r}}\|_{0,{\Omega}}},$$ where, in this case, $({\boldsymbol{\sigma}}_h,{\boldsymbol{r}}_h)\in {\boldsymbol{\mathcal{W}}}_h\times {\boldsymbol{\mathcal{Q}}}_h$ is the solution of . Accordingly, the experimental rates of convergence of the DG scheme are given by $$\label{EdO}
\texttt{r}_d^{\kappa}({\boldsymbol{\sigma}}) := \frac{\log (\texttt{e}_d^{\kappa}({\boldsymbol{\sigma}})/
\hat{\texttt{e}}_d^{\kappa}({\boldsymbol{\sigma}}))}{\log (h/\hat{h})}\,,\qquad
\texttt{r}_d^{\kappa}({\boldsymbol{r}}) :=
\frac{\log (\texttt{e}_d^{\kappa}({\boldsymbol{r}})/ \hat{\texttt{e}}_d^{\kappa}({\boldsymbol{r}}))}{\log (h/\hat{h})}.$$
We begin by testing the convergence order of the continuous Galerkin method for the range of values $\kappa = 4, 8, 16, 32$ of the wave number. We report in Tables \[table:sCG2\], \[table:rCG2\], \[table:sCG3\], \[table:rCG3\] the relative errors and the convergence orders obtained in the cases $k=2$ and $k= 4$, respectively. It is clear that the correct quadratic and quartic convergence rates of the errors are attained in each variable and for each fixed wave number $\kappa$.
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_c^4(\boldsymbol{\sigma})$ & $\texttt{r}_c^4(\boldsymbol{\sigma})$ & $\verb"e"_c^8(\boldsymbol{\sigma})$ & $\texttt{r}_c^8(\boldsymbol{\sigma})$ & $\verb"e"_c^{16}(\boldsymbol{\sigma})$ & $\texttt{r}_c^{16}(\boldsymbol{\sigma})$ & $\verb"e"_c^{32}(\boldsymbol{\sigma})$ & $\texttt{r}_c^{32}(\boldsymbol{\sigma})$
$8$ & 6.90e$-$02 & $-$ & 2.96e$-$01 & $-$ & 1.64e$+$00 & $-$ & 2.43e$+$00 & $-$
$16$ & 1.79e$-$02 & 1.94 & 6.80e$-$02 & 2.12 & 3.09e$-$01 & 2.41 & 1.11e$+$00 & 1.12
$32$ & 4.53e$-$03 & 1.99 & 1.77e$-$02 & 1.94 & 6.78e$-$02 & 2.19 & 3.01e$-$01 & 1.89
$64$ & 1.13e$-$03 & 2.00 & 4.46e$-$03 & 1.99 & 1.76e$-$02 & 1.94 & 6.77e$-$02 & 2.16
$128$ & 2.84e$-$04 & 2.00 & 1.12e$-$03 & 2.00 & 4.44e$-$03 & 1.99 & 1.76e$-$02 & 1.94
$256$ & 7.10e$-$05 & 2.00 & 2.80e$-$04 & 2.00 & 1.11e$-$03 & 2.00 & 4.44e$-$03 & 1.99
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_c^4({\boldsymbol{r}})$ & $\texttt{r}_c^4({\boldsymbol{r}})$ & $\verb"e"_c^8({\boldsymbol{r}})$ & $\texttt{r}_c^8({\boldsymbol{r}})$ & $\verb"e"_c^{16}({\boldsymbol{r}})$ & $\texttt{r}_c^{16}({\boldsymbol{r}})$ & $\verb"e"_c^{32}({\boldsymbol{r}})$ & $\texttt{r}_c^{32}({\boldsymbol{r}})$
$8$ & 1.96e$-$01 & $-$ & 7.64e$-$01 & $-$ & 9.97e$+$00 & $-$ & 1.59e$+$01 & $-$
$16$ & 5.32e$-$02 & 1.88 & 1.95e$-$01 & 1.98 & 9.32e$-$01 & 3.42 & 6.80e$+$00 & 1.23
$32$ & 1.36e$-$02 & 1.97 & 5.28e$-$02 & 1.88 & 1.95e$-$01 & 2.26 & 7.80e$-$01 & 3.12
$64$ & 3.43e$-$03 & 1.99 & 1.35e$-$02 & 1.97 & 5.27e$-$02 & 1.89 & 1.95e$-$01 & 2.00
$128$ & 8.60e$-$04 & 2.00 & 3.39e$-$03 & 1.99 & 1.34e$-$02 & 1.97 & 5.27e$-$02 & 1.89
$256$ & 2.15e$-$04 & 2.00 & 8.48e$-$04 & 2.00 & 3.38e$-$03 & 1.99 & 1.34e$-$02 & 1.97
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_c^4(\boldsymbol{\sigma})$ & $\texttt{r}_c^4(\boldsymbol{\sigma})$ & $\verb"e"_c^8(\boldsymbol{\sigma})$ & $\texttt{r}_c^8(\boldsymbol{\sigma})$ & $\verb"e"_c^{16}(\boldsymbol{\sigma})$ & $\texttt{r}_c^{16}(\boldsymbol{\sigma})$ & $\verb"e"_c^{32}(\boldsymbol{\sigma})$ & $\texttt{r}_c^{32}(\boldsymbol{\sigma})$
$8$ & 9.33e$-$04 & $-$ & 1.67e$-$02 & $-$ & 2.34e$-$01 & $-$ & 1.50e$+$00 & $-$
$16$ & 5.95e$-$05 & 3.97 & 8.92e$-$04 & 4.23 & 1.70e$-$02 & 3.79 & 1.16e$-$01 & 3.69
$32$ & 3.74e$-$06 & 3.99 & 5.68e$-$05 & 3.97 & 8.81e$-$04 & 4.27 & 1.69e$-$02 & 2.78
$64$ & 2.34e$-$07 & 4.00 & 3.57e$-$06 & 3.99 & 5.61e$-$05 & 3.97 & 8.78e$-$04 & 4.27
$128$ & 1.46e$-$08 & 4.00 & 2.23e$-$07 & 4.00 & 3.53e$-$06 & 3.99 & 5.60e$-$05 & 3.97
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_c^4({\boldsymbol{r}})$ & $\texttt{r}_c^4({\boldsymbol{r}})$ & $\verb"e"_c^8({\boldsymbol{r}})$ & $\texttt{r}_c^8({\boldsymbol{r}})$ & $\verb"e"_c^{16}({\boldsymbol{r}})$ & $\texttt{r}_c^{16}({\boldsymbol{r}})$ & $\verb"e"_c^{32}({\boldsymbol{r}})$ & $\texttt{r}_c^{32}({\boldsymbol{r}})$
$8$ & 1.88e$-$03 & $-$ & 3.31e$-$02 & $-$ & 1.52e$+$00 & $-$ & 8.98e$+$00 & $-$
$16$ & 1.23e$-$04 & 3.93 & 1.77e$-$03 & 4.23 & 3.64e$-$02 & 5.38 & 5.24e$-$01 & 4.10
$32$ & 7.80e$-$06 & 3.98 & 1.15e$-$04 & 3.94 & 1.75e$-$03 & 4.38 & 3.41e$-$02 & 3.94
$64$ & 4.90e$-$07 & 3.99 & 7.27e$-$06 & 3.98 & 1.13e$-$04 & 3.95 & 1.74e$-$03 & 4.29
$128$ & 3.27e$-$08 & 3.91 & 4.56e$-$07 & 4.00 & 7.15e$-$06 & 3.99 & 1.13e$-$04 & 3.95
The subsequent numerical tests are for the discontinuous Galerkin scheme . We present throughout Tables \[table:sDG4\], \[table:rDG4\], \[table:sDG6\] and \[table:rDG6\] results corresponding to $k=4$ with a range of wave numbers given by $\kappa = 4, 8, 16, 32$. We also show results corresponding to $k=6$ with $\kappa = 16, 28, 32, 40$. For both polynomial degrees ($k=4, 6$) we take a stabilization parameter $\texttt{a}=100$. The expected rates of convergence are attained in all the cases. We notice that the higher the value of the wave number $\kappa$ is, the smaller is the mesh size needed to reduce the error below a given tolerance.
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^4(\boldsymbol{\sigma})$ & $\texttt{r}_d^4(\boldsymbol{\sigma})$ & $\verb"e"_d^8(\boldsymbol{\sigma})$ & $\texttt{r}_d^8(\boldsymbol{\sigma})$ & $\verb"e"_d^{16}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{16}(\boldsymbol{\sigma})$ & $\verb"e"_d^{32}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{32}(\boldsymbol{\sigma})$
$8$ & 9.41e$-$04 & $-$ & 1.68e$-$02 & $-$ & 2.35e$-$01 & $-$ & 1.49e$+$00 & $-$
$16$ & 6.01e$-$05 & 3.97 & 9.00e$-$04 & 4.22 & 1.70e$-$02 & 3.79 & 1.17e$-$01 & 3.68
$32$ & 3.78e$-$06 & 3.99 & 5.75e$-$05 & 3.97 & 8.89e$-$04 & 4.26 & 1.70e$-$02 & 2.78
$64$ & 2.36e$-$07 & 4.00 & 3.61e$-$06 & 3.99 & 5.68e$-$05 & 3.97 & 8.87e$-$04 & 4.26
$128$ & 1.48e$-$08 & 4.00 & 2.26e$-$07 & 4.00 & 3.57e$-$06 & 3.99 & 5.67e$-$05 & 3.97
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^4({\boldsymbol{r}})$ & $\texttt{r}_d^4({\boldsymbol{r}})$ & $\verb"e"_d^8({\boldsymbol{r}})$ & $\texttt{r}_d^8({\boldsymbol{r}})$ & $\verb"e"_d^{16}({\boldsymbol{r}})$ & $\texttt{r}_d^{16}({\boldsymbol{r}})$ & $\verb"e"_d^{32}({\boldsymbol{r}})$ & $\texttt{r}_d^{32}({\boldsymbol{r}})$
$8$ & 1.88e$-$03 & $-$ & 3.30e$-$02 & $-$ & 1.52e$+$00 & $-$ & 8.84e$+$00 & $-$
$16$ & 1.23e$-$04 & 3.93 & 1.77e$-$03 & 4.22 & 3.63e$-$02 & 5.38 & 5.22e$-$01 & 4.08
$32$ & 7.81e$-$06 & 3.98 & 1.15e$-$04 & 3.94 & 1.75e$-$03 & 4.38 & 3.40e$-$02 & 3.94
$64$ & 4.90e$-$07 & 3.99 & 7.27e$-$06 & 3.98 & 1.13e$-$04 & 3.95 & 1.74e$-$03 & 4.28
$128$ & 3.14e$-$08 & 3.97 & 4.56e$-$07 & 4.00 & 7.15e$-$06 & 3.99 & 1.13e$-$04 & 3.95
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^{16}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{16}(\boldsymbol{\sigma})$ & $\verb"e"_d^{28}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{28}(\boldsymbol{\sigma})$ & $\verb"e"_d^{32}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{32}(\boldsymbol{\sigma})$ & $\verb"e"_d^{40}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{40}(\boldsymbol{\sigma})$
$8$ & 1.46e$-$02 & $-$ & 4.41e$-$01 & $-$ & 4.89e$-$01 & $-$ & 1.24e$+$00 & $-$
$16$ & 3.69e$-$04 & 5.30 & 8.02e$-$03 & 5.78 & 9.42e$-$03 & 5.70 & 5.25e$-$02 & 4.56
$32$ & 4.80e$-$06 & 6.27 & 1.30e$-$04 & 5.95 & 3.70e$-$04 & 4.67 & 1.05e$-$03 & 5.65
$40$ & 4.29e$-$07 & 5.96 & 1.19e$-$05 & 5.89 & 2.62e$-$05 & 6.53 & 9.74e$-$05 & 5.86
$56$ & 1.71e$-$07 & 5.97 & 4.77e$-$06 & 5.93 & 1.05e$-$05 & 5.91 & 3.93e$-$05 & 5.88
$64$ & 7.68e$-$08 & 5.98 & 2.16e$-$06 & 5.94 & 4.77e$-$06 & 5.93 & 1.79e$-$05 & 5.90
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^{16}({\boldsymbol{r}})$ & $\texttt{r}_d^{16}({\boldsymbol{r}})$ & $\verb"e"_d^{28}({\boldsymbol{r}})$ & $\texttt{r}_d^{28}({\boldsymbol{r}})$ & $\verb"e"_d^{32}({\boldsymbol{r}})$ & $\texttt{r}_d^{32}({\boldsymbol{r}})$ & $\verb"e"_d^{40}({\boldsymbol{r}})$ & $\texttt{r}_d^{40}({\boldsymbol{r}})$
$8$ & 8.13e$-$03 & $-$ & 2.08e$+$00 & $-$ & 2.65e$+$00 & $-$ & 3.93e$+$00 & $-$
$16$ & 8.08e$-$04 & 6.65 & 2.49e$-$02 & 6.38 & 3.09e$-$02 & 6.42 & 1.67e$-$01 & 4.56
$32$ & 9.71e$-$06 & 6.38 & 2.62e$-$04 & 6.58 & 7.85e$-$04 & 5.30 & 2.09e$-$03 & 6.32
$40$ & 8.68e$-$07 & 5.96 & 2.40e$-$05 & 5.89 & 5.29e$-$05 & 6.65 & 1.96e$-$04 & 5.84
$56$ & 3.45e$-$07 & 5.97 & 9.64e$-$06 & 5.93 & 2.13e$-$05 & 5.91 & 7.94e$-$05 & 5.87
$64$ & 1.55e$-$07 & 5.99 & 4.36e$-$06 & 5.94 & 9.64e$-$06 & 5.93 & 3.61e$-$05 & 5.90
To test the locking-free character of the method in the nearly incompressible case, we consider now Lamé coefficients $\lambda$ and $\mu$ corresponding to a Poisson ratio $\nu=0.499$ and a Young modulus $E = 10$. We fix the polynomial degree to $k=2$, take a stabilization parameter $\texttt{a} = 50$ and report in Tables \[table:s\_incompressible\] and \[table:r\_incompressible\] the experimental rates of convergence for $\kappa=4,8,16,32$. We observe that the method is thoroughly robust for nearly incompressible materials. However, it seems that the pre-asymptotic region increases in this case for big values of $\kappa$.
We now study the influence of $\kappa$ on the choice of the stabilization parameter $\texttt{a}$ of the discontinuous Galerkin scheme . To this end, we present in Figure \[fig:a0k\] different approximations corresponding to $\kappa=2,4,8,16,32$, obtained with the mesh $h=1/32$ and a polynomial degree $k=3$. In each case, we represent in a double logarithmic scale the errors versus the parameter $\texttt{a}$. Clearly, $\texttt{a}$ is not sensible to the variations of $\kappa$. However, higher polynomial degrees $k$ require higher values for the stabilization parameter $\texttt{a}$. This is made clear in Figure \[fig:a0m\] where the polynomial degrees $k=1,\cdots,7$ are considered on a fixed mesh $h=1/32$, with a fixed wave number $\kappa=16$. In each case, the errors are depicted versus $\texttt{a}$.
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^4(\boldsymbol{\sigma})$ & $\texttt{r}_d^4(\boldsymbol{\sigma})$ & $\verb"e"_d^8(\boldsymbol{\sigma})$ & $\texttt{r}_d^8(\boldsymbol{\sigma})$ & $\verb"e"_d^{16}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{16}(\boldsymbol{\sigma})$ & $\verb"e"_d^{32}(\boldsymbol{\sigma})$ & $\texttt{r}_d^{32}(\boldsymbol{\sigma})$
$8$ & 6.90e$-$02 & $-$ & 3.27e$-$01 & $-$ & 3.58e$-$02 & $-$ & 1.80e$-$02 & $-$
$16$ & 1.80e$-$02 & 1.94 & 6.83e$-$02 & 2.26 & 3.27e$-$01 & $-$ & 1.83e$-$02 & $-$
$32$ & 4.54e$-$03 & 1.99 & 1.78e$-$02 & 1.94 & 6.81e$-$02 & 2.27 & 3.27e$-$01 & $-$
$64$ & 1.14e$-$03 & 2.00 & 4.49e$-$03 & 1.99 & 1.77e$-$02 & 1.94 & 6.80e$-$02 & 2.27
$128$ & 2.84e$-$04 & 2.00 & 1.13e$-$03 & 2.00 & 4.48e$-$03 & 1.99 & 1.77e$-$02 & 1.94
=0.12cm
[c | cc | cc | cc | cc ]{} $1/h$ & $\verb"e"_d^4({\boldsymbol{r}})$ & $\texttt{r}_d^4({\boldsymbol{r}})$ & $\verb"e"_d^8({\boldsymbol{r}})$ & $\texttt{r}_d^8({\boldsymbol{r}})$ & $\verb"e"_d^{16}({\boldsymbol{r}})$ & $\texttt{r}_d^{16}({\boldsymbol{r}})$ & $\verb"e"_d^{32}({\boldsymbol{r}})$ & $\texttt{r}_d^{32}({\boldsymbol{r}})$
$8$ & 1.00e$+$00 & $-$ & 1.15e$+$01 & $-$ & 5.19e$+$00 & $-$ & 3.60e$+$00 & $-$
$16$ & 1.38e$-$01 & 2.87 & 1.08e$-$00 & 3.42 & 1.26e$+$01 & $-$ & 2.92e$+$00 & $-$
$32$ & 2.08e$-$02 & 2.73 & 1.37e$-$01 & 2.98 & 1.13e$+$00 & 3.48 & 1.32e$+$01 & $-$
$64$ & 3.94e$-$03 & 2.40 & 2.04e$-$02 & 2.75 & 1.38e$-$01 & 3.03 & 1.16e$+$00 & 3.50
$128$ & 8.91e$-$04 & 2.15 & 3.88e$-$03 & 2.40 & 2.04e$-$02 & 2.76 & 1.39e$-$01 & 3.06
[9]{} <span style="font-variant:small-caps;">D. N. Arnold</span>, *An interior penalty finite element method with discontinuous elements*, SIAM J. Numer. Anal., 19 (1982), pp. 742–760.
<span style="font-variant:small-caps;">D. N. Arnold, R. S. Falk, and R. Winther</span>, *Finite element exterior calculus, homological techniques, and applications*, Acta Numerica, 15 (2006), pp. 1–155.
<span style="font-variant:small-caps;">D. N. Arnold, R. S. Falk, and R. Winther</span>, *Mixed finite element methods for linear elasticity with weakly imposed symmetry*, Math. Comp. 76 (2007), pp. 1699-1723.
<span style="font-variant:small-caps;">D. Boffi, F. Brezzi, and M. Fortin</span>, *Reduced symmetry elements in linear elasticity*, Comm. Pure Appl. Anal., 8 (2009), pp. 1–28.
<span style="font-variant:small-caps;">F. Brezzi, J. Douglas, Jr., and L. D. Marini</span>, *Two families of mixed finite elements for second order elliptic problems*, Numer. Math., 47 (1985), pp. 217–235.
, *Remarks on the discretization of some non-coercive operator with applications to heterogeneous Maxwell equations*, SIAM J. Numer. Anal., 43 (2005), pp. 1–18.
, *A new elasticity element made for enforcing weak stress symmetry*, Math. Comp. 79 (2010), pp. 1331–1349.
, Mathematical Aspects of Discontinuous Galerkin Methods. Springer-Verlag Berlin Heidelberg 2012.
, in: Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, of Lecture Notes in Physics, vol. 58, Springer, Berlin, 1976.
Automated Solution of Differential Equations by the Finite Element Method. Springer 2012.
, *Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in three dimensions*, SIAM J. Numer. Anal. 50 (2012), pp. 1648–1674.
, *Finite element analysis of a time harmonic Maxwell problem with an impedance boundary condition*, IMA J. Numer. Anal. 32 (2012), pp. 534–552.
, *A second elasticity element using the matrix bubble*, IMA J. Numer. Anal. 32 (2012), pp. 352–372.
, *Interior penalty method for the indefinite time-harmonic Maxwell equations*, Numer. Math. 100 (2005), pp. 485–518.
, *Finite element spectral analysis for the mixed formulation of the elasticity equations*, SIAM J. Numer. Anal., 51 (2013) pp. 1041–1063.
, [*A family of mixed finite elements for the elasticity problem*]{}. Numerische Mathematik, 53, (1988), pp. 513–538.
[^1]: Departamento de Construcción e Ingeniería de Fabricación, Universidad de Oviedo, Oviedo, España, e-mail: [amarquez@uniovi.es]{}
[^2]: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España, e-mail: [salim@uniovi.es]{}
[^3]: School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia, e-mail: [thanh.tran@unsw.edu.au]{}
[^4]: Partially supported by Spain’s Ministry of Education Project MTM2013-43671-P and the Australian Research Council Grant DP120101886.
|
---
abstract: 'The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.'
---
[ **Exactly solvable reaction diffusion models on a Cayley tree**]{}
2 cm
[Laleh Farhang Matin[[^1]]{}, Amir Aghamohammadi[[^2]]{}, & Mohammad Khorrami[[^3]]{} ]{} 5 mm *Department of Physics, Alzahra University, Tehran 1993891167, Iran.*
[**PACS numbers:**]{} 05.40.-a, 02.50.Ga
[**Keywords:**]{} reaction-diffusion, Cayley tree
Introduction
============
Reaction-diffusion systems have been studied using various methods, including analytical techniques, approximation methods, and simulation. Approximation methods are generally different in different dimensions, as for example the mean field techniques, working good for high dimensions, generally do not give correct results for low-dimensional systems. A large fraction of analytical studies belong to low-dimensional (specially one-dimensional) systems, as solving low-dimensional systems should in principle be easier. [@ScR; @ADHR; @KPWH; @HS1; @PCG; @HOS1; @HOS2; @AL; @AKK; @AKK2; @AM1].
The Cayley tree is a tree (a lattice having no loops) where every site is connected to $\xi$ nearest neighbor sites. This no-loops property may allow exact solvability for some models, for general coordination number $\xi$. Reaction diffusion models on the Cayley tree have been studied in, for example [@VNH; @jK; @SNPVP; @MKMS; @SNM; @ACVP]. In [@VNH; @jK; @SNM] diffusion-limited aggregations, and in [@SNPVP] two-particle annihilation reactions for immobile reactants have been studied. There are also some exact results for deposition processes on the Bethe lattice [@ACVP].
The empty interval method (EIM) has been used to analyze the one dimensional dynamics of diffusion-limited coalescence [@BDb; @BDb1; @BDb2; @BDb3]. Using this method, the probability that $n$ consecutive sites are empty has been calculated. This method has been used to study a reaction-diffusion process with three-site interactions [@HH]. EIM has been also generalized to study the kinetics of the $q$-state one-dimensional Potts model in the zero-temperature limit [@MB]. In [@BDb; @BDb1; @BDb2; @BDb3], one-dimensional diffusion-limited processes have been studied using EIM. There, some of the reaction rates have been taken infinite, and the models have been worked out on continuum. For the cases of finite reaction-rates, some approximate solutions have been obtained.
In [@AKA; @AM3], all the one dimensional reaction-diffusion models with nearest neighbor interactions which can be exactly solved by EIM have been found and studied. Conditions have been obtained for the systems with finite reaction rates to be solvable via EIM, and then the equations of EIM have been solved. In [@AKA], general conditions were obtained for a single-species reaction-diffusion system with nearest neighbor interactions, to be solvable through EIM. Here solvability means that evolution equation for $E_n$ (the probability that $n$ consecutive sites be empty) is closed. It turned out there, that certain relations between the reaction rates are needed, so that the system is solvable via EIM. The evolution equation of $E_n$ is a recursive equation in terms of $n$, and is linear. It was shown that if certain reactions are absent, namely reactions that produce particles in two adjacent empty sites, the coefficients of the empty intervals in the evolution equation of the empty intervals are $n$-independent, so that the evolution equation can be easily solved. The criteria for solvability, and the solution of the empty-interval equation were generalized to cases of multi-species systems and multi-site interactions in [@KAA; @AAK; @AK].
In this article the most general single-species reaction-diffusion model with nearest-neighbor interactions on a Cayley tree is investigated, which can be solved exactly through the empty interval method. The scheme of the paper is as follows. In section 2, the most general reaction-diffusion model with nearest-neighbor interactions on a Cayley tree is studied, which can be solved exactly through EIM. The evolution equation of $E_n$ is also obtained. In section 3 the stationary solution of such models, as well as their dynamics are discussed. Finally, section 4 is devoted to concluding remarks.
(170,170)
(170,170)
Models solvable through the empty interval method on a Cayley tree
==================================================================
The Cayley tree is a tree (a lattice without loops) where each site is connected to $\xi$ sites (fig. 1). Two sites are called neighbors iff they are connected through a link. Consider a system of particles on a Cayley tree. Each site is either empty or occupied by one particle. The interaction (of particles and vacancies) is nearest neighbor. The probability that a connected collection of $n$ sites be empty is denoted by $E_n$. It is assumed that this quantity does not depend on the choice of the collection. An example is a tree where the probability that a site is occupied is $\rho$ and is independent of the states of other sites. Then $$\label{ca.01}
E_n=(1-\rho)^n.$$ The following graphical representations help express various relations in a more compact form. An empty (occupied) site is denoted by $\circ$ ($\bullet$). A connected collection of $n$ empty sites is denoted by $\bo_n$.
There is no loop in a Cayley tree, so each site can only be connected to a single existing cluster site, by a single link. For $\xi\geq 3$ (the case we are interested in here) the closedness of the evolution equation for $E_n$ requires that the rate of creating an empty site be zero. The reason is that if it is not the case, then an empty $n$-cluster can be created from two disjoint empty clusters joined by a single occupied site [@bAG]. This shows that if the evolution of the empty clusters is to be closed, then the only possible reactions are the following, with the rates indicated. $$\begin{aligned}
\label{ca.02}
\bullet\circ\to&\bullet\bullet,\quad r_1\cr
\circ\circ\to&\circ\bullet,\quad r_2\cr
\circ\circ\to&\bullet\bullet,\quad r_3.\end{aligned}$$ (There is no distinction between left and right, of course.) This means that the reactants are immobile, and the coagulation and diffusion rates are zero.
Using these, one arrives at the following time evolution for $E_n$: $$\begin{aligned}
\label{ca.03}
\frac{\d E_n}{\d t}=&-R_n\,r_1\,P(\bullet\hskip-0.14cm-
\hskip-0.14cm\bo_n)-R_n\,(r_2+r_3)\,P(\circ\hskip-0.14cm-
\hskip-0.14cm\bo_n)\cr & \cr &
-(n-1)\,(2\,r_2+r_3)\,P(\bo_n),\end{aligned}$$ where $R_n$ is the number of sites adjacent to a collection of $n$ connected sites. A simple induction shows that $$\label{ca.04}
R_n=n\,(\xi -2)+2.$$ One has $$\label{ca.05}
P(\bullet\hskip-0.14cm- \hskip-0.14cm\bo_n)+P(\circ\hskip-0.14cm-
\hskip-0.14cm\bo_n)=P(\bo_n),$$ from which $$\label{ca.06}
P(\bullet\hskip-0.14cm- \hskip-0.14cm\bo_n)=E_n-E_{n+1}.$$ Using this, one arrives at $$\label{ca.07}
\frac{\d E_n}{\d t}=R_n\,[-r_1\,(E_n-E_{n+1})-(r_2+r_3)\,E_{n+1}]
-(n-1)\,(2\,r_2+r_3)\,E_n.$$ Throughout the paper, it is assumed that $r_1$, $r_2$, and $r_3$ are all nonzero.
The solution
============
The stationary solution of the system ($E^{\mathrm{s}}$, for which the time derivative vanishes), satisfies $$\label{ca.08}
R_n\,[-r_1\,(E^{\mathrm{s}}_n-E^{\mathrm{s}}_{n+1})-(r_2+r_3)
\,E^{\mathrm{s}}_{n+1}]-(n-1)\,(2\,r_2+r_3)\,E^{\mathrm{s}}_n=0.$$ As $E_n$’s are nonnegative and nonincreasing in $n$, it is easy to see that the only solution to is $$\label{ca.09}
E^{\mathrm{s}}_n=0.$$ This means that in the stationary configuration, all of the sites are occupied, which is not a surprise since in all reactions particles are created.
Regarding dynamics, one question is to obtain the spectrum of the evolution Hamiltonian. This is equivalent to finding solutions with exponential time dependence: $$\label{ca.10}
E^{\mathcal{E}}_n(t)=E^{\mathcal{E}}_n\,\exp(\mathcal{E}\,t).$$ Putting this in , one arrives at $$\label{ca.11}
-[R_n\,r_1+(n-1)\,(2\,r_2+r_3)+\mathcal{E}]\,E^{\mathcal{E}}_n
+R_n\,(r_1-r_2-r_3)\,E^{\mathcal{E}}_{n+1}=0.$$ From this, $$\label{ca.12}
E^{\mathcal{E}}_{n+1}=\zeta_n\,E^{\mathcal{E}}_n,$$ where $$\label{ca.13}
\zeta_n:=\frac{R_n\,r_1+(n-1)\,(2\,r_2+r_3)+\mathcal{E}}
{R_n\,(r_1-r_2-r_3)}.$$ It is seen that $$\label{ca.14}
\lim_{n\to\infty}\zeta_n=\frac{(\xi-2)\,r_1+2\,r_2+r_3}{(\xi-2)\,(r_1-r_2-r_3)}.$$ The right-hand side is either negative or greater than one. So if all $E^{\mathcal{E}}_n$’s are nonzero, then $E^{\mathcal{E}}_n$’s either are not all nonnegative or blow up for large $n$’s. Such $E^{\mathcal{E}}_n$’s are not acceptable as probabilities. To see the reason, consider $\mathcal{E}_1$ (the largest $\mathcal{E}$). for large times, only $E_n$’s corresponding to this eigenvalue survive. But these should be nonincreasing with respect to $n$, and nonnegative, which is not the case. So $E^{\mathcal{E}_1}_n$’s must be identically zero for $n$ larger than a certain integer (say $n_1$). A similar reasoning can then be made for $\mathcal{E}_2$ (the next largest value of $\mathcal{E}$), and the values of $E^{\mathcal{E}_2}_n$ for $n>n_1$, to show that there should be another integer $n_2$ so that $E^{\mathcal{E}_2}_n$ vanishes for $n>n_2$. This argument can be continued to show that for all $\mathcal{E}$’s, there must be an integer so that $E^{\mathcal{E}}_n$’s are identically zero for $n$ larger than that integer. This shows that $\zeta_n$ must be zero for some positive $n$, which gives the allowed values of $\mathcal{E}$: $$\label{ca.15}
\mathcal{E}_k=-\xi\,r_1-(k-1)\,\beta,\qquad k\geq 1,$$ where $$\label{ca.16}
\beta:=(\xi-2)\,r_1+2\,r_2+r_3.$$ This spectrum is discrete, and there is a gap between the largest eigenvalue and zero, which means that the system evolves towards its stationary configuration with a relaxation time. This relaxation time is $$\label{ca.17}
\tau=\frac{1}{\xi\,r_1}.$$ One can also find $E^{\mathcal{E}}_n$’s. Denoting $E^{\mathcal{E}_k}_n$ by $E^k_n$, and using and , one arrives at $$\label{ca.18}
E_n^k=\frac{\displaystyle{\Gamma\left(k+\frac{2}{\xi-2}\right)\,\alpha^{k-n}}}
{\displaystyle{\Gamma\left(n+\frac{2}{\xi-2}\right)\,(k-n)!}},$$ where $$\label{ca.19}
\alpha:=\frac{(\xi-2)\,(r_2+r_3-r_1)}{(\xi-2)\,r_1+2\,r_2+r_3}.$$
The general solution to is then $$\label{ca.20}
E_n(t)=\sum_{k=1}^\infty c_k\,E^k_n\,\exp(\mathcal{E}_k\,t),$$ where $c_k$’s are to be determined from the initial condition.
A special solution to is of the form $$\label{ca.21}
E_n(t)=E_1(t)\,[b(t)]^{n-1}.$$ Putting this in , one arrives at $$\begin{aligned}
\label{ca.22}
\frac{\d b}{\d t}&=-\beta\,b-\beta\,\alpha\,b^2,\cr \frac{\d
E_1}{\d t}&=-
\left(\xi\,r_1+\frac{\xi}{\xi-2}\,\alpha\,\beta\,b\right)\,E_1.\end{aligned}$$ These are readily solved and one obtains $$\begin{aligned}
\label{ca.23}
b(t)=&\frac{b(0)\exp(-\beta\,t)}{1+\alpha\,b(0)\,[1-\exp(-\beta\,t)]},\cr
&\cr
E_1(t)=&E_1(0)\exp(-\xi\,r_1\,t)\,
\left\{\frac{1}{1+\alpha\,b(0)\,[1-\exp(-\beta\,t)]}\right\}
^{\frac{\xi}{\xi-2}}.\end{aligned}$$ Using these, one obtains $$\label{ca.24}
E_n(t)=E_n(0)\exp[-\xi\,r_1\,t-(n-1)\,\beta\,t]\,
\left\{\frac{1}{1+\alpha\,b(0)\,[1-\exp(-\beta\,t)]}\right\}
^{\frac{\xi}{\xi-2}+n-1}.$$ It is seen that for large times, all $E_n$’s tend to zero. In fact they decay like $$\label{ca.25}
E_n(t)\sim \,\exp[-\xi\,r_1\,t-(n-1)\,\beta\,t].$$ One notes that in fact $E_n(t)$ decays like $\exp(-\mathcal{E}_n\,t)$, and this is expected, as $E_n^k$ is zero for $k<n$.
A special case where the ansatz works is the case of initially uncorrelated-sites, so that each site is occupied with probability $\rho$ regardless of other sites. One has then $$\label{ca.26}
E_n(0)=(1-\rho)^n,$$ so that $$\begin{aligned}
\label{ca.27}
E_1(0)=&1-\rho,\cr b(0)=&1-\rho.\end{aligned}$$
The special case $\xi=2$ can be treated directly or as a limiting case of the general problem. The results corresponding to and would be $$\label{ca.28}
\mathcal{E}_k=-2\,r_1-(k-1)\,(2\,r_2+r_3),\qquad \xi=2,$$ and $$\label{ca.29}
E^k_n=\frac{1}{(k-n)!}\,\left[\frac{2\,(r_2+r_3-r_1)}{2\,r_2+r_3}\right]^{k-n},
\qquad \xi=2.$$ Finally, the solutions corresponding to the ansatz would be $$\label{ca.30}
b(t)=b(0)\,\exp[-(2\,r_2+r_3)\,t],\qquad \xi=2,$$ and $$\begin{aligned}
\label{ca.31}
E_1(t)=&\
E_1(0)\,\exp\left\{\frac{2\,(r_1-r_2-r_3)}{2\,r_2+r_3}\,b(0)\,
\Big[1-\exp[-(2\,r_2+r_3)\,t]\Big]\right\}\cr &\
\times\exp(-2\,r_1\,t),\qquad \xi=2,\end{aligned}$$ so that $$\begin{aligned}
\label{ca.32}
E_n(t)=&\
E_n(0)\,\exp\left\{\frac{2\,(r_1-r_2-r_3)}{2\,r_2+r_3}\,b(0)\,
\Big[1-\exp[-(2\,r_2+r_3)\,t]\Big]\right\}\cr &\
\times\exp\{-[2\,r_1+(n-1)\,(2\,r_2+r_3)]\,t\},\qquad \xi=2,\end{aligned}$$
Concluding remarks
==================
The most general single-species exclusion model on a Cayley tree was considered, for which the evolution of the empty-intervals is closed. It was shown that in the stationary configuration of such models all sites are occupied. The dynamics of such systems were also studied and it was shown that the spectrum of the evolution Hamiltonian is discrete. The time evolution of the initially uncorrelated system was also obtained. Among the questions remaining, one can mention the problem of Cayley trees with boundaries, with injection and extraction at the boundaries.\
\
**Acknowledgement**: The authors would like to thank Daniel ben-Avraham for his very useful comments. This work was partially supported by the research council of the Alzahra University.
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[^1]: laleh.matin@alzahra.ac.ir
[^2]: mohamadi@alzahra.ac.ir
[^3]: mamwad@mailaps.org
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---
author:
- |
Andrey G. Grozin$^{a,b}$, Maik Höschele$^b$, Jens Hoff$^b$ and Matthias Steinhauser$^b$\
$^a$ Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia\
$^b$ Institut für Theoretische Teilchenphysik, Karlsruher Institut für Technologie,\
D-76128 Karlsruhe, Germany\
E-mail: , , and
title: Simultaneous decoupling of bottom and charm quarks
---
Introduction {#S:Intro}
============
QCD where all six quark flavours are treated as active degrees of freedom is rarely used in practical applications. If the characteristic energy scale is below some heavy-flavour masses, it is appropriate to construct a low-energy effective theory without those heavy flavours. The Lagrangian of this theory has the same form as the one of QCD plus corrections suppressed by powers of heavy-quark masses. Usually, heavy flavours are decoupled one at a time which results in a tower of effective theories, each of them differ from the previous one by integrating out a single heavy flavour. The parameters of the Lagrangian of such an effective low-energy QCD ($\alpha_s(\mu)$, the gauge fixing parameter $a(\mu)$, light-quark masses $m_i(\mu)$) are related to the parameters of the underlying theory (including the heavy flavour) by so-called decoupling relations. The same holds for the light fields (gluon, ghost, light quarks) which exist in both theories. QCD decoupling constants are known at two- [@Bernreuther:1981sg; @Larin:1994va; @Chetyrkin:1997un], three- [@Chetyrkin:1997un] and even four-loop order [@Schroder:2005hy; @Chetyrkin:2005ia].
The conventional approach just described ignores power corrections in ratios of heavy-quark masses. Let us, e.g., consider the relation between $\alpha_s^{(3)}$ and $\alpha_s^{(5)}$ (the superscript denotes the number of active flavours). Starting from three loops, there are diagrams containing both $b$- and $c$-quark loops which depend on $m_c/m_b$. The power correction $\sim(\alpha_s/\pi)^3\,(m_c/m_b)^2$ is not taken into account in the standard approach, although, it might be comparable with the four-loop corrections of order $(\alpha_s/\pi)^4$. In the present paper, we consider $(m_c/m_b)^n$ power corrections at three loops by decoupling $b$ and $c$ quarks in a single step.
Of course, the results presented in this paper are generic and apply to any two flavours which are decoupled simultaneously from the QCD Lagrangian. Our full theory is QCD with $n_l$ light flavours, $n_c$ flavours with mass $m_c$, and $n_b$ flavours with mass $m_b$ (in the real world $n_c=n_b=1$). Furthermore we introduce the total number of quarks $n_f=n_l+n_c+n_b$. We study the relation of full QCD to the low-energy effective theory containing neither $b$ nor $c$.
The bare gluon, ghost and light-quark fields in the effective theory are related to the bare fields in the full theory by $$A_0^{(n_l)} = \left(\zeta_A^0\right)^{1/2} A_0^{(n_f)}\,,\quad
c_0^{(n_l)} = \left(\zeta_c^0\right)^{1/2} c_0^{(n_f)}\,,\quad
q_0^{(n_l)} = \left(\zeta_q^0\right)^{1/2} q_0^{(n_f)}\,,
\label{Intro:fields0}$$ where the bare decoupling constants are computed in the full theory via [@Chetyrkin:1997un] $$\begin{aligned}
\zeta_A^0(\alpha_{s0}^{(n_f)},a_0^{(n_f)}) &=& 1 + \Pi_A(0)
= \left[Z_A^{\rm os}\right]^{-1}\,,
\nonumber\\
\zeta_c^0(\alpha_{s0}^{(n_f)},a_0^{(n_f)}) &=& 1 + \Pi_c(0)
= \left[Z_c^{\rm os}\right]^{-1}\,,
\nonumber\\
\zeta_q^0(\alpha_{s0}^{(n_f)},a_0^{(n_f)}) &=& 1 + \Sigma_V(0)
= \left[Z_q^{\rm os}\right]^{-1}\,,
\label{Intro:zetafields0}\end{aligned}$$ with $\alpha_{s0}=g_0^2/(4\pi)^{1-\varepsilon}$; $\Pi_A(q^2)$, $\Pi_c(q^2)$ and $\Sigma(q) = \rlap/q \Sigma_V(q^2)+m_{q0} \Sigma_S(q^2)$ are the (bare) gluon, ghost and light-quark self-energies (we may set all light-quark masses to 0 in $\Sigma_V$ and $\Sigma_S$). The fields renormalized in the on-shell scheme coincide in both theories; therefore, the bare decoupling coefficients (\[Intro:zetafields0\]) are the ratio of the on-shell renormalization constants of the fields. In the effective theory all the self-energies vanish at $q=0$ (they contain no scale), and the on-shell $Z$ factors are exactly 1. In the full theory, only diagrams with at least one heavy-quark loop survive.[^1]
Next to the fields also the parameters of the full and effective QCD Lagrangian are related by decoupling constants $$\alpha_{s0}^{(n_l)} = \zeta_{\alpha_s}^0 \alpha_{s0}^{(n_f)}\,,\quad
a_0^{(n_l)} = \zeta_A^0 a_0^{(n_f)}\,,\quad
m_{q0}^{(n_l)} = \zeta_m^0 m_{q0}^{(n_f)}\,,
\label{Intro:params0}$$ where $a$ is the gauge parameter defined through the gluon propagator $$D_{\mu\nu}(k) = -\frac{i}{k^2}\, \left( g_{\mu\nu} - (1-a)\,
\frac{k_\mu k_\nu}{k^2} \right)\,.
\label{eq::gluon_prop}$$ The bare decoupling constants in Eq. (\[Intro:params0\]) are computed with the help of [@Chetyrkin:1997un] $$\begin{aligned}
\zeta_{\alpha_s}^0(\alpha_{s0}^{(n_f)}) &=&
\left(1+\Gamma_{A\bar{c}c}\right)^2 \left(Z_c^{\rm os}\right)^2 Z_A^{\rm os} =
\left(1+\Gamma_{A\bar{q}q}\right)^2\left(Z_q^{\rm os}\right)^2 Z_A^{\rm os} =
\left(1+\Gamma_{AAA}\right)^2
\left(Z_A^{\rm os}\right)^3\,,
\nonumber\\
\zeta_m^0(\alpha_{s0}^{(n_f)}) &=& Z_q^{\rm os} \left[1 - \Sigma_S(0)\right]\,.
\label{Intro:zetaparams0}\end{aligned}$$ The $A\bar{c}c$, $A\bar{q}q$ and $AAA$ proper vertex functions are expanded in their external momenta, and only the leading non-vanishing terms are retained. In the low-energy theory they get no loop corrections, and are given by the tree-level vertices of dimension-4 operators in the Lagrangian. In full QCD (with the heavy flavours) they have just one colour and tensor (and Dirac) structure, namely, that of the tree-level vertices (if this were not the case, the Lagrangian of the low-energy theory would not have the usual QCD form[^2]). Therefore, we have the tree-level vertices times $(1+\Gamma_i)$, where loop corrections $\Gamma_i$ contain at least one heavy-quark loop. The various versions in the first line of Eq. (\[Intro:zetaparams0\]) are obtained with the help of the QCD Ward identities involving three-particle vertices. In our calculation we restrict ourselves for convenience to the ghost–gluon vertex. Note that the gauge parameter dependence cancels in $\zeta_{\alpha_s}^0$ and $\zeta_m^0$ whereas the individual building blocks in Eq. (\[Intro:zetaparams0\]) still depend on $a$. This serves as a check of our calculation.
The $\overline{\mbox{MS}}$ renormalized parameters and fields in the two theories are related by $$\begin{aligned}
\alpha_s^{(n_l)}(\mu') &=& \zeta_{\alpha_s}(\mu',\mu) \alpha_s^{(n_f)}(\mu)\,,\quad
a^{(n_l)}(\mu') = \zeta_A(\mu',\mu) a^{(n_f)}(\mu)\,,
\nonumber\\
m_q^{(n_l)}(\mu') &=& \zeta_m(\mu',\mu) m_q^{(n_f)}(\mu)\,,\quad
A^{(n_l)}(\mu') = \zeta_A^{1/2}(\mu',\mu) A^{(n_f)}(\mu)\,,
\nonumber\\
c^{(n_l)}(\mu') &=& \zeta_c^{1/2}(\mu',\mu) c^{(n_f)}(\mu)\,,\quad
q^{(n_l)}(\mu') = \zeta_q^{1/2}(\mu',\mu) q^{(n_f)}(\mu)\,,
\label{Intro:ren}\end{aligned}$$ where we allow for two different renormalization scales in the full and effective theory. The finite decoupling constants are obtained by renormalizing the fields and parameters in Eqs. (\[Intro:zetafields0\]) and (\[Intro:params0\]) which leads to $$\begin{aligned}
\zeta_{\alpha_s}(\mu',\mu) &=&
\left(\frac{\mu}{\mu'}\right)^{2\varepsilon}
\frac{Z_{\alpha}^{(n_f)}\left(\alpha_s^{(n_f)}(\mu)\right)}{Z_{\alpha}^{(n_l)}\left(\alpha_s^{(n_l)}(\mu')\right)}
\zeta_{\alpha_s}^0\left(\alpha_{s0}^{(n_f)}\right)\,,
\nonumber\\
\zeta_m(\mu',\mu) &=&
\frac{Z_m^{(n_f)}\left(\alpha_s^{(n_f)}(\mu)\right)}{Z_m^{(n_l)}\left(\alpha_s^{(n_l)}(\mu')\right)}
\zeta_m^0\left(\alpha_{s0}^{(n_f)}\right)\,,
\nonumber\\
\zeta_A(\mu',\mu) &=&
\frac{Z_A^{(n_f)}\left(\alpha_s^{(n_f)}(\mu),a^{(n_f)}(\mu)\right)}{Z_A^{(n_l)}\left(\alpha_s^{(n_l)}(\mu'),a^{(n_l)}(\mu')\right)}
\zeta_A^0\left(\alpha_{s0}^{(n_f)},a_0^{(n_f)}\right)\,,
\nonumber\\
\zeta_q(\mu',\mu) &=&
\frac{Z_q^{(n_f)}\left(\alpha_s^{(n_f)}(\mu),a^{(n_f)}(\mu)\right)}{Z_q^{(n_l)}\left(\alpha_s^{(n_l)}(\mu'),a^{(n_l)}(\mu')\right)}
\zeta_q^0\left(\alpha_{s0}^{(n_f)},a_0^{(n_f)}\right)\,,
\nonumber\\
\zeta_c(\mu',\mu) &=&
\frac{Z_c^{(n_f)}\left(\alpha_s^{(n_f)}(\mu),a^{(n_f)}(\mu)\right)}{Z_c^{(n_l)}\left(\alpha_s^{(n_l)}(\mu'),a^{(n_l)}(\mu')\right)}
\zeta_c^0\left(\alpha_{s0}^{(n_f)},a_0^{(n_f)}\right)\,,
\label{Intro:zetaren}\end{aligned}$$ where $Z_i^{(n_f)}$ are the $\overline{\rm MS}$ renormalization constants in $n_f$-flavour QCD which we need up to three-loop order.
Calculation {#S:Calc}
===========
Our calculation is automated to a large degree. In a first step we generate all Feynman diagrams with [QGRAF]{} [@Nogueira:1991ex]. The various diagram topologies are identified and transformed to [FORM]{} [@Vermaseren:2000nd] with the help of [q2e]{} and [exp]{} [@Harlander:1997zb; @Seidensticker:1999bb] (these topologies have been investigated in [@Bekavac:2007tk]). Afterwards we use the program [FIRE]{} [@Smirnov:2008iw] to reduce the two-scale three-loop integrals to four master integrals which can be found in analytic form in Ref. [@Bekavac:2009gz].
As a cross check we apply the asymptotic expansion (see, e.g., Ref. [@Smirnov:2002pj]) in the limit $m_c\ll m_b$ and evaluate five expansion terms in $(m_c/m_b)^2$. The asymptotic expansion is automated in the program [exp]{} which provides output that is passed to the package [MATAD]{} [@Steinhauser:2000ry] performing the actual calculation.
In the following we present explicit results for the two-point functions and $\Gamma_{A\bar{c}c}$ needed for the construction of the decoupling constants. Other vertex functions can be easily reconstructed from the bare decoupling coefficient $\zeta_{\alpha_s}^0$ in Section \[S:as\] (see Eq. (\[Intro:zetaparams0\])).
Gluon self-energy
-----------------
The bare gluon self-energy at $q^2=0$ in the full theory can be cast in the following form[^3] $$\begin{aligned}
\Pi_A(0) &=& \frac{1}{3}
\left( n_b m_{b0}^{-2\varepsilon} + n_c m_{c0}^{-2\varepsilon} \right)
T_F \frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)
\nonumber\\
&&{} + P_h \left( n_b m_{b0}^{-4\varepsilon} + n_c m_{c0}^{-4\varepsilon} \right)
T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^2
\nonumber\\
&&{} + \biggl[ \left(P_{hg} + P_{hl} T_F n_l\right)
\left( n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon} \right)
+ P_{hh} T_F
\left( n_b^2 m_{b0}^{-6\varepsilon} + n_c^2 m_{c0}^{-6\varepsilon} \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ P_{bc}\left(\frac{m_{c0}}{m_{b0}}\right)
T_F n_b n_c \left(m_{b0} m_{c0}\right)^{-3\varepsilon} \biggr]
T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3
+ \cdots
\label{Calc:Pi0}\end{aligned}$$ where the exact dependence on $\varepsilon=(4-d)/2$ ($d$ is the space-time dimension) of the bare two-loop result is given by $$P_h = \frac{1}{4 (2-\varepsilon) (1+2\varepsilon)} \left[
- C_F \frac{\varepsilon}{3} (9+7\varepsilon-10\varepsilon^2)
+ C_A \frac{3+11\varepsilon-\varepsilon^2-15\varepsilon^3+4\varepsilon^5}{2 (1-\varepsilon) (3+2\varepsilon)} \right]
\label{Calc:Ph}$$ ($C_F=(N_C^2-1)/(2N_C)$ and $C_A=N_C$ are the eigenvalues of the quadratic Casimir operators of the fundamental and adjoint representation of $SU(N_C)$, respectively, and $T_F=1/2$ is the index of the fundamental representation). The three-loop quantities $P_{hg}$, $P_{hl}$ and $P_{hh}$ are only available as an expansion in $\varepsilon$. The analytic results read $$\begin{aligned}
P_{hg} &=& C_F^2 \frac{\varepsilon^2}{24} \left[ 17
- \frac{1}{8} \left( 95 \zeta_3 + \frac{274}{3} \right) \varepsilon + \cdots \right]
\nonumber\\
&&{} - C_F C_A \frac{\varepsilon}{288} \left[ 89
- \left( 36 \zeta_3 - \frac{785}{6} \right) \varepsilon
- 9 \left( 4 B_4 - \frac{\pi^4}{5} + \frac{1957}{24} \zeta_3 - \frac{10633}{162} \right)
\varepsilon^2 + \cdots \right]
\nonumber\\
&&{} + \frac{C_A^2}{1152} \Biggl[ 3 \xi + 41
- \frac{1}{2} \left( 21 \xi - \frac{781}{3} \right) \varepsilon
- \left( 108 \zeta_3 - \frac{137}{4} \xi - \frac{3181}{12} \right) \varepsilon^2
\nonumber\\
&&\hphantom{{}+\frac{C_A^2}{1152}\Biggl[\Biggr.}
\!- \left( 72 B_4 - \frac{27}{5} \pi^4 - \left( 24 \xi - \frac{1805}{4} \right) \zeta_3
+ \frac{1}{24} \left( 3577 \xi + \frac{42799}{9} \right) \right) \varepsilon^3
+ \cdots \Biggr]\,,
\nonumber\\
P_{hl} &=& \frac{5}{72} C_F \varepsilon \left[ 1 - \frac{31}{30} \varepsilon
+ \frac{971}{180} \varepsilon^2 + \cdots \right]
\nonumber\\
&&{} - \frac{C_A}{72} \left[ 1 + \frac{5}{6} \varepsilon + \frac{101}{12} \varepsilon^2
+ \left( 8 \zeta_3 - \frac{3203}{216} \right) \varepsilon^3 + \cdots \right]\,,
\nonumber\\
P_{hh} &=& C_F \frac{\varepsilon}{18} \left[ 1 - \frac{5}{6} \varepsilon
+ \frac{1}{32} \left( 63 \zeta_3 + \frac{218}{9} \right) \varepsilon^2 + \cdots \right]
\nonumber\\
&&{} - \frac{C_A}{144} \left[ 1 + \frac{35}{6} \varepsilon + \frac{37}{12} \varepsilon^2
- \frac{1}{8} \left( 287 \zeta_3 - \frac{6361}{27} \right) \varepsilon^3 + \cdots \right]\,,
\label{Calc:P3}\end{aligned}$$ where $\xi=1-a_0^{(n_f)}$, and [@Broadhurst:1991fi] $$B_4 = 16 {\mathop{\mathrm{Li}}\nolimits_{4}}\left(\frac{1}{2}\right) + \frac{2}{3} \log^2 2 (\log^2 2 - \pi^2)
- \frac{13}{180} \pi^4\,.$$
A new result obtained in this paper is the analytic expression for $P_{bc}(x)$ which arises from diagrams where $b$ and $c$ quarks are simultaneously present in the loops (see Fig. \[F:Glue\] for typical diagrams). The analytic expression is given by $$\begin{aligned}
P_{bc}(x) &=& C_F \frac{\varepsilon}{9} \left[ 1 - \frac{5}{6} \varepsilon
+ p_F(x) \varepsilon^2 + \cdots \right]
\nonumber\\
&&{} - \frac{C_A}{72} \left[ 1 + \frac{35}{6} \varepsilon
+ \left( \frac{9}{2} L^2 + \frac{37}{12} \right) \varepsilon^2
+ p_A(x) \varepsilon^3 + \cdots \right]\,,
\label{Calc:Pbc}\end{aligned}$$ with $L=\log x$, $$\begin{aligned}
p_F(x) &=& \frac{9}{128} \Biggl[
\frac{(1+x^2)(5-2x^2+5x^4)}{x^3} L_-(x)\\
&&{} - \frac{5-38x^2+5x^4}{x^2} L^2
+ 10 \frac{1-x^4}{x^2} L
- 10 \frac{(1-x^2)^2}{x^2} \Biggr]
+ \frac{109}{144}\,,\\
p_A(x) &=& 24 L_+(x)
- \frac{3}{4} \frac{(1+x^2)(4+11x^2+4x^4)}{x^3} L_-(x)\\
&&{} + \frac{(1+6x^2)(6+x^2)}{2x^2} L^2
- 6 \frac{1-x^4}{x^2} L
+ 6 \frac{(1-x^2)^2}{x^2}
+ 8 \zeta_3 + \frac{6361}{216}\,,\end{aligned}$$ where the functions $L_\pm(x)$ are defined in (\[Ix:L\]). The function $P_{bc}(x)$ satisfies the properties $$P_{bc}(x^{-1}) = P_{bc}(x)\,,\quad
P_{bc}(1) = 2 P_{hh}\,,
\label{Calc:testPi}$$ which are a check of our result. For $x\to0$, the hard contribution to $P_{bc}(x) x^{-3\varepsilon}$ is given by $P_{hl}$. However, there is also a soft contribution, and it is not possible to obtain a relation between $P_{bc}(x\to0)$ and $P_{hl}$ if they are expanded in $\varepsilon$ (this would be possible for a non-zero $\varepsilon<0$, cf. (\[Ix:0\])).
Ghost self-energy
-----------------
The bare ghost self-energy at $q^2=0$ can be cast in the form $$\begin{aligned}
\Pi_c(0) &=&
C_h \left(n_b m_{b0}^{-4\varepsilon} + n_c m_{c0}^{-4\varepsilon}\right)
C_A T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^2
\nonumber\\
&&{} + \biggl[ \left(C_{hg} + C_{hl}T_F n_l\right)
\left( n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon} \right)
+ C_{hh}T_F
\left( n_b^2 m_{b0}^{-6\varepsilon} + n_c^2 m_{c0}^{-6\varepsilon} \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ C_{bc}\left(\frac{m_{c0}}{m_{b0}}\right)
T_F n_b n_c \left(m_{b0} m_{c0}\right)^{-3\varepsilon} \biggr]
C_A T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3
+ \cdots\,,
\label{Calc:Ghost}\end{aligned}$$ where the two-loop term is given by $$C_h = - \frac{(1+\varepsilon) (3-2\varepsilon)}{16 (1-\varepsilon) (2-\varepsilon) (1+2\varepsilon) (3+2\varepsilon)}\,,
\label{Calc:Ch}$$ and the $\varepsilon$ expansions of the single-scale three-loop coefficients read $$\begin{aligned}
C_{hg} &=& C_F \frac{\varepsilon}{64} \left[ 5
- \left( 4 \zeta_3 + \frac{9}{2} \right) \varepsilon
- \left( 4 B_4 - \frac{\pi^4}{5} + \frac{57}{2} \zeta_3 - \frac{157}{4} \right)
\varepsilon^2 + \cdots \right]
\nonumber\\
&&{} + \frac{C_A}{2304} \Biggl[ 3 \xi - 47
- \frac{1}{2} \left( 9 \xi + \frac{83}{3} \right) \varepsilon
+ \left( 108 \zeta_3 + \frac{131}{4} \xi - \frac{9083}{36} \right) \varepsilon^2
\nonumber\\
&&\hphantom{{}+\frac{C_A}{2304}\Biggl[\Biggr.}
+ \left( 72 B_4 - \frac{27}{5} \pi^4 + (24 \xi + 407) \zeta_3
- \frac{1}{24} \left( 2239 \xi - \frac{49795}{9} \right) \right)
\varepsilon^3 + \cdots \Biggr]\,,
\nonumber\\
C_{hl} &=& \frac{1}{144} \left[ 1 - \frac{5}{6} \varepsilon + \frac{337}{36} \varepsilon^2
+ \left( 8 \zeta_3 - \frac{5261}{216} \right) \varepsilon^3 + \cdots \right]\,,
\nonumber\\
C_{hh} &=& \frac{1}{72} \left[ 1 - \frac{5}{6} \varepsilon + \frac{151}{36} \varepsilon^2
- \left( 7 \zeta_3 + \frac{461}{216} \right) \varepsilon^3 + \cdots \right]\,.
\label{Calc:C3}\end{aligned}$$
The function $C_{bc}(x)$ is obtained from the diagram of Fig. \[F:Ghost\] and can be written as $$C_{bc}(x) = - \frac{3-2\varepsilon}{64(2-\varepsilon)} I(x)\,,
\label{Calc:Cbc}$$ with $$\int \frac{\Pi_b(k^2) \Pi_c(k^2)}{(k^2)^2} d^d k =
i T_F^2 \frac{\alpha_{s0}^2}{16 \pi^\varepsilon} \Gamma^3(\varepsilon)
(m_{b0} m_{c0})^{-3\varepsilon} I\left(\frac{m_{c0}}{m_{b0}}\right)\,,
\label{Calc:Idef}$$ where $\Pi_{b}(k^2)$ and $\Pi_{c}(k^2)$ are the $b$- and $c$-loop contributions to the gluon self-energy. The integral $I(x)$ is discussed in Appendix \[S:Ix\] where an analytic result is presented. In analogy to Eq. (\[Calc:testPi\]), we have $$C_{bc}(x^{-1}) = C_{bc}(x)\,,\quad
C_{bc}(1) = 2 C_{hh}\,.$$ For a non-zero $\varepsilon<0$, $C_{bc}(x\to0)\to C_{hl} x^{3\varepsilon}$ (only the hard part survives in (\[Ix:0\])).
Light-quark self-energy
-----------------------
The parts of the light-quark self-energy $\Sigma_V(0)$ and $\Sigma_S(0)$ (with vanishing light-quark masses) are conveniently written in the form $$\begin{aligned}
\Sigma_V(0) &=&
V_h \left(n_b m_{b0}^{-4\varepsilon} + n_c m_{c0}^{-4\varepsilon}\right)
C_F T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^2
\nonumber\\
&&{} + \biggl[ \left(V_{hg} + V_{hl} T_F n_l\right)
\left( n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon} \right)
+ V_{hh} T_F
\left( n_b^2 m_{b0}^{-6\varepsilon} + n_c^2 m_{c0}^{-6\varepsilon} \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ V_{bc}\left(\frac{m_{c0}}{m_{b0}}\right)
T_F n_b n_c \left(m_{b0} m_{c0}\right)^{-3\varepsilon} \biggr]
C_F T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3
+ \cdots\,,
\nonumber\\
\Sigma_S(0) &=&
S_h \left(n_b m_{b0}^{-4\varepsilon} + n_c m_{c0}^{-4\varepsilon}\right)
C_F T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^2
\nonumber\\
&&{} + \biggl[ \left(S_{hg} + S_{hl} T_F n_l\right)
\left( n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon} \right)
+ S_{hh} T_F
\left( n_b^2 m_{b0}^{-6\varepsilon} + n_c^2 m_{c0}^{-6\varepsilon} \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ S_{bc}\left(\frac{m_{c0}}{m_{b0}}\right)
T_F n_b n_c \left(m_{b0} m_{c0}\right)^{-3\varepsilon} \biggr]
C_F T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3
+ \cdots\,,
\label{Calc:Sigma}\end{aligned}$$ where $$V_h = - \frac{\varepsilon (1+\varepsilon) (3-2\varepsilon)}{8 (1-\varepsilon) (2-\varepsilon) (1+2\varepsilon) (3+2\varepsilon)}\,,\quad
S_h = - \frac{(1+\varepsilon) (3-2\varepsilon)}{8 (1-\varepsilon) (1+2\varepsilon) (3+2\varepsilon)}\,,
\label{Calc:VSh}$$ and $$\begin{aligned}
V_{hg} &=& - C_F \frac{\varepsilon}{96} \left[ 1 - \frac{39}{2} \varepsilon
+ \left( 12 \zeta_3 + \frac{335}{12} \right) \varepsilon^2 + \cdots \right]
\nonumber\\
&&{} + \frac{C_A}{192} \Biggl[ \xi - 1
- \left( 3 \xi + \frac{10}{3} \right) \varepsilon
+ \frac{1}{3} \left( 35 \xi - \frac{227}{3} \right) \varepsilon^2
\nonumber\\
&&\hphantom{{}+\frac{C_A}{192}\Biggl[\Biggr.}
+ \left( 8 (\xi + 2) \zeta_3
- \frac{1}{9} \left( 407 \xi - \frac{1879}{6} \right)
\right) \varepsilon^3 + \cdots \Biggr]\,,
\nonumber\\
V_{hl} &=& \frac{\varepsilon}{72} \left[ 1 - \frac{5}{6} \varepsilon
+ \frac{337}{36} \varepsilon^2 + \cdots \right]\,,
\nonumber\\
V_{hh} &=& \frac{\varepsilon}{36} \left[ 1 - \frac{5}{6} \varepsilon
+ \frac{151}{36} \varepsilon^2 + \cdots \right]\,,
\nonumber\\
S_{hg} &=& C_F \frac{\varepsilon}{16} \left[ 5
- \left( 4 \zeta_3 + \frac{23}{3} \right) \varepsilon
- \left( 4 B_4 - \frac{\pi^4}{5} + \frac{53}{2} \zeta_3 - \frac{257}{6} \right)
\varepsilon^2 + \cdots \right]
\nonumber\\
&&{} + \frac{C_A}{576} \Biggl[ - 3 \xi - 41
+ \left( 9 \xi - \frac{124}{3} \right) \varepsilon
+ \left( 144 \zeta_3 - 35 \xi - \frac{836}{9} \right) \varepsilon^2
\nonumber\\
&&\hphantom{{}+\frac{C_A}{576}\Biggl[\Biggl.}
+ \left( 72 B_4 - \frac{36}{5} \pi^4 - (24 \xi - 581) \zeta_3
+ \frac{1}{3} \left( 407 \xi - \frac{9751}{9} \right) \right) \varepsilon^3
+ \cdots \Biggr]\,,
\nonumber\\
S_{hl} &=& \frac{1}{36} \left[ 1 - \frac{4}{3} \varepsilon
+ \frac{88}{9} \varepsilon^2
+ 8 \left( \zeta_3 - \frac{98}{27} \right) \varepsilon^3
+ \cdots \right]\,,
\nonumber\\
S_{hh} &=& \frac{1}{18} \left[ 1 - \frac{4}{3} \varepsilon
+ \frac{83}{18} \varepsilon^2
- \left( 7 \zeta_3 + \frac{457}{108} \right) \varepsilon^3
+ \cdots \right]\,.
\label{Calc:V3}\end{aligned}$$ Exact $d$-dimensional expressions for these coefficients have been obtained in [@Grozin:2006xm].
The quantities $V_{bc}(x)$ and $S_{bc}(x)$ arise from diagrams similar to Fig. \[F:Ghost\] and can be expressed in terms of $I(x)$: $$V_{bc}(x) = - \frac{\varepsilon(3-2\varepsilon)}{32(2-\varepsilon)} I(x)\,,\quad
S_{bc}(x) = - \frac{3-2\varepsilon}{32} I(x)\,.
\label{Calc:VSbc}$$ They satisfy the relations analogous to Eq. (\[Calc:testPi\]) which again serves as a welcome check of our calculation. Retaining only the hard part of (\[Ix:0\]) for $x\to0$, we reproduce $V_{hl}$, $S_{hl}$. $V_{bc}$ has been calculated up to $\mathcal{O}(\varepsilon^3)$ in Ref. [@Bekavac:2009zc].
Ghost–gluon vertex
------------------
The two-loop correction vanishes in the arbitrary covariant gauge exactly in $\varepsilon$, see Appendix \[S:Ghost\]. For the same reasons, the three-loop correction contains only diagrams with a single quark loop (bottom or charm), and vanishes in Landau gauge: $$\begin{aligned}
&&\Gamma_{A\bar{c}c} = 1 + \Gamma_3 (1-\xi)
(n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon}) C_A^2 T_F
\left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3 + \cdots\,,
\label{Calc:GAcc}\\
&&\Gamma_3 = - \frac{1}{384} \left[ 1
- \frac{5}{2} \varepsilon
+ \frac{67}{6} \varepsilon^2
+ \left(8 \zeta_3 - \frac{727}{18}\right) \varepsilon^3
+ \cdots \right]\,.
\nonumber\end{aligned}$$
Decoupling for $\alpha_s$ {#S:as}
=========================
The gauge parameter dependence cancels in the bare decoupling constant (\[Intro:zetaparams0\]) (which relates $\alpha_{s0}^{(n_l)}$ to $\alpha_{s0}^{(n_f)}$, see Eq. (\[Intro:params0\])). Since the result is more compact we present analytical expressions for $\left(\zeta_{\alpha_s}^0\right)^{-1}$ which reads $$\begin{aligned}
\left(\zeta_{\alpha_s}^0\right)^{-1} &=& 1
+ \frac{1}{3} \left(n_b m_{b0}^{-2\varepsilon} + n_c m_{c0}^{-2\varepsilon}\right)
T_F \frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)
\nonumber\\
&&{} + Z_h \varepsilon T_F
(n_b m_{b0}^{-4\varepsilon} + n_c m_{c0}^{-4\varepsilon})
\left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^2
\nonumber\\
&&{} + \biggl[ \left(Z_{hg} + Z_{hl} T_F n_l\right)
\left( n_b m_{b0}^{-6\varepsilon} + n_c m_{c0}^{-6\varepsilon} \right)
+ Z_{hh} T_F
\left( n_b^2 m_{b0}^{-6\varepsilon} + n_c^2 m_{c0}^{-6\varepsilon} \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ Z_{bc}\left(\frac{m_{c0}}{m_{b0}}\right)
T_F n_b n_c \left(m_{b0} m_{c0}\right)^{-3\varepsilon} \biggr] \varepsilon
T_F \left(\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon)\right)^3
+ \cdots\,,
\label{as:bare}\end{aligned}$$ where $$\begin{aligned}
Z_h &=& \frac{1}{4 (2-\varepsilon) (1+2\varepsilon)} \left[
- \frac{1}{3} C_F (9+7\varepsilon-10\varepsilon^2)
+ \frac{1}{2} C_A
\frac{10+11\varepsilon-4\varepsilon^2-4\varepsilon^3}{3+2\varepsilon}
\right]\,,\\
Z_{hg} &=& \frac{C_F^2 \varepsilon}{24}
\left[17 - \frac{1}{4} \left(\frac{95}{2} \zeta_3 + \frac{137}{3}\right) \varepsilon
+ \cdots\right]\\
&&{} - \frac{C_F C_A}{72}
\left[11 + \frac{257}{6} \varepsilon
- \frac{1}{16} \left(\frac{3819}{2} \zeta_3 - \frac{8549}{9}\right) \varepsilon^2
+ \cdots\right]\\
&&{} + \frac{C_A^2}{216}
\left[19 + \frac{359}{24} \varepsilon
+ \frac{1}{32} \left(\frac{45}{2} \zeta_3 - \frac{3779}{3}\right) \varepsilon^2
+ \cdots\right]\,,\\
Z_{hl} &=& \frac{C_F}{72}
\left[5 - \frac{31}{6} \varepsilon + \frac{971}{36} \varepsilon^2 + \cdots\right]
- \frac{C_A}{216}
\left[5 - \frac{17}{6} \varepsilon + \frac{343}{12} \varepsilon^2 + \cdots\right]\,,\\
Z_{hh} &=& \frac{C_F}{18}
\left[1 - \frac{5}{6} \varepsilon
+ \frac{1}{16} \left(\frac{63}{2} \zeta_3 + \frac{109}{9}\right) \varepsilon^2
+ \cdots\right]\\
&&{} - \frac{C_A}{108}
\left[5 - \frac{113}{24} \varepsilon
- \frac{1}{16} \left(\frac{189}{2} \zeta_3 - 311\right) \varepsilon^2
+ \cdots\right]\,,\\
Z_{bc}(x) &=& \frac{C_F}{9} \left[1 - \frac{5}{6} \varepsilon
+ z_F(x) \varepsilon^2 + \cdots\right]
- \frac{C_A}{54} \left[5 - \frac{113}{24} \varepsilon
+ z_A(x) \varepsilon^2 + \cdots\right]\,,\\
z_F(x) &=& \frac{9}{64} \Biggl[
\frac{(1+x^2) (5-2x^2+5x^4)}{2 x^3} L_-(x)\\
&&{} - \frac{5-38x^2+5x^4}{2 x^2} L^2
+ 5 \frac{1-x^4}{x^2} L
- 5 \frac{(1-x^2)^2}{x^2} \Biggr]
+ \frac{109}{144}\,,\\
z_A(x) &=& \frac{3}{16} \Biggl[
- 9 \frac{(1+x^2) (1+x^4)}{2 x^3} L_-(x)\\
&&{} + \frac{9+92x^2+9x^4}{2 x^2} L^2
- 9 \frac{1-x^4}{x^2} L
+ 9 \frac{(1-x^2)^2}{x^2} \Biggr]
+ \frac{311}{16}\,.\end{aligned}$$ Note that $Z_{bc}(x^{-1})=Z_{bc}(x)$, $Z_{bc}(1)=2Z_{hh}$. If desired, the vertices $\Gamma_{A\bar{q}q}$ and $\Gamma_{AAA}$ can be reconstructed using Eq. (\[Intro:zetaparams0\]).
In order to relate the renormalized couplings $\alpha_s^{(n_f)}(\mu)$ and $\alpha_s^{(n_l)}(\mu)$, we first express all bare quantities in the right-hand side of the equation $$\alpha_{s0}^{(n_l)} = \zeta_{\alpha_s}^0(\alpha_{s0}^{(n_f)},m_{b0},m_{c0}) \alpha_{s0}^{(n_f)}$$ via the $\overline{\mbox{MS}}$ renormalized ones [@vanRitbergen:1997va; @Czakon:2004bu; @Chetyrkin:1997dh; @Vermaseren:1997fq] $$\begin{aligned}
&&\frac{\alpha_{s0}^{(n_f)}}{\pi} \Gamma(\varepsilon) =
\frac{\alpha_s^{(n_f)}(\mu)}{\pi \varepsilon}
Z_\alpha^{(n_f)}\left(\alpha_s^{(n_f)}(\mu)\right)
e^{\gamma_E \varepsilon} \Gamma(1+\varepsilon)
\mu^{2\varepsilon}\,,
\label{as:MSbar}\\
&&m_{b0} = Z_m^{(n_f)}\left(\alpha_s^{(n_f)}(\mu)\right) m_b(\mu)
\label{as:massren}\end{aligned}$$ (and similarly for $m_{c0}$). This leads to an equation where $\alpha_{s0}^{(n_l)}$ is expressed via the $n_f$-flavour $\overline{\mbox{MS}}$ renormalized quantities[^4] $\alpha_s^{(n_f)}(\mu)$, $m_c(\mu)$ and $m_b(\mu)$. In a next step we invert the series $$\frac{\alpha_{s0}^{(n_l)}}{\pi} \Gamma(\varepsilon) =
\frac{\alpha_s^{(n_l)}(\mu')}{\pi \varepsilon}
Z_\alpha^{(n_l)}\left(\alpha_s^{(n_l)}(\mu')\right)
e^{\gamma_E \varepsilon} \Gamma(1+\varepsilon)
\left(\mu^\prime\right)^{2\varepsilon}$$ to express $\alpha_s^{(n_l)}(\mu')$ via $\alpha_{s0}^{(n_l)}$, and substitute the series for $\alpha_{s0}^{(n_l)}$ derived above.
In order to obtain compact formulae it is convenient to set $\mu=\bar{m}_b$ where $\bar{m}_b$ is defined as the root of the equation $m_b(\bar{m}_b) = \bar{m}_b$. Furthermore, we choose $\mu'=m_c(\bar{m}_b)$ and thus obtain $\alpha_s^{(n_l)}(m_c(\bar{m}_b))$ as a series in $\alpha_s^{(n_f)}(\bar{m}_b)$ with coefficients depending on $$x = \frac{m_c(\bar{m}_b)}{\bar{m}_b}\,.
\label{as:x}$$ We obtain ($L=\log x$) $$\zeta_{\alpha_s}(m_c(\bar{m}_b),\bar{m}_b)\!=\!
e^{-2L\varepsilon} \left[ 1
+ d_1 \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
+ d_2 \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
+ d_3 \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3\!\! + \cdots
\right],
\label{as:renorm}$$ where $$\begin{aligned}
d_1 &=& - \left[ 11 C_A - 4 T_F (n_l + n_c) \right] \frac{L}{6}
+ \left\{ \left[ 11 C_A - 4 T_F (n_l + n_c) \right] L^2
- T_F (n_b + n_c) \frac{\pi^2}{6} \right\} \frac{\varepsilon}{6}\\
&&{} - \left\{ \left[ 11 C_A - 4 T_F (n_l + n_c) \right] L^3
- T_F n_c \frac{\pi^2}{2} L - T_F (n_b + n_c) \zeta_3 \right\} \frac{\varepsilon^2}{9}
+ \mathcal{O}(\varepsilon^3)\,,\\
d_2 &=& \left[11 C_A - 4 T_F (n_l+n_c) \right]^2 \frac{L^2}{36}
- \left[ 17 C_A^2 - 6 C_F T_F (n_l - n_c) - 10 C_A T_F (n_l + n_c) \right] \frac{L}{12}\\
&&{} - \frac{(39 C_F - 32 C_A) T_F (n_b+n_c)}{144}\\
&&{} + \biggl\{
- \left[11 C_A - 4 T_F (n_l+n_c) \right]^2 \frac{L^3}{18}\\
&&\hphantom{{}+\biggl\{\biggr.}
+ \left[ 17 C_A^2 - 6 C_F T_F (n_l - 2 n_c) - 10 C_A T_F (n_l + n_c) \right] \frac{L^2}{6}\\
&&\hphantom{{}+\biggl\{\biggr.}
+ T_F \left[ \frac{13}{12} C_F n_c
+ \frac{C_A}{9} \left( \frac{11}{12} \pi^2 (n_b + n_c) - 8 n_c \right)
- T_F \frac{\pi^2}{27} (n_b + n_c)(n_l + n_c)
\right] L\\
&&\hphantom{{}+\biggl\{\biggr.}
+ \left[ \frac{C_F}{4} \left( \pi^2 + \frac{35}{2} \right)
- \frac{C_A}{3} \left( \frac{5}{4} \pi^2 + \frac{43}{3} \right) \right]
\frac{T_F (n_b + n_c)}{12}
\biggr\} \varepsilon + \mathcal{O}(\varepsilon^2)\,,\\
d_3 &=& - \frac{\left[11 C_A - 4 T_F (n_l+n_c) \right]^3}{216} L^3\\
&&{} + \biggl[ \frac{935}{24} C_A^3
- \frac{55}{4} C_F C_A T_F (n_l-n_c)
- \frac{445}{12} C_A^2 T_F (n_l+n_c)\\
&&\hphantom{{}+\biggl[\biggr.}
+ 5 C_F T_F^2 (n_l^2 - n_c^2)
+ \frac{25}{3} C_A T_F^2 (n_l+n_c)^2
\biggr] \frac{L^2}{6}\\
&&{} + \biggl[ - \frac{2857}{1728} C_A^3
- C_F^2 T_F \frac{n_l - 9 n_c}{16}
+ \frac{C_F C_A T_F}{48} \left( \frac{205}{6} n_l - 19 n_c + \frac{143}{3} n_b \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A^2 T_F}{27} \left( \frac{1415}{32} n_l + \frac{359}{32} n_c - 22 n_b \right)
- C_F T_F^2 \frac{(n_l + n_c) (11 n_l + 30 n_c) + 26 n_l n_b}{72}\\
&&\hphantom{{}+\biggl[\biggr.}
- C_A T_F^2 \frac{(n_l + n_c) (79 n_l - 113 n_c) - 128 n_l n_b}{432}
\biggr] L\\
&&{} + \biggl[
\frac{C_F^2}{96} \left( \frac{95}{2} \zeta_3 - \frac{97}{3} \right)
- \frac{C_F C_A}{96} \left( \frac{1273}{8} \zeta_3 - \frac{2999}{27} \right)
- \frac{C_A^2}{768} \left( \frac{5}{2} \zeta_3 - \frac{11347}{27} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{41}{162} C_F T_F n_l
- \frac{C_F T_F (n_b+n_c)}{16} \left( \frac{7}{4} \zeta_3 - \frac{103}{81} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_A T_F n_l}{2592}
- \frac{7}{64} C_A T_F (n_b+n_c) \left( \frac{1}{2} \zeta_3 - \frac{35}{81} \right)
\biggr] T_F (n_b+n_c)\\
&&{} + T_F^2 n_b n_c \left( C_F d_F(x) + C_A d_A(x) \right)
+ \mathcal{O}(\varepsilon)\,.\end{aligned}$$ The functions $$\begin{aligned}
d_F(x) &=& - \frac{(1 + x^2) (5 - 2 x^2 + 5 x^4)}{128 x^3} L_-(x)
+ \frac{7}{32} \zeta_3\\
&&{} + \left[ \frac{5}{4} \frac{(1-x^2)^2}{x^2} + \frac{11}{3} \right] \frac{L^2}{32}
- \frac{5}{4} \left[ \frac{1-x^4}{16 x^2} + \frac{1}{3} \right] L
+ \frac{5}{64} \frac{(1-x^2)^2}{x^2}\,,\\
d_A(x) &=& - \frac{(1+x^2) (1+x^4)}{64 x^3} L_-(x)
+ \frac{7}{64} \zeta_3\\
&&{} + \left[ \frac{(1-x^2)^2}{2 x^2} + \frac{5}{3} \right] \frac{L^2}{32}
- \left[ \frac{1-x^4}{2 x^2} - \frac{113}{27} \right] \frac{L}{16}
+ \frac{(1-x^2)^2}{32 x^2}\end{aligned}$$ are defined in such a way that $d_{F,A}(1)=0$. Thus for $x=1$ Eq. (\[as:renorm\]) reduces to the ordinary decoupling of $n_b+n_c$ flavours with the same mass [@Chetyrkin:1997un]. For $x\ll1$ the functions $d_{F}(x)$ and $d_{A}(x)$ become $$\begin{aligned}
d_F(x) &=& - \frac{1}{36} \left( 13 L - \frac{89}{12} \right) + \frac{7}{32} \zeta_3
+ \left( 2 L + \frac{13}{30} \right) \frac{x^2}{15} + \cdots
\nonumber\\
d_A(x) &=& \frac{1}{27} \left( 8 L - \frac{41}{16} \right) + \frac{7}{64} \zeta_3
- \left( \frac{1}{2} L^2 - \frac{121}{30} L + \frac{19}{225} \right)
\frac{x^2}{60} + \cdots\,.
\label{as:x0}\end{aligned}$$
An expression for $\alpha_s^{(n_f)}(\bar{m}_b)$ via $\alpha_s^{(n_l)}(m_c(\bar{m}_b))$ can be obtained by inverting the series (\[as:renorm\]). If one wants to express $\alpha_s^{(n_l)}(\mu_c)$ as a truncated series in $\alpha_s^{(n_f)}(\mu_b)$ (without resummation) for some other choice of $\mu_b\sim m_b$ and $\mu_c\sim m_c$, this can be easily done in three steps: $(i)$ run from $\mu_b$ to $\bar{m}_b$ in the $n_f$-flavour theory (without resummation); $(ii)$ use Eq. (\[as:renorm\]) for the decoupling; and $(iii)$ run from $m_c(\bar{m}_b)$ to $\mu_c$ in the $n_l$-flavour theory (without resummation). After that, relating $\alpha_s^{(n_l)}(\mu')$ and $\alpha_s^{(n_f)}(\mu)$ for any values of $\mu$ and $\mu'$ (possibly widely separated from $m_b$ and $m_c$) can be done in a similar way: $(i)$ run from $\mu$ to $\mu_b$ in the $n_f$-flavour theory (with resummation); $(ii)$ use the decoupling relation derived above; and $(iii)$ run from $\mu_c$ to $\mu'$ in the $n_l$-flavour theory (with resummation). The steps $(i)$ and $(iii)$ can conveniently be performed using the program [RunDec]{} [@Chetyrkin:2000yt].
In the case of QCD ($T_F=1/2$, $C_A=3$, $C_F=4/3$, $n_b=n_c=1$) the decoupling constant in Eq. (\[as:renorm\]) reduces to (for $\varepsilon=0$) $$\begin{aligned}
&&\zeta_{\alpha_s}(m_c(\bar{m}_b),\bar{m}_b) = 1
+ \frac{2 n_l - 31}{6} L \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
\nonumber\\
&&{} + \left[
\frac{(2 n_l - 31)^2}{36} L^2
+ \frac{19 n_l - 142}{12} L
+ \frac{11}{36} \right]
\left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
\nonumber\\
&&{} + \biggl[
\frac{(2 n_l - 31)^3}{216} L^3
+ \left( \frac{95}{9} n_l^2 - \frac{485}{2} n_l
+ \frac{58723}{48} \right) \frac{L^2}{8}
\nonumber\\
&&\hphantom{{}+\biggl\{\biggr.}
- \left( \frac{325}{6} n_l^2 - \frac{15049}{6} n_l + 12853 \right) \frac{L}{288}
- \frac{(1+x^2) (19 - 4 x^2 + 19 x^4)}{768 x^3} L_-(x)
\nonumber\\
&&\hphantom{{}+\biggl\{\biggr.}
+ \frac{19}{768} \left( \frac{(1 - x^2)^2}{x^2} (L^2 + 2) - 2 \frac{1 - x^4}{x^2} L \right)
\nonumber\\
&&\hphantom{{}+\biggl\{\biggr.}
- \frac{1}{1728} \left( \frac{82043}{8} \zeta_3 + \frac{2633}{9} n_l
- \frac{572437}{36} \right) \biggr]
\left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3 + \cdots\,.
\label{as:su3}\end{aligned}$$ For $x\ll1$ the coefficient of $(\alpha_s/\pi)^3$ becomes $$\begin{aligned}
&&\frac{(2 n_l - 31)^3}{216} L^3
+ \frac{5 (2 n_l - 31) (19 n_l - 142)}{144} L^2
- \frac{325 n_l^2 - 15049 n_l + 77041}{1728} L\\
&&{} - \frac{1}{1728} \left( \frac{82043}{8} \zeta_3 + \frac{2633}{9} n_l
- \frac{563737}{36} \right)
- \left( L^2 - \frac{683}{45} L - \frac{926}{675} \right) \frac{x^2}{160}
+ \mathcal{O}(x^4)\,.\end{aligned}$$
Decoupling for the light-quark masses {#S:m}
=====================================
The bare quark mass decoupling coefficient $\zeta_m^0$ of Eq. (\[Intro:zetafields0\]) is determined by $\Sigma_{V}(0)$ and $\Sigma_{S}(0)$, see Eq. (\[Calc:Sigma\]); it is gauge parameter independent. The renormalized decoupling constant $\zeta_m$ in Eq. (\[Intro:zetaren\]) (see [@Chetyrkin:1997dh; @Vermaseren:1997fq] for the mass renormalization constants) can be obtained by re-expressing $\alpha_s^{(n_l)}$ in the denominator via $\alpha_s^{(n_f)}$ (cf. Sect. \[S:as\]; note that in $\zeta_{\alpha_s}$ positive powers of $\varepsilon$ should be kept). Our result reads $$\zeta_m(m_c(\bar{m}_b),\bar{m}_b) = 1
+ d^m_1 C_F \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
+ d^m_2 C_F \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
+ d^m_3 C_F \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3
+ \cdots
\,,
\label{m:ren}$$ where $$\begin{aligned}
d^m_1 &=& - \frac{3}{2} L
\left( 1 - L \varepsilon + \frac{2}{3} L^2 \varepsilon^2
+ \mathcal{O}(\varepsilon^3) \right)\,,\\
d^m_2 &=&
\left[ 9 C_F + 11 C_A - 4 T_F (n_l + n_c) \right] \frac{L^2}{8}
- \left[ 9 C_F + 97 C_A - 20 T_F (n_l + n_c) \right] \frac{L}{48}\\
&&{} + \frac{89}{288} T_F (n_b + n_c)\\
&&{} + \biggl\{
- \left[9 C_F + 11 C_A - 4 T_F (n_l + n_c)\right] \frac{L^3}{4}
+\left[9 C_F + 97 C_A - 20 T_F (n_l + n_c)\right] \frac{L^2}{24}
\nonumber\\
&&\hphantom{{}+\biggl\{\biggr.}
+ \frac{3 \pi^2 n_b - 89 n_c}{72} T_F L
- \left( 5 \pi^2 + \frac{869}{6} \right) T_F\frac{n_b + n_c}{288}
\biggr\} \varepsilon
+ \mathcal{O}(\varepsilon^2)
\,,\\
d^m_3 &=&
\biggl[
- \frac{(9 C_F + 11 C_A) (9 C_F + 22 C_A)}{16}
+ \frac{27 C_F + 44 C_A}{4} T_F (n_l + n_c)\\
&&\hphantom{\biggl[\biggr.}
- T_F^2 \bigl( 2 (n_l + n_c)^2 - n_b n_c \bigr)
\biggr] \frac{L^3}{9}\\
&&{} + \biggl[
\frac{9}{4} C_F^2 + 27 C_F C_A + \frac{1373}{36} C_A^2
- \left( 9 C_F + \frac{197}{9} C_A \right) T_F (n_l + n_c)\\
&&\hphantom{{}+\biggl[\biggr.}
+ T_F^2 \frac{20 (n_l + n_c)^2 - 29 n_b n_c}{9}
\biggr] \frac{L^2}{8}\\
&&{} + \biggl[
- 129 C_F \left( C_F - \frac{C_A}{2} \right) - \frac{11413}{54} C_A^2
- 96 (C_F - C_A)T_F (n_l + n_c) \zeta_3\\
&&\hphantom{{}+\biggl[\biggr.}
+ 4 C_F T_F \left( 23 n_l + \frac{67}{12} n_c - \frac{11}{12} n_b \right)
+ \frac{8}{3} C_A T_F \left( \frac{139}{9} n_l - \frac{47}{4} n_c - 8 n_b \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{8}{27} T_F^2 \left( (n_l + n_c) (35 n_l + 124 n_c) + 124 n_b n_c \right)
\biggr] \frac{L}{64}\\
&&{} + \biggl[
\frac{C_F}{4}
\left( B_4 - \frac{\pi^4}{20} + \frac{57}{8} \zeta_3 - \frac{683}{144} \right)
- \frac{C_A}{8}
\left( B_4 - \frac{\pi^4}{10} + \frac{629}{72} \zeta_3 - \frac{16627}{1944} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{T_F}{18} \left( - \bigl( 4 n_l - 7 (n_b + n_c) \bigr) \zeta_3
+ \frac{2654 n_l - 1685 (n_b + n_c)}{432}
\right)
\biggr] T_F (n_b + n_c)\\
&&{} + \biggl[ - 64 L_+(x) + \frac{(1 + x^2) (5 + 22 x^2 + 5 x^4)}{x^3} L_-(x)
- 96 \zeta_3\\
&&\hphantom{{}+\biggl[\biggr.}
- 5 \left( \frac{(1 - x^2)^2}{x^2} (L^2 + 2) - 2 \frac{1 - x^4}{x^2} L \right)
\biggr] \frac{T_F^2 n_b n_c}{96}
+ \mathcal{O}(\varepsilon)\,.\end{aligned}$$ At $x=1$ this result reduces to the ordinary decoupling of $n_b+n_c$ flavours with the same mass [@Chetyrkin:1997un].
Specifying to QCD leads to (for $\varepsilon=0$) $$\begin{aligned}
&&\zeta_m(m_c(\bar{m}_b),\bar{m}_b) = 1
- 2 L \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
\nonumber\\
&&{} + \left[
- \left( n_l - \frac{43}{2} \right) \frac{L^2}{3}
+ \left( 5 n_l - \frac{293}{2} \right) \frac{L}{18}
+ \frac{89}{216}
\right] \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
\nonumber\\
&&{} + \biggl[
- 2 \left( n_l^2 - 40 n_l + \frac{1589}{4} \right) \frac{L^3}{27}
+ \left( \frac{5}{3} n_l^2 - \frac{679}{6} n_l + \frac{2497}{2} \right) \frac{L^2}{18}
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ \left( 5 \zeta_3 (n_l + 1)
+ \frac{1}{72} \left( \frac{35}{3} n_l^2 + 607 n_l - \frac{103771}{12} \right)
\right) \frac{L}{3}
- \frac{2}{9} L_+(x)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{(1 + x^2) (5 + 22 x^2 + 5 x^4)}{288 x^3} L_-(x)
- \frac{5}{288} \left( \frac{(1 - x^2)^2}{x^2} (L^2 + 2) - 2 \frac{1 - x^4}{x^2} L \right)
\nonumber\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{1}{18} \biggl( B_4 - \frac{\pi^4}{2} + \frac{8}{3} \zeta_3 n_l - \frac{439}{24} \zeta_3
- \frac{1327}{324} n_l - \frac{21923}{648} \biggr)
\biggr] \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3
+ \cdots\,,
\nonumber\\
\label{m:su3}\end{aligned}$$ where for $x\ll1$ the coefficient of $(\alpha_s/\pi)^3$ takes the form $$\begin{aligned}
&&- 2 \left( n_l^2 - 40 n_l + \frac{1591}{4} \right) \frac{L^3}{27}
+ \left( 5 n_l^2 - \frac{679}{2} n_l + \frac{15011}{4} \right) \frac{L^2}{54}\\
&&{} \left[ 5 \zeta_3 (n_l + 1)
+ \frac{1}{72} \left( \frac{35}{3} n_l^2 + 607 n_l - \frac{104267}{12} \right)
\right] \frac{L}{3}\\
&&{} - \frac{1}{18} \left( B_4 - \frac{\pi^4}{2}
+ \frac{8}{3} \zeta_3 n_l + \frac{439}{24} \zeta_3
- \frac{1327}{324} n_l - \frac{24935}{648} \right)\\
&&{} - \left( 2 L - \frac{47}{30} \right) \frac{x^2}{15}
+ \mathcal{O}(x^4)\,.\end{aligned}$$
Decoupling for the fields {#S:fields}
=========================
Gluon field and the gauge parameter {#S:a}
-----------------------------------
Decoupling of the gluon field and the gauge fixing parameter are given by the same quantity $\zeta_A^0$ (cf. (\[Intro:zetafields0\])): $$a_0^{(n_l)} = a_0^{(n_f)} \zeta_A^0(\alpha_{s0}^{(n_f)},a_0^{(n_f)},m_{b0},m_{c0})\,.
\label{a:bare}$$ In a first step we replace the bare quantities in the right-hand side via the renormalized ones using Eqs. (\[as:MSbar\]), (\[as:massren\]), and [@Larin:1993tp; @Chetyrkin:2004mf; @Czakon:2004bu] $$a_0^{(n_f)} = Z_A^{(n_f)}\left(\alpha_s^{(n_f)}(\mu),a^{(n_f)}(\mu)\right) a^{(n_f)}(\mu)\,,
\label{a:rena}$$ and thus we express $a_0^{(n_l)}$ via the $n_f$-flavour renormalized quantities. In a next step we can find $a^{(n_l)}(\mu')$ in terms of $a_0^{(n_l)}$ by solving the equation $$a_0^{(n_l)} = Z_A^{(n_l)}\left(\alpha_s^{(n_l)}(\mu'),a^{(n_l)}(\mu')\right) a^{(n_l)}(\mu')
\label{a:renl}$$ iteratively. The result reads $$\zeta_A(m_c(\bar{m}_b),\bar{m}_b) = 1
+ d^A_1 \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
+ d^A_2 \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
+ d^A_3 \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3 + \cdots
\,,
\label{s:renorm}$$ where $$\begin{aligned}
d^A_1 &=& - \frac{C_A (3 a - 13) + 8 T_F (n_l + n_c)}{12} L\\
&&{} + \left\{
\left[ C_A (3 a - 13) + 8 T_F (n_l + n_c) \right] L^2
+ T_F (n_b + n_c) \frac{\pi^2}{3}
\right\} \frac{\varepsilon}{12}\\
&&{} - \left\{
\left[ C_A (3 a - 13) + 8 T_F (n_l + n_c) \right] L^3
+ T_F n_c \pi^2 L
+ 2 T_F (n_b + n_c) \zeta_3
\right\} \frac{\varepsilon^2}{18} + \mathcal{O}(\varepsilon^3)\,,\\
d^A_2 &=& C_A \frac{2 a + 3}{96}
\left[ C_A (3 a - 13) + 8 T_F (n_l+n_c) \right] L^2\\
&&{} - \left[ C_A^2 \frac{2 a^2 + 11 a - 59}{64}
+ C_F T_F \frac{n_l-n_c}{2} + \frac{5}{8} C_A T_F (n_l+n_c)\right] L\\
&&{} + \frac{13}{192} (4 C_F - C_A) T_F (n_b+n_c)\\
&&{} + \biggl\{
- C_A \frac{2 a + 3}{48} \left[ C_A (3 a - 13) + 8 T_F (n_l + n_c) \right] L^3\\
&&\hphantom{{}+\biggl\{\biggr.}
+ \left[ C_A^2 \frac{2 a^2 + 11 a - 59}{32} + C_F T_F (n_l - 2 n_c)
+ \frac{5}{4} C_A T_F (n_l + n_c) \right] L^2\\
&&\hphantom{{}+\biggl\{\biggr.}
- T_F \left[ 13 C_F n_c
+ C_A \frac{\pi^2 \bigl( n_c (a + 3) + n_b a \bigr) - 39 n_c}{12} \right]
\frac{L}{12}\\
&&\hphantom{{}+\biggl\{\biggr.}
- \left[C_F (2 \pi^2 + 35) - \frac{C_A}{2} \left( 5 \pi^2 + \frac{169}{6} \right) \right]
\frac{T_F (n_b + n_c)}{96}
\biggr\} \varepsilon + \mathcal{O}(\varepsilon^2)\,,\\
d^A_3 &=& \frac{C_A}{18} \biggl[
- C_A^2 \frac{(3 a - 13) (6 a^2 + 18 a + 31)}{64}
- C_A T_F (n_l + n_c) \frac{6 a^2 + 15 a + 44}{8}\\
&&\hphantom{C_A\biggl[\biggr.}
+ T_F^2 \bigl( (n_l + n_c)^2 + n_b n_c \bigr)
\biggr] L^3\\
&&{} + \biggl[
\frac{C_A^3}{128} \left( \frac{5}{2} a^3 + \frac{29}{3} a^2 - 17 a - \frac{3361}{18} \right)
+ C_F C_A T_F \frac{6 a (n_l - n_c) + 31 n_l - 49 n_c}{48}\\
&&\hphantom{C_A\biggl[\biggr.}
+ \frac{C_A^2 T_F (n_l + n_c)}{16} \left( \frac{a^2}{3} + 3 a + \frac{401}{18} \right)
- \frac{C_F T_F^2}{6} \left( n_l^2 - n_c^2 + \frac{11}{16} n_b n_c \right)\\
&&\hphantom{C_A\biggl[\biggr.}
- \frac{C_A T_F^2}{18} \left( 5 (n_l + n_c)^2 + \frac{73}{16} n_b n_c \right)
\biggr] L^2\\
&&{} + \biggl[
- \frac{C_A^3}{1024} \left( 6 \zeta_3 (a+1) (a+3)
+ 7 a^3 + 33 a^2 + 167 a - \frac{9965}{9} \right)
+ C_F^2 T_F \frac{n_l - 9 n_c}{16}\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_F C_A T_F}{4} \left( 3 \zeta_3 (n_l + n_c) + \frac{13}{48} a (n_b + n_c)
+ \frac{1}{36} \left( \frac{5}{4} n_l - 227 n_c \right) \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A^2 T_F}{16} \biggl( 9 \zeta_3 (n_l + n_c)
+ a \left( n_l + \frac{61}{48} n_c - \frac{25}{72} n_b \right)\\
&&\hphantom{{}+\biggl[{}+\frac{C_A^2T_F}{16}\biggl(\biggr.\biggr.}
- \frac{1}{36} \left( 911 n_l + \frac{3241}{4} n_c - \frac{1157}{12} n_b \right)
\biggr)\\
&&\hphantom{{}+\biggl[\biggr.}
+ C_F T_F^2 \frac{(n_l + n_c) (11 n_l + 4 n_c) + 4 n_b n_c}{72}\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A T_F^2}{32}
\left( \frac{(n_l + n_c) (76 n_l + 63 n_c)}{9}
+ n_b \left( 7 n_c - \frac{178}{54} n_l \right) \right)
\biggr] L\\
&&{} + \biggl[
- \frac{C_F^2}{12} \left( \frac{95}{2} \zeta_3 - \frac{97}{3} \right)
+ C_F C_A \left( B_4 - \frac{\pi^4}{20} + \frac{1957}{96} \zeta_3 - \frac{36979}{2592} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_A^2}{2}
\left( B_4 - \frac{3 \pi^4}{40} + \frac{\zeta_3 a}{3} + \frac{1709}{288} \zeta_3
- \frac{677}{432} a + \frac{22063}{3888} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{164}{81} C_F T_F n_l
+ C_F T_F (n_b + n_c) \left( \frac{7}{8} \zeta_3 - \frac{103}{162} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_A T_F n_l}{9} \left( 8 \zeta_3 - \frac{665}{54} \right)
+ \frac{C_A T_F (n_b + n_c)}{18} \left( \frac{287}{8} \zeta_3 - \frac{605}{27} \right)
\biggr] \frac{T_F (n_b + n_c)}{8}\\
&&{} + T_F^2 n_b n_c \biggl[ - \frac{C_A}{3} L_+(x)
+ \frac{1+x^2}{32 x^3} \left( C_F \frac{5 - 2 x^2 + 5 x^4}{4}
+ C_A \frac{4 + 11 x^2 + 4 x^4}{3} \right) L_-(x)\\
&&\hphantom{{}+T_F^2n_bn_c\biggl[\biggr.}
- \frac{14 C_F + 39 C_A}{64} \zeta_3\\
&&\hphantom{{}+T_F^2n_bn_c\biggl[\biggr.}
- \left( \frac{5}{16} C_F + \frac{C_A}{3} \right)
\left( \frac{(1-x^2)^2}{8 x^2} \left( L^2 + 2 \right)
- \frac{1-x^4}{4 x^2} L \right) \biggr]
+ \mathcal{O}(\varepsilon)\,,\end{aligned}$$ with $a\equiv a^{(n_f)}(\bar{m}_b)$. The easiest way to express $a^{(n_f)}(\bar{m}_b)$ via $a^{(n_l)}(m_c(\bar{m}_b))$ is to re-express $\alpha_s^{(n_f)}(\bar{m}_b)$ via $\alpha_s^{(n_l)}(m_c(\bar{m}_b))$ in the right-hand side of the equation $a^{(n_l)}(m_c(\bar{m}_b)) = a^{(n_f)}(\bar{m}_b) \zeta_A(\bar{m}_b,m_c(\bar{m}_b))$ and then solve it for $a^{(n_f)}(\bar{m}_b)$ iteratively.
Light-quark fields {#S:q}
------------------
The bare decoupling coefficient $\zeta_q^0$ of Eq. (\[Intro:zetafields0\]) is determined by $\Sigma_V(0)$ (cf. Eq. (\[Calc:Sigma\])). The renormalized version $\zeta_q$ (\[Intro:zetaren\]) can be obtained (see Refs. [@Larin:1993tp; @Chetyrkin:1999pq; @Czakon:2004bu] for the three-loop wave function renormalization constant) by re-expressing $\alpha_s^{(n_l)}$ and $a^{(n_l)}$ in the denominator via the $n_f$-flavour quantities (see Sects. \[S:as\] and \[S:a\]; note that positive powers of $\varepsilon$ should be kept). The result can be cast in the form $$\zeta_q(m_c(\bar{m}_b),\bar{m}_b) = 1
+ d^q_1 C_F \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
+ d^q_2 C_F \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
+ d^q_3 C_F \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3 + \cdots
\,,
\label{q:renorm}$$ where $$\begin{aligned}
d^q_1 &=& - \frac{a}{2} L \left( 1 - L \varepsilon + \frac{2}{3} L^2 \varepsilon^2
+ \mathcal{O}(\varepsilon^3) \right)\,,\\
d^q_2 &=& \frac{a}{16} \left[ 2 C_F a + C_A (a + 3) \right] L^2
+ \left( 6 C_F - C_A (a^2 + 8 a + 25) + 8 T_F (n_l + n_c) \right) \frac{L}{32}\\
&&{} + \frac{5}{96} T_F (n_b + n_c)\\
&&{} - \biggl[ a \left[ 2 C_F a + C_A (a + 3) \right] L^3
+ \left( 6 C_F - C_A (a^2 + 8 a + 25) + 8 T_F (n_l + n_c) \right) \frac{L^2}{2}\\
&&\hphantom{{}-\biggl[\biggr.}
+ \frac{5}{3} T_F n_c L
+ \frac{T_F (n_b + n_c)}{12} \left( \pi^2 + \frac{89}{6} \right)
\biggr] \frac{\varepsilon}{8} + \mathcal{O}(\varepsilon^2)\,,\\
d^q_3 &=& \frac{a}{8} \biggl[
- C_F^2 \frac{a^2}{6} - C_F C_A \frac{a (a+3)}{4} - C_A^2 \frac{2 a^2 + 9 a + 31}{24}
+ C_A T_F \frac{n_l + n_c}{3} \biggr] L^3\\
&&{} + \biggl[
- \frac{3}{32} C_F^2 a
+ C_F C_A \frac{a^3 + 8 a^2 + 25 a - 22}{64}
+ \frac{C_A^2}{64}
\left( a^3 + \frac{25}{4} a^2 + \frac{343}{12} a + \frac{275}{3} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- T_F \frac{n_l + n_c}{8} \left( C_F (a-1) + C_A \frac{13 a + 94}{12} \right)
+ T_F^2 \frac{(n_l + n_c)^2}{6}
\biggr] L^2\\
&&{} + \biggl[ - \frac{3}{64} C_F^2
- \frac{C_F C_A}{8} \left( 3 \zeta_3 -\frac{143}{16} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_A^2}{512} \left( 6 \zeta_3 (a^2 + 2 a - 23)
+ 5 a^3 + \frac{39}{2} a^2 + \frac{263}{2} a + \frac{9155}{9} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{C_F T_F}{32} \left( \frac{5}{6} (n_b + n_c) a - 3 (n_l + 5 n_c) \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A T_F}{288} \left( \frac{153 (n_l+n_c) - 89 n_b}{4} a + 287 n_l + 232 n_c \right)
- \frac{5}{72} T_F^2 n_l (n_l + n_c)
\biggr] L\\
&&{} + \biggl[
- C_F \left( 3 \zeta_3 + \frac{155}{48} \right)
- C_A \left( \zeta_3 (a-3)
- \frac{1}{72} \left( \frac{2387}{8} a + \frac{1187}{3} \right) \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{35}{2592} T_F (2 n_l + n_b + n_c)
\biggr] \frac{T_F (n_b + n_c)}{24} + \mathcal{O}(\varepsilon)\,.\end{aligned}$$ Note that the power corrections in $x$ drop out in the sum of all diagrams. For $x=1$ this result reduces to the ordinary decoupling of $n_b+n_c$ flavours with the same mass [@Chetyrkin:1997un] (see Ref. [@Grozin:2006xm] for an expression in terms of $C_A$ and $C_F$).
Ghost field {#S:c}
-----------
The bare decoupling coefficient $\zeta_c^0$ in Eq. (\[Intro:zetafields0\]) is determined by $\Pi_c(0)$ as given in Eq. (\[Calc:Ghost\]). The renormalized decoupling constant $\zeta_c$ of Eq. (\[Intro:zetaren\]) is given by (see Refs. [@Chetyrkin:2004mf; @Czakon:2004bu] for the corresponding renormalization constant) $$\zeta_c(m_c(\bar{m}_b),\bar{m}_b) = 1
+ d^c_1 C_A \frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}
+ d^c_2 C_A \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^2
+ d^c_3 C_A \left(\frac{\alpha_s^{(n_f)}(\bar{m}_b)}{\pi}\right)^3 + \cdots
\,,
\label{c:renorm}$$ where $$\begin{aligned}
d^c_1 &=& - \frac{a-3}{8} L
\left( 1 - L \varepsilon + \frac{2}{3} L^2 \varepsilon^2
+ \mathcal{O}(\varepsilon^3) \right)\,,\\
d^c_2 &=&
\left[ C_A \frac{3 a^2 - 35}{16} + T_F (n_l + n_c) \right] \frac{L^2}{8}
+ \left[ C_A \frac{3 a + 95}{8} - 5 T_F (n_l + n_c) \right] \frac{L}{48}\\
&&{} - \frac{89}{1152} T_F (n_b + n_c)\\
&&{} + \biggl\{
- \left[ C_A \frac{3 a^2 - 35}{16} + T_F (n_l + n_c) \right] \frac{L^3}{4}
- \left[ C_A \frac{3 a + 95}{8} - 5 T_F (n_l + n_c) \right] \frac{L^2}{24}\\
&&\hphantom{{}+\biggl\{\biggr.}
- T_F \frac{3 \pi^2 n_b - 89 n_c}{288} L
+ \frac{T_F (n_b + n_c)}{1152} \left( 5 \pi^2 + \frac{869}{6} \right)
\biggr\} \varepsilon + \mathcal{O}(\varepsilon^2)\,,\\
d^c_3 &=& \biggl[
- \frac{C_A^2}{256} \left( 5 a^3 + 9 a^2 - \frac{35}{3} a - \frac{2765}{9} \right)
- C_A T_F (n_l + n_c) \frac{3 a + 149}{144}\\
&&\hphantom{\biggl[\biggr.}
+ \frac{T_F^2}{9} \bigl( 2 (n_l + n_c)^2 - n_b n_c \bigr)
\biggr] \frac{L^3}{4}\\
&&{} + \biggl[
\frac{C_A^2}{16} \left( a^3 + \frac{9}{2} a^2 - \frac{11}{3} a - \frac{5773}{18} \right)
+ \left( 3 C_F + C_A \frac{3 a + 545}{36} \right) T_F (n_l + n_c)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{T_F^2}{9} \bigl( 20 (n_l + n_c)^2 - 29 n_b n_c \bigr)
\biggr] \frac{L^2}{32}\\
&&{} + \biggl[
\frac{C_A^2}{128} \left( 3 \zeta_3 (a+1) (a+3)
- \frac{3}{2} a^3 - 3 a^2 - 17 a + \frac{15817}{54} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ C_F T_F \left( 3 \zeta_3 (n_l + n_c)
- \frac{45 n_l + 25 n_c + 13 n_b}{16} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A T_F}{32} \left( - 72 \zeta_3 (n_l + n_c)
+ \frac{252 n_l + 341 n_c - 89 n_b}{36} a
- \frac{194}{27} n_l + \frac{695 n_c + 167 n_b}{12}
\right)\\
&&\hphantom{{}+\biggl[\biggr.}
- \frac{T_F^2}{27} \left( \frac{(n_l + n_c) (35 n_l + 124 n_c)}{4} + 31 n_b n_c \right)
\biggr] \frac{L}{8}\\
&&{} + \biggl[
- \frac{C_F}{2} \left( B_4 - \frac{\pi^4}{20} + \frac{57}{8} \zeta_3 - \frac{481}{96} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{C_A}{4} \left( B_4 - \frac{3 \pi^4}{40} - \frac{\zeta_3 a}{3} + \frac{431}{72} \zeta_3
+ \frac{685}{864} a - \frac{5989}{1944} \right)\\
&&\hphantom{{}+\biggl[\biggr.}
+ \frac{4}{9} T_F n_l \left( \zeta_3 - \frac{1327}{864} \right)
- \frac{T_F (n_b + n_c)}{9} \left( 7 \zeta_3 - \frac{1685}{432} \right)
\biggr] \frac{T_F (n_b + n_c)}{8}\\
&&{} + \frac{T_F^2 n_b n_c}{6} \biggl[ L_+
- \frac{(1 + x^2) (5 + 22 x^2 + 5 x^4)}{64 x^3} L_-
+ \frac{3}{2} \zeta_3\\
&&\hphantom{{}+\frac{T_F^2n_bn_c\biggl[\biggr.}{}}
+ \frac{5}{64} \left( \frac{(1 - x^2)^2}{x^2} (L^2 + 2) - 2 \frac{1 - x^4}{x^2} L \right)
\biggr] + \mathcal{O}(\varepsilon)\,.\end{aligned}$$
Phenomenological applications {#S:Phen}
=============================
In this section we study the numerical consequences of the decoupling relations computed in the previous sections. For convenience we use in this Section the decoupling relations in terms of on-shell heavy quark masses (see Appendix \[S:OS\] and the [Mathematica]{} file which can be downloaded from [@progdata]) which we denote by $M_c$ and $M_b$.
$\alpha_s^{(5)}(M_Z)$ from $\alpha_s^{(3)}(M_\tau)$
---------------------------------------------------
Let us in a first step check the dependence on the decoupling scales which should become weaker after including higher order perturbative corrections. We consider the relation between $\alpha_s^{(3)}(M_\tau)$ and $\alpha_s^{(5)}(M_Z)$. $\alpha_s^{(3)}(M_\tau)$ has been extracted from experimental data using perturbative results up to order $\alpha_s^4$ [@Baikov:2008jh]. Thus it is mandatory to perform the transition from the low to the high scale with the highest possible precision. In the following we compare the conventional approach with the single-step decoupling up to three-loop order.
For our analysis we use for convenience the decoupling constants expressed in terms of on-shell quark masses. In this way the mass values are fixed and they are not affected by the running from $M_\tau$ to $M_Z$. In our analysis we use $M_c=1.65$ GeV and $M_b=4.7$ GeV. Furthermore, $\alpha_s^{(3)}(M_\tau) = 0.332$ [@Baikov:2008jh] is used as starting value of our analysis.
In Fig. \[fig::asMz\](a) we show $\alpha_s^{(5)}(M_Z)$ as a function of $\mu_{\rm dec}$, the scale where the $c$ and $b$ quarks are simultaneously integrated out. In a first step $\alpha_s^{(3)}(M_\tau)$ is evolved to $\alpha_s^{(3)}(\mu_{\rm dec})$ using the $N$-loop renormalization group equations. Afterwards the $(N-1)$-loop decoupling relation is applied and finally $N$-loop running is employed in order to arrive at $\alpha_s^{(5)}(M_Z)$. One observes a strong dependence on $\mu_{\rm dec}$ for $N=1$ (short-dashed line) which becomes rapidly weaker when increasing $N$ leading to a reasonably flat curve for $N=4$ (longer dashes correspond to larger values of $N$).
Comparison of one- and two-step decoupling approach
---------------------------------------------------
In the step-by-step decoupling approach we have two decoupling scales $\mu_{{\rm dec},c}$ and $\mu_{{\rm dec},b}$ which can be chosen independently. First we choose[^5] $\mu_{{\rm dec},c}=3$ GeV and identify $\mu_{{\rm dec},b}$ with $\mu_{{\rm dec}}$. The result for $N=4$ is shown in Fig. \[fig::asMz\](b) together with the four-loop curve from Fig. \[fig::asMz\](a) as dash-dotted line (long dashes). One observes a significantly flatter behaviour as for the one-step decoupling which can be explained by the occurrence of $\log(\mu^2/M_c^2)$ terms in the one-step formula which might become large for large values of $\mu=\mu_{{\rm dec}}$. Alternatively it is also possible to study the dependence on $\mu_{{\rm dec},c}$, i.e., identify $\mu_{{\rm dec},c}$ with $\mu_{{\rm
dec}}$, set $\mu_{{\rm dec},b}=10$ GeV and compare to the one-step decoupling. The results are also shown in Fig. \[fig::asMz\](b) as dash-dotted line (short dashes) where only values $\mu_{{\rm dec}}\le 10$ are considered.
For comparison we show in Fig. \[fig::asMz\](b) also the result of the two-step five-loop analysis as dotted line where the four-loop decoupling relation is taken from Refs. [@Schroder:2005hy; @Chetyrkin:2005ia]. The (unknown) five-loop coefficient of the $\beta$ function, $\beta_4$, is set to zero.[^6] If one restricts to scales $\mu_{\rm dec}$ between 2 GeV and 10 GeV it seems that the four-loop decoupling constant is numerically more relevant than the power-suppressed terms included by construction in the one-step decoupling procedure. Thus, from these considerations one tends to prefer the two-step decoupling over the one-step approach as it seems that the resummation of $\log(\mu^2/M_{c,b}^2)$ is more important than the inclusion of power-suppressed corrections.
Let us in a next step restrict ourselves to decoupling scales which are of the order of the respective quark masses. In Tab. \[tab::mub\] we compare the value for $\alpha_s^{(5)}(M_Z)$ as obtained from the one- and two-step decoupling where two variants of the former are used: $\zeta_{\alpha_s}$ which directly relates $\alpha_s^{(3)}(\mu_c)$ and $\alpha_s^{(5)}(\mu_b)$ as given in Eq. (\[Intro:zetaren\]) with $\mu^\prime=\mu_c$ and $\mu=\mu_b$ ($\zeta_{\alpha_s}(\mu_c,\mu_b)$; see also [@progdata]) and the version with only one decoupling scale where $\mu^\prime=\mu$ has been set ($\zeta_{\alpha_s}(\mu)$). We thus define two deviations $$\begin{aligned}
\delta\alpha_s^{(a)} &=&
\alpha_s^{(5)}(M_Z)\Big|_{\zeta_{\alpha_s}(\mu_c,\mu_b)}
- \alpha_s^{(5)}(M_Z)\Big|_{\mbox{\scriptsize 2-step}}
\,,\nonumber\\
\delta\alpha_s^{(b)} &=&
\alpha_s^{(5)}(M_Z)\Big|_{\zeta_{\alpha_s}(\mu)}
- \alpha_s^{(5)}(M_Z)\Big|_{\mbox{\scriptsize 2-step}}
\,,
\label{eq::deltaalpha}\end{aligned}$$ where the scale $\mu$ in the second equation is either identified with $\mu_c$ (right part of Tab. \[tab::mub\]) or $\mu_b$ (left part), respectively.
[cc]{}
--------- ----------------------- ------------------------ ------------------------
$\mu_b$ $\alpha_s^{(5)}(M_Z)$ $\delta\alpha_s^{(a)}$ $\delta\alpha_s^{(b)}$
(GeV) $\times10^3$ $\times10^3$
$(\mu=\mu_b)$
2 0.11985 $-0.28$ 0.18
5 0.11977 0.23 $-0.16$
7 0.11974 0.36 $-0.26$
10 0.11970 0.19 $-0.36$
--------- ----------------------- ------------------------ ------------------------
: \[tab::mub\] Decoupling scale $\alpha_s^{(5)}(M_Z)$ as obtained from the four-loop analysis of the two-step approach, and the deviations as defined in the text. In the left table $\mu_c=3$ GeV and in the right one $\mu_b=10$ GeV has been chosen.
&
--------- ----------------------- ------------------------ ------------------------
$\mu_c$ $\alpha_s^{(5)}(M_Z)$ $\delta\alpha_s^{(a)}$ $\delta\alpha_s^{(b)}$
(GeV) $\times10^3$ $\times10^3$
$(\mu=\mu_c)$
2 0.11984 $-4.02$ $0.20$
3 0.11970 0.19 $0.14$
4 0.11961 0.33 $0.10$
5 0.11955 0.26 $0.06$
--------- ----------------------- ------------------------ ------------------------
: \[tab::mub\] Decoupling scale $\alpha_s^{(5)}(M_Z)$ as obtained from the four-loop analysis of the two-step approach, and the deviations as defined in the text. In the left table $\mu_c=3$ GeV and in the right one $\mu_b=10$ GeV has been chosen.
It is interesting to note that (except for the choice $\mu_c=2$ GeV and $\mu_b=10$ GeV) the deviations presented in Tab. \[tab::mub\] amount to about 30% to 50% of the uncertainty of the world average for $\alpha_s(M_Z)$ which is given by $\delta\alpha_s=0.7\cdot 10^{-3}$ [@Nakamura:2010zzi].
Improving the two-step approach by power-suppressed terms
---------------------------------------------------------
From the previous considerations it is evident that the resummation of logarithms of the form $[\alpha_s \log(\mu_c/\mu_b)]^k$, which is automatically incorporated in the two-step approach, is numerically more important than power-suppressed terms in $M_c/M_b$. Thus it is natural to use the two-step approach as default method and add the power-corrections afterwards. This is achieved in the following way: In a first step we invert $\zeta_{\alpha_s}(\mu_c,\mu_b)$ (cf. Eq. (\[Intro:zetaren\])) and express it in terms of $\alpha_s^{(3)}(\mu_c)$ in order to arrive at the equation $\alpha_s^{(5)}(\mu_b)=\zeta^{-1}_{\alpha_s}(\mu_c,\mu_b)
\alpha_s^{(3)}(\mu_c)$. Now an expansion is performed for $M_c/M_b\to 0$ to obtain the leading term which is then subtracted from $\zeta^{-1}_{\alpha_s}(\mu_c,\mu_b)$ since it is part of the two-step decoupling procedure. The result is independent of $\mu_c$ and $\mu_b$ and has following series expansion $$\begin{aligned}
\delta\zeta_{\alpha_s}^{-1} &=& \left(\frac{\alpha_s^{(3)}(\mu_c)}{\pi}\right)^3
\left[
\frac{\pi^2}{18}x
+ \left(-\frac{6661}{18000} - \frac{1409}{21600}L +
\frac{1}{160}L^2 \right) x^2
+ {\cal O}(x^3)
\right]
\nonumber\\
&\approx& 0.170 \left(\frac{\alpha_s^{(3)}(\mu_c)}{\pi}\right)^3
\,,
\label{eq::deltazeta}\end{aligned}$$ where the numerical value in the second line has been obtained with the help of the exact dependence on $x$. Note that the linear term in $x$ arises from the $\overline{\rm MS}$–on-shell quark mass relation. The quantity $\delta\zeta_{\alpha_s}^{-1}$ is used in order to compute an additional contribution to $\alpha_s^{(5)}(\mu_b)$ as obtained from the two-step method: $$\begin{aligned}
\delta\alpha_s^{(5)}(\mu_b) &=& \delta\zeta_{\alpha_s}^{-1} \alpha_s^{(3)}(\mu_c)
\,.\end{aligned}$$ Inserting numerical values leads to shifts which are at most a few times $10^{-5}$ and are thus beyond the current level of accuracy. It is in particular more than an order of magnitude smaller than the four-loop decoupling term which is shown as dotted curve in Fig. \[fig::asMz\](b).
Note that as far as the strong coupling in Eq. (\[eq::deltazeta\]) is concerned both the number of flavours and the renormalization scale of $\alpha_s$ are not fixed since power-suppressed terms appear for the first time at this order. However, the smallness of the contribution is not affected by the choices made in Eq. (\[eq::deltazeta\]).
One-step decoupling of the bottom quark with finite charm quark mass
--------------------------------------------------------------------
An alternative approach to implement power-suppressed corrections in $m_c/m_b$ in the decoupling procedure is as follows: We consider the step-by-step decoupling and use at the scale $\mu_{\rm dec,c}$ the standard formalism for the decoupling of the charm quark as implemented in [RunDec]{} [@Chetyrkin:2000yt]. At the scale $\mu_{\rm dec,b}$, however, we consider the matching of five- to four-flavour QCD where we keep the charm quark massive. This requires a modification of the formulae in Eqs. (\[Intro:zetafields0\]) and (\[Intro:zetaparams0\]) to ($n_f^\prime=n_f-1$) $$\begin{aligned}
&&\zeta_A^0 = \frac{1 + \Pi_A^{(n_f)} (0)}{1 + \Pi_A^{(n_f^\prime)}
(0)}\,, \quad
\zeta_c^0 = \frac{1 + \Pi_c^{(n_f)} (0)}{1 + \Pi_c^{(n_f^\prime)}
(0)}\,, \quad
\zeta_q^0 = \frac{1 + \Pi_q^{(n_f)} (0)}{1 + \Pi_q^{(n_f^\prime)}
(0)}\,, \nonumber\\
&&\zeta_m^0 = (\zeta_q^0)^{-1} \frac{1 - \Sigma_S^{(n_f)} (0)}{1
- \Sigma_S^{(n_f^\prime)} (0)}\,, \quad
\zeta_{\alpha_s}^0 = (\zeta_c^0)^{-2} (\zeta_A^0)^{-1} \frac{\left(1 +
\Gamma_{A \bar{c} c}^{(n_f)} \right)^2}{\left(1 + \Gamma_{A
\bar{c} c}^{(n_f^\prime)} \right)^2}\,,
\label{eq::2step_mv_c}\end{aligned}$$ where the $n_f$-flavour quantities contain contributions form massive charm and bottom quarks. They are identical to the one-step decoupling procedure described above. In the $n_f^\prime$-flavour quantities appearing in the denominators those diagrams have to be considered which contain a charm quark. Note that they depend on the bare parameters of the effective theory ($\alpha_{s 0}^{(n_f^\prime)}$, $a_0^{(n_f^\prime)}$, $m_{c 0}^{(n_f^\prime)}$) and thus they have to be decoupled iteratively in order to express all quantities on the r.h.s. of the above equations by the same parameters ($\alpha_{s 0}^{(n_f)}$, $a_0^{(n_f)}$, $m_{c 0}^{(n_f)}$). In the standard approach the $n_f^\prime$-flavour quantities vanish since only scale-less integrals are involved.
As a cross check we have verified that we reobtain the analytical result for the single-step decoupling if we apply the formalism of Eq. (\[eq::2step\_mv\_c\]) and the subsequent decoupling of the charm quark at the same scale.
We have incorporated the finite charm quark mass effects in the two-step decoupling approach (cf. Fig. \[fig::asMz\]) and observe small numerical effects. A minor deviation from the $m_c=0$ curve can only be seen for decoupling scales of the order of 1 GeV which confirms the conclusions reached above that the power-suppressed terms are numerically negligible. Thus we both refrain from explicitly presenting numerical results and analytical formulae for the renormalized decoupling coefficients as obtained from Eqs. (\[eq::2step\_mv\_c\]).
Decoupling effects in the strange quark mass
--------------------------------------------
In analogy to the strong coupling we study in the following the relation of the strange quark mass $m_s(\mu)$ defined with three and five active quark flavours, respectively. The numerical analysis follows closely the one for $\alpha_s$: $N$-loop running is accompanied by $(N-1)$-loop decoupling relations. It is, however, slightly more involved since besides $m_s(\mu)$ also $\alpha_s(\mu)$ has to be known for the respective renormalization scale and number of active flavours. We organized the calculation in such a way that we simultaneously solve the renormalization group equations for $m_s(\mu)$ and $\alpha_s(\mu)$ (truncated to the considered order) using [Mathematica]{}.
In Fig. \[fig::mqMz\] we show $m_s^{(5)}(M_Z)$ as a function of $\mu_{\rm dec}$ and again compare the single-step (dashed lines) to the two-step (dash-dotted lines) approach. For our numerical analysis we use in addition to the parameters specified above $m_s(2~\mbox{GeV})=100$ MeV. The same conclusion as for $\alpha_s$ can be drawn: The difference between the two approaches becomes smaller with increasing loop order. At the same time the prediction for $m_s^{(5)}(M_Z)$ becomes more and more independent of $\mu_{\rm dec}$. The results again suggest that the power-corrections $M_c/M_b$ are small justifying the application of the two-step decoupling.
Effective coupling of the Higgs boson to gluons {#S:Higgs}
===============================================
The production and decay of an intermediate-mass Higgs boson can be described to good accuracy by an effective Lagrange density where the top quark is integrated out. It contains an effective coupling of the Higgs boson to gluons given by $$\begin{aligned}
{\cal L}_{\rm eff} &=& -\frac{\phi}{v} C_1 {\cal O}_1
\,,
\label{eq::leff}\end{aligned}$$ with ${\cal O}_1 = G_{\mu\nu} G^{\mu\nu}$. $C_1$ is the coefficient function containing the remnant contributions of the top quark, $G^{\mu\nu}$ is the gluon field strength tensor, $\phi$ denotes the CP-even Higgs boson field and $v$ is the vacuum expectation value.
The effective Lagrange density in Eq. (\[eq::leff\]) can also be used for theories beyond the Standard Model like supersymmetric models or extensions with further generations of heavy quarks. In all cases the effect of the heavy particles is contained in the coefficient function $C_1$.
In Ref. [@Chetyrkin:1997un] a low-energy theorem has been derived which relates the effective Higgs-gluon coupling $C_1$ to the decoupling constant for $\alpha_s$. In this Section we apply this theorem to an extension of the Standard Model containing additional heavy quarks which couple to the Higgs boson via a top quark-like Yukawa coupling. Restating Eq. (39) of Ref. [@Chetyrkin:1997un] in our notation and for the case of several heavy quarks leads to $$\begin{aligned}
C_1 &=& -\frac{1}{2} \sum_{i=1}^{N_h} M_i^2 \frac{{\rm d}}{{\rm d} M_i^2}
\log\zeta_{\alpha_s}
\,,\end{aligned}$$ where $N_h$ is the number of heavy quarks with on-shell masses $M_i$. Using $\zeta_{\alpha_s}$ from Eq. (\[Intro:zetaren\]) (see also [@progdata]) we obtain for $C_1$ the following result[^7] $$\begin{aligned}
C_1 &=& \frac{\alpha_s^{\rm (full)}(\mu)}{\pi} \left( - T_F \frac{N_h}{6} \right)
+ \left( \frac{\alpha_s^{\rm (full)}(\mu)}{\pi} \right)^2 \left( \frac{C_F T_F}{8}
- C_A T_F \frac{5}{24} + T_F^2 \frac{\Sigma_h}{18} \right) N_h \nonumber\\
&&{} + \left( \frac{\alpha_s^{\rm (full)}(\mu)}{\pi} \right)^3
\left\{ - C_F^2 T_F \frac{9}{64} N_h
+ C_F C_A T_F \left[ \frac{25}{72} N_h
+ \frac{11}{96} \Sigma_h \right] \right. \nonumber\\
&&{} + C_F T_F^2 \left[ \frac{5}{96} N_h n_l + \frac{17}{288} N_h^2
- \Sigma_h \left( \frac{N_h}{8} + \frac{n_l}{12} \right) \right]
- C_A^2 T_F \left[ \frac{1063}{3456} N_h
+ \frac{7}{96} \Sigma_h \right] \nonumber\\
&&{} \left. + C_A T_F^2 \left[ \frac{47}{864} n_l - \frac{49}{1728} N_h
+ \frac{5}{24} \Sigma_h \right] N_h
- T_F^3 \Sigma_h^2 \frac{N_h}{54} \right\}
\,,\end{aligned}$$ where $\alpha_s^{\rm (full)}$ is the strong coupling in the full theory with $n_l+N_h$ active quark flavours and $\Sigma_h = \sum_{i=1}^{N_h} \log(\mu^2/M_i^2)$. After expressing $\alpha_s^{\rm (full)}$ in terms of $\alpha_s^{(5)}$ and specifying the colour factors to SU(3) we reproduce the result of Ref. [@Anastasiou:2010bt] which has been obtained by an explicit calculation of the Higgs-gluon vertex corrections. For $N_h=1$ the result obtained in Ref. [@Chetyrkin:1997un] is reproduced. It is remarkable that although $\zeta_{\alpha_s}$ contains di- and tri-logarithms there are only linear logarithms present in $C_1$.
Conclusion {#S:Conc}
==========
The main result of this paper is the computation of a decoupling constant relating the strong coupling defined with three active flavours to the one in the five-flavour theory. At three-loop order Feynman diagrams with two mass scales, the charm and the bottom quark mass, have to be considered. The corresponding integrals have been evaluated exactly and analytical results have been presented. The new results can be used in order to study the effect of power-suppressed terms in $M_c/M_b$ which are neglected in the conventional approach [@Chetyrkin:1997un]. Various analyses are performed which indicate that the mass corrections present in the one-step approach are small as compared to $\log(\mu^2/M_{c,b}^2)$ which are resummed using the conventional two-step procedure.
Using a well-known low-energy theorem [@Chetyrkin:1997un] we can use our result for the decoupling constant in order to obtain the effective gluon-Higgs boson coupling for models containing several heavy quarks which couple to the Higgs boson via the same mechanism as the top quark. This constitutes a first independent check of the result presented in Ref. [@Anastasiou:2010bt] where the matching coefficient has been obtained by a direct evaluation of the Higgs-gluon-gluon vertex diagrams.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Deutsche Forschungsgemeinschaft through the SFB/TR-9 “Computational Particle Physics”. We are grateful to K.G. Chetyrkin for useful discussions.
Integral $I(x)$ {#S:Ix}
===============
With the help of `FIRE` [@Smirnov:2008iw] we can express the integral $I(x)$ as defined in Eq. (\[Calc:Idef\]) as a linear combination of master integrals $$\begin{aligned}
&&I(x) = I(x^{-1}) =
\frac{1}{(d-1) (d-4) (d-6) (d-8) (d-10)}
\label{Ix:I}\\
&&{}\times\biggl[ \frac{1}{4} \left( c_{10} + c_{11} (x^{-2}+x^2) + c_{12} (x^{-4}+x^4) \right) I_1(x)
\nonumber\\
&&\hphantom{{}\times\biggl[\biggr.}
+ \frac{3}{16} (d-2) (x^{-1}+x) \left( c_{20} + c_{21} (x^{-2}+x^2) \right) I_2(x)
\nonumber\\
&&\hphantom{{}\times\biggl[\biggr.}
- \frac{c_{-1} (x^{2+\varepsilon} + x^{-2-\varepsilon})
+ c_0 (x^\varepsilon + x^{-\varepsilon})
+ c_1 (x^{-2+\varepsilon} + x^{2-\varepsilon})
+ c_2 (x^{-4+\varepsilon} + x^{4-\varepsilon})}{(d-2)^2 (d-3) (d-5) (d-7)} \biggr]\,.
\nonumber\end{aligned}$$ $I_1$ and $I_2$ are master integrals with four massive lines (see Fig. \[F:Master\]) which are given by $$\begin{aligned}
&&I_1(x) = I_1(x^{-1}) =
\frac{(m_b m_c)^{-2+3\varepsilon}}{(i \pi^{d/2})^3 \Gamma^3(\varepsilon)}
\int \frac{d^d k_1\,d^d k_2\,d^d k_3}{D_1 D_2 D_3 D_4}\,,
\nonumber\\
&&I_2(x) = I_2(x^{-1}) =
\frac{(m_b m_c)^{-3+3\varepsilon}}{(i \pi^{d/2})^3 \Gamma^3(\varepsilon)}
\int \frac{N\,d^d k_1\,d^d k_2\,d^d k_3}{D_1 D_2 D_3 D_4}\,,
\nonumber\\
&&D_1 = m_b^2 - k_1^2\,,\quad
D_2 = m_b^2 - k_2^2\,,\quad
D_3 = m_c^2 - k_3^2\,,
\nonumber\\
&&D_4 = m_c^2 - (k_1-k_2+k_3)^2\,,\quad
N = - (k_1-k_2)^2\,,
\label{Ix:I12}\end{aligned}$$ and $c_i$ and $c_{ij}$ are coefficients depending on $d=4-2\varepsilon$ $$\begin{aligned}
&&c_{10} = (d-1) (5 d^4 - 104 d^3 + 73 d^2 - 2116 d + 2086)\,,\\
&&c_{11} = (d-1) (2d-7) (2 d^3 - 35 d^2 + 180 d - 256)\,,\\
&&c_{12} = (d-9) (2d-5) (2d-7) (2d-9)\,,\\
&&c_{20} = 2 (d^4 - 22 d^3 + 165 d^2 - 491 d + 487)\,,\\
&&c_{21} = (d-9) (2d-7) (2d-9)\,,\\
&&c_{-1} = (d-3) (d-5) (d-7) (d-9) (2d-5) (2d-7) (2d-9)\,,\\
&&c_0 = (d-1) (d-3) (4 d^5 - 108 d^4 + 1090 d^3 - 5009 d^2 + 9838 d - 5335)\,,\\
&&c_1 = (d-1) (d-7) (2 d^5 - 46 d^4 + 384 d^3 - 1423 d^2 + 2158 d - 739)\,,\\
&&c_2 = (d-1) (d-5) (d-7) (d-9) (2d-7) (2d-9)\,.\end{aligned}$$
The master integrals used in Ref. [@Bekavac:2009gz] are related to $I_{1,2}$ by $$\begin{aligned}
I_{4.3} &=& (m_b m_c)^{2-3\varepsilon} \Gamma^3(\varepsilon) I_1(x)\,,
\nonumber\\
I_{4.3a} &=& (m_b m_c)^{1-3\varepsilon} \Gamma^3(\varepsilon) \frac{x}{1-x^2}
\nonumber\\
&&{}\times\left[ - \frac{1}{4} \left(d-3 - (2d-5) x^2\right) I_1(x)
+ \frac{3}{16} (d-2) x I_2(x)
+ \frac{x^\varepsilon + x^{2-\varepsilon}}{(d-2)^2} \right]\,.
\label{Ix:4.3}\end{aligned}$$ Using their expansions in $\varepsilon$ [@Bekavac:2009gz] we obtain $$I(x) = - \frac{32}{27} \left[ 1 - \frac{2}{3} \varepsilon
+ \frac{1}{2} \left( \frac{25}{3} + 3 L^2 \right) \varepsilon^2
+ B \varepsilon^3 + \cdots \right]\,,
\label{Ix:e}$$ where $$\begin{aligned}
&&\frac{32}{3} B = 64 L_+(x)
- \frac{(1+x^2)(5+22x^2+5x^4)}{x^3} L_-(x)
\nonumber\\
&&{} + \frac{5+18x^2+5x^4}{x^2} L^2
- 10 \frac{1-x^4}{x^2} L
+ 10 \frac{(1-x^2)^2}{x^2} + \frac{64}{3} \zeta_3 - \frac{1256}{81}\,,
\label{Ix:B}\end{aligned}$$ and $$\begin{aligned}
L_\pm(x) &=& L_\pm(x^{-1}) =
{\mathop{\mathrm{Li}}\nolimits_{3}}(x) - L {\mathop{\mathrm{Li}}\nolimits_{2}}(x) - \frac{L^2}{2} \log(1-x) + \frac{L^3}{12}
\nonumber\\
&&{} \pm \left[ {\mathop{\mathrm{Li}}\nolimits_{3}}(-x) - L {\mathop{\mathrm{Li}}\nolimits_{2}}(-x) - \frac{L^2}{2} \log(1+x) + \frac{L^3}{12} \right]\,,
\label{Ix:L}\end{aligned}$$ with $L=\log x$. Note that the functions $L_\pm(x)$ are analytical from 0 to $+\infty$.
For $x=1$, $I_2(1)$ is not independent [@Broadhurst:1991fi]: $$I_2(1) = - \frac{4}{3} \left( I_1(1) + \frac{8}{(d-2)^3} \right)\,.
\label{Ix:1}$$ The expansion of $I_1(1)$ in $\varepsilon$ has been studied in Refs. [@Broadhurst:1991fi; @Broadhurst:1996az]. Using the explicit formulas (3.2) and (2.3) from [@Bekavac:2009gz], it is easy to get $$I(1) = - \frac{32}{27} \left[ 1 - \frac{2}{3} \varepsilon + \frac{25}{6} \varepsilon^2
- \left( 7 \zeta_3 + \frac{157}{108} \right) \varepsilon^3 + \cdots \right]\,,
\label{Ix:1a}$$ in agreement with (\[Ix:e\]).
For $x\to0$, two regions [@Smirnov:2002pj] contribute to $I(x)$ (see Eq. (\[Calc:Idef\])), the hard ($k\sim m_b$) and and the soft ($k\sim m_c$) one. The result for the leading term is given by $$\begin{aligned}
I(x) &=& I_h x^{3\varepsilon} \left[1 + \mathcal{O}(x^2)\right]
+ I_s x^{-\varepsilon} \left[1 + \mathcal{O}(x^2)\right]\,,
\label{Ix:0}\\
I_h &=& \frac{8}{3}
\frac{d-5}{(d-1) (d-3) (2d-9) (2d-11)}
\frac{\Gamma(1-\varepsilon) \Gamma^2(1+2\varepsilon) \Gamma(1+3\varepsilon)}{\Gamma^2(1+\varepsilon) \Gamma(1+4\varepsilon)}\,,
\nonumber\\
I_s &=& \frac{8}{3}
\frac{d-6}{(d-2) (d-5) (d-7)}\,.
\nonumber\end{aligned}$$ Expanding this formula in $\varepsilon$ we reproduce Eq. (\[Ix:e\]) for $x\to0$.
Ghost–gluon vertex at two loops {#S:Ghost}
===============================
We need this vertex expanded in the external momenta up to the linear terms. Let us consider the right-most vertex on the ghost line: $$\raisebox{-7mm}{\begin{picture}(27,18)
\put(13.5,9.5){\makebox(0,0){\includegraphics{figs/ghr.eps}}}
\put(23.5,0){\makebox(0,0)[b]{$p$}}
\put(12.5,18){\makebox(0,0)[tl]{$\mu$}}
\put(22,4.5){\makebox(0,0)[bl]{$\nu$}}
\end{picture}}
= A^{\mu\nu} p_\nu\,.$$ The tensor $A^{\mu\nu}$ may be calculated at zero external momenta, hence $A^{\mu\nu}=Ag^{\mu\nu}$. Therefore all loop diagrams have the Lorentz structure of the tree vertex, as expected.
Now let us consider the left-most vertex: $$\raisebox{-7mm}{\begin{picture}(27,18)
\put(13.5,9.5){\makebox(0,0){\includegraphics{figs/ghl.eps}}}
\put(3.5,0){\makebox(0,0)[b]{$0$}}
\put(8.5,0){\makebox(0,0)[b]{$k$}}
\put(5,10){\makebox(0,0){$k$}}
\end{picture}}\,.$$ It gives $k^\lambda$, thus singling out the longitudinal part of the gluon propagator. Therefore, all loop corrections vanish in Landau gauge. Furthermore, diagrams with self-energy insertions into the left-most gluon propagator vanish in any covariant gauge: $$\raisebox{-9mm}{\includegraphics{figs/gh0.eps}} =
\raisebox{-7mm}{\includegraphics{figs/gh3.eps}} = 0\,.$$
In the diagrams including a quark triangle, the contraction of $k^\lambda$ transfers the gluon propagator to a spin 0 propagator and a factor $k^\rho$ which contracts the quark-gluon vertex. After decomposing $k\!\!\!/$ into a difference of the involved fermion denominators one obtains in graphical form $$\begin{aligned}
&&\raisebox{-7mm}{\includegraphics{figs/gh1.eps}}
= a_0 \left[ \raisebox{-7mm}{\includegraphics{figs/gha.eps}}
- \raisebox{-7mm}{\includegraphics{figs/ghb.eps}} \right]\,,\\
&&\raisebox{-7mm}{\includegraphics{figs/gh2.eps}}
= a_0 \left[ \raisebox{-7mm}{\includegraphics{figs/ghb.eps}}
- \raisebox{-7mm}{\includegraphics{figs/gha.eps}} \right]\,.\end{aligned}$$ The diagrams with a massless triangle vanish. The non-vanishing diagrams contain the same Feynman integral, but differ by the order of the colour matrices along the quark line, thus leading to a commutator of two Gell-Mann matrices.
The remaining diagram contains a three-gluon vertex with a self energy inserted in the right-most gluon propagator. The contraction of $k^\lambda$ with the three-gluon vertex cancels the gluon propagator to the right of the three-gluon vertex: $$\raisebox{-7mm}{\includegraphics{figs/gh4.eps}}
= a_0 \raisebox{-7mm}{\includegraphics{figs/gha.eps}}\,.$$ The colour structure of the three-gluon vertex is identical to the commutator above, however with opposite sign. Therefore, after summing all contributions the result is zero.
Decoupling at on-shell masses {#S:OS}
=============================
For some applications it is convenient to parametrize the decoupling constants in terms of the on-shell instead of $\overline{\mbox{MS}}$ quark masses. The corresponding counterterm relation reads $$m_{b0} = Z_{m_b}^{{\rm os}(n_f)}\left(\alpha_{s0}^{(n_f)}\right) M_b\,,\quad
m_{c0} = Z_{m_c}^{{\rm os}(n_f)}\left(\alpha_{s0}^{(n_f)}\right) M_c\,,
\label{OS:mass}$$ where in our application $Z_{m_b}^{{\rm os}(n_f)}$ and $Z_{m_c}^{{\rm os}(n_f)}$ are needed to two-loop accuracy. They have been calculated in Ref. [@Gray:1990yh] (see also [@Davydychev:1998si; @Bekavac:2007tk]). Note that the two-loop coefficients of $Z_{m_b}^{{\rm os}(n_f)}$ and $Z_{m_c}^{{\rm os}(n_f)}$ are non-trivial functions of $m_c/m_b$; a compact expression can be found in Ref. [@Bekavac:2007tk].
The advantage of using on-shell masses is that they are identical in all theories (with any number of flavours). Furthermore their numerical value does not depend on the renormalization scale. However, it is well known that usually the coefficients of perturbative series for physical quantities grow fast when expressed via on-shell quark masses and hence the ambiguities of the mass values (extracted from those observable quantities) are quite large. Nevertheless, using on-shell masses in intermediate theoretical formulae (at any finite order of perturbation theory) can be convenient.
The decoupling relations are particularly compact if $\alpha_s^{(n_l)}(M_c)$ is expressed as a series in $\alpha_s^{(n_f)}(M_b)$ since then the coefficients only depend on $x_{\rm os} = M_c/M_b$ (see results in [@progdata]).
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K.G. Chetyrkin, J.H. Kühn, M. Steinhauser, *`RunDec`: A `Mathematica` package for running and decoupling of the strong coupling and quark masses*, \[arXiv:\]. S.A. Larin, J.A.M. Vermaseren, *The three-loop QCD $\beta$-function and anomalous dimensions*, \[arXiv:\].
K.G. Chetyrkin, *Four-loop renormalization of QCD: Full set of renormalization constants and anomalous dimensions*, \[arXiv:\]. K.G. Chetyrkin, A. Rétey, *Renormalization and running of quark mass and field in the regularization invariant and $\overline{\mbox{MS}}$ schemes at three loops and four loops*, \[arXiv:\].
`http://www-ttp.particle.uni-karlsruhe.de/Progdata/ttp11/ttp11-07/`
P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, *Order $\alpha^4_s$ QCD corrections to $Z$ and $\tau$ decays*, \[ \[hep-ph\]\]. J.H. Kühn, M. Steinhauser, C. Sturm, *Heavy quark masses from sum rules in four-loop approximation*, \[arXiv:\]. K. Nakamura [*et al.*]{} \[Particle Data Group\], *Review of particle physics*, . C. Anastasiou, R. Boughezal, E. Furlan, *The NNLO gluon fusion Higgs production cross-section with many heavy quarks*, \[ \[hep-ph\]\]. D.J. Broadhurst, *On the enumeration of irreducible $k$-fold Euler sums and their roles in knot theory and field theory*, arXiv:. N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, *Three loop relation of quark $\overline{\mbox{MS}}$ and pole masses*, . A.I. Davydychev, A.G. Grozin, *Effect of $m_c$ on $b$ quark chromomagnetic interaction and on-shell two-loop integrals with two masses*, \[arXiv:\].
[^1]: At low $q\ne0$, the self-energies in the full theory are given by sums of contributions from various integration regions, see, e.g., [@Smirnov:2002pj]; the contribution we need comes from the completely hard region, where all loop momenta are of order of heavy-quark masses.
[^2]: The $A\bar{q}q$ vertex at 0-th order in its external momenta obviously has only the tree-level structure. For the $A\bar{c}c$ vertex at the linear order in external momenta, this statement is proven in Appendix \[S:Ghost\]. The $AAA$ vertex at the linear order in its external momenta can have, in addition to the tree-level structure, one more structure: $d^{a_1 a_2 a_3}(g^{\mu_1 \mu_2}k_3^{\mu_3}+\mbox{cycle})$; however, the Slavnov–Taylor identity ${\langle}T\{\partial^\mu A_\mu(x),\partial^\nu A_\nu(y),\partial^\lambda A_\lambda(z)\}{\rangle}=0$ leads to $\Gamma^{a_1 a_2 a_3}_{\mu_1 \mu_2 \mu_3} k_1^{\mu_1} k_2^{\mu_2} k_3^{\mu_3} = 0$ (see Ref. [@Pascual:1984zb]), thus excluding this second structure.
[^3]: Note that $\Gamma(\varepsilon)=1/\varepsilon + {\cal
O}(1)$.
[^4]: Note that the masses $m_c(\mu)$ and $m_b(\mu)$ (and $m_{c0}$, $m_{b0}$) are those in the full $n_f$-flavour QCD. They do not exist in the low-energy $n_l$-flavour QCD, and therefore we do not assign a superscript $n_f$ to these masses.
[^5]: It has been argumented in Refs. [@Kuhn:2007vp] that in the case of charm the scale $\mu=m_c$ is too small leading to a value of $\alpha_s$ which is too large. Thus $m_c(3~\mbox{GeV})$ has been proposed as reference value.
[^6]: For $\beta_4>0$ the dotted curve in Fig. \[fig::asMz\](b) moves towards the four-loop curve.
[^7]: Note that up to three-loop order there are only diagrams with at most two different quark flavours. Thus it is possible to obtain the result for $C_1$ for $N_h$ heavy quarks.
|
---
abstract: 'We study the interaction of electromagnetic (EM) radiation with single-layer graphene and a stack of parallel graphene sheets at arbitrary angles of incidence. It is found that the behavior is qualitatively different for transverse magnetic (or $p-$polarized) and transverse electric (or $s-$polarized) waves. In particular, the absorbance of single-layer graphene attains minimum (maximum) for $p$ ($s$) polarization, at the angle of total internal reflection when the light comes from a medium with a higher dielectric constant. In the case of equal dielectric constants of the media above and beneath graphene, for grazing incidence graphene is almost 100% transparent to $p-$polarized waves and acts as a tunable mirror for the $s-$polarization. These effects are enhanced for the stack of graphene sheets, so the system can work as a broad band polarizer. It is shown further that a periodic stack of graphene layers has the properties of an one-dimensional photonic crystal, with gaps (or stop–bands) at certain frequencies. When an incident EM wave is reflected from this photonic crystal, the tunability of the graphene conductivity renders the possibility of controlling the gaps, and the structure can operate as a tunable spectral–selective mirror.'
author:
- 'Yu. V. Bludov, N. M. R. Peres, M. I. Vasilevskiy'
bibliography:
- 'arbit\_angle.bib'
title: Unusual reflection of electromagnetic radiation from a stack of graphene layers at oblique incidence
---
Introduction
============
Electromagnetic (EM) metamaterial engineering yields specific optical properties which do not exist in natural materials[@Engheta2006]. These properties include EM energy concentration in sub-wavelength regions and radiation guiding[@Bozhevolnyi2013], enhanced absorption[@Kravets2008], reflection[@Joannopoulos2008] and transmission[@GarciadeAbajo2007], colour filtering[@TingXu2010], [*etc.*]{} An important and prominent example of metamaterials and their specific properties are photonic crystals (PCs)[^1], where the propagation of electromagnetic waves of certain frequencies, belonging to gaps (or stop-bands) in the spectrum, can be prohibited, or allowed in certain directions only[@Joannopoulos2008]. Thus, the so called three-cylinder structure[@Yablonovitch1991] was the first experimental realization of full photonic band gap, where the propagation of electromagnetic waves is not possible in any direction. The photonic band-gap structure of PC resembles and appears in full analogy with the electronic one in solid-state crystals.
In the metamaterials engineering it is useful to implement some tools for adjusting their EM properties, thus achieving the *tunability*. Tunable metamaterials allow for continuous variation of their properties through a certain external influence (for review see, e.g. Refs.[@Shalaev2011; @Vendik2012; @Zheludev2012]). Among the possible instruments to achieve the PCs dynamical tunability we can mention the optical beam intensity in a nonlinear material [@Chen2011], electric field in ferroelectrics[@Figotin1991], applied voltage in liquid crystals[@ozaki2002; @Li2007], magnetic field in ferromagnets or ferrimagnets [@Figotin1991; @PRB2000], and mechanical force for changing the PC period[@APL2004]. There are also possible tuning mechanisms in crystalline colloidal arrays of high refractive index particles [@advmat2012], magnetic fluids[@magFluid2012] or superconductors [@savelev2005; @savelev2006].
The two-dimensional carbon material graphene possesses a number of unique and extraordinary properties, such as high charge carrier mobility, electronic energy spectrum without a gap between the conduction and valence bands, and frequency-independent absorption of EM radiation. The optical properties of graphene have been extensively studied both theoretically [@nmrPRB06; @falkovsky; @stauberBZ; @stauberphonons; @StauberGeim; @carbotte; @Juan; @Mishchenko; @rmp; @rmpPeres; @LiLouie; @PRL; @aires; @APLPhotonic] and experimentally [@nair; @kuzmenko; @mak; @Crommieopt; @kuzmenko2; @kuzmenkoFaraday; @NatureLoh; @Li2008]. Since the carrier concentration in graphene (and, hence, its frequency–dependent conductivity) can be effectively tuned in wide limits by applying an external gate voltage [@Li2008], it is a perspective material for tunable photonic components. For example, in the area of plasmonics it is possible to make devices such as tunable graphene-based switch[@BluVasPer2010], polarizer[@Polarizer2012], and polaritonic crystal[@PolCrys2012]. Moreover, using two[@Hwang2009; @Svintsov2012; @Stauber2012; @PRIMER2013] or more[@Eberlein2008; @Jovanovic2011; @Hajian2013; @Metal2011; @Sreekanth2012; @Iorsh2012] parallel sheets of graphene can result in an unusual optical response of the structure owing to the interaction of charge carriers in the different layers by means of EM waves. Alternatively, for the same purposes it is possible to use an array of graphene ribbons[@Ju2011; @nikitin_ribbon2012; @Hipolito2012], two-[@Yan_disks2012; @Thongrattanasiri2012] or three-dimensional[@Berman2010; @Yan2012] arrays of graphene disks, or a two-dimensional array of antidots[@nikitin2012].
The aim of the present work is twofold. On the one hand, in the studies considering the transmittance of radiation through graphene[@LiLouie; @aires; @nair; @kuzmenkoFaraday] several authors analyzed only the case of normal incidence of the radiation on the graphene sheet. By restraining themselves to this particular case, these studies overlooked the unusual reflection and transmission properties taking place at oblique incidence. In this paper we discuss the transmission of EM radiation through a graphene sheet when the impinging beam makes an arbitrary angle, $\theta $, with the normal to the interface. We will show that, at grazing incidence (i.e. for $\theta $ close to $90^{o}$) and when the graphene layer is cladded by two identical dielectrics, it behaves like a mirror for $s$-polarized waves and is almost transparent for $p$-polarized waves. (In contrast, if the dielectric constants of the media below and above the graphene sheet are different, the interface reflects almost totally both $s$- and $p$-polarised waves as it is usual at grazing incidence). Furthermore, for $\theta $ close to the total internal reflection angle, a single sheet of monolayer graphene strongly absorbs $s$-polarized waves, while there is almost no $p$-polarized absorption in these conditions. On the other hand, we theoretically investigate the reflection of EM radiation, in the THz to far-infrared (FIR) range and for arbitrary $\theta$, from a periodic stack of parallel graphene sheets which constitute a semi-infinite one-dimensional (1D) photonic crystal. As it will be shown, this PC is highly reflective within certain frequency intervals corresponding to the gaps in its spectrum, and the widths of these gaps can be tuned by varying the gate voltage giving the possibility to create a tunable mirror. Moreover, when the angle of incidence for a $p$-polarized wave exceeds that of total internal reflection, it is possible to excite a surface EM mode supported by the semi-infinite photonic crystal.
Single-layer graphene\[sec:Single-layer-graphene\]
==================================================
Let us first consider a single flat sheet of monolayer graphene located at the plane $z=0$ (so, the $z$ axis is perpendicular to it) and cladded by two semi-infinite dielectrics, a substrate with a dielectric permittivity $\varepsilon_{1}>0$ and a capping medium with $\varepsilon_{2}>0$, occupying the half-spaces $z>0$ and $z<0$, respectively \[see Fig. \[fig:scheme\](a)\]. If the EM field is uniform along the $y$ direction ($\partial/\partial y\equiv0$), it can be decomposed into two separate waves with different polarizations. Thus, a $p$-polarized (or TM) wave with the magnetic field perpendicular to the plane of incidence ($xz$), possesses the electromagnetic field components $\vec{E}=\left\{ E_{x},0,E_{z}\right\} ,$ $\vec{H}=\left\{ 0,H_{y},0\right\} $, while an $s$-polarized (or TE) wave is described by electromagnetic field components $\vec{E}=\left\{ 0,E_{y},0\right\} ,$ $\vec{H}=\left\{ H_{x},0,H_{z}\right\} $, with the electric field perpendicular to the plane of incidence.
![Schematic representation of the systems considered in Sec. \[sec:Single-layer-graphene\]) (a) and \[sec:Graphene-multilayer-PC\] (b), showing $p-$ or $s-$polarized incident and reflected waves.[]{data-label="fig:scheme"}](graphene6.png){width="8.5cm"}
The temporal dependence of the fields is assumed of the form, $\vec{E}^{(j)},\vec{H}^{(j)}\sim\exp(-i\omega t)$, where $\omega$ is the angular frequency of the radiation and the superscripts $j=1,2$ correspond to the electromagnetic field in the substrate and the capping dielectric, respectively. Maxwell equations written explicitly for TE and TM waves in this particular situation can be found in several previous publications[@BluVasPer2010; @Polarizer2012; @PolCrys2012; @PRIMER2013]. We reckon that the EM wave falls on the interface $z=0$ from the capping dielectric side. In this case the solution of the Maxwell equations for a $p$-polarized wave can be written as follows: $$\begin{aligned}
H_{y}^{(2)}(x,z)=\left[H_{i}^{p}\exp(ik_{2,z}z)+H_{r}^{p}\exp(-ik_{2,z}z)\right]\times\nonumber \\
\exp(ik_{x}x),\label{eq:hrp}\qquad \\
H_{y}^{(1)}(x,z)=H_{t}^{p}\exp(ik_{x}x+ik_{1,z}z),\qquad \label{eq:htp}\\
E_{x}^{(2)}(x,z)=\frac{k_{2,z}}{\kappa\varepsilon_{2}}[H_{i}^{p}\exp(ik_{2,z}z)-H_{r}^{p}\exp(-ik_{2,z}z)]\times\nonumber \\
\exp(ik_{x}x),\qquad \label{eq:erp}\\
E_{x}^{(1)}(x,z)=\frac{k_{1,z}}{\kappa\varepsilon_{1}}H_{t}^{p}\exp(ik_{x}x+ik_{1,z}z),\qquad \label{eq:etp}\end{aligned}$$ where $$\begin{aligned}
k_{j,z}=\left(\kappa^{2}\varepsilon_{j}-k_{x}^{2}\right)^{1/2},\nonumber \\
k_{x}=\kappa\sqrt{\varepsilon_{2}}\sin\theta\label{eq:Kx}\:,\end{aligned}$$ $\kappa=\omega/c$, $c$ is the velocity of light in vacuum.
At the same time, for an $s$-polarized wave we have: $$\begin{aligned}
E_{y}^{(2)}(x,z)=\left[E_{i}^{s}\exp(ik_{2,z}z)+E_{r}^{s}\exp(-ik_{2,z}z)\right]\times\nonumber \\
\exp(ik_{x}x),\qquad \label{eq:ers}\\
E_{y}^{(1)}(x,z)=E_{t}^{s}\exp(ik_{x}x+ik_{1,z}z),\qquad \label{eq:ets}\\
H_{x}^{(2)}(x,z)=-\frac{k_{2,z}}{\kappa}[E_{i}^{s}\exp(ik_{2,z}z)-E_{r}^{s}\exp(-ik_{2,z}z)]\times\nonumber \\
\exp(ik_{x}x),\qquad \label{eq:hrs}\\
H_{x}^{(1)}(z)=-\frac{k_{1,z}}{\kappa}E_{t}^{s}\exp(ik_{x}x+ik_{1,z}z).\qquad \label{eq:hts}\end{aligned}$$ In Eqs. (\[eq:hrp\])–(\[eq:etp\]) and (\[eq:ers\])–(\[eq:hts\]), $H_{i}^{p} \left (E_{i}^{s}\right )$, $H_{r}^{p} \left (E_{r}^{s}\right )$ and $H_{t}^{p} \left (E_{t}^{s}\right )$ denote the amplitudes of the incident, reflected and transmitted $p$- ($s$)-polarized waves, respectively.
In order to find the transmittance and the reflectance of the structure we apply boundary conditions at $z=0$, which include the continuity of the tangential component of the electric field, $E_{x}^{(2)}(x,0)=E_{x}^{(1)}(x,0)$; $E_{y}^{(2)}(x,0)=E_{y}^{(1)}(x,0)$, and the discontinuity of the tangential component of the magnetic field caused by the induced surface currents in graphene, $H_{y}^{(1)}(x,0)-H_{y}^{(2)}(x,0)=-(4\pi/c)j_{x}=-(4\pi/c)\sigma_{g}E_{x}(x,0)$, $H_{x}^{(1)}(x,0)-H_{x}^{(2)}(x,0)=(4\pi/c)j_{y}=(4\pi/c)\sigma_{g}E_{y}(x,0)$, where $\sigma_{g}$ is the graphene conductivity. Matching the solutions for $z<0$ and $z>0$ using these boundary conditions, we obtain the amplitudes of the reflected and transmitted waves, $$\begin{aligned}
H_{r}^{p}=\frac{\varepsilon_{1}k_{2,z}-\varepsilon_{2}k_{1,z}+\frac{4\pi}{\omega}\sigma_{g}k_{2,z}k_{1,z}}{\varepsilon_{1}k_{2,z}+\varepsilon_{2}k_{1,z}+\frac{4\pi}{\omega}\sigma_{g}k_{2,z}k_{1,z}}H_{i}^{p},\label{eq:hr}\\
H_{t}^{p}=\frac{2\varepsilon_{1}k_{2,z}H_{i}^{p}}{\varepsilon_{1}k_{2,z}+\varepsilon_{2}k_{1,z}+\frac{4\pi}{\omega}\sigma_{g}k_{2,z}k_{1,z}},\label{eq:ht}\end{aligned}$$ for $p$-polarization and $$\begin{aligned}
E_{r}^{s}=-\frac{k_{1,z}-k_{2,z}+\frac{4\pi\omega}{c^{2}}\sigma_{g}}{k_{1,z}+k_{2,z}+\frac{4\pi\omega}{c^{2}}\sigma_{g}}E_{i}^{s}\label{eq:er}\\
E_{t}^{s}=\frac{2k_{2,z}E_{i}^{s}}{k_{1,z}+k_{2,z}+\frac{4\pi\omega}{c^{2}}\sigma_{g}}.\label{eq:et}\end{aligned}$$ for $s$-polarization. The transmittance (reflectance) is expressed as the ratio of the Poynting vector $z$-components of the transmitted (reflected) and the incident waves, $$\begin{aligned}
R_{p}=\left|\frac{H_{r}^{p}}{H_{i}^{p}}\right|^{2},\qquad T_{p}=\frac{k_{1,z}\varepsilon_{2}}{k_{2,z}\varepsilon_{1}}\left|\frac{H_{t}^{p}}{H_{i}^{p}}\right|^{2},\\
R_{s}=\left|\frac{E_{r}^{s}}{E_{i}^{s}}\right|^{2},\qquad T_{s}=\frac{k_{1,z}}{k_{2,z}}\left|\frac{E_{t}^{s}}{E_{i}^{s}}\right|^{2}.\end{aligned}$$
Since $R$ and $T$ are determined by the conductivity $\sigma_{g}$ [\[]{}see Eqs. (\[eq:hr\])–(\[eq:et\])[\]]{}, we briefly consider its frequency dependence. The frequency–dependent (optical) conductivity of graphene is a sum of two contributions: (i) a Drude term describing intra-band processes, and (ii) a term taking into account inter-band transitions. At zero temperature the optical conductivity has a simple analytical expression[@nmrPRB06; @falkovsky; @rmp; @rmpPeres; @StauberGeim]. The inter-band contribution has the form $\sigma_{I}=\sigma_{I}'+i\sigma_{I}''$, where $$\begin{aligned}
\sigma_{I}'=\sigma_{0}\left(1+\frac{1}{\pi}\arctan\frac{\hbar\omega-2E_{F}}{\hbar\Gamma}\right.\nonumber \\
-\left.\frac{1}{\pi}\arctan\frac{\hbar\omega+2E_{F}}{\hbar\Gamma}\right)\,,\end{aligned}$$ and $$\sigma_{I}''=-\sigma_{0}\frac{1}{2\pi}\ln\frac{(2E_{F}+\hbar\omega)^{2}+\hbar^{2}\Gamma^{2}}{(2E_{F}-\hbar\omega)^{2}+\hbar^{2}\Gamma^{2}}\,,$$ where $\sigma_{0}=\pi e^{2}/(2h)$ is the so called universal conductivity of graphene. The Drude conductivity term is $$\sigma_{D}=\sigma_{0}\frac{4E_{F}}{\pi}\frac{1}{\hbar\Gamma-i\hbar\omega}\,,\label{eq_sigma_xx_semiclass}$$ where $\Gamma$ is the inverse of the momentum relaxation time and $E_{F}>0$ is the Fermi level position with respect to the Dirac point. The total conductivity is $$\sigma_{g}=\sigma_{I}'+i\sigma_{I}''+\sigma_{D}\,.$$ We can write $\sigma_{g}=\sigma_{0}f(\omega)$, where $f(\omega)$ is a dimensionless function. In Fig. \[fig\_conductivity\] we depict the Drude and inter-band contributions to the total optical conductivity of graphene.
![Optical conductivity of uniform graphene: Drude (left) and inter-band (right) contributions. We assumed $E_{F}=0.23$ eV and $\Gamma=2.6$ meV. The solid (dashed) line stands for the real (imaginary) part of the conductivity.[]{data-label="fig_conductivity"}](Fig_optical_conductivity_2){width="7cm"}
It is evident that at low-frequencies (left panel in Fig. \[fig\_conductivity\]) the Drude term significantly exceeds the interband one (both real and imaginary parts), while in the high-frequency range (right panel in Fig. \[fig\_conductivity\]) the interband term dominates. Moreover, in the vicinity of the threshold frequency, $\omega=2E_{F}/\hbar$, the real part of the conductivity increases drastically and achieves the universal value, $\sigma_{0}$ (onset of interband transitions), while the imaginary part is minimal, negative and of the order of several universal conductivities in modulus. As a result, at low frequencies the presence of graphene at the interface between two dielectrics influences significantly the reflectance and the transmittance of the structure. This effect, owing to the high value of the Drude conductivity, is clearly seen in Figs. \[fig:transmittance\] and \[fig:absorbance\](a–c), where the low-frequency region is characterized by the lower transmittance [\[]{}see Figs. \[fig:transmittance\](a), \[fig:transmittance\](c), \[fig:transmittance\](e) and the respective insets for $\omega=0.01E_F$[\]]{}, higher reflectance [\[]{}Figs. \[fig:transmittance\](b), \[fig:transmittance\](d), \[fig:transmittance\](f) and the respective insets for $\omega=0.01E_F$[\]]{} and enhanced absorbance [\[]{}Figs. \[fig:absorbance\](a)–\[fig:absorbance\](c)[\]]{} for all parameters’ values. In fact, the transmittance and the reflectance are mainly determined by the real part of the conductivity, except when the imaginary part is large in modulus (at low frequencies and near $\hbar \omega=2E_{F}$).
![Transmittance $T_{p,s}$ (left column) and reflectance $R_{p,s}$ (right column) of graphene cladded by two semi–infinite dielectrics [*versus*]{} angle of incidence $\theta$ and frequency $\omega$. In all cases $\varepsilon_{1}=3.9$ (SiO$_2$) and $\Gamma=2.6\,$meV. Other parameters are: $\varepsilon_{2}=3.9$, $E_{F}=0.157\,$eV (upper row), $\varepsilon_{2}=1.0$, $E_{F}=0.1\,$eV (middle row), or $\varepsilon_{2}=11.9$ (Si), $E_{F}=0.25\,$eV (lower row). In each panel, the angular dependences for two fixed frequencies ($\hbar \omega=0.01E_F$ and $\hbar \omega=0.1E_F$) are depicted in the insets.[]{data-label="fig:transmittance"}](TR_single_2){width="8cm"}
At normal incidence ($\theta=0$), we have (for any polarization): $$\begin{aligned}
T_{p}=T_{s}=\sqrt{\varepsilon_{2}\varepsilon_{1}}\left|\frac{2}{\sqrt{\varepsilon_{1}}+\sqrt{\varepsilon_{2}}+\pi\alpha f(\omega)}\right|^{2},\label{eq:Tzero}\\
R_{p}=R_{s}=\left|\frac{\sqrt{\varepsilon_{1}}-\sqrt{\varepsilon_{2}}+\pi\alpha f(\omega)}{\sqrt{\varepsilon_{1}}+\sqrt{\varepsilon_{2}}+\pi\alpha f(\omega)}\right|^{2}\:,
\label{eq:Rzero}\end{aligned}$$ where $\alpha$ is the fine structure constant. For oblique incidence ($\theta\ne0$), the dependencies of the transmittance and the reflectance on $\omega $ are strongly affected by the relation between the dielectric permittivities of the substrate and the capping dielectric. Therefore, we considered all three possible situations: (i) $\varepsilon_{1}=\varepsilon_{2}$ [\[]{}Figs. \[fig:transmittance\](a), \[fig:transmittance\](b)[\]]{}, (ii) $\varepsilon_{1}>\varepsilon_{2}$ [\[]{}Figs. \[fig:transmittance\](c), \[fig:transmittance\](d)[\]]{}, and (iii) $\varepsilon_{1}<\varepsilon_{2}$ [\[]{}Figs. \[fig:transmittance\](e), \[fig:transmittance\](f)[\]]{}. In the “symmetric” case of $\varepsilon_{1}=\varepsilon_{2}=\varepsilon$, the transmittance and the reflectance can be expressed by simple formulae, $$\begin{aligned}
R_{p}=\left|\frac{\pi\alpha f(\omega)\cos\theta/\sqrt{\varepsilon }}{2+\pi\alpha f(\omega)\cos\theta/\sqrt{\varepsilon }}\right|^{2},\label{eq:rp}\\
T_{p}=\left|\frac{2}{2+\pi\alpha f(\omega)\cos\theta/\sqrt{\varepsilon }}\right|^{2},\\
R_{s}=\left|\frac{\pi\alpha f(\omega)/\sqrt{\varepsilon }}{2\cos\theta+\pi\alpha f(\omega)\cos\theta/\sqrt{\varepsilon }}\right|^{2},\\
T_{s}=\left|\frac{2\cos\theta}{2\cos\theta+\pi\alpha f(\omega)/\sqrt{\varepsilon }}\right|^{2}\!.\label{eq:ts}\end{aligned}$$ Note that the factor $4\pi\sigma_{0}/c=\pi\alpha$ multiplying the dimensionless function $f(\omega )$, which represents the frequency dependence of the graphene conductivity, is a small number (=0.023). Thus, unless the absolute value of $f(\omega )$ is large, the term related to graphene in Eqs. (\[eq:hr\])–(\[eq:et\]) and, accordingly, in the above expressions for $T$ and $R$ is small. Therefore, the reflectance and the transmittance of the structure are close to the values defined by usual Fresnel’s expressions, except for $\omega \rightarrow 0$ and $\omega \approx 2E_F/\hbar $. In particular, the reflectance is proportional to $(\pi\alpha )^2$.
It should also be noticed that in Eqs.(\[eq:Tzero\]), (\[eq:Rzero\]) and (\[eq:rp\])–(\[eq:ts\]) the effect of graphene is stronger for lower dielectric constants and is maximal for free standing graphene ($\varepsilon_{1,2}=1$). As $\theta$ increases [\[]{}see Figs. \[fig:transmittance\](a), \[fig:transmittance\](b) and the insets[\]]{}, the transmittance $T_{s}$ decreases and attains zero for $\theta=\pi/2$, while the reflectance $R_{s}$ increases and tends to unity at $\theta \rightarrow \pi/2$. In contrast, a $p$-polarized wave is “totally transmitted” at $\theta \rightarrow \pi/2$ ($R_{p}=0$, $T_{p} \rightarrow 1$). With the electric field perpendicular to the graphene sheet (TM wave), no charge oscillations are induced at the interface and the EM field is not perturbed. Also, the low-frequency absorbance at grazing incidence is a decreasing function of the angle for both $p$- and $s$-polarized waves and the limit $\theta=\pi/2$ corresponds to zero absorbance [\[]{}see Fig. \[fig:absorbance\](a)[\]]{}. Note that the absorption is entirely related to graphene because the dielectrics are assumed dispersionless.
![(a-c) Absorbance, $A_{p,s}$, at frequency $\omega=0.01E_{F}$ for $p-$ and $s-$polarizations as function of the angle of incidence. The parameters of panels (a), (b), and (c) are the same as the same as for the upper, middle, and lower rows in Fig. \[fig:transmittance\], respectively. (d) Frequency dependence of the *quasi*-Brewster angle, $\theta_{br}$, for $\varepsilon_{2}=1$, $\varepsilon_{1}=3.9$ and three different values of $E_F$ as marked on the plot. The dashed horizontal line depicts the conventional Brewster angle $\theta_{br}^0$ of the interface without graphene. The inset shows the low-frequency region magnified.[]{data-label="fig:absorbance"}](A_single_1){width="8cm"}
The situation is different when the dielectric constants of the substrate and capping dielectric are not equal because in this case there would be reflection even in the absence of graphene. Here, at grazing incidence the reflectance (transmittance) is close to unity(zero) for *both* polarizations [\[]{}see Figs. \[fig:transmittance\](c)–\[fig:transmittance\](f)[\]]{}, just like for a “normal” interface between two dielectrics (without graphene). In the case (ii), although the angular dependence of the low-frequency absorbance [\[]{}Fig. \[fig:absorbance\](b)[\]]{} is qualitatively similar to the “symmetric” case (i), it is higher for TM waves [\[]{}cf. Fig. \[fig:absorbance\](b) and Fig. \[fig:absorbance\](a)[\]]{}, in contrast with the case (i). The particularity of the case (ii) ($\varepsilon_{1}>\varepsilon_{2}$, e.g. uncovered graphene on a substrate) is that the angular dependence of the TM-wave reflectance [\[]{}see Fig. \[fig:transmittance\](d)[\]]{} possesses a minimum at a certain $\theta$ close to the Brewster angle for the two dielectrics considered, $\theta_{br}^0=\mbox {arctg}[(\varepsilon_{1}/\varepsilon_{2})^{1/2}]$[@Born-Wolf]. Owing to the non-zero imaginary part of the graphene conductivity, the TM wave reflectance at this angle is finite, while it would reach zero in the case without graphene. Therefore we call this $\theta$ the “*quasi*-Brewster” angle, $\theta_{br}$. It depends upon the frequency and the Fermi energy of graphene; this dependence is depicted in Fig. \[fig:absorbance\](d). At low frequencies, where the conductivity $\sigma_{g}$ is high owing to the Drude term, the quasi-Brewster angle of the structure exceeds the conventional Brewster angle $\theta_{br}^0$ of the interface without graphene. This effect is more pronounced for higher values of the Fermi energy – compare the three curves in the plot. When the frequency grows (and the Drude term in $\sigma_{g}$ becomes smaller) $\theta_{br}$ approaches the value of the conventional Brewster angle $\theta_{br}^0$. However, at $\omega\sim 2E_{F}$ the quasi-Brewster angle jumps up because of the Fermi step in the real part of $\sigma_{g}$ (onset of interband transitions). The difference between $\theta_{br}$ and the conventional Brewster angle, $\theta_{br}^0$, characteristic of graphene–free interface between the same two dielectrics, can be used for visualization of graphene, which constitutes a considerable problem [@Schellenberg]. At low frequencies, $\theta_{br}-\theta_{br}^0$ can reach easily detectable values of $\sim 5^o$.
In the last possible situation (iii), there is a critical angle of total internal reflection \[$\approx35^{o}$ for the parameters of Figs. \[fig:transmittance\](e) and \[fig:transmittance\](f)\]. Above this critical angle, $\theta _c$, the transmittance vanishes and the reflectance is close to unity [\[]{}see Figs. \[fig:transmittance\](e) and \[fig:transmittance\](f)[\]]{}. Notice that the value of $\theta _c$ does not depend upon the graphene parameters. In the case under consideration, the low-frequency absorbance [\[]{}Fig. \[fig:absorbance\](c)[\]]{} exhibits the most interesting behavior: in the vicinity of the total internal reflection angle the absorbance of the $p-$polarized wave is almost zero, while that of the $s-$polarized wave reaches its maximum of $\approx75\%$.
Graphene multilayer photonic crystal\[sec:Graphene-multilayer-PC\]
==================================================================
Now we shall consider an external EM wave falling on the periodic multilayer structure \[see Fig. \[fig:scheme\](b)\] consisting of an infinite number of parallel monolayer graphene sheets separated by dielectric slabs of thickness $d$; in practical terms few graphene layers play the same role as an infinite number of them [@EPL2013]. The geometry of the problem is similar to considered in Sec.II, but with graphene layers (for which we shall assume the same Fermi energy) located at positions $z=md$, $m\in\left[0,\infty\right)$. Thus, the considered structure is a semi-infinite 1D PC, terminated by the graphene layer at $z=0$. We shall still consider a capping dielectric (with arbitrary real $\varepsilon_{2}$) on top of it.
In order to find the reflectance of this PC, we notice, that the structure of the electromagnetic field in the capping dielectric is the same, as represented by Eqs. (\[eq:hrp\]),(\[eq:erp\]) and (\[eq:ers\]),(\[eq:hrs\]) for $p$- and $s$-polarized waves, respectively. At the same time, the fields in the substrate should be considered separately in each layer between adjacent graphene sheets at planes $z=md$ and $z=(m+1)d$. Namely, solutions of the Maxwell equations at spatial domain $md\leq z\leq(m+1)d$ can be represented as $$\begin{aligned}
H_{y}^{(1)}(x,z)=\left\{ H_{+}^{(m)}\exp\left[ik_{1,z}\left(z-md\right)\right]+\right.\label{eq:htp-sl}\\
\left.H_{-}^{(m)}\exp\left[-ik_{1,z}\left(z-md\right)\right]\right\} \exp(ik_{x}x),\nonumber \\
E_{x}^{(1)}(x,z)=\frac{k_{1,z}}{\kappa\varepsilon_{1}}\left\{ H_{+}^{(m)}\exp\left[ik_{1,z}\left(z-md\right)\right]-\right.\label{eq:etp-sl}\\
\left.H_{-}^{(m)}\exp\left[-ik_{1,z}\left(z-md\right)\right]\right\} \exp(ik_{x}x)\nonumber\end{aligned}$$ and $$\begin{aligned}
E_{y}^{(1)}(x,z)=\left\{ E_{+}^{(m)}\exp\left[ik_{1,z}\left(z-md\right)\right]+\right.\label{eq:ets-sl}\\
\left.E_{-}^{(m)}\exp\left[-ik_{1,z}\left(z-md\right)\right]\right\} \exp(ik_{x}x),\nonumber \\
H_{x}^{(1)}(x,z)=-\frac{k_{1,z}}{\kappa}\left\{ E_{+}^{(m)}\exp\left[ik_{1,z}\left(z-md\right)\right]-\right.\label{eq:hts-sl}\\
\left.E_{-}^{(m)}\exp\left[-ik_{1,z}\left(z-md\right)\right]\right\} \exp(ik_{x}x).\nonumber\end{aligned}$$ Here $H_{\pm}^{(m)}$ are the amplitudes for forward (sign ”+”) or backward (sign –) propagating TM waves. Correspondingly, $E_{\pm}^{(m)}$ represent the amplitudes of the TE waves. The amplitudes $H_{\pm}^{(m+1)}$ can be related to $H_{\pm}^{(m)}$ by matching boundary conditions at $z=(m+1)d$ on graphene (similar to that, used in Sec. \[sec:Single-layer-graphene\]), namely: $$\begin{aligned}
\left(\begin{array}{c}
H_{+}^{(m+1)}\\
H_{-}^{(m+1)}
\end{array}\right)=\hat{M}_{p}\left(\begin{array}{c}
H_{+}^{(m)}\\
H_{-}^{(m)}
\end{array}\right),\label{eq:h-mp}\\
\hat{M}_{p}=\left(\begin{array}{cc}
e^{ik_{1,z}d}\left[1-\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}\right] & e^{-ik_{1,z}d}\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}\\
-e^{ik_{1,z}d}\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g} & e^{-ik_{1,z}d}\left[1+\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}\right]
\end{array}\right).\nonumber\end{aligned}$$ Similarly, for $s-$polarization, $$\begin{aligned}
\left(\begin{array}{c}
E_{+}^{(m+1)}\\
E_{-}^{(m+1)}
\end{array}\right)=\hat{M}_{s}\left(\begin{array}{c}
E_{+}^{(m)}\\
E_{-}^{(m)}
\end{array}\right),\label{eq:e-ms}\\
\hat{M}_{s}=\left(\begin{array}{cc}
e^{ik_{1,z}d}\left[1-\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g}\right] & -e^{-ik_{1,z}d}\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g}\\
e^{ik_{1,z}d}\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g} & e^{-ik_{1,z}d}\left[1+\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g}\right]
\end{array}\right).\nonumber\end{aligned}$$ Since the considered structure is periodic [^2], it is possible to use the Bloch theorem, which determines the proportionality between the field amplitudes in the adjacent periods through the Bloch wavevector $q$: $$\begin{aligned}
H_{\pm}^{(m+1)} & = & \exp\left(iqd\right)H_{\pm}^{(m)},\\
E_{\pm}^{(m+1)} & = & \exp\left(iqd\right)E_{\pm}^{(m)}.\end{aligned}$$ After substitution of these relations into Eqs.(\[eq:h-mp\]),(\[eq:e-ms\]), the compatibility condition of the resulting linear equations requires: $$\begin{aligned}
\mathrm{Det}\left|\hat{M}_{p}-\exp\left(iqd\right)\hat{I}\right|=0,\label{eq:disp-p-mat}\\
\mathrm{Det}\left|\hat{M}_{s}-\exp\left(iqd\right)\hat{I}\right|=0,\label{eq:disp-s-mat}\end{aligned}$$ where $\hat{I}$ is the unity matrix. Equations (\[eq:disp-p-mat\], \[eq:disp-s-mat\]) yield the dispersion relations for $p-$ and $s-$polarized EM waves in the graphene multilayer PC: $$\cos\left(qd\right)-\cos\left(k_{1,z}d\right)+i\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}\sin\left(k_{1,z}d\right)=0\:,
\label{eq:disp-p}$$ and $$\cos\left(qd\right)-\cos\left(k_{1,z}d\right)+i\frac{2\pi\omega}{c^{2}k_{1,z}}\sigma_{g}\sin\left(k_{1,z}d\right)=0\:.
\label{eq:disp-s}$$ We note that similar expressions have been obtained in Ref.[@Hajian2013]
![(a) TM wave eigenfrequencies *versus* $x-$component of the wavevector, $k_{x}$, and Bloch wavevector, $q$; (b) Eigenfrequencies $\omega$ *vs* $q$ for fixed $k_{x}=0.1\,\mu\mathrm{m}^{-1}$; (c) Eigenfrequencies $\omega$ *vs* $k_{x}$, dashed zones correspond to the allowed bands with the boundaries determined by $q=0$ and $q=\pi/d$. Other parameters are: $d=40\,\mu$m, $E_{F}=0.157\,$eV, $\varepsilon_{1}=3.9$, $\Gamma=0$. The numbers designate different allowed bands, 1 for surface mode, 2, 3, [*etc*]{} for bulk modes. \[fig:sl-p\] ](sl_p_3){width="8.5cm"}
![The same as in Fig. \[fig:sl-p\], but for TE waves. There is no allowed surface mode in this case. \[fig:sl-s\]](sl_s_3){width="8.5cm"}
Before considering the dispersion properties of $p-$ and $s-$polarized waves in detail, it should be noticed, that dispersion curves depicted in Figs. \[fig:sl-p\] and \[fig:sl-s\] have been calculated for zero damping, $\Gamma=0$, when the graphene conductivity possesses only the imaginary part. As a result, the eigenfrequencies, the in-plane component of the wavevector $k_{x}$, and the Bloch wavevector $q$ are real values. In the case of nonzero $\Gamma$ the eigenfrequencies will be complex values with imaginary part characterizing the mode damping. The calculated spectra exhibit the band structure of a photonic crystal for both $p-$ (Fig. \[fig:sl-p\]) and $s-$polarized waves (Fig. \[fig:sl-s\]). In particular, there are gaps in the spectra [\[]{}see Figs. \[fig:sl-p\](a) and \[fig:sl-s\](a)[\]]{} that appear in the center ($q=0$) and in the edges ($q=\pm \pi/d$) of the first Brillouin zone, and the widths of gaps decrease with the increase of $k_{x}$ [\[]{}see Figs. \[fig:sl-p\](c) and \[fig:sl-s\](c)[\]]{}. However, the main feature of the $p-$polarization spectrum is the presence of *surface* mode with purely imaginary $k_{1,z}$ \[marked by 1 (red color) in Figs. \[fig:sl-p\](b) and \[fig:sl-p\](c)\] together with *bulk* modes with purely real $k_{1,z}$ [\[]{}marked by 2 (green), 3 (blue), 4 (orange) and 5 (pink) colors in Figs. \[fig:sl-p\](b) and \[fig:sl-p\](c)[\]]{}. The former is a “Bloch surface plasmon-polariton”, with the electric and magnetic fields strongly localized at the graphene sheets [@PRIMER2013] but with a *real* Bloch wavevector. In the case of $s$ polarization such a surface mode does not exist and only bulk modes are present in the spectrum [\[]{}see Figs. \[fig:sl-s\](b) and \[fig:sl-s\](c)[\]]{}.
![Eigenfrequencies for $p-$ (a) and $s-$polarized (b) waves *vs* graphene Fermi energy for a fixed $k_{x}$($=0.05\,\mu\mathrm{m}^{-1}$). Other parameters are: $\varepsilon_{1}=3.9$, $\Gamma=0$, $d=40\,\mu$m. As in Figs. \[fig:sl-p\] and \[fig:sl-s\], dashed zones correspond to allowed bands. \[fig:var-gap-eigen\]](var_gap_eigen_2){width="8.5cm"}
Particular solutions of the dispersion relation for $p$ polarization, Eq. (\[eq:disp-p\]), for $q=\pi/d$ [\[]{}so called “Bragg modes”[\]]{} can be represented as $$k_{1,z}=\left(\frac{\omega}{c^{2}}^{2}\varepsilon_{1}-k_{x}^{2}\right)^{1/2}=\left(2n+1\right)\pi/d,\qquad n\in\left[0,\infty\right).\label{eq:w-g-pi}$$ For $q=0$ we have: $$k_{1,z}=\left(\frac{\omega}{c^{2}}^{2}\varepsilon_{1}-k_{x}^{2}\right)^{1/2}=2n^{\prime}\pi/d,\qquad n^{\prime}\in\left[0,\infty\right).\label{eq:w-g-0}$$ $S-$polarized Bragg modes [\[]{}solutions of (\[eq:disp-s\])[\]]{} for $q=\pi/d$ are exactly the same as (\[eq:w-g-pi\]), but for $q=0$ they are similar to (\[eq:w-g-0\]) (except that $n^{\prime}\neq 0$). The modes with Bragg wavevectors, (\[eq:w-g-pi\]) and (\[eq:w-g-0\]), have nodes at graphene layers and, therefore, these solutions do not involve the graphene conductivity, $\sigma_g$. As a matter of fact, these solutions correspond to $H_{+}^{(m)}=H_{-}^{(m)}$ for $p$ polarization and $E_{+}^{(m)}=-E_{-}^{(m)}$ for $s$ polarization. It implies zero in-plane components of the electric field in both cases, consequently, no electric current is induced in graphene sheets located at $z=md$ [\[]{}see Eqs. (\[eq:etp-sl\]) and (\[eq:ets-sl\])[\]]{}.
Secondly, in the case of $p$ polarization $k_{1,z}=0$ is the solution that implies arbitrary $H_{+}^{(m)}$ and $H_{-}^{(m)}$, and, as a result, $H_{y}^{(1)}$ independent upon $z$ as well as $E_{x}^{(1)}\equiv0$. At the same time, for $s$ polarization the solution $k_{1,z}=0$ corresponds to a trivial solution of the Maxwell equations with zero electric and magnetic fields. For $p-$polarization, the line $k_{1,z}=0$ is crossed by another dispersion curve at the point $k_{x}=\sqrt {4\alpha E_{F}/(\hbar cd)}$ [\[]{}see Fig. \[fig:sl-p\](c)[\]]{}, where there is no gap between the surface and bulk mode bands. Below this point, the solution $k_{1,z}=0$ corresponds to the top of the surface mode band, while above this $k_{x}$ it corresponds to the bottom of the bulk mode band. Similarly, the upper bands depicted in Figs. \[fig:sl-p\](c) and Fig. \[fig:sl-s\](c) are delimited by the Bragg modes, (\[eq:w-g-pi\]) and (\[eq:w-g-0\]).
![$P-$polarization reflectance, $R$, for a semi-infinite graphene multilayer PC, plotted against the frequency, $\omega$, and the angle of incidence, $\theta$ (top row), or *vs* frequency $\omega$ (lower row) at fixed angles of incidence: $\theta=30^{o}$ [\[]{}dashed line in panel (d)[\]]{}, $\theta=30^{o}$ [\[]{}dashed line in panel (e)[\]]{}, $\theta=69.324^{o}$ [\[]{}dash-dotted line in panel (e)[\]]{}, $\theta=10^{o}$ [\[]{}dashed line in panel (f)[\]]{}, $\theta=40^{o}$ [\[]{}dash-dotted line in panel (f)[\]]{}. Other parameters are: $\varepsilon_{1}=3.9$, $\Gamma=2.6\,$meV, $\varepsilon_{2}=3.9$, $E_{F}=0.157\,$eV, $d=40\,\mu\mathrm{m}$ (left column), $\varepsilon_{2}=1.0$, $E_{F}=0.1\,$eV, $d=60\,\mu\mathrm{m}$ (middle column), or $\varepsilon_{2}=11.9$, $E_{F}=0.25\,$eV, $d=4\,\mu\mathrm{m}$ (right column). Notice that the plots (d), (e), and (f) represent the variations along the corresponding vertical lines in panels (a), (b), and (c), respectively. \[fig:reflec-p\]](reflec_p_1){width="8.5cm"}
At the same time, changing the graphene Fermi energy, $E_{F}$, e.g. by varying an external gate voltage, it is possible to tune the width of the gaps, as it can be seen from Figs. \[fig:var-gap-eigen\](a) and \[fig:var-gap-eigen\](b) (for $p$ and $s$ polarizations, respectively). In particular, the gaps vanish when the Fermi level coincides with the Dirac point. At the same time, the waveguide modes defined by Eqs. (\[eq:w-g-pi\]) and (\[eq:w-g-0\]) remain unchanged because of their above-mentioned independence upon the graphene conductivity.
In order to obtain the expression for the reflectance of an EM wave from the graphene multilayer stack, we notice that, by virtue of Eqs. (\[eq:disp-p-mat\]) and (\[eq:disp-s\]), the amplitudes $H_{\pm}^{(m)}$ and $E_{\pm}^{(m)}$ are related by: $$H_{-}^{(m)}=\rho_{p}H_{+}^{(m)},\qquad E_{-}^{(m)}=-\rho_{s}E_{+}^{(m)}\:,$$ where $$\begin{aligned}
\rho_{p}=\frac{\exp\left(ik_{1,z}d\right)\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}}{\exp\left(-ik_{1,z}d\right)\left[1+\frac{2\pi k_{1,z}}{\omega\varepsilon_{1}}\sigma_{g}\right]-\exp\left(iqd\right)},\label{eq:rho-p}\\
\rho_{s}=\frac{\exp\left(ik_{1,z}d\right)\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g}}{\exp\left(-ik_{1,z}d\right)\left[1+\frac{2\pi\omega}{c{}^{2}k_{1,z}}\sigma_{g}\right]-\exp\left(iqd\right)},\label{eq:rho-s}\end{aligned}$$ and Bloch wavevector $q$ for $\rho_{p}$ and $\rho_{s}$ is obtained from Eqs.(\[eq:disp-p\]) and (\[eq:disp-s\]), respectively. It should be pointed out that, if the graphene conductivity is complex, so is the Bloch wavevector in Eqs. (\[eq:rho-p\]) and (\[eq:rho-s\]). Then, applying the above-mentioned boundary conditions at $z=0$, one can obtain expressions for the amplitude of the reflected wave in the form: $$\begin{aligned}
H_{r}^{p}=\frac{\varepsilon_{1}k_{2,z}\frac{1+\rho_{p}}{1-\rho_{p}}-\varepsilon_{2}k_{1,z}+\frac{4\pi}{\omega}\sigma_{g}k_{2,z}k_{1,z}}{\varepsilon_{1}k_{2,z}\frac{1+\rho_{p}}{1-\rho_{p}}+\varepsilon_{2}k_{1,z}+\frac{4\pi}{\omega}\sigma_{g}k_{2,z}k_{1,z}}H_{i}^{p},\label{eq:hr-sl}\\
E_{r}^{s}=-\frac{k_{1,z}\frac{1+\rho_{s}}{1-\rho_{s}}-k_{2,z}+\frac{4\pi\omega}{c^{2}}\sigma_{g}}{k_{1,z}\frac{1+\rho_{s}}{1-\rho_{s}}+k_{2,z}+\frac{4\pi\omega}{c^{2}}\sigma_{g}}E_{i}^{s}.\label{eq:er-sl}\end{aligned}$$ Notice that, when $\rho_{p}=0$, Eq. (\[eq:hr-sl\]) coincides with Eq. (\[eq:hr\]) for the single graphene layer structure. Similarly, when $\rho_{s}=0$, Eq. (\[eq:er-sl\]) turns into Eq. (\[eq:er\]).
![Same as in Fig. \[fig:reflec-p\], but for $s-$polarized wave. \[fig:reflec-s\]](reflec_s_1){width="8.5cm"}
An incident wave with $\omega$ and $k_{x}$ inside one of the allowed bands of the photonic crystal is (partially) transmitted into the structure. This effect is clearly seen in Figs. \[fig:reflec-p\] and \[fig:reflec-s\] for $p-$ and $s-$polarized waves, respectively). Thus, when $\omega$ and $k_{x}$ of the incident wave match one of the bands, the reflectance of the graphene multilayer photonic crystal resembles that of the single-layer graphene [\[]{}compare Figs. \[fig:reflec-p\](a), \[fig:reflec-s\](a) with \[fig:transmittance\](b), as well as Figs. \[fig:reflec-p\](b), \[fig:reflec-s\](b) with \[fig:transmittance\](d) and Figs. \[fig:reflec-p\](c), \[fig:reflec-s\](c) with \[fig:transmittance\](f)[\]]{}.
On the contrary, incident EM waves with $\omega$ and $k_x$ belonging to the gaps of the PC band structure induce evanescent waves (characterized by imaginary Bloch wavevector $q$, in contrast with the PC surface mode with real $q$ and imaginary $k_{1,z}$), and are nearly totally reflected from it. The graphene-multilayer PC reflectance is considerably higher than that of single-layer graphene heterostructure, and at certain frequencies can achieve unity [\[]{}see panels (d),(e) and (f) in Figs. \[fig:reflec-p\] and \[fig:reflec-s\][\]]{}.
Perhaps the most interesting effects take place when $\varepsilon_{2}<\varepsilon_{1}$ [\[]{}panels (b) and (e) in Figs. \[fig:reflec-p\] and \[fig:reflec-s\][\]]{}. As expected, in the vicinity of the Brewster angle of the interface without graphene ($\theta _{br}^0 \approx 63.124^{o}$), the $s-$polarization reflectance exceeds significantly that of $p-$ polarized waves for all frequencies inside the band [\[]{}compare dash-dotted lines in Figs. \[fig:reflec-p\](e) and \[fig:reflec-s\](e)[\]]{}, similar to the case of single graphene layer. However, it is not so for $\omega$ and $k_x$ inside the gaps. Here both polarizations exhibit an enhanced reflectance. Furthermore, we find some features specific for TM waves. As it has been shown in the previous section, the presence of graphene at the interface modifies the angle at which the reflectivity minimum in $p-$polarization occurs and this quasi-Brewster angle ($\theta _{br}$) is frequency-dependent [\[]{}see Fig. \[fig:absorbance\](d)[\]]{}. What happens to the minimum reflectivity angle, $\theta_{min}$, when the wave is reflected from the graphene multilayer PC instead of the single interface? The answer follows from Fig. \[fig:min\_angle\_sl\]. When $\omega$ and $k_{x}$ belong to a band of allowed modes, $\theta_{min}$ oscillates around the conventional Brewster angle ($\theta _{br}^0$, dashed horizontal line in the plot), except for the very low-frequency range ($\hbar \omega <3$ meV), where the frequency dependence of the difference, $\theta _{min} -\theta _{br}^0$, resembles that for the single graphene layer structure [\[]{}compare to Fig. \[fig:absorbance\](d)[\]]{}. The most striking feature in Fig. \[fig:min\_angle\_sl\] is the divergence of $\theta_{min}$ for the frequencies corresponding to the stop–bands of the photonic crystal (compare to the middle column of Fig. \[fig:reflec-p\]).
The particularity of the situation $\varepsilon_{2}>\varepsilon_{1}$ [\[]{}see panels (c) and (f) in Figs. \[fig:reflec-p\] and \[fig:reflec-s\][\]]{} is the possibility to excite the $p-$polarized surface mode. If the angle of incidence is below the critical one ($\theta _c \approx 35^{o}$), the excitation of bulk PC modes takes place, while for $\theta > \theta _c$, only the surface mode can be excited, as it can be seen by the low-frequency minima in the reflectivity spectra [\[]{}see Figs. \[fig:reflec-p\](c) and \[fig:reflec-p\](f)[\]]{}. A similar spectral shape has been observed experimentally in Ref. [@Sreekanth2012]. One can say that the interface between the PC and the capping dielectric acts as an attenuated total internal reflection structure for single graphene layer, described in Ref. [@BluVasPer2010]. It should be emphasized that the origin of the low-frequency minimum observed for $s-$polarized waves [\[]{}Figs. \[fig:reflec-s\](c) and \[fig:reflec-s\](f)[\]]{} is completely different. The former is a photonic crystal effect, while the latter exists also in the single-layer case [\[]{}see Fig. \[fig:transmittance\](f)[\]]{} and is unrelated to any PC surface mode.
![Frequency dependence of the angle of incidence corresponding to the minimal reflectance of $p-$polarized waves, $\theta_{min}$, for two values of the Fermi level, $E_{F}=0.1\,$eV (solid lines) and $E_{F}=0.2\,$eV (dashed lines). Other parameters are the same as for the middle column of Fig. \[fig:reflec-p\] ($\varepsilon_{2}<\varepsilon_{1}$). The dashed horizontal line depicts the conventional Brewster angle. \[fig:min\_angle\_sl\]](min_angle_sl_1){width="7.5cm"}
![(a,b) Reflectance *versus* $E_{F}$ and frequency for $p-$ (a) and $s-$polarized (b) waves, for the angle if incidence $\theta=30^{o}$. (c) Reflectance *versus* Fermi level for $p-$ and $s-$polarized waves, for $\hbar \omega=9.5\,$meV \[subtracted from panels (a) and (b) along respective horizontal lines\]. Other parameters are the same as for the left column of Fig. \[fig:reflec-p\]. \[fig:var-gap-sl\]](var_gap_sl_2){width="8cm"}
The possibility to change gap widths in the graphene multilayer PC spectrum by changing the Fermi level of graphene layers [\[]{}see Fig.\[fig:var-gap-eigen\][\]]{} has an important consequence, the reflectance of the PC can be dynamically varied through the electrostatic gating, by changing the voltage applied to the graphene layers. This effect is depicted in Figs. \[fig:var-gap-sl\](a) and \[fig:var-gap-sl\](b) for $p-$ and $s-$polarized waves, respectively. It can be used to design a tunable mirror. One has to choose the frequency of the incident wave inside one of the allowed bands, for a low Fermi energy, and inside the gap for a large $E_F$. Then the reflectance of the structure can be varied in a broad range, as shown in Fig. \[fig:var-gap-sl\](c). The dependence $R\left(E_{F}\right)$ can be made even more abrupt using graphene layers with a smaller damping parameter $\Gamma$.
![Reflectance *vs* frequency for $p-$ (a) and $s-$polarized (b) waves falling on a finite PC containing 5 (thick blue lines) or 20 (thin red lines) graphene layers. Other parameters are: $\varepsilon_{1}=3.9$, $\Gamma=2.6\,$meV, $\varepsilon_{2}=3.9$, $E_{F}=0.157\,$eV, $d=40\,\mu\mathrm{m}$, $\theta=30^{o}$ \[panel (a)\], or $\varepsilon_{2}=11.9$, $E_{F}=0.25\,$eV, $d=4\,\mu\mathrm{m}$, $\theta=40^{o}$ \[panel (b)\]. In both panels dashed lines correspond to the case of infinite number of graphene layers in PC for the same parameters. \[fig:reflec\_fin\_lay\]](reflec_fin_lay_2){width="8cm"}
All the above results have been obtained for an *infinite* periodic stack of graphene layers. In reality, of course, PCs consist of a *finite* number ($N$) of layers. How does the value of $N$ affect the mode eigenfrequencies and the frequency dependence of the reflectance? As known from the band theory of crystalline solids, the eigenmode spectrum is quantized and corresponds to a discrete set of “allowed” Bloch wavevectors, $q_m=(\pi/d) (m/(N+1));\:m=[1,N]$ obtained from the usual Born–von Karman boundary conditions. For $N\rightarrow \infty $, the Bloch wavevector varies in a quasicontinuous way within the interval $q\in[0,\pi/d]$ and Eqs. (\[eq:disp-p\]) and (\[eq:disp-s\]) hold with a very high precision. For relatively small values of $N$, say, $N\sim 10$, the eigenmode band structure is washed away although the density of states retains a qualitative similarity with the case of $N\rightarrow \infty $. The “stop-bands” are broadened and correspond to the maximum reflectivity well below the unity \[see Fig. \[fig:reflec\_fin\_lay\](b)\], however, the latter increases rapidly with the number of layers, as known for periodically stratified media [@Born-Wolf]. Already for $N=20$, the reflectance for $s-$polarized waves is very similar to that for infinite PC \[compare dashed and thin solid lines in Fig. \[fig:reflec\_fin\_lay\](b)\].
Fig. \[fig:reflec\_fin\_lay\](a) shows the finite size effect on the frequency dependence of reflectance related to the surface mode for $p$ polarization. No qualitative difference between the cases of $N=5$ and $N\rightarrow \infty $ is seen \[compare thick solid and dashed lines in Fig. \[fig:reflec\_fin\_lay\](a)\], which can be understood by the low dispersion of the surface-type PC mode with respect to the Bloch wavevector \[see Fig. \[fig:sl-p\](b)\]. This mode is, in fact, a Bloch-type surface plasmon-polariton (SPP) excitation induced by the incident wave when the attenuated total reflection conditions are met [@PRIMER2013]. The flatness of the $\omega (q)$ dependence for this PC surface mode originates from the small overlap of the amplitudes of the SPP excitations in the different graphene layers.
Conclusions
===========
In conclusion, there are several interesting effects related to the optical properties of graphene, which are revealed at oblique incidence. Some of them are expected already for a single graphene layer or just few of them. Under total internal reflection conditions at an interface between two dielectrics, the presence of graphene leads to EM energy absorption only for $s-$polarized waves. The absorbance attains its maximum exactly at the critical angle of incidence for $s$ polarization (and the maximum value is higher when the graphene conductivity is large), while it vanishes for $p$ polarization \[Fig. \[fig:absorbance\](c)\]. The minimum reflectance of $p-$polarized waves occurs at a (frequency dependent) quasi-Brewster angle that can differ by several degrees from the conventional Brewster angle for the same pair of dielectrics. Close to grazing incidence, graphene (when dielectric constants of substrate and capping layer are equal) is fully transparent to $p-$polarized waves and behaves like a mirror for $s$ polarization. This effect can be used for polarization-selective guidance of EM radiation. We have shown that a periodic stack of equally spaced parallel layers of graphene has the properties of a 1D photonic crystal, with narrow stop–bands that are nearly periodic in frequency. The PC properties are revealed also at oblique incidence. In particular, the stop–bands correspond to singularities of the minimum $p-$polarized reflection angle calculated as the function of frequency, which is an effect of potential interest for optical switching. We investigated the finite PC size effect and found that about 20 periods are sufficient to get the properties very close to those of the infinite PC. Finally, we should stress the possibility of tuning of the gaps’ (stop–bands’) width by changing the graphene conductivity via electrostatic gating, that would allow for dynamical variation of the reflectance at specific selected frequencies.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by FEDER through the COMPTETE Program and by the Portuguese Foundation for Science and Technology (FCT) through Strategic Project PEst-C/FIS/UI0607/2011.
[^1]: Although photonic crystals are different from the “classical” metamaterials (devices with negative refractive index) since the wavelength of operation is comparable to their structural period, the term “metamaterial” nowadays is applied to any artificial structure with designed material (in our case, optical) properties.
[^2]: The surface at $z=0$, of course, breaks the translational symmetry. This symmetry breaking can give rise to local (surface) modes, known as Tamm states in the case of electronic structure of crystals (not to be confused with the Bloch-type TM surface wave discussed below!). However, the band structure of the spectrum is preserved and the surface EM mode can be excited only under special conditions, enhancing the in-plane wavevector [@BluVasPer2010; @PRIMER2013].
|
---
abstract: |
The basic theorem of the Lagrangian formulation for general superfield theory of fields (GSTF) is proved. The gauge transformations of general type (GTGT) and gauge algebra of generators of GTGT (GGTGT) as the consequences of the above theorem are studied.
It is established the gauge algebra of GGTGT contains the one of generators of gauge transformations of special type (GGTST) as one’s subalgebra. In the framework of Lagrangian formulation for GSTF the nontrivial superfield model generalizing the model of Quantum Electrodynamics and belonging to the class of gauge theory of general type (GThGT) with Abelian gauge algebra of GGTGT is constructed.
author:
- 'A.A. Reshetnyak[^1]'
date: |
*Department of Mathematics, Seversk State Technological Institute,\
Seversk, [636036]{}, Russia*
title: |
Basic Theorem, Gauge Algebra, $\theta$-superfield QED\
in the Lagrangian Formulation of\
General Superfield Theory of Fields
---
[PACS codes: 11.10.Ef, 11.15.-q, 12.20.-m, 03.50.-z\
Keywords: Lagrangian quantization, Gauge theory, Superfields.]{}
Introduction
============
The Lagrangian and Hamiltonian formulations for GSTF of the superfield (with respect to odd time $\theta$) models description, suggested in the papers \[1,2\], had permitted to solve the problem of constructing the superfield Lagrangian (in usual sense) quantization method for general gauge theories in the framework of general superfield quantization method (GSQM) in the Lagrangian formalism \[3\].
GSQM permits by means of the path integral method to quantize the ordinary gauge models of the quantum field theory extended, in a natural way, to the superfield GSTF models. GSQM contains the BV quantization method for gauge theories \[4\] as the particular case under special choice of the generating equation \[3\]. By the main resulting GSQM feature it appears the Ward identities form for generating functionals (superfunctions) of Green’s functions, including the effective action, which reflect the fact of these superfunctions invariance under their translation with respect to variable $\theta$ along integral curve of solvable \[1,2\] Hamiltonian system constructed with respect to quantum gauge fixed action superfunction $S^{\Psi}_{H}(\theta,
\hbar)$. It is the above Hamiltonian system with which the standard BRST symmetry transformations are associated \[3\] under corresponding notations providing the $\theta$-superfield realization of that symmetry.
Theorem 1, formulated in Ref.\[1\], on reduction of the 1st order with respect to differentiation on $\theta $ system of $N$ ordinary differential equations (ODE) to generalized normal form (GNF) in the case of its linear (functional) dependence appears by the key one in GSTF \[1,2\] and in GSQM \[3\] construction on the whole. The paper is devoted to its proof, to the investigation of the GTGT and a gauge algebra of the GGTGT, to the connection of the latters with GTST and a gauge algebra of the GGTST, to the demonstration of the efficiency of these results on the example of superfield (on $\theta$) quantum electrodynamics model.
In work the definitions, conventions and notations introduced in Ref.\[1\] are made use unless otherwise stated.
Proof of the Basic Theorem
==========================
Consider the 2nd order with respect to derivatives on $\theta$ system of $N$ ODE in normal form (NF) $$\begin{aligned}
{\stackrel{\;\circ\circ}{g}\,}{}^i(\theta) = f^i\bigl(g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)\;,\; f^i(\theta) \in
C^{1}\bigl( T_{odd}{\cal N} \times \{\theta\} \bigr)\,,$$ in a some domain of the supermanifold ${\cal N}$ parametrized by local coordinates $g^i(\theta), i = 1,\ldots,$ $N =(N_+, N_-)$ $(g^i(\theta)=g^i_0+
g^i_1\theta)$ being by unknown superfunctions[^2]. Grassmann parities $\varepsilon_P, \varepsilon_{\bar{J}}, \varepsilon$ of quantities $g^i(\theta), g^i_0, g^i_1, f^i(\theta)$ are given by the formula ($
\varepsilon = \varepsilon_P + \varepsilon_{\bar{J}}$ \[1\]) $$\begin{aligned}
{} & (\varepsilon_P, \varepsilon_{\bar{J}}, \varepsilon)b(\theta) =
(\varepsilon_P(g^i_1)+1, \varepsilon_{\bar{J}}(g^i_1), \varepsilon(g^i_1)+1)=
(\varepsilon_P(g^i(\theta)), \varepsilon_{\bar{J}}(g^i(\theta)),
\varepsilon(g^i(\theta)))\,,{} &
\nonumber \\
{} & b(\theta)\in \{g^i(\theta), g^i_0, f^i(\theta)\}\,. {} &
$$ Eqs.(2.1) are equivalent to the following system of $2N$ ODE at most of the 2nd order with respect to $\theta$ $$\begin{aligned}
{} & {} &{\stackrel{\;\circ\circ}{g}\,}{}^i(\theta) \equiv
\displaystyle\frac{d^2 g^i(\theta)}{d\theta^2\phantom{xxx}} \equiv
\displaystyle\frac{d\phantom{}}{d\theta}
\displaystyle\frac{d g^i(\theta)}{d\theta\phantom{xxx}}=0\,, \\ {} & {} & f^i\bigl(g(\theta), {\stackrel{\;\circ}{g}}(\theta),\theta\bigr)
=0\,,
$$ so that the Cauchy problem setting for (2.1) is controlled by differential constraints which are the subsystem of the 1st order on $\theta$ $N$ ODE (2.4) \[1\]. In a general case the Eqs.(2.4) appear by (functionally) dependent. Singling from (2.4) the independent subsystem of the 1st order on $\theta$ ODE is effectively realized in fulfilling of the following assumptions \[1\]: $$\begin{aligned}
1)\hspace{0.2cm}\left(\tilde{g}{}^{i}(\theta),{\stackrel{\;\circ}{\tilde{g}}}{
}^i(\theta) \right) = (0,0) \in T_{odd}\Phi =
\left\{\left(g^{i}(\theta),{\stackrel{\;\circ}{g}}{}^i(\theta) \right)
\vert f^i\bigl(g(\theta), {\stackrel{\;\circ}{g}}(\theta),\theta\bigr) \equiv
0 \right\}\,, \hspace{2.1cm} $$ 2) $f^i\bigl(g(\theta), {\stackrel{\;\circ}{g}}(\theta),\theta\bigr)=0$ determines the 1st order smooth surface $T_{odd}\Phi$ in $T_{odd}{\cal N}$ which the condition holds on $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}}
\left\|\frac{\delta_l f^{i}\bigl( g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)} {\delta
g^j(\theta_1)\phantom{xxxxxxxx}}\right\|_{\mid T_{odd} \Phi} \leq
N \equiv [f^{i}]\,.$$ Notion of the rank for supermatrix of the form (2.6) with respect to $\varepsilon_{\bar{J}}$ grading for $f^i(\theta)$, $g^j(\theta_1)$ was defined in \[1,3\] and $\frac{\delta_l\phantom{xxx}}{\delta g^j(\theta_1)}$ denotes the left superfield variational derivative with respect to superfunction $g^j(\theta_1)$ ($\theta_1\ne \theta$).
System of the 1st order on $\theta$ $N$ ODE with respect to $g^i(\theta)$ (2.4) subject to conditions (2.5), (2.6) being unsolvable with respect to ${\stackrel{\;\circ}{g}}{}^i(\theta)$ is reduced to equivalent system of independent equations in GNF under following nondegenerate parametrization for $g^i(\theta)=
(\alpha^{\bar{i}}(\theta),\beta^{\underline{i}}(\theta),\gamma^{
\sigma}(\theta))$, $i=(\bar{i},\underline{i},\sigma)$ $$\begin{aligned}
{\stackrel{\;\circ}{\alpha}}{}^{\bar{i}}(\theta) =
\varphi^{\bar{i}}\bigl({\alpha}(\theta) , {\gamma}(\theta),
{\stackrel{\;\circ}{\gamma}}(\theta), \theta\bigr),\ \;
{\beta}^{\underline{i}}(\theta) =
\kappa^{\underline{i}}\bigl({\alpha}(\theta) , {\gamma}(\theta),
\theta\bigr)\;, $$ with arbitrary superfunctions ${\gamma}^{\sigma}(\theta)$ and $\varphi^{\bar{i}}(\theta), \kappa^{\underline{i}}(\theta)$ $\in$ $
C^1(T_{odd}{\cal N}\times\{\theta\})$. The number of $[{\gamma}^{\sigma}]$ coincides with one of differential identities among Eqs.(2.4) $$\begin{aligned}
\int d\theta f^{i}\bigl( g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta\bigr) \check{{\cal R}}_{i
\sigma}\bigl( g(\theta), {\stackrel{\;\circ}{g}}(\theta), \theta;
{\theta}^{\prime}\bigr) = 0\;,$$ where operators $\check{{\cal R}}_{i \sigma}\bigl( g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta; {\theta}^{\prime}\bigr)$ are a) local on $\theta$ and b) functionally independent ones $$\begin{aligned}
{\rm a})\hspace{0.3cm} \check{{\cal R}}_{i \sigma}\bigl( g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta; {\theta}^{\prime}\bigr) \equiv
\check{{\cal R}}_{i \sigma}(\theta; {\theta}^{\prime}) =
\sum_{k=0}^{1}\left(\left(\frac{d}{d\theta}\right)^{k}\delta(\theta -
\theta')\right) \check{{\cal R}}^{k}_{{}i \sigma}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)\,,
\hspace{0.9cm}$$ b) functional equation $$\begin{aligned}
\int d{\theta}'\check{{\cal R}}_{i \sigma}(\theta;
{\theta}^{\prime})u^{\sigma}\bigl( g(\theta'),
{\stackrel{\;\circ}{g}}(\theta'), {\theta}^{\prime}\bigr) = 0,\
u^{\sigma}(\theta)\in C^1(T_{odd}{\cal N}\times\{\theta\}) $$ has unique trivial solution.
includes the investigation scheme of the corresponding system of the 1st and 2nd orders with respect to even derivatives on $t\in{\bf
R}$ \[5\] because one can regard that $t\in i$ \[1\].
[**1)**]{} In correspondence with (2.6) let us assume that $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}}
\left\|\frac{\delta_l f^{i}(\theta)}{\delta{\stackrel{\;\circ}{
g}}{}^j(\theta_1)\phantom{}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid T_{odd} \Phi}=
N-M \Longleftrightarrow
{\rm corank}_{\varepsilon_{\bar{J}}}
\left\|\frac{\delta_l f^{i}(\theta)}{\delta{\stackrel{\;\circ}{
g}}{}^j(\theta_1)\phantom{}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid T_{odd} \Phi}= M = (M_+,M_-)\,.$$ Then $f^i(\theta)$ as the functions of ${\stackrel{\;\circ}{g}}{}^j(\theta)$ are dependent ones and from (2.11) it follows the possibility of the representation $$\begin{aligned}
{} & {} & f^i(\theta) = ({\cal P}^a_1(\theta), p^A_1(\theta)),\;
a=1,\ldots,M; A=M+1,\ldots,N\,,\\{} & {} & \hspace{-1em}
{\rm rank}_{\varepsilon_{\bar{J}}} \left\|
\displaystyle\frac{\partial_l p^{A}_1\bigl(g(\theta),{\stackrel{\;\circ}{
g}}(\theta), \theta\bigr)}{\partial
{\stackrel{\;\circ}{g}}{}^j(\theta)\phantom{xxxxxxxxx}}\right\|{\hspace{-0.5em}
\phantom{\Bigr)}}_{\mid T_{odd}\Phi}= N-M \Longleftrightarrow
{\rm corank}_{\varepsilon_{\bar{J}}}
\left\|\displaystyle\frac{\partial_l p^{A}_1(\theta)}{\partial
{\stackrel{\;\circ}{g}}{}^j(\theta)\phantom{x}}\right\|{\hspace{-0.5em}
\phantom{\Bigr)}}_{\mid T_{odd} \Phi}= M\,,\\{} & {} & {\cal P}^a_1\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr) = p^{A}_1\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr)
\alpha_1^a{}_A\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr) +
\Delta^a(g(\theta),\theta)\,. $$ The superfunctions $\Delta^a(\theta)$ may be dependent ones, i.e. $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}}\left\|\frac{\partial_l
\Delta^{a}(g(\theta),\theta)}{\partial
{g}^j(\theta)\phantom{xxxxx}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid\Phi}= M-K \leq M,\ 0\leq K=(K_+, K_-)
\,.$$ It means the superfunctions $\delta^{(1){}a_1}(g(\theta),\theta)$, $a_1=K+1,
\ldots,M$ exist that the condition holds $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}}
\left\|\frac{\partial_l \delta^{(1){}a_1}(\theta)}{\partial
{g}^j(\theta)\phantom{xxx}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid\Phi}= M-K\,.$$ In (2.11), (2.13)–(2.16) the left partial superfield derivatives with respect to ${\stackrel{\;\circ}{g}}{}^j(\theta), g^j(\theta)$ are denoted as $\frac{\partial_l\phantom{xxx}}{\partial{\stackrel{\;\circ}{g}}{}^j(
\theta)}$, $\frac{\partial_l\phantom{xxx}}{\partial g^j(\theta)}$ respectively \[1\]. By virtue of the assumption (2.5) for $\Delta^a(\theta)$, ${\cal P}^a_1(\theta)$ the following representation is valid $$\begin{aligned}
{} & {} & \Delta^{a}(g(\theta),\theta) =
\delta^{(1){}a_1}(g(\theta),\theta)\beta_1^a{}_{a_1}(g(\theta),\theta),\
{\rm rank}_{\varepsilon_{\bar{J}}}
\left\|\beta_1^a{}_{a_1}(\theta)\right\|_{\mid\Phi}= M-K\,,
\nonumber \\
{} & {} &
{\cal P}^a_1\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr) =
p^{A}_1(\theta)
\alpha_1^a{}_A(\theta) + \delta^{(1){}a_1}(\theta)\beta_1^a{}_{a_1}(\theta)
\,.$$ Divide the all ${\cal P}^a_1(\theta)$ onto 2 groups: ${\cal P}^a_1(\theta)$ = $(\overline{\cal P}{}^{A_1}_1(\theta),
\underline{\cal P}{}^{a_1}_1(\theta))$, $A_{1}=1,\ldots,K$; $a_{1}=K+1,
\ldots, M, a=(A_{1}, a_{1})$ $$\begin{aligned}
{} \hspace{-0.5em}\underline{\cal P}{}^{a_1}_1\bigl(g(\theta),{\stackrel{
\;\circ}{g}}(\theta),
\theta\bigr)\hspace{-0.1em}=\hspace{-0.1em} p_1^A(\theta)
\underline{\alpha}{}^{a_1}_1{}_A\bigl(g(\theta),{\stackrel{\;\circ}{g}}(
\theta), \theta\bigr)\hspace{-0.1em}+ \hspace{-0.1em}
\delta^{(1){}b_1}(\theta)\underline{\beta}{}_1^{a_1
}{}_{b_1}(g(\theta),\theta),\, {\rm sdet}\left\|\underline{\beta}{}_1^{a_1
}{}_{b_1}(\theta)\right\| \neq 0. $$
Then from (2.11)–(2.18) it follows that $f^i(\theta)=
(\overline{\cal P}{
}^{A_1}_1(\theta)$, $\underline{\cal P}{}^{a_1}_1(\theta)$, $p^A_1(\theta))$ are connected with $f^{j}_{(1)}\bigl(g(\theta), {\stackrel{\;\circ}{g
}}(\theta), \theta\bigr)$ = $(\overline{\cal P}{
}^{B_1}_1(\theta)$, $\delta^{(1)b_1}(\theta), p^B_1(\theta))$ by means of the nondegenerate supermatrix $K^{0,1}(\theta)$ = $\left\|K^{0,1}{}^i{}_j\bigl(
g(\theta),
{\stackrel{\;\circ}{g}}(\theta),\theta\bigr)\right\|$ in $T_{odd}V$ $\supset$ $T_{odd}\Phi$ for some $V\subset{\cal N}$
$$\begin{aligned}
f^i(\theta) = f^{j}_{(1)}(\theta)K^{0,1}{}^i{}_j(\theta),\
\bigl(K^{0,1}(\theta)\bigr)^{-1} = K^{1,0}(\theta)\,,$$
providing the equivalence of Eqs.(2.4) and $f^{j}_{(1)}(\theta)=0$. Consider the superfunctions $\overline{\cal P}{}^{A_1}_1(\theta)$ among $
{\cal P}^{a}_1(\theta)$ in (2.17). They are the identities for superfunctions $f^{j}_{(1)}(\theta)$ or $f^i(\theta)$ and can be written by means of two equivalent expressions $$\begin{aligned}
\int d\theta f^{j}_{(1)}\bigl(g(\theta), {\stackrel{\;\circ}{g}}(\theta),
\theta\bigr){\cal R}^{(1)}_{jA_1}\bigl(g(\theta), {
\stackrel{\;\circ}{g}}(\theta), \theta;\theta'\bigr) = 0\,;$$
$$\begin{aligned}
\setcounter{lyter}{1}
\overline{\cal P}{}^{A_1}_1(\theta) & = &
\int d\theta' \underline{f}_{(1)}^{(a_1,A)}\bigl(g(\theta'), {\stackrel{\;
\circ}{g}}(\theta'),
\theta'\bigr)\Lambda_{(1){}(a_1,A)}{}^{A_1}\bigl(g(\theta'),
{\stackrel{\;\circ}{g}}(\theta'), \theta';\theta\bigr)
\,,\\\setcounter{equation}{21}
\setcounter{lyter}{2}
f^{j}_{(1)}(\theta) & \equiv &
\left(\overline{\cal P}{}^{B_1}_1(\theta),
\underline{f}_{(1)}^{(b_1,B)}\bigl(g(\theta), {\stackrel{\;\circ}{g}}(\theta),
\theta\bigr)\right) \,.$$
Therefore, among $f^{i}_{(1)}(\theta)=0$ the only $\underline{f}{}^{(a_1,A)}_{(1)}(\theta)=0$ are essential. It is the latter equations are equivalent to (2.4).
[**2)**]{} In its turn the superfunctions ${\stackrel{\;\circ}{\delta}}{}^{
(1){}a_1}(\theta)$, $p_1^A(\theta)$ may be dependent ones with respect to ${\stackrel{\;\circ}{g}}{}^j(\theta)$
$$\begin{aligned}
{} & {\rm rank}_{\varepsilon_{\bar{J}}}\left\|
\displaystyle\frac{\partial_l
\bigl({\stackrel{\;\circ}{\delta}}{}^{(1){}a_1}(\theta),
p_1^A (\theta)\bigr)}{\partial
{\stackrel{\;\circ}{g}}{}^j(\theta)\phantom{xxxxxxxxx}}\right\|{
\hspace{-0.5em}\phantom{\Bigr)}}_{\mid
T_{odd}\Phi}\hspace{-0.5em}= N-K -K_1 <[\delta^{(1)}(\theta)] +
[p_1(\theta)]\,, {} & \nonumber \\
{} & N-K-K_1 \geq [\delta^{(1)}(\theta)], K_1=(K_{1+}, K_{1-})\,.{} &$$
It follows from (2.22) the representability of $p_1^A(\theta)$ in the form $$\begin{aligned}
{} & p_1^A(\theta)=
\bigl({\cal P}_2^{a_{11}}, p_2^{A_{11}}\bigr)(\theta),
a_{11} = M+1,\ldots,M+K_1; A_{11} = M+K_1+1,\ldots,N\,, {} &
\\{} & {\cal P}_2^{a_{11}}\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr) = p_2^{A_{11}}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)
{\alpha_2}^{a_{11}}{}_{A_{11}}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr) +
{} & \nonumber \\
{} & \hspace{-1.0em} {\stackrel{\;\circ}{\delta}}{}^{(1){}a_1}(\theta)
{\nu_1}^{a_{11}}{}_{a_1}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr) +
\delta^{(1){}a_1}(\theta){\mu_1}^{a_{11}}{}_{a_1}
\bigl(g(\theta), \theta\bigr) +
\delta^{(2){}a_2}(g(\theta), \theta){\beta_2}^{a_{11}}{}_{a_2}
\bigl(g(\theta),\theta\bigr),{} & \\{} & {\rm rank}_{\varepsilon_{\bar{J}}}
\left\|{\beta_2}^{a_{11}}{}_{a_2}(\theta)\right\|=[\delta^{(2)}(
\theta)] = M_1=(M_{1+},M_{1-})\,,{} & \nonumber \\
{} & a_2= M+K_1 - M_1 +1,\ldots,M+K_1\,,{} &\\{} &
{\rm rank}_{\varepsilon_{\bar{J}}} \left\|\displaystyle\frac{
\partial_l \bigl({\stackrel{\;\circ}{\delta}}{}^{(1){}a_1}(\theta),
p_2^{A_{11}}(\theta)\bigr)}{\partial
{\stackrel{\;\circ}{g}}{}^j(\theta)\phantom{xxxxxxxxx}}\right\|{
\hspace{-0.5em}\phantom{\Bigr)}}_{\mid
T_{odd}\Phi}= [\delta^{(1)}(\theta)]+ [p_2(\theta)] = N-K-K_1\,,{} &$$ where $\delta^{(2)a_2}(\theta)$ are the superfunctions being independent ones on $\delta^{(1){}a_1}(\theta)$ $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}} \left\|
\frac{\partial_l \bigl({\delta}^{(1){}a_1}(\theta),
\delta^{(2){}a_{2}}(\theta)\bigr)}{\partial
{g}^j(\theta)\phantom{xxxxxxxxxxx}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid\Phi}= [\delta^{(1)}(
\theta)]+ [\delta^{(2)}(\theta)] = M+M_1-K\,.$$ According to (2.24)–(2.27) divide ${\cal P}_2^{a_{11}}(\theta)$ onto 2 groups $$\begin{aligned}
{} & \hspace{-1.0em}{\cal P}_2^{a_{11}}(\theta) =
\bigl(\overline{\cal P}{}_2^{A_2},
\underline{\cal P}{}_2^{a_2}\bigr)\bigl(g(\theta),{\stackrel{\;\circ}{g}}(
\theta), \theta\bigr),
A_2 = M+1,\ldots,M+K_1-M_1, {} &\\{} & \underline{\cal P}{}_2^{a_2}(\theta) =
p_2^{A_{11}}(\theta)
{\underline{\alpha}{}_2}^{a_2}{}_{A_{11}}(\theta) +
{\stackrel{\;\circ}{\delta}}{}^{(1){}a_1}(\theta)
{\underline{\nu}{}_1}^{a_2}{}_{a_1}(\theta)+ {} & \nonumber \\
{} & {\delta}^{(1){}a_1}(\theta){\underline{\mu}{}_1}^{a_2}{}_{a_1}(\theta)+
\delta^{(2){}b_2}(\theta){\underline{\beta}{}_2}^{a_2}{}_{b_2}(\theta),\;
{\rm sdet}\left\|{\underline{\beta}{}_2}^{a_2}{}_{b_2}(\theta\right\|\ne 0
\,.{} & $$ and define the set of superfunctions $f^{j}_{(2)}\bigl(g(\theta),{\stackrel{
\;\circ}{g}}(\theta),\theta\bigr)$ = $(\overline{\cal P}{}^{A_1}_1(\theta)$, $\overline{\cal P}{}^{A_2}_2(\theta)$, $\delta^{(1)a_1}(\theta)$, $\delta^{(2)
a_2}(\theta)$, $p^{A_{11}}_2(\theta))$ connected with $f^{i}_{(1)}\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr)$ = $(\overline{\cal P}{
}^{B_1}_1(\theta)$, $\delta^{(1)b_1}(\theta)$, $\overline{\cal P}{}^{B_2}_2(
\theta)$, $\underline{\cal P}{}^{b_2}_2(\theta)$, $p^{B_{11}}_2(\theta))$ (and therefore with $f^j(\theta)$ (2.4)) by the nondegenerate supermatrix $K^{1,2}(\theta)$ = $\left\|K^{1,2}{}^i{}_j
\bigl(g(\theta),
{\stackrel{\;\circ}{g}}(\theta),\theta\bigr)\right\|$ in $T_{odd}V$ $$\begin{aligned}
{} & {} & f^{i}_{(1)}(\theta) = f^{j}_{(2)}(\theta)K^{1,2}{}^i{}_j(\theta),\
K^{1,2}(\theta) = \left\|
\begin{array}{ccccc}
\delta^{B_1}{}_{A_1} & 0 & 0 & 0 & 0 \\
0 & 0 & \delta^{B_2}{}_{A_2} & 0 & 0 \\
0 & \delta^{b_1}{}_{a_1} & 0 & A^{b_2}{}_{a_1}(\theta) & 0 \\
0 & 0 & 0 & B^{b_2}{}_{a_2}(\theta) & 0 \\
0 & 0 & 0 & C^{b_2}{}_{A_{11}}(\theta) & \delta^{B_{11}}{}_{A_{11}} \\
\end{array}\right\|,\nonumber \\
{} & {} & A^{b_2}{}_{a_1}(\theta) = \displaystyle\frac{\stackrel{
\leftarrow}{d}}{d\theta}\underline{\nu}{}_1{}^{b_2}{}_{a_1}(\theta) +
\underline{\mu}{}_1{}^{b_2}{}_{a_1}(\theta),
B^{b_2}{}_{a_2}(\theta)=\underline{\beta}{}_2{}^{b_2}{}_{a_2}(\theta),
C^{b_2}{}_{A_{11}}(\theta) = \underline{\alpha}{}_2{}^{b_2}{}_{A_{11}}(
\theta)\,.$$ Its inverse supermatrix has the form $$\begin{aligned}
K^{2,1}(\theta) =(K^{1,2}(\theta))^{-1} =
\left\|
\begin{array}{ccccc}
\delta_{B_1}{}^{C_1} & 0 & 0 & 0 & 0 \\
0 & 0 & \delta_{b_1}{}^{c_1} & -(A B^{-1})_{b_1}{}^{c_2}(\theta) & 0 \\
0 & \delta_{B_2}{}^{C_2} & 0 & 0 & 0 \\
0 & 0 & 0 & (B^{-1})_{b_2}{}^{c_2}(\theta) & 0 \\
0 & 0 & 0 & - (C B^{-1})_{B_{11}}{}^{c_2}(\theta) & \delta_{B_{11}}{}^{C_{11}}
\\
\end{array}\right\|\,.$$ From (2.30), (2.31) it follows both $K^{1,2}(\theta)$ and $K^{2,1}(\theta)$ appear by the local differentiation operators with respect to $\theta$. In its turn from (2.22)–(2.24) it implies the additional, already differential on $\theta$, identities exist among $f^{j}_{(2)}(\theta)$ $\equiv$ $(
\overline{\cal P}{}^{A_1}_1(\theta)$, $\underline{f}{}^{(a_1,A)}_{(1)}(
\theta))$ besides of ones (2.20) $$\begin{aligned}
\int d\theta \underline{f}{}_{(1)}^{(a_1, A)}\bigl(g(\theta),
{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr){\cal R}^{(2)}_{(a_1, A){}A_2}\bigl(g(\theta),
{\stackrel{\;\circ}{g}}(\theta), \theta;\theta'\bigr) = 0
\,.$$ One can equivalently represent them in the form
$$\begin{aligned}
\setcounter{lyter}{1}
\overline{\cal P}{}^{A_2}_2(\theta) & = &
\int d\theta' \underline{f}_{(2)}^{(a_1,a_2, A_{11})}\bigl(g(\theta'),
{\stackrel{\;\circ}{g}}(\theta'), \theta'\bigr)
\Lambda_{(2){}(a_1,a_2,A_{11})}{}^{A_2}\bigl(g(\theta'),
{\stackrel{\;\circ}{g}}(\theta'), \theta';\theta\bigr)
\,,\\\setcounter{equation}{33}
\setcounter{lyter}{2}
f^{j}_{(2)}(\theta) & \equiv &
\left(\overline{\cal P}{}^{A_1}_1(\theta),\overline{\cal P}{}^{A_2}_2(\theta),
\underline{f}{}_{(2)}^{(a_1,a_2,A_{11})}(\theta)\right)\,.$$
Thus the only $\underline{f}{}^{(a_1,a_2,A_{11})}_{(2)}(\theta)=0$ are the essential equations from $\underline{f}^{(a_1,A)}_{(1)}(\theta)=0$. It is the former equations are completely equivalent to the initial ones (2.4).
[**3)**]{} Let us assume the set of superfunctions $p_2^{A_{11}}(\theta)$, ${\stackrel{\;\circ}{\delta}}{}^{(1)a_1}(\theta)$, ${\stackrel{\;\circ}{
\delta}}{}^{(2)a_2}(\theta)$ are dependent with respect to variables ${\stackrel{\;\circ}{g}}{}^j(\theta)$. Then, analogously to above case the new constraints $\delta^{(3)}(g(\theta),\theta)$ arise being by independent on $\delta^{(1)}(\theta), \delta^{(2)}(\theta)$. In view of the finiteness of the discrete part of index $i$ and in ignoring of the covariance requirement with respect to $i$ it follows from induction principle the existence of the $l$th step ($l\leq N$) of the iterative procedure the such that the equations
$$\begin{aligned}
{} &\underline{f}{}_{(l)}^{((a)_l,A_{l-1{}1})}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)\equiv
\underline{f}{}_{(l)}^{(a_1,\ldots,a_l,A_{l-1{}1})}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)= 0\,, {} &
\nonumber \\
{} & \underline{f}{}_{(l)}^{((a)_l,A_{l-1{}1})}(\theta) =
\Bigl(\delta^{(1){}a_1}(g(\theta),\theta),\ldots,\delta^{(l){}a_l}(g(\theta),
\theta), p_l^{A_{l-1{}1}} \bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta),
\theta\bigr)\Bigr) {} &
$$
are equivalent to Eqs.(2.4) and $$\begin{aligned}
{} &\hspace{-1.1em}{\rm rank}_{\varepsilon_{\bar{J}}}\hspace{-0.3em}\left\|
\displaystyle\frac{\partial_l
\bigl({\stackrel{\;\circ}{\delta}}{}^{(1){}a_1}(\theta),
\ldots, {\stackrel{\;\circ}{\delta}}{}^{(l){}a_l}(\theta), p_l^{A_{l-1{}1}}
(\theta)\bigr)}{\partial
{\stackrel{\;\circ}{g}}{}^j(\theta)\phantom{xxxxxxxxxxxxxxxxxxx}}
\right\|{\hspace{-0.5em}\phantom{\Bigr)}}_{\mid
T_{odd}\Phi}\hspace{-1.2em}
=\hspace{-0.15em} N\hspace{-0.15em}-\hspace{-0.15em} K\hspace{-0.15em} +
\hspace{-0.15em}\sum\limits_{s=1}^{l-1}(M_{s}-K_s)\hspace{-0.15em} =
\hspace{-0.15em}[p_l]\hspace{-0.15em} + \hspace{-0.15em}\sum\limits_{
s=1}^{l}[\delta^{(s)}],{} &\\{} & {\rm rank}_{\varepsilon_{\bar{J}}} \left\|
\displaystyle\frac{\partial_l \bigl({\delta}^{(1){}a_1}(\theta),\ldots,
\delta^{(l){}a_{l}}(\theta)\bigr)}{\partial
{g}^j(\theta)\phantom{xxxxxxxxxxxxxx}}\right\|{\hspace{-0.5em}\phantom{\Bigr)}
}_{\mid\Phi}=
\sum\limits_{s=1}^{l}[\delta^{(s)}] = M-K+\sum\limits_{s=1}^{l-1}M_{s}
\,,{} &\\{} & \hspace{-1em}
[p_l] = N-M-\sum\limits^{l-1}_{k=1}K_k,
[\delta^{(s)}]=M_s, s=\overline{1,l}\,.{} & $$ Formally, the constructed algorithm of system (2.4) reduction to GNF can be written as follows $$\begin{aligned}
f^i(\theta) = f^{j}_{(1)}(\theta)K^{0,1}{}^i{}_j(\theta)\,,$$
$$\begin{aligned}
\setcounter{lyter}{1}
{} & {} & \hspace{-2.5em}
f^{i}_{(1)}(\theta) = \bigl(\overline{\cal P}{}_1^{A_1}(\theta),
\underline{f}{}_{(1)}^{(a_1,A)}(\theta)\bigr)\,, \\\setcounter{equation}{39}
\setcounter{lyter}{2}
{} & {} &\hspace{-2.5em}
\overline{\cal P}{}^{A_1}_1(\theta) =
\int d\theta' \underline{f}{}_{(1)}^{(a_1,A)}(\theta')
\Lambda_{(1){}(a_1,A)}{}^{A_1}(\theta';\theta)\Longleftrightarrow
\int d\theta f^{j}_{(1)}(\theta){\cal R}^{(1)}_{j{}A_1}(\theta;\theta') = 0\,;
\\\setcounter{equation}{39}
\setcounter{lyter}{3}
{} & {} & \hspace{-3.0em}
f^{i}_{(1)}(\theta) = f^{j}_{(2)}(\theta)K^{1,2}{}^i{}_j(\theta)
\,,\\\setcounter{equation}{40}
\setcounter{lyter}{1}
{} & {} & \hspace{-3.0em}
f^{i}_{(2)}(\theta) = \bigl(\overline{\cal P}{}_1^{A_1}(\theta),
\overline{\cal P}{}_2^{A_2}(\theta),
\underline{f}{}_{(2)}^{(a_1,a_2,A_{11})}(\theta)\bigr)=
\bigl(\overline{\cal P}{}_1^{A_1}(\theta),
\underline{f}{}_{(1)}^{(a_1,A)}(\theta)\bigr)
\,,\\\setcounter{equation}{40}
\setcounter{lyter}{2}
{} & {} & \hspace{-3.0em}
\overline{\cal P}{}_2^{A_2}(\theta)\hspace{-0.1em} =\hspace{-0.2em}
\int\hspace{-0.2em} d\theta'\hspace{-0.1em} \underline{f}{}_{(2)}^{((a)_2,A_{
11})}(\theta')
\Lambda_{(2){}((a)_2,A_{11})}{}^{A_2}(\theta';\theta)
\hspace{-0.2em}\Longleftrightarrow\hspace{-0.2em}
\int\hspace{-0.2em} d\theta\hspace{-0.1em}
\underline{f}{}_{(1)}^{(a_1,A)}(\theta){\cal R}^{(2)}_{(a_1,A_1){}A_2}(
\theta;\theta')\hspace{-0.1em} =\hspace{-0.1em} 0;\\\setcounter{equation}{40}
\setcounter{lyter}{3}
{} & {} & \hspace{-3.0em}
f^{i}_{(2)}(\theta) = f^{j}_{(3)}(\theta)K^{2,3}{}^i{}_j(\theta)\,,$$
$$\begin{aligned}
\ldots & \ldots & \ldots \nonumber\end{aligned}$$ $$\begin{aligned}
\setcounter{equation}{41}
\setcounter{lyter}{1}
{} & {} & \hspace{-3.5em}
f^{i}_{(l)}(\theta) =
\bigl(\overline{\cal P}{}_1^{A_1}(\theta),
\ldots, \overline{\cal P}{}_{l}^{A_{l}}(\theta),
\underline{f}{}_{(l)}^{((a)_{l},A_{l-1{}1})}(\theta)\bigr)\hspace{-0.1em}=
\hspace{-0.1em}
\bigl(\overline{\cal P}{}_1^{A_1}(\theta),\ldots,\overline{\cal P}{}_{l-1}^{
A_{l-1}}(\theta),\underline{f}{}_{(l-1)}^{((a)_{l-1},A_{l-2{}1})}(
\theta)\bigr),\\\setcounter{equation}{41}
\setcounter{lyter}{2}
{} & {} &\hspace{-3.0em}
\overline{\cal P}{}_l^{A_l}(\theta) =
\int d\theta' \underline{f}{}_{(l)}^{((a)_l,A_{l-1{}1})}(\theta')
\Lambda_{(l){}((a)_l,A_{l-1{}1})}{}^{A_l}(\theta';\theta)
\Longleftrightarrow \nonumber \\
{} & {} & \hspace{-0.5em} \int d\theta
\underline{f}{}_{(l-1)}^{((a)_{l-1},A_{l-2{}1})}(\theta){\cal
R}^{(l)}_{((a)_{l-1},A_{l-2{}1}){}A_l}(\theta;\theta') = 0\,.$$ Thus, the Eqs.(2.4) have been reduced to equivalent ones in GNF (2.34) being by functionally independent. Comparison of (2.34) with (2.7) means by virtue of (2.35), (2.36) that
$$\begin{aligned}
{} & {\stackrel{\;\circ}{\alpha}}{}^{\bar{i}}(\theta) -
\varphi^{\bar{i}}\bigl({\alpha}(\theta) , {\gamma}(\theta),
{\stackrel{\;\circ}{\gamma}}(\theta), \theta\bigr)=0\Longleftrightarrow
p_l^{A_{l-1{}1}}
\bigl(g(\theta),{\stackrel{\;\circ}{g}}(\theta), \theta\bigr)=0,\
\bar{i}=A_{l-1{}1}\,,{} &\\{} & {\beta}^{\underline{i}}(\theta) -
\kappa^{\underline{i}}\bigl({\alpha}(\theta) , {\gamma}(\theta),
\theta\bigr)=0\Longleftrightarrow
\delta^{(k){}a_k}(g(\theta),\theta)=0,\ k=1,\ldots,l, \underline{i}=(a_1,
\ldots,a_l)
\,.{} & $$
The number of arbitrary superfunctions $\gamma(\theta)$ in (2.7) is equal to $$\begin{aligned}
[\gamma(\theta)] = [g^i(\theta)]-[\alpha^{\bar{i}}(\theta)]-
[{\beta}^{\underline{i}}(\theta)]=[f^i(\theta)] - [\underline{f}{}^{((a)_l,
A_{l-1{}1})}_{(l)}(\theta)]=K+ \sum\limits^{l-1}_{s=1}(K_s - M_s)
$$ and coincides with one of the identities in (2.39b), (2.40b)$,\ldots,$ (2.41b).
As far as the supermatrices $K^{s,s-1}(\theta)$ = $(K^{s-1,s}(\theta))^{-1}$ are the local ones on $\theta$, $s=1,\ldots,l$, then one can write the identities in terms of the initial equations (2.4) which have the form (2.8) with local on $\theta$ and functionally independent operators $\check{\cal R}_{i\sigma}(\theta;\theta')$ being by polynomials with respect to $K^{s,s-1}(\theta)$. : The above proof have not concerned the possible complicated structure of index $i$ (see footnote 1). The locality of operators $\check{\cal R}_{i\sigma}(\theta;\theta')$ with respect to other continual parts of the indices $i, \sigma$ may be shown in the analogous way. However, the requirement of functional independence of $\check{\cal R}_{i\sigma}(\theta;\theta')$ leads, in general case, to the loss of covariance for these quantities.
From the Theorem 1 proof the validity of its consequence \[1\] easily follows (with use of the integer-valued functions of degree and least degree: ${\rm deg}_{c(\theta)}$, ${\rm min\,deg}_{c(\theta)}$, $c(\theta)\in$ $\{g^i(\theta)$, ${\stackrel{\;\circ}{g}}{}^i(\theta)$, $g^i(\theta){
\stackrel{\;\circ}{g}}{}^j(\theta),\ldots\}$ \[1\]).
If $f^i(\theta)$ (2.4) are the holonomic constraints $$\begin{aligned}
{\rm deg}_{{\stackrel{\;\circ}{g}}(\theta)}f^i(\theta) =0\,,$$ then for $f^i(g(\theta),\theta)$ under following parametrization of superfunctions $g^i(\theta)$ $\mapsto$ $g'{}^i(g(\theta))$ $$\begin{aligned}
g'{}^i(\theta) = (\alpha^A(\theta),\gamma^{\sigma}(\theta)),\ i=(A, \sigma),
\sigma=1,\ldots,[\gamma], A=1,\ldots,[\alpha]
$$ there exists the equivalent system of the holonomic constraints $$\begin{aligned}
\Phi^A(\alpha(\theta),\gamma(\theta),\theta)=0\,.$$ The number $[\gamma]$ coincides with one of algebraic (in the sense of differentiation with respect to $\theta$) identities among $f^i(\theta)$ $$\begin{aligned}
f^i(g(\theta),\theta)\check{\cal R}{}^{(0)}_{i\sigma}(g(\theta),\theta)=0
$$ being obtained from (2.8) by means of integration on $\theta$ with allowance made for validity of the type (2.9) connection of $\check{\cal R}_{i\sigma}(
\theta,\theta')$ with algebraic (on $\theta$) operators $\check{\cal R}{}^{(0)}_{i\sigma}(\theta)$ $$\begin{aligned}
\check{\cal R}_{i\sigma}(g(\theta),\theta;\theta') = \delta(\theta-\theta')
\check{\cal R}{}^{(0)}_{i\sigma}(g(\theta),\theta)(-1)^{\varepsilon(g^i(
\theta))}\,.$$ A dependence upon ${\stackrel{\;\circ}{g}}{}^i(\theta)$ in (2.49) may be only parametric one.
Application to GSTF in Lagrangian Formulation
=============================================
Consider as ${\cal N}$ the supermanifold ${\cal M}_{cl}$ parametrized by the classical superfields ${\cal A}^{\imath}(\theta)$ $${\cal A}^{\imath}(\theta)=A^{\imath} + \lambda^{\imath}\theta,\
(\varepsilon_P, \varepsilon_{\bar{J}}, \varepsilon){\cal A}^{\imath}(\theta)=
((\varepsilon_P)_{\imath}, (\varepsilon_{\bar{J}})_{\imath},
\varepsilon_{\imath}),\ \imath=1,\ldots,n = (n_+,n_-)\,,$$ being by superfunctions defined on ${\cal M}=\tilde{\cal M}\times\tilde{P}$, in its turn to be the quotient space of the supergroup $J=\bar{J}\times P$ = $(\overline{M}{\mbox{$\times \hspace{-1em}\supset$}}\bar{J}_{\tilde{A}}) \times P$: ${\cal M}=J/{J}_{
\tilde{A}}$ with one-parametric subsupergroup $P$ generated by the Grassmann nilpotent variable $\theta$ \[1\]. Superspace ${\cal M}$ may be parametrized by sets of supernumbers $(z^a,\theta)=(x^{\mu},\theta^{Ak},
\theta)$, if the representation for $\tilde{\cal M}$ is valid \[1\] $$\tilde{\cal M} = {\bf R}^{1,D-1\vert L{}c},\ \mu=0,1,\ldots,D-1,\,
A=1,\ldots,c=2^{[D/2]},\, k=\overline{1,L}\,,$$ meaning that $\tilde{\cal M}$ appears by the real $D$-dimensional Minkowski superspace with $L$ supersymmetries (if $\bar{J}$ is the corresponding group of the usual $L$-extended supersymmetry). Superfields ${\cal A}^{\imath}(\theta)$ are considered by belonging to the special Berezin superalgebra $\tilde{\Lambda}_{D\vert Lc+1}(z^{a},\theta;{\bf K})$, ${\bf K}=({\bf R}$ or ${\bf C})$ \[1–3\].
The $\Lambda_1(\theta,{\bf R})$-valued superfunction $S_L(\theta)$ $\equiv$ $S_L\bigl({\cal A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta),\theta\bigr)$ $\in$ $C^k(T_{odd}{\cal M}_{cl}\times\{\theta\})$, $k\leq\infty$, $(\varepsilon_P$, $\varepsilon_{\bar{J}}$, $\varepsilon)S_L(\theta)$ = $(0,0,0)$, $T_{odd}{\cal M}_{cl}$ = $\bigl\{\bigl({\cal A}^{\imath}(\theta)$, ${
\stackrel{\ \circ}{\cal A}}{}^{\imath}(\theta)\bigr)\vert{\cal A}^{\imath}(
\theta)$ $\in$ ${\cal M}_{cl}\bigr\}$ and superfunctional $Z[{\cal A}]$ = $\int d\theta S_L(\theta)$, $Z[{\cal A}]$ $\in$ $C_{F}$, $(\varepsilon_P$, $\varepsilon_{\bar{J}}$, $\varepsilon)Z[{\cal A}]$ = $(1,0,1)$ are the central objects in the Lagrangian formulation for GSTF characterizing the superfield model on this stage \[1\] of investigation. Dynamics of the model follows from a variational principle for $Z[{\cal A}]$ and is described by Euler-Lagrange equations \[1\] $$\begin{aligned}
{\cal L}_{\imath}^{l}(\theta) S_{L}(\theta)\equiv
\left(\frac{\partial_l \phantom{xxx}}{
\partial{\cal A}^{\imath}(\theta)}
-(-1)^{\varepsilon_{\imath}}\frac{d}{d\theta}\frac{\partial_l
\phantom{xxx}}{\partial{\stackrel{\ \circ}{\cal
A}}{}^{\imath}(\theta)}\right)S_{L}(\theta) =
\frac{\delta_l Z[{\cal A}]}{\delta{\cal A}^{\imath}(\theta)} = 0\,,
$$ being represented equivalently by virtue of (2.1), (2.3), (2.4) by the Lagrangian system (LS) $$\begin{aligned}
{} & {} &
{\stackrel{\,\circ\circ}{\cal A}}{}^{\jmath}(\theta) \displaystyle\frac{
\partial^{2}_{l} S_{L}(\theta) \phantom{xxxx}}
{{\partial{\stackrel{\ \circ}{\cal
A}}{}^{\imath}(\theta)}{\partial{\stackrel{\ \circ}{\cal A}}{}^{\jmath}
(\theta)}} = 0
\,,\\{} & {} & {\Theta}_{\imath}\bigl( {{\cal A}}(\theta), {\stackrel{\
\circ}{{\cal A}}}(\theta), \theta \bigr) \equiv
\displaystyle\frac{
\partial_l S_{L}(\theta) }{\partial{\cal A}^{\imath}(\theta)}
-(-1)^{\varepsilon_{\imath}}\left(\frac{\partial_l}{\partial\theta}\frac{
\partial_l
S_{L}(\theta)}{{\partial{\stackrel{\ \circ}{\cal A}}{}^{\imath}(\theta)}} +
{\stackrel{\ \circ}{\cal A}}{}^{\jmath}(\theta)\displaystyle\frac{
\partial_l \phantom{xxx}}{\partial{\cal A}^{\jmath}(\theta)}
\displaystyle\frac{\partial_l S_{L}(\theta)}
{{\partial{\stackrel{\ \circ}{\cal A}}{}^{\imath}(\theta)}} \right) = 0
\,.$$ Subsystem (3.3), for ${\rm deg}_{{\stackrel{\ \circ}{\cal A}}(\theta)}\Theta_{
\imath}(\theta) \ne 0$ called the differential constraints in Lagrangian formalism (DCLF) (for ${\rm deg}_{{\stackrel{\ \circ}{\cal A}}(\theta)}\Theta_{\imath}(
\theta) = 0$ the holonomic constraints in Lagrangian formalism (HCLF)), restricts an arbitrariness in a choice of $2n$ initial conditions $\left(\bar{{\cal A}}^{\imath}(0), \bar{\stackrel{
\ \circ}{\cal A}\;}{}^{\imath}(0)\right)$ for $\theta = 0$ in setting of Cauchy problem. Subsystem (3.2) are not written in NF with respect to ${\stackrel{\,\circ\circ}{\cal A}}{}^{\imath}(\theta)$ and possibility to pass to NF depends on the nondegeneracy of the supermatrix $K(\theta)$ = $
\left\|\frac{\partial_{r}\phantom{xxx}}{{\partial{\stackrel{\ \circ}{\cal
A}}{}^{\jmath}(\theta)}} \frac{\partial_l
S_{L}(\theta)}{{\partial{\stackrel{\ \circ}{\cal A}}{}^{\imath}(\theta)}}
\right\|$.
DCLF themselves appear, in general, by dependent system of the 1st order on $\theta$ $n$ ODE with respect to unknowns ${\cal A}^{\imath}(\theta)$. Reduction of $\Theta_{\imath}(\theta)$ to GNF is realized independently on subsystem of the 2nd order on $\theta$ $n$ ODE (3.2) in the result of Theorem 1 application directly to (3.3). To this end let us adapt the assumption (2.5), (2.6) to the case of the Lagrangian GSTF \[1,3\] only in terms of $Z[{\cal A}]$: $$\begin{aligned}
{}\hspace{-4em} 1) {}\hspace{4em} \exists
\left({{\cal A}}^{\imath}_{0}(\theta),
{\stackrel{\circ}{{\cal A}^{\imath}_{0}}}(\theta)\right)
\in T_{odd}{\cal M}_{cl}:
{\Theta}_{\imath}(\theta)_{\Bigl| \left({\cal
A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta)\right) = \left( {\cal
A}_0(\theta), {\stackrel{\ \circ}{\cal A}}_0(\theta)\right)} = 0\,;
$$ $$\begin{aligned}
{} & {}\hspace{-2em}
2) {}\hspace{2em} \exists\Sigma \subset {\cal M}_{cl}\; \mbox{(
$\Sigma$--smooth supersurface)}: \left( {{\cal
A}}^{\imath}_{0}(\theta), {\stackrel{\ \circ}{\cal A}}{}^{\imath}_{0}(
\theta)\right) \in T_{odd}\Sigma,\ \Theta_{\imath}(\theta)_{\mid T_{
odd}\Sigma} = 0\,, {} & \\ {} & {\rm dim}_{\varepsilon}{\Sigma}=m=(m_+,m_-), {\rm dim}_{\varepsilon
}{T_{odd}\Sigma}\equiv
{\rm dim}{T_{odd}\Sigma}= (m_+ +m_-,m_- +m_+)\,; {} & $$ index $\imath$ can be divided $\imath=(A,\alpha)$, $A=1,\ldots,n-m$, $\alpha=n-m+1,\ldots,n$ in a such way that the condition holds $$\begin{aligned}
{\rm rank}_{\varepsilon_{\bar{J}}} \left\| \frac{\delta_l \phantom {xxxx}}{
\delta{\cal A}^{\jmath}(\theta_1)} \frac{\delta_l Z[{\cal A}]}{\delta{\cal
A}^{\imath}(\theta)}\right\|_{\textstyle\mid\Sigma} = {\rm rank}_{
\varepsilon_{\bar{J}}} \left\|
\frac{\delta_l \phantom {xxxx}}{\delta{\cal A}^{\jmath}(\theta_1)}
\frac{\delta_l Z[{\cal A}]}{\delta{\cal
A}^{A}(\theta)}\right\|_{\textstyle\mid\Sigma} = n-m\,.
$$ Remind \[1\], in the first place, that $\Sigma$ is considered as local supersurface and, in the second, the following representation is true for DCLF in terms of the superfields $\tilde{\cal A}^{\imath}(\theta)$ = ${\cal A}^{\imath}(
\theta)$ $-$ ${\cal A}^{\imath}_0(\theta)$: $\tilde{\cal A}^{\imath}_0(\theta)=0$ $\in$ $\Sigma$ $$\begin{aligned}
{} & {} & {\Theta}_{\imath} \bigl(
{\cal A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta), \theta \bigr) =
{\Theta}_{\imath\; {\rm lin}} \bigl(\tilde{\cal A}(\theta),
{\stackrel{\ \circ}{\tilde{\cal A}}}(\theta),
\theta \bigr) + {\Theta}_{\imath\;{\rm nl}} \bigl( \tilde{\cal A}(\theta),
{\stackrel{\ \circ}{\tilde{\cal A}}} (\theta),
\theta \bigr)\,,\nonumber \\
{} & {} & \Bigl(\min{\rm deg}_{\tilde{\cal A}(\theta){\stackrel{\
\circ}{\tilde{\cal A}}}(\theta)}, {\rm deg}_{\tilde{{\cal
A}}(\theta){\stackrel{\ \circ}{\tilde{\cal
A}}}(\theta)}\Bigr){\Theta}_{\imath\;{\rm lin}}(\theta) = (1, 1),\ \min{\rm
deg}_{\tilde{{\cal A}}(\theta){\stackrel{\
\circ}{\tilde{\cal A}}}(\theta)} {\Theta}_{\imath\;{\rm nl}}(\theta) \geq 2
\,.$$ Whereas the assumption $2$ gives the possibility to present $\Theta_{\imath
}(\theta)$ in the form of two special subsystems in formal ignoring of the requirements of locality and covariance with respect to index $\imath$ relative to restriction of the superfield representation $T$ onto subsupergroup $\bar{J}: T_{\vert\bar{J}}$.
Reduction of DCLF to equivalent system of the 1st order on $\theta$ ODE in GNF immediately follows from the Theorem 1 application \[1\] in the form of
A nondegenerate parametrization for ${\cal A}^{\imath}(\theta)$ exists $$\begin{aligned}
{} & {} & \hspace{-4em}{{\cal A}}^{\imath}(\theta) = \bigl(
\delta^{\bar{\imath}}(\theta), \beta^{\underline{\imath}} (\theta),
\xi^{\alpha}(\theta)\bigr) \equiv \bigl(\varphi^{A}(\theta), \xi^{
\alpha}(\theta)\bigr),\ \imath = (\bar{\imath}, {\underline{\imath}}, \alpha)
\equiv (A, \alpha),\ \bar{\imath} = 1,\ldots, n - \underline{m}, \nonumber \\
{} & {} & \hspace{-4em} {\underline{\imath}} = n- \underline{m}+1, \ldots,
n-m,\;\underline{m}=(\underline{m}_+ , \underline{m}_-),\
A=1,\ldots,n-m,\;\alpha = n-m+1,\ldots,n, $$ the such that $\Theta_{\imath}(\theta)$ (3.3) are equivalent to the system of independent ODE in GNF $$\begin{aligned}
{\stackrel{\;\circ}{\delta}}{}^{\bar{\imath}}(\theta) =
\phi^{\bar{\imath}}
\bigl( \delta(\theta),{\stackrel{\;\circ}{\xi}}(\theta), \xi(\theta),
\theta\bigr),\ \;
{\beta}^{\underline{\imath}}(\theta) = \kappa^{\underline{\imath}}
\bigl( \delta(\theta), \xi(\theta), \theta\bigr)\,,
$$ with $\phi^{\bar{\imath}}(\theta), \kappa^{\underline{\imath}}(\theta)\in
C^k(T_{odd}{\cal M}_{cl}\times\{\theta\})$ and arbitrary superfields $\xi^{\alpha}(\theta)$: $[\xi^{\alpha}(\theta)]$ = $m
<n$. Their $(\xi^{\alpha}(\theta))$ number coincides with one of differential identities among $\Theta_{\imath}(\theta)$ $$\begin{aligned}
{}\hspace{-1em}\int
\hspace{-0.4em} d\theta {}\displaystyle\frac{\delta Z[{\cal A} ]}{\delta
{\cal A}^{\imath}(\theta)}{}{\hat{{\cal R}}}^{\imath}_{\alpha} \bigl(\theta;
{\theta}' \bigr)\hspace{-0.1em}= \hspace{-0.1em}
0,\ (\varepsilon_{P}, \varepsilon_{\bar{J}}, \varepsilon) {\hat{{\cal
R}}}^{\imath}_{\alpha}(\theta; {\theta}') \hspace{-0.1em}=\hspace{-0.1em} (1
\hspace{-0.1em}+\hspace{-0.1em}
(\varepsilon_P)_{\imath}, (\varepsilon_{\bar{J}})_{\imath}\hspace{-0.1em}+
\hspace{-0.1em}\varepsilon_{\alpha} , \varepsilon_{\imath} \hspace{-0.1em}+
\hspace{-0.1em}\varepsilon_{\alpha} \hspace{-0.1em}+\hspace{-0.1em}1)
$$ with a) local and b) functionally independent operators ${\hat{\cal R}}^{\imath}_{\alpha}
\bigl({\cal A}(\theta)$, ${\stackrel{\ \circ}{\cal A}}(\theta),
\theta;\theta'\bigr)$ $\equiv$ ${\hat{\cal R}}^{\imath}_{\alpha}(\theta;
{\theta}')$: $$\begin{aligned}
{}\hspace{-6.5cm} {\rm a)}
\hspace{3.0cm}{\hat{\cal R}}^{\imath}_{\alpha}(\theta; {\theta}') =
\sum\limits_{k=0}^{1} \left(\left(\displaystyle\frac{d}{d\theta} \right)^k
\delta(\theta - \theta')\right)
{\hat{ {\cal R}}}_{k}{}^{\imath}_{\alpha}
\bigl( {\cal A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta), \theta\bigr),\
\\ \hspace{-3cm}
\phantom{\rm a)} \hspace{2.5cm}(\varepsilon_{P}, \varepsilon_{\bar{J}},
\varepsilon) {\hat{{\cal R}}}_{k}{}^{\imath}_{ \alpha}(\theta) =
(\delta_{1k} +
(\varepsilon_P)_{\imath}, (\varepsilon_{\bar{J}})_{\imath}+
\varepsilon_{\alpha} ,
\varepsilon_{\imath}+ \varepsilon_{\alpha} + \delta_{1k}),\;k=0,1\;, $$ b) functional equation $$\begin{aligned}
\int d\theta' {\hat{{\cal R}}}^{\imath}_{\alpha}(\theta;
{\theta}') u^{\alpha}\bigl( {\cal A}(\theta'), {\stackrel{\
\circ}{\cal A}}(\theta'), \theta'\bigr) = 0,\ u^{\alpha}(\theta) \in
C^k(T_{odd}{\cal M}_{cl}\times\{\theta\}) $$ has the unique vanishing solution.
It literally follows from Theorem 2, after change of corresponding symbols, the consequence being analogous to Corollary 1 for HCLF $\Theta_{\imath}({\cal A}(\theta),\theta)$ with ${\cal R}_{0}{}^{\imath}_{ \alpha}({\cal A}(\theta),\theta)$ = ${\hat{
\cal R}}_{0}{}^{\imath}_{\alpha}({\cal A}(\theta),
\theta)$ \[1\].
The interpretation for operators ${\hat{\cal R}}^{\imath}_{\alpha}(\theta;{\theta}')$, ${\cal R}_{0}{}^{\imath}_{ \alpha}({\cal A}(\theta),\theta)$ as the GGTGT, GGTST respectively had been given in Ref.\[1\]. It had been shown the complete sets of the GGTGT, GGTST appear by the bases in affine $C^k(T_{odd}{\cal M}_{cl}\times\{\theta\})$-module $Q(Z)$ = ${\rm Ker}\{
\frac{\delta_l Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{\imath}(\theta)}\}$ and affine $C^k({\cal M}_{cl}\times\{\theta\})$-module $Q(S_L)$ = ${\rm Ker}\{\Theta_{\imath}({\cal A}(\theta),\theta)\}$ respectively. By realization of the mentioned consequence for Theorem 2 it appears the Corollary 2.2 from Ref.\[1\] in the framework of which a GSTF model is the almost natural system $$\begin{aligned}
{} & S_L \bigl( {\cal A}(\theta), {\stackrel{\
\circ}{\cal A}}(\theta), \theta\bigr) = T \bigl( {\cal A}(\theta),
{\stackrel{\ \circ}{\cal A}}(\theta)\bigr) - S\bigl( {\cal A}(\theta),
\theta\bigr),\ \min{\rm deg}_{{\cal A}(\theta)}S(\theta) = 2 \;, & {}
\\ {} & T\bigl( {\cal A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta)\bigr)
= T_1\bigl({\stackrel{\ \circ}{\cal A}}(\theta)\bigr) +
{\stackrel{\;\circ}{{\cal A}^{\jmath}}}(\theta) T_{\jmath}\bigl( {\cal A}
(\theta)\bigr),\ T_{\jmath}\bigl( {\cal A}(\theta)\bigr) = g_{\jmath
\imath}(\theta){\cal A}^{\imath} (\theta)\ , & {} \nonumber\\
{} & g_{\jmath
\imath}(\theta) = (-1)^{\varepsilon_{\jmath}\varepsilon_{\imath}}g_{\imath
\jmath}(\theta),\ g_{\jmath \imath}(\theta) = P_0(\theta)g_{\jmath
\imath}(\theta),\ \min{\rm deg}_{{\stackrel{\;\circ}{\cal
A}}(\theta)}T_1(\theta) = 2\,, & {} $$ so that the HCLF and condition (3.7) have the form $$\begin{aligned}
{\Theta}_{\imath}
\bigl({\cal A}(\theta), \theta \bigr) = - S,_{\imath} \bigl( {{\cal
A}}(\theta), \theta \bigr)(-1)^{ \varepsilon_{\imath}} = 0,\
{\rm rank}_{\varepsilon_{\bar{J}}}\left\|S,_{\imath
\jmath} \bigl( {{\cal A}}(\theta),\theta)\right\|_{\textstyle \mid \Sigma}
= n-m\,, $$ Corresponding GTGT, GTST have the form \[1\] $$\begin{aligned}
{} &
{\cal A}^{\imath}(\theta) \mapsto {\cal A}'^{\imath
}(\theta) = {\cal A}^{\imath}(\theta) + \delta_g{\cal A}^{\imath}(\theta);\
\delta_g{\cal A}^{\imath}(\theta) = \displaystyle\int d\theta'
\hat{\cal R}^{\imath}_{\alpha}(\theta;\theta')\xi^{\alpha}(\theta')
\,,{} &
\\{} & {\cal A}^{\imath}(\theta) \mapsto {\cal A}'^{\imath}(\theta) =
{\cal A}^{\imath}(\theta) + \delta{\cal A}^{\imath}(\theta);\
\delta{\cal A}^{\imath}(\theta) =
{\cal R}_0{}^{\imath}_{\alpha}({\cal A}(\theta),\theta)\xi_0^{\alpha}(
\theta){} &
$$ and appear by infinitesimal invariance transformations with arbitrary superfunctions $\xi^{\alpha}(\theta)$, $\xi_0^{\alpha}(\theta)$ \[$(\varepsilon_{P}, \varepsilon_{\bar{J}}, \varepsilon)\xi^{\alpha}(\theta)$ = $(0, \varepsilon_{\alpha}, \varepsilon_{\alpha})$\] for $
Z[{\cal A}]$, $S({\cal A}(\theta), \theta)$ respectively.
In addition for local with respect to $z^a$ models the GGTGT, GGTST can be represented by the local differential operators with respect to $z^a$. At least in ignoring of the requirement of covariance on index $\imath$ the GGTGT, GGTST, as it follows from Theorem 2, appear by independent and hence define the irreducible GSTF models in the Lagrangian formulation, called the irreducible GThGT and GThST (gauge theory of the special type) respectively \[1\]. In general case the conservation of mentioned conditions on locality and covariance for ${\hat{\cal R}}^{\imath}_{\alpha}(\theta; {\theta}')$, ${\cal R}_{0}{}^{\imath}_{ \alpha}(\theta)$ leads to modification of the Theorem 2 conclusion concerning of the solution for Eq.(3.14) and its analog for ${\cal R}_{0}{}^{\imath}_{ \alpha}(\theta)$. Namely, for the last equations the nonvanishing solutions may exist as well. By definition the GThGT (GThST) with the such property are called the reducible GThGT (GThST) with functionally dependent GGTGT (GGTST).
Gauge Algebra of GGTGT
======================
Following to Refs.\[6,7\] let us investigate the GTGT (3.18) and algebraic structures connected with them. From GTGT in a more than one-valued form one can construct the finite transformations of invariance for $Z[{\cal A}]$ $$\begin{aligned}
{} & {\cal A}^{\imath}(\theta) \mapsto {\cal A}^{\imath}_f(\theta)=
G^{\imath}({\cal A}(\theta)\vert \xi(\theta)),\
G^{\imath}({\cal A}(\theta)\vert 0)={\cal A}^{\imath}(\theta)\,, {} &
\\{} & \hspace{-0.5em}\displaystyle\frac{\delta_l G^{\imath}({
\cal A}(\theta)\vert
\xi(\theta))}{\delta\xi^{\alpha}(\theta')\phantom{xxxxxx}}{\hspace{-0.5em}
\phantom{\Bigr)}}_{\mid\xi^{\alpha}(\theta)=0}=
\hat{\cal R}^{\imath}_{\alpha}(\theta;\theta'):
Z[{\cal A}_f] = Z[{\cal A}]
,\,(\varepsilon_{P}, \varepsilon_{\bar{J}}, \varepsilon)\displaystyle\frac{
\delta_l\phantom{xxx}}{\delta\xi^{\alpha}(\theta)}\hspace{-0.1em}=
\hspace{-0.1em}
(1, \varepsilon_{\alpha}, \varepsilon_{\alpha}+1)\,.{} & $$ As $G^{\imath}(\theta)$ one can make use, for instance, the superfields satisfying to the $\theta$-superfield condition \[6\] $$\begin{aligned}
\frac{\partial }{\partial\tau}G^{\imath}\bigl({\cal A}(\theta)\vert\tau
\xi(\theta)\bigr)=
\int d\theta' \xi^{\alpha}(\theta'){\hat{{\cal R}}}^{\imath}_{\alpha}(\theta;
{\theta}')_{\mid
{\cal A}^{\imath}(\theta) =
G^{\imath}({\cal A}(\theta)\mid\tau\xi(\theta))},\ \tau \in
{\bf R}\,.
$$ Really, having denoted $Z_{\tau}\equiv Z[{\cal A}]_{\mid{\cal A}=G({\cal A}
\vert\tau\xi)}$, obtain from (3.11), (4.3) the relationships $$\begin{aligned}
\frac{\partial }{\partial\tau}Z_{\tau}=
\int d\theta \frac{\delta Z[{\cal A} ]}{\delta
{\cal A}^{\imath}(\theta)}
\int d\theta' \xi^{\alpha}(\theta'){\hat{{\cal R}}}^{
\imath}_{\alpha}(\theta;{\theta}'){\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid{\cal A}^{\imath}(\theta) =
G^{\imath}({\cal A}(\theta)\vert\tau\xi(\theta))}=0\,,$$ from which it follows $$\begin{aligned}
Z_{\tau=0}=Z_{\tau=1} \Longleftrightarrow Z[{\cal A}]=Z[{\cal A}_f]\,.$$ Formal solution for Eq.(4.3) with initial condition from (4.1) has the form $$\begin{aligned}
{} & G^{\imath}\bigl({\cal A}(\theta)\vert\xi(\theta)\bigr)=
{\rm exp}\bigl\{
\displaystyle\int\hspace{-0.4em} d\theta'\xi^{\alpha}(\theta')\hat{\Gamma}_{
\alpha}(\theta')\bigr\}{\cal A}^{\imath}(\theta)\,,{} & \\{} & \hat{\Gamma}_{\alpha}(\theta')F[{\cal A}]=
\displaystyle\int d\theta
\frac{\delta F[{\cal A} ]}{\delta
{\cal A}^{\imath}(\theta)}{\hat{{\cal R}}}^{\imath}_{\alpha}(\theta;\theta'),\
(\varepsilon_{P}, \varepsilon_{\bar{J}}, \varepsilon)
\hat{\Gamma}_{\alpha}(\theta)= (1, \varepsilon_{\alpha},
\varepsilon_{\alpha}+1),\; F[{\cal A} ] \in C_F \,.$$ Really, for an arbitrary superfunctional $F[{\cal A}]$, having the polynomial series expansion with respect to ${\cal A}^{\imath}(\theta)$, the operatorial formula is valid $$\begin{aligned}
F[G]\equiv F[{\cal A}_f]= {\rm exp}\bigl\{
\int d\theta'\xi^{\alpha}(\theta')\hat{\Gamma}_{\alpha}(\theta')
\bigr\}F[{\cal A}]\,.$$ Choosing $F[G]\equiv G^{\imath}({\cal A}(\theta)\vert\xi(\theta))$ obtain the solution of Eqs.(4.2) in the form (4.6).
From differential consequences of the identities (3.11) $$\begin{aligned}
\left(\int d\theta\Bigl[\Bigl(
\frac{\delta\phantom{xxx}}{\delta{\cal A}^{\imath}(\theta)}
\frac{\delta Z[{\cal A}]\phantom{
x}}{\delta{\cal A}^{\jmath}(\theta'')}\Bigr)
{\hat{{\cal R}}}^{\imath}_{\alpha}(\theta;\theta')(-1)^{\varepsilon_{\jmath}
\varepsilon_{\alpha}} - \frac{\delta Z[{\cal A}]\phantom{
x}}{\delta{\cal A}^{\imath}(\theta)}
\frac{\delta{\hat{{\cal R}}}^{\imath}_{\alpha}(\theta;\theta') }{
\delta{\cal A}^{\jmath}(\theta'')\phantom{xx}}\Bigr]\right)(-1)^{
\varepsilon_{\imath}}=0
$$ it follows the transformation rule for $\frac{\delta Z[{\cal A}]\phantom{}
}{\delta{\cal A}^{\imath}(\theta)}$ under finite GTGT (4.1), (4.6) $$\begin{aligned}
\frac{\delta Z[{\cal A}]\phantom{}}{\delta{\cal A}^{\imath}(\theta)}{
\hspace{-0.5em}\phantom{\Bigr)}}_{\mid {\cal A}^{\imath} =
G^{\imath}({\cal A}\vert\xi)} =
\int d\theta' Q_{\imath}{
}^{\jmath}\bigl({\cal A}(\theta), {\stackrel{\ \circ}{\cal A}}(\theta),
\theta; \theta'\bigr)
\frac{\delta Z[{\cal A}]\phantom{}}{\delta{\cal A}^{\jmath}(\theta')}
$$ with nondegenerate supermatrix $Q_{\imath}{}^{\jmath}(\theta;\theta')$ $\in$ $C^k(T_{odd}{\cal M}_{cl}\times\{\theta,\theta'\})$ in a some neighbourhood of $\xi^{\alpha}(\theta)$ = $0$: $
Q_{\imath}{}^{\jmath}(\theta;\theta')_{\mid\xi(\theta)=0}$ = $\delta(\theta'-
\theta)\delta_{\imath}{}^{\jmath}$.
Investigation of the gauge algebra of GTGT properties is based on the study of properties of the supercommutator of the 1st order differential operators $\hat{\Gamma}_{\alpha}(\theta)$, with respect to variational superfield derivatives on ${\cal A}^{\imath}(\theta)$ (4.7), having $Z[{\cal A}]$ as the eigensuperfunction with zero eigenvalue. By definition, the supercommutator $[\hat{\Gamma}_{\alpha}(\theta_1), \hat{\Gamma}_{
\beta}(\theta_2)]_s$ possesses by the same property as well. Its value in calculating on arbitrary $F[{\cal A}]$ $\in$ $C_F$ is equal to $$\begin{aligned}
{} &
[\hat{\Gamma}_{\alpha}(\theta_1), \hat{\Gamma}_{\beta}(\theta_2)]_s
F[{\cal A}]\hspace{-0.1em}=\hspace{-0.1em}
\hat{\Gamma}_{\alpha}(\theta_1)\left(\hat{\Gamma}_{\beta}(\theta_2)F[{\cal A}]
\right)\hspace{-0.1em} -\hspace{-0.1em}
(-1)^{(\varepsilon_{\alpha}+1)(\varepsilon_{\beta}+1)}\bigl((\alpha,\theta_1)
\hspace{-0.1em}
\longleftrightarrow\hspace{-0.1em} (\beta,\theta_2)\bigr)\hspace{-0.1em} =
\hspace{-0.1em}{} &\nonumber \\
{} &
\displaystyle\int\hspace{-0.2em} d\theta_2'
\displaystyle\frac{\delta F[{\cal A}]\phantom{
x}}{\delta{\cal A}^{\jmath}(\theta_2')}\displaystyle\int\hspace{-0.2em}
d\theta_1'\left(\Bigl(\frac{\delta{\hat{\cal R}}^{\jmath}_{\beta}(\theta_2';
\theta_2)}{\delta{\cal A}^{\imath}(\theta_1')\phantom{xx}}\Bigr)
{\hat{\cal R}}^{\imath}_{\alpha}(\theta_1';\theta_1)\hspace{-0.2em} -
\hspace{-0.2em}
(-1)^{(\varepsilon_{\alpha}+1)(\varepsilon_{\beta}+1)}\bigl((\alpha,\theta_1)
\hspace{-0.2em}\longleftrightarrow \hspace{-0.2em}
(\beta,\theta_2)\bigr) \hspace{-0.2em}\right) {} &
\nonumber \\
{} &
=\hspace{-0.2em}
\displaystyle\int d\theta_2'\displaystyle\frac{\delta F[{\cal A}]\phantom{
x}}{\delta{\cal A}^{\jmath}(\theta_2')}
\hat{y}^{\jmath}_{\beta\alpha}\bigl({\cal A}(\theta_2'), {\stackrel{\
\circ}{\cal A}}(\theta_2'), \theta_2'; \theta_2, \theta_1\bigr)\,,{} &
\\{} & \hspace{-0.7em}
\hat{y}^{\imath}_{\beta\alpha}(\theta_1'; \theta_2, \theta_1)\hspace{-0.2em}=
\hspace{-0.2em}-\hspace{-0.1em} (-1)^{(\varepsilon_{\alpha}+1)(\varepsilon_{
\beta}+1)}\hat{y}^{\imath}_{\alpha\beta}(\theta_1'; \vec{\theta}_2),\,
(\varepsilon_{P}, \hspace{-0.1em}\varepsilon_{\bar{J}}, \hspace{-0.1em}
\varepsilon)\hat{y}^{\imath}_{
\beta\alpha}\hspace{-0.15em}=\hspace{-0.15em}
((\varepsilon_P)_{\imath}, (\varepsilon_{\bar{J}})_{\imath}\hspace{-0.15em} +
\hspace{-0.15em}\varepsilon_{\alpha}
\hspace{-0.15em}+\hspace{-0.15em} \varepsilon_{\beta},
\varepsilon_{\imath}\hspace{-0.15em} +
\hspace{-0.15em}\varepsilon_{\alpha} \hspace{-0.15em}+ \hspace{-0.15em}
\varepsilon_{\beta}),{} &\nonumber \\
{} & \hat{y}^{\imath}_{\alpha\beta}\bigl({\cal A}(\theta_1'), {\stackrel{\
\circ}{\cal A}}(\theta_1'), \theta_1'; \theta_1,\theta_2\bigr) \equiv
\hat{y}^{\imath}_{\alpha\beta}(\theta_1'; \vec{\theta}_2) \in
C^k(T_{odd}{\cal M}_{cl}\times\{\theta_1',\vec{\theta}_2\}),
\vec{\theta}_k \equiv \theta_1,\ldots,\theta_k\,.{} &
$$ Superfunctions $\hat{y}^{\imath}_{\alpha\beta}(\theta'_1;\vec{\theta}_2)$ appear by local operators of differentiation on $\theta$ (and with respect to $z^a$, if ${\hat{\cal R}}^{\imath}_{\alpha}(\theta; {\theta}')$ are the same). By virtue of completeness of the GGTGT $
{\hat{\cal R}}^{\imath}_{\alpha}(\theta; {\theta}')$ the quantities $\hat{y}^{\imath}_{\alpha\beta}$ must be expressed through GGTGT and the trivial GGTGT $\hat{\tau}^{\imath}_{\alpha\beta}(\theta'_1;\vec{\theta
}_2)$ \[1\] $$\begin{aligned}
\hat{y}^{\imath}_{\alpha\beta}(\theta_1';\vec{\theta}_2) =
(-1)^{\varepsilon_{\imath}}\int d\theta_3
{\hat{\cal R}}^{\imath}_{\gamma}(\theta_1';\theta_3)
\hat{\cal F}^{\gamma}_{\alpha\beta}(\theta_3;\vec{\theta}_2) +
\int d\theta_2'\frac{\delta Z[{\cal A}]\phantom{
x}}{\delta{\cal A}^{\jmath}(\theta_2')}
\hat{\cal M}^{\imath\jmath}_{\alpha\beta}(\vec{\theta}{}_2';\vec{\theta}_2)
(-1)^{\varepsilon_{\jmath}}\,.{} & $$ Superfunctions $\hat{\cal F}^{\gamma}_{\alpha\beta}(\theta_3;\vec{\theta
}_2)$, $\hat{\cal M}^{\imath\jmath}_{\alpha\beta}(\vec{\theta}{}'_2;
\vec{\theta}_2)$ $\in$ $C^k\bigl(T_{odd}{\cal M}_{cl}\times\{\vec{\theta}_3,
\vec{\theta}{}'_2\}\bigr)$ possess by the properties $$\begin{aligned}
{} & \begin{array}{l|ccc}
{}& \varepsilon_P & \varepsilon_{\bar{J}} & \varepsilon \\ \hline
\hat{\cal F}^{\gamma}_{\alpha\beta}(\theta_3;\vec{\theta}_2) & 0 &
\varepsilon_{\gamma} + \varepsilon_{\alpha}+ \varepsilon_{\beta} &
\varepsilon_{\gamma} + \varepsilon_{\alpha}+ \varepsilon_{\beta} \\
\hat{\cal M}^{\imath\jmath}_{\alpha\beta}(\vec{\theta}{}'_2;\vec{\theta}_2) &
1+(\varepsilon_P)_{\imath}+(\varepsilon_P)_{\jmath} &
(\varepsilon_{\bar{J}})_{\imath} + (\varepsilon_{\bar{J}})_{\jmath} +
\varepsilon_{\alpha}+
\varepsilon_{\beta} & 1+
\varepsilon_{\imath} + \varepsilon_{\jmath} + \varepsilon_{\alpha}+
\varepsilon_{\beta}
\end{array}, {} & \\{} & \hat{\cal F}^{\gamma}_{\alpha\beta}(\theta_3;\vec{\theta}_2) \equiv
\hat{\cal F}^{\gamma}_{\alpha\beta}\bigl({\cal A}(\theta_3), {\stackrel{\
\circ}{\cal A}}(\theta_3), \theta_3; \vec{\theta}_2\bigr) =
- (-1)^{(\varepsilon_{\alpha}+1)(\varepsilon_{\beta}+1)}
\hat{\cal F}^{\gamma}_{\beta\alpha}(\theta_3;\theta_2,\theta_1)\,,{} &
\nonumber \\
{} &
\hat{\cal M}^{\imath\jmath}_{\alpha\beta}(\vec{\theta}{}'_2;\vec{\theta}_2)
\equiv
\hat{\cal M}^{\imath\jmath}_{\alpha\beta}
\bigl({\cal A}(\theta_1'), {\stackrel{\
\circ}{\cal A}}(\theta_1'), \vec{\theta}{}'_2;\vec{\theta}_2\bigr) =
- (-1)^{(\varepsilon_{\imath}+1)(\varepsilon_{\jmath}+1)}
\hat{\cal M}^{\jmath\imath}_{\alpha\beta}(\theta'_2,\theta'_1;
\vec{\theta}_2) {} & \nonumber\\
{} &
= - (-1)^{(\varepsilon_{\alpha}+1)(\varepsilon_{\beta}+1)}
\hat{\cal M}^{\imath\jmath}_{\beta\alpha}(\vec{\theta}{}'_2;{\theta}_2,
{\theta}_1) {} & $$ and can be chosen by $\theta$-local ones.
The explicit form of $\hat{\cal F}^{\gamma}_{\alpha\beta}$, $\hat{\cal M}^{\imath\jmath}_{\alpha\beta}$ and their properties are based on the analysis of the general solution for equation $$\begin{aligned}
\int d\theta
\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{\imath}(\theta)}
\hat{y}^{\imath}\bigl( {\cal A}(\theta), {\stackrel{\ \circ}{\cal A}
}(\theta),\theta\bigr)=0
\,.$$ :
General solution of the Eq.(4.16) for irreducible GGTGT satisfying to completeness condition has the form in the superalgebra $C^k\bigl(T_{odd}{\cal M}_{cl}\times\{\theta\})$ $$\begin{aligned}
{} &
\hat{y}^{\imath}(\theta) \equiv
\hat{y}^{\imath}\bigl( {\cal A}(\theta), {\stackrel{\ \circ}{\cal
A}}(\theta),\theta\bigr) = \displaystyle\int d\theta'\Bigl[
{\hat{\cal R}}^{\imath}_{\alpha}(\theta;\theta'){\hat{\Phi}}^{\alpha}(\theta'
)(-1)^{\varepsilon_{\imath}} +
\displaystyle\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal
A}^{\jmath}(\theta')}\hat{E}^{\imath\jmath}(\theta,\theta')(-1)^{\varepsilon_{
\jmath}}\Bigr]\,,{} &
\\{} & \hspace{-1.0em}{\hat{\Phi}}^{\alpha}(\theta) \hspace{-0.1em}\equiv
\hspace{-0.1em}{\hat{\Phi}}^{\alpha}
\bigl( {\cal A}(\theta), {\stackrel{\ \circ}{\cal A}
}(\theta),\theta\bigr),\;
\hat{E}^{\imath\jmath}(\theta,\theta') \equiv
\hat{E}{}^{\imath\jmath}\bigl({\cal A}(\theta), {\stackrel{\ \circ}{\cal
A}}(\theta),\theta,\theta'\bigr) = -(-1)^{(\varepsilon_{\imath} +1)(
\varepsilon_{\jmath} +1)}\hat{E}^{\jmath\imath}(\theta',\theta)\,, {} &
\nonumber \\
{} &
\begin{array}{l|cccc}
{} & \varepsilon_P & \varepsilon_{\bar{J}}& \varepsilon & {}\\ \hline
{\hat{\Phi}}^{\alpha}(\theta) &
\varepsilon_{P}(\hat{y}^{\imath})+ (\varepsilon_{P})_{\imath} &
\varepsilon_{\bar{J}}(\hat{y}^{\imath}) +
(\varepsilon_{\bar{J}})_{\imath} + \varepsilon_{\alpha} &
\varepsilon(\hat{y}^{\imath}) + \varepsilon_{\imath} +
\varepsilon_{\alpha} & {} \\
{\hat{E}}^{\imath\jmath}(\theta,\theta') & \varepsilon_{P}(\hat{y}^{
\imath})+ (\varepsilon_{P})_{\jmath}+ 1 &
\varepsilon_{\bar{J}}(\hat{y}^{\imath}) + (\varepsilon_{\bar{J}})_{\jmath} &
\varepsilon(\hat{y}^{\imath}) + \varepsilon_{\jmath} + 1 & .\\
\end{array} $$ Assumption (3.7) permits to represent Eq.(4.16) and identities (3.11) in the form respectively $$\begin{aligned}
{} & {} &
\displaystyle\int d\theta\Bigl[
\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{A}(\theta)}
\hat{y}^{A}(\theta) +
\displaystyle\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal
A}^{\alpha}(\theta)}\hat{y}^{\alpha}(\theta)\Bigr]=0\,,
\\{} & {} &
\displaystyle\int d\theta\Bigl[
\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{A}(\theta)}
\hat{\cal R}^{A}_{\beta}(\theta;\theta') + \displaystyle\frac{\delta
Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{\alpha}(\theta)}
\hat{\cal R}^{\alpha}_{\beta}(\theta;\theta')\Bigr]=0
\,,\\{} & {} &
\hat{\cal R}^{\imath}_{\beta}(\theta;\theta') \equiv \left(\hat{\cal
R}^{A}_{\beta}(\theta;\theta'), \hat{\cal R}^{\alpha}_{\beta}(\theta;\theta')
\right),\
{\rm rank}_{{\varepsilon}_{\bar{J}}}\left\|\hat{\cal R}^{\alpha}_{\beta}(
\theta;\theta')\right\|_{\mid\Sigma}\equiv \nonumber \\
{} & {} &
{\rm rank}_{{\varepsilon}_{\bar{J}}}\left\|\sum\limits_{k\geq 0}{\hat{
\cal R}_k}{}^{\alpha}_{\beta}(
\theta)\left(\frac{d}{d\theta}\right)^k\right\|_{\mid\Sigma}\delta(\theta-
\theta'):
{\rm rank}_{{\varepsilon}_{\bar{J}}}\left\|\sum\limits_{k\geq 0}{
\hat{\cal R}_k}{}^{
\alpha}_{\beta}(\theta)\left(\frac{d}{d\theta}\right)^k\right\|_{\mid\Sigma}=m
\,,\\{} & {} & \exists \bigl(\hat{\cal R}^{-1}\bigr)^{\beta}{}_{\gamma}(\theta;
\theta_1): \,\displaystyle\int d\theta'
\hat{\cal R}^{\alpha}_{\beta}(\theta;\theta')
\bigl(\hat{\cal R}^{-1}\bigr)^{\beta}{}_{\gamma}(\theta';\theta_1)=\delta^{
\alpha}{}_{\gamma}\delta(\theta-\theta_1)\,.$$ From (4.19)–(4.22) it follows the equivalent representation for (3.11), (4.16) $$\begin{aligned}
{} & \displaystyle\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal
A}^{\gamma}(\theta_1)} = \displaystyle\int d\theta' d\theta
\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{B}(\theta)}
\hat{\cal R}^{B}_{\alpha}(\theta;\theta')
\bigl(\hat{\cal R}^{-1}\bigr)^{\alpha}{}_{\gamma}(\theta';\theta_1)
\,,{} & \\{} & \displaystyle\int d\theta d\theta_1
\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal
A}^{A}(\theta_1)} \hat{z}^A(\theta,\theta_1)(-1)^{\varepsilon_{A}}=0\,,{} &
\nonumber \\
{} & \hat{z}^A(\theta,\theta_1) = \delta(\theta_1-\theta)\hat{y}^A(\theta_1)
- \displaystyle\int d\theta'\hat{\cal R}^{A}_{\alpha}(\theta_1;\theta')
\bigl(\hat{\cal R}^{-1}\bigr)^{\alpha}{}_{\gamma}(\theta';\theta)
\hat{y}^{\gamma}(\theta)\,.{} & $$ Condition (3.7) guarantees the existence of the special parametrization for ${\cal A}^{\imath}(\theta)$ $$\begin{aligned}
{\cal A}^{\imath}(\theta)\mapsto \tilde{\cal A}{}^{\imath}(\theta)=
\left(\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal A}^{A}(\theta)},
{\cal A}^{\alpha}(\theta)\right)\equiv ({\cal F}_A(\theta),
{\cal A}^{\alpha}(\theta))\,,$$ in terms of which Eq.(4.24) is written in the form $$\begin{aligned}
\int d\theta d\theta_1{\cal F}_A(\theta_1)
\hat{z}^A\bigl(\tilde{\cal A}(\theta),{\stackrel{\ \circ}{\tilde{\cal A}}}(
\theta), \theta,\theta_1\bigr)(-1)^{\varepsilon_{A}}=0\,.$$ Calculating the variational superfield derivative of expression (4.26) with respect to ${\cal F}_B(\theta_2)$ we obtain $$\begin{aligned}
{} & \hat{z}^B(\theta,\theta_2) + \displaystyle\int d\theta_1
{\cal F}_A(\theta_1)
\displaystyle\frac{\delta_l \hat{z}^B(\theta,\theta_2)}{\delta{\cal
F}_{A}(\theta_1)\phantom{x}} =
\displaystyle\int d\theta_1{\cal F}_A(\theta_1){\cal P}^{BA}(
\theta;{\theta}_2,{\theta}_1)\,,\nonumber \\
{} & \displaystyle\frac{\delta_l {\cal F}_B(\theta_1)}{\delta{\cal
F}_{C}(\theta)\phantom{x}} = \delta_B{}^C\delta(\theta-\theta_1)(-1)^{
\varepsilon_B}\,,{} &\\{} & {\cal P}^{BA}(\theta;{\theta}_2,{\theta}_1)\equiv
{\cal P}^{BA}\bigl(\tilde{\cal A}(\theta),{\stackrel{\ \circ}{\tilde{\cal A}}
}(\theta), \theta;{\theta}_2,{\theta}_1)=
\displaystyle\frac{\delta_l \hat{z}^B(\theta,\theta_2)}{\delta{\cal
F}_{A}(\theta_1)\phantom{x}} - (-1)^{(\varepsilon_A +1)(\varepsilon_B +1)}
\times{} & \nonumber \\
{} & \left((A,\theta_1)\leftrightarrow (B,\theta_2)\right),\
{\cal P}^{AB}(\theta;\vec{\theta}_2) =
- (-1)^{(\varepsilon_A +1)(\varepsilon_B +1)}{\cal P}^{BA}(
\theta;{\theta}_2,{\theta}_1)\,.$$ After scaled transformation ${\cal F}_A(\theta)\mapsto\tau{\cal F}_A(\theta)$, $\tau\in [0,1]$ $\subset {\bf R}$ in (4.27) this equation will pass into system of the 1st order on $\tau$ ODE $$\begin{aligned}
{} & \displaystyle\frac{d}{d\tau}\left(
\tau\hat{z}^B\bigl(
\tau{\cal F}(\theta), {\cal A}^{\alpha}(\theta),
\tau{\stackrel{\,\circ}{\cal F}}(\theta),{\stackrel{\ \circ}{\cal A}
}{}^{\alpha}(\theta), \theta,\theta_1\bigr)\right)=
\displaystyle\int d\theta_2 \tau{\cal F}_A(\theta_2) \times \nonumber \\
{} &{\cal P}^{BA}\bigl(
\tau{\cal F}(\theta), {\cal A}^{\alpha}(\theta),
\tau{\stackrel{\,\circ}{\cal F}}(\theta),{\stackrel{\ \circ}{\cal A}
}{}^{\alpha}(\theta),\theta;\vec{\theta}_2\bigr)\,.{} & $$ By direct integration of Eq.(4.29) with respect to $\tau$ along the segment $[0,1]$ we obtain (the integral is regarded as improper one) $$\begin{aligned}
{} & \hat{z}^B\bigl(\tilde{\cal A}(\theta),{\stackrel{\ \circ}{\tilde{\cal
A}}}(\theta), \theta,\theta_1\bigr) - \lim\limits_{\tau\to 0}\tau
\hat{z}^B\bigl(
\tau{\cal F}(\theta), {\cal A}^{\alpha}(\theta),
\tau{\stackrel{\circ}{\cal F}}(\theta),{\stackrel{\ \circ}{\cal A}
}{}^{\alpha}(\theta),\theta,\theta_1\bigr) = {} & \nonumber \\
{} &
\displaystyle\int d\theta_2{\cal F}_A(\theta_2)
\displaystyle\int\limits_0^1 d\tau \tau
{\cal P}^{BA}\bigl(
\tau{\cal F}(\theta), {\cal A}^{\alpha}(\theta),
\tau{\stackrel{\,\circ}{\cal F}}(\theta),{\stackrel{\ \circ}{\cal A}
}{}^{\alpha}(\theta),\theta;\vec{\theta}_2\bigr)\,.{} & $$ The boundedness of the solution for Eq.(4.16) near ${\cal F}_A(\theta)$ = $0$ and existence of the integral from right-hand side by hypothesis of the Lemma mean the limit in the left of (4.30) is equal to $0$ and the general solution for (4.16) taking account of (4.24) has the form $$\begin{aligned}
{} & \hat{y}^A(\theta) = \displaystyle\int d\theta_1
\left[\displaystyle\int d\theta'\hat{\cal R}^{A}_{\alpha}(\theta;\theta')
\bigl(\hat{\cal R}^{-1}\bigr)^{\alpha}{}_{\gamma}(\theta';\theta_1)
\hat{y}^{\gamma}(\theta_1)-\hat{z}^A(\theta_1,\theta)\right]= {} &
\nonumber\\
{} & \hspace{-1em}\displaystyle\int d\theta_1\left[
{\hat{\cal R}}^{A}_{\alpha}(\theta;\theta_1)
{\hat{\Phi}}^{\alpha}\bigl({\cal A}(\theta_1),{\stackrel{\ \circ}{{\cal
A}}}(\theta_1),\theta_1\bigr)
(-1)^{\varepsilon_{A}} +
\displaystyle\frac{\delta Z[{\cal A}]\phantom{x}}{\delta{\cal
A}^{B}(\theta_1)}
\hat{E}^{AB}\bigl({\cal A}(\theta),{\stackrel{\ \circ}{{\cal
A}}}(\theta),\theta,\theta_1\bigr)(-1)^{\varepsilon_{B}}\right],{} &
\\{} & \hat{E}^{AB}\bigl({\cal A}(\theta),{\stackrel{\ \circ}{{\cal
A}}}(\theta),\theta,\theta_1\bigr)=
\displaystyle\int d\theta_2\int\limits_0^1 d\tau \tau {\cal P}^{AB}\bigl(
\tau{\cal F}(\theta_2), {\cal A}^{\alpha}(\theta_2),
\tau{\stackrel{\,\circ}{\cal F}}(\theta_2),{\stackrel{\ \circ}{\cal A}
}{}^{\alpha}(\theta_2),\theta_2;{\theta},{\theta}_1\bigr)\,,{} &
\\{} &
{\hat{\Phi}}^{\alpha}\bigl({\cal A}(\theta),{\stackrel{\ \circ}{{\cal
A}}}(\theta),\theta\bigr) = (-1)^{\varepsilon_{\alpha}}
\displaystyle\int d\theta'
\bigl(\hat{\cal R}^{-1}\bigr)^{\alpha}{}_{\gamma}(\theta;\theta')
\hat{y}^{\gamma}(\theta') {} &$$ with arbitrary superfunctions $\hat{y}^{\gamma}(\theta)$.
Setting $$\hat{E}^{\alpha B}(\vec{\theta}_2) = \hat{E}^{A\beta}(\vec{\theta}_2) =
\hat{E}^{\alpha\beta}(\vec{\theta}_2) = 0 $$ we arrive to validity of the formula (4.17) with the properties (4.18). :
[**1)**]{} Let us call the identities (3.11), expressions for supercommutator GGTGT (4.13) the of the 1st and 2nd orders respectively of the . Call the superfunctional $Z[{\cal A}]$; superfunctions ${\hat{\cal R}}^{\imath}_{
\alpha}(\theta; {\theta}')$; $\hat{\cal F}^{\gamma}_{\alpha\beta}(\theta$; $\vec{\theta}_2)$, $
\hat{\cal M}^{\imath\jmath}_{\alpha\beta}(\vec{\theta}{}'_2$ ;$\vec{\theta
}_2)$ by the of zero; 1st; 2nd orders respectively. The set of quantities $Z[{\cal A}]$, ${\hat{\cal R}}^{\imath}_{\alpha}$, $\hat{\cal F}^{\gamma}_{\alpha\beta}$, $\hat{\cal M}^{\imath\jmath}_{
\alpha\beta}$ and so on together with structural equations let us call the gauge algebra of GTGT on $Q(Z)$. Thus, the order of the structural superfunction and equation is equal to the number of free lower indices in the nonzero function and equation respectively.
[**2)**]{} The rank $R$ of the gauge algebra of GTGT, by definition, is given by the maximal number of free upper indices for the such structural superfunction from the set of all structural superfunctions that corresponding numbers for other elements of this set not greater than given one $(R\in {\bf Z}$, $R \leq \infty)$. For $R=0$ the GSTF model appears by nondegenerate theory of general type (ThGT) \[1\].
Further structural equations and superfunctions of the gauge algebra are deduced from systematic use of the definitions of GThGT, Lemma 1 and all preceding structural equations and functions including their differential consequences in analyzing of supercommutators of the form $[\hat{\Gamma}_{\alpha_1}(
\theta_1)$, $[\hat{\Gamma}_{\alpha_2}(\theta_2)$, $[\ldots[\hat{\Gamma}_{
\alpha_{k-1}}(\theta_{k-1})$, $\hat{\Gamma}_{\alpha_{k}}(\theta_{k})]
\ldots]]]$, $k\geq 3$. This investigation remains out the paper’s scope. Let us only point out the maximal numbers of different with respect to set of upper indices on the structural functions and equations in the fixed $k$th order of the gauge algebra of GTGT are equal to $[\frac{k}{2}]+1$ and $
[\frac{k+1}{2}]$ respectively.
Note the non-invariance of the definition for structural superfunctions and rank of gauge algebra because of GGTGT are defined by ambiguously \[1\] with accuracy up to equivalence transformations and in view of the fact that the form of structural functions and equations depends on a choice of parametrization for superfields ${\cal A}^{\imath}(\theta)$.
Connection with Gauge Algebra of Irreducible
=============================================
It had been shown in \[1\] the $Q(S_L)$ appears by the $C^k({\cal M}_{cl}\times\{\theta,
\theta'\})$-submodule of the affine $C^k(T_{odd}{\cal M}_{cl}\times
\{\theta,\theta'\})$-module $Q(Z)$. By analogy with Sec.4 one can deduce the basic relationships by means of the literal change of corresponding symbols and operations and find the quantities defining a gauge algebra for irreducible GThST on $Q(S_L)$ $\equiv$ $Q(S)$ $\equiv$ ${\rm Ker}\{S,_{
\imath}(\theta)\}$ with $S(\theta)$ (3.15) satisfying to (3.17). Namely, the identities (3.11), the finite invariance transformations for $S({\cal A}(\theta), \theta)$ constructed from infinitesimal GTST (3.19) analogously to scheme of Sec.4 (relationships (4.1)–(4.8)) and transformation rule for HCLF $\Theta_{\imath}({\cal A}(\theta),\theta)$ under finite GTST have the form respectively
$$\begin{aligned}
{} & {} & \hspace{-2.5em}
{S,}_{\imath}(\theta){{\cal R}_0}^{\imath}_{\alpha}(
{\cal A}(\theta),\theta)=0\,,\\{} & {} & \hspace{-2.5em}
{\cal A}^{\imath}(\theta) \mapsto {\cal A}^{\imath}_{fin}(\theta)=
G^{\imath}_0({\cal A}(\theta)\vert \xi(\theta)),\
G^{\imath}_0({\cal A}(\theta)\vert 0)={\cal A}^{\imath}(\theta)\,,\\{} & {} & \hspace{-2.5em}
\displaystyle\frac{\partial_l G^{\imath}_0({\cal A}(\theta)\vert
\xi(\theta))}{\partial\xi^{\alpha}(\theta)\phantom{xxxxxxx}}{\hspace{-0.5em}
\phantom{\Bigr)}}_{\mid\xi^{\alpha}(\theta)=0}=
{{\cal R}_0}^{\imath}_{\alpha}({\cal A}(\theta),\theta)
\,,\\{} & {} & \hspace{-2.5em}
S({\cal A}_{fin}(\theta), \theta) = S({\cal A}(\theta),\theta)
\,,\\{} & {} & \hspace{-2.5em}
G^{\imath}_0({\cal A}(\theta)\vert \xi(\theta)) = {\rm exp}\bigl\{
\xi^{\alpha}(\theta)\Gamma_{0{}\alpha}(\theta)\bigr\}{\cal A}^{\imath}(
\theta)\,,\\{} & {} &\hspace{-2.5em}
\Gamma_{0{}\alpha}(\theta){\cal F}({\cal A}(\theta),\theta)=
{{\cal F},}_{\imath}({\cal A}(\theta),\theta)
{\cal R}_0{}^{\imath}_{\alpha}({\cal A}(\theta),\theta),\
(\varepsilon_P,\varepsilon_{\bar{J}},\varepsilon)\Gamma_{0{}\alpha}(\theta) =
(0,\varepsilon_{\alpha},\varepsilon_{\alpha})\,,\\{} & {} &\hspace{-2.5em}
{\cal F}_{\imath}(G_0(\theta),\theta) \equiv {\cal F}_{\imath}({\cal
A}_{fin}(\theta),\theta) = {\rm exp}\bigl\{\xi^{\alpha}(\theta)
\Gamma_{0{}\alpha}(\theta)\bigr\}{\cal F}({\cal A}(\theta),\theta),\;
{\cal F}(\theta) \in C^k({\cal M}_{cl}\times\{\theta\})\,,\\{} & {} &\hspace{-2.5em}
{S,}_{\imath}(\theta){\hspace{-0.5em}\phantom{\Bigr)}}_{\mid
{\cal A}^{\imath}(\theta) =
G^{\imath}_0({\cal A}(\theta)\vert \xi(\theta))}\hspace{-0.1em}
=\hspace{-0.1em} Q_0{}_{\imath}{}^{\jmath}({\cal A}(\theta),\theta)
{S,}_{\jmath}(\theta),\;{\rm sdet}\left\|Q_0{}_{\imath}{}^{\jmath}(\theta)
\right\|\hspace{-0.1em}\ne\hspace{-0.1em} 0,
Q_0{}_{\imath}{}^{\jmath}(\theta){\hspace{-0.5em}\phantom{\Bigr)}}_{
\mid\xi(\theta)=0}\hspace{-0.1em}=
\hspace{-0.1em}\delta_{\imath}{}^{\jmath}.$$
Vector fields $\Gamma_{0{}\alpha}(\theta)$ on ${\cal M}_{cl}\times
\{\theta\}$, annulling $S({\cal A}(\theta),\theta)$, play the role of operators $\hat{\Gamma}_{\alpha}(\theta)$ in on $T_{odd}{\cal M}_{cl}
\times \{\theta\}$. Their supercommutator possesses by the same property that according to (4.11)–(4.15) means respectively $$\begin{aligned}
{} & [\Gamma_{0{}\alpha}(\theta),\Gamma_{0{}\beta}(\theta)]_s
{\cal F}(\theta) = {{\cal F},}_{\jmath}(\theta)\Bigl(
{\cal R}_{0}{}^{\jmath}_{\beta},_{\imath}(\theta)
{\cal R}_{0}{}^{\imath}_{\alpha}(\theta) - (-1)^{\varepsilon_{
\alpha}\varepsilon_{\beta}}(\alpha \leftrightarrow \beta)\Bigr)\,,{}&\\ {} &
{\cal R}_{0}{}^{\jmath}_{\beta},_{\imath}(\theta)
{\cal R}_{0}{}^{\imath}_{\alpha}(\theta) - (-1)^{\varepsilon_{
\alpha}\varepsilon_{\beta}}(\alpha \leftrightarrow \beta)=
- {\cal R}_{0}{}^{\jmath}_{\gamma}(\theta)
{\cal F}_{0}{}^{\gamma}_{\beta\alpha}({\cal A}(\theta),\theta) -
S,_{\imath}(\theta)
{\cal M}_{0}{}^{\jmath\imath}_{\beta\alpha}({\cal A}(\theta),\theta)
\,,{} & \nonumber \\
{} &
{\cal F}_{0}{}^{\gamma}_{\beta\alpha}({\cal A}(\theta),\theta)\equiv
{\cal F}_{0}{}^{\gamma}_{\beta\alpha}(\theta),
{\cal M}_{0}{}^{\jmath\imath}_{\beta\alpha}({\cal A}(\theta),\theta)\equiv
{\cal M}_{0}{}^{\jmath\imath}_{\beta\alpha}(\theta) \in
C^k({\cal M}_{cl}\times\{\theta\})\,, {} &
\\{} & \begin{array}{l|cccl}
{}& \varepsilon_P & \varepsilon_{\bar{J}} & \varepsilon &{}\\ \hline
{\cal F}_{0}{}^{\gamma}_{\alpha\beta}(\theta) & 0 &
\varepsilon_{\gamma} + \varepsilon_{\alpha}+ \varepsilon_{\beta} &
\varepsilon_{\gamma} + \varepsilon_{\alpha}+ \varepsilon_{\beta} & {}\\
{\cal M}_{0}{}^{\imath\jmath}_{\alpha\beta}(\theta) & (\varepsilon_P)_{\imath}
+ (\varepsilon_P)_{\jmath} &
(\varepsilon_{\bar{J}})_{\imath} + (\varepsilon_{\bar{J}})_{\jmath} +
\varepsilon_{\alpha} + \varepsilon_{\beta} &
\varepsilon_{\imath} + \varepsilon_{\jmath} + \varepsilon_{\alpha}+
\varepsilon_{\beta} & ,
\end{array} {} &
\\{} & \hspace{-1.5em} {\cal F}_{0}{}^{\gamma}_{\alpha\beta}(\theta) = -
(-1)^{\varepsilon_{\alpha} \varepsilon_{\beta}}{\cal F}_{0}{}^{\gamma}_{
\beta\alpha}(\theta),\
{\cal M}_{0}{}^{\imath\jmath}_{\alpha\beta}(\theta) = -
(-1)^{\varepsilon_{\imath} \varepsilon_{\jmath}}
{\cal M}_{0}{}^{\jmath\imath}_{\alpha\beta}(\theta) = -
(-1)^{\varepsilon_{\alpha} \varepsilon_{\beta}}
{\cal M}_{0}{}^{\imath\jmath}_{\beta\alpha}(\theta).{} &$$ The explicit form of the superfunctions ${\cal F}_0{}^{\gamma}_{\alpha\beta}(
\theta)$, ${\cal M}_0{}^{\imath\jmath}_{\alpha\beta}(\theta)$ and their properties are based on the being easily proved analog of Lemma 1 in question.
:
General solution of the equation $$\begin{aligned}
{S,}_{\imath}(\theta)y_0^{\imath}({\cal A}(\theta),\theta)=0$$ for irreducible GGTST satisfying to condition of completeness has the form in $C^k({\cal M}_{cl}\times\{\theta\})$ $$\begin{aligned}
{} & y_0^{\imath}(\theta)\equiv y_0^{\imath}({\cal A}(\theta),\theta) =
{\cal R}_{0}{}^{\imath}_{\gamma}({\cal A}(\theta),\theta)\Phi_0^{\gamma}({\cal A}(\theta),\theta) + {S,}_{\jmath}(\theta)E_0^{\imath\jmath}({\cal
A}(\theta),\theta)\,,{} & \\{} & E_0^{\imath\jmath}({\cal A}(\theta),\theta)\equiv
E_0^{\imath\jmath}(\theta)=- (-1)^{\varepsilon_{\imath}\varepsilon_{
\jmath}} E_0^{\jmath\imath}(\theta),\
\Phi_0^{\gamma}({\cal A}(\theta),\theta)\equiv \Phi_0^{\gamma}(\theta)\,,
{} & \nonumber \\
{} & \begin{array}{l|cccl}
{}& \varepsilon_P & \varepsilon_{\bar{J}} & \varepsilon &{}\\\hline
\Phi_{0}^{\gamma}(\theta) & \varepsilon_P(y_0^{\imath}) &
\varepsilon_{\bar{J}}(y_0^{\imath}) + (\varepsilon_{\bar{J}})_{\imath}+
\varepsilon_{\gamma} &
\varepsilon(y_0^{\imath}) + \varepsilon_{\imath} + \varepsilon_{\gamma} & {}\\
E_0^{\imath\jmath}(\theta) & \varepsilon_P(y_0^{\imath})+(\varepsilon_P)_{
\jmath} &
\varepsilon_{\bar{J}}(y_0^{\imath}) + (\varepsilon_{\bar{J}})_{\jmath} &
\varepsilon(y_0^{\imath}) + \varepsilon_{\jmath} & .
\end{array} {} &
$$
Identities (5.1), expression (5.10) are called in correspondence with Sec.4 and Ref.\[7\] by the structural equations of the 1st and 2nd orders of a respectively. Superfunctions $S(\theta)$; ${\cal R}_0{}^{\imath}_{\alpha}(\theta)$; $
{\cal F}_0{}^{\gamma}_{\alpha\beta}(\theta)$, ${\cal M}_0{}^{\imath\jmath}_{\alpha\beta}(\theta)$ are called the structural functions of zero; $1$st; $2$nd orders respectively. The set of $S(\theta)$, ${\cal R}_0{}^{\imath}_{\alpha}(\theta)$, $
{\cal F}_0{}^{\gamma}_{\alpha\beta}(\theta)$, ${\cal M}_0{}^{\imath\jmath}_{\alpha\beta}(\theta)$ and so on together with corresponding structural equations is called the gauge algebra of GTST on $Q(S)$.
The all other concepts and remarks in the end of Sec.4 are literally transferred onto $Q(S)$. A dependence upon ${\stackrel{\ \circ}{\cal A}}{}^{
\imath}(\theta)$ in the structural functions and equations may be only by parametric one. The results of Sec.5 on the gauge algebra of GTST can be obtained from gauge algebra of ordinary (not superfield on $\theta$) irreducible gauge theory \[4,6,7\] by continuation of the component fields $A^{\imath}$ to the superfields ${\cal A}^{\imath}(\theta)$ and simultaneously by deformation on $\theta$ of the all structural functions and equations (in the sense of their explicit dependence on $\theta$).
Gauge algebra of GTST for GThST on the whole can be efficiently described by means of generating equations for superfunction $S(\Gamma_{min}(\theta),\theta)$ $\in$ $C^{k}(T^{\ast}_{odd}{\cal M}_{min}$ $\times$ $\{\theta\})$, $k \le \infty$, $T_{odd}^{\ast}{\cal M}_{min}$ = $\{(\Phi_{min}^A(\theta)$, $\Phi_{A{}min}^{\ast}(\theta))\vert$ $\Phi_{min}^A(\theta)$ = $({\cal A}^{
\imath}(\theta), C^\alpha(\theta))$ $\in$ ${\cal M}_{min}$ = ${\cal M}_{cl}
\times {\cal M}_C$, $\Phi_{A{}min}^{\ast}(\theta)$ = $({\cal A}_{\imath}^{
\ast}(\theta)$, $C_\alpha^{\ast}(\theta))$, $A = (\imath,\alpha)\}$ which in contrast to its analog $S_{H{}min}(\Gamma_{min}(\theta))$ in \[3\] depends upon $\theta$ explicitly and is not restricted by requirement of ordinary ghost number vanishing.
Not any GThST appears by part of a given GThGT just as not arbitrary GThGT contains a nontrivial GThST (see corollary 2.1, 2.2 for Theorem 2 from Ref.\[1\]). However if the GThST with $S(\theta)$ is embedded into the GThGT with $Z[{
\cal A}]$ (in representing of $S_{L}(\theta)$ in the form (3.15), (3.16)) then the corresponding gauge algebra for GThST is the gauge subalgebra in the corresponding gauge algebra for GThGT. Really, the vector fields $\Gamma_{0{}\alpha}(\theta)$ (5.6) are connected with ones $\hat{\Gamma}'_{
\alpha}(\theta)$ of the type (4.7) given and acting on $C^k({\cal M}_{cl} \times \{\theta\})$ by the formulae $$\begin{aligned}
{} &\left(\Gamma_{0{}\alpha}(\theta_1'){\cal F}(\theta_1')\right)\delta(
\theta_1'
-\theta_1)= \hat{\Gamma}'_{\alpha}(\theta_1){\cal F}(\theta_1'),\
{\cal F}(\theta)\in C^k({\cal M}_{cl} \times \{\theta\})\,, {} & \\
{} &
\hat{\Gamma}'_{\alpha}(\theta_1)F[{\cal A}]= \displaystyle\int \hspace{-0.4em}
d\theta\frac{\delta F[{\cal A}]}{\delta{\cal A}^{\imath}(\theta)}
\hat{\cal R}'{}_{\alpha}^{\imath}({\cal A}(\theta), \theta;{\theta}_1),\;
\hat{\cal R}'{}_{\alpha}^{\imath}({\cal A}(\theta), \theta;{\theta}_1)=
{\cal R}_{0}{}^{\imath}_{\alpha}({\cal A}(\theta), \theta)\delta(\theta-
\theta_1).{} & $$ The structural functions $S({\cal A}(\theta), \theta)$; ${\cal R}_{0}{}^{\imath}_{\alpha}({\cal A}(\theta), \theta)$; ${\cal F}_0{}^{\gamma}_{\alpha\beta}({\cal A}(\theta), \theta)$, ${\cal M}_0{}^{\imath\jmath}_{\alpha\beta}({\cal A}(\theta),\theta)$ of zero, $1$st, $2$nd orders of the gauge algebra of GTST are connected with corresponding ones $Z_{0}[{\cal A}]$; $\hat{\cal R}'{}_{\alpha}^{\imath}(\theta;{\theta}')$; $
\hat{\cal F}^{\prime}{}_{\alpha\beta}^{\gamma}(\theta;{\theta}',{\theta}'_1)$, $\hat{\cal M}'{}_{\alpha
\beta}^{\imath\jmath}(\theta,\theta_1;{\theta}', {\theta}'_1)$ of zero, $1$st, $2$nd orders of the gauge algebra of GTGT by the relationships in addition to the 2nd expression in (5.17) $$\begin{aligned}
{} & Z_{0}[{\cal A}]= - \displaystyle\int d\theta S({\cal A}(\theta),
\theta)
\,,{} &\\ {} &\hat{\cal F}^{\prime}{}_{\alpha\beta}^{\gamma}({\cal A}(\theta),\theta;
\vec{\theta}_2) = (-1)^{\varepsilon_{\gamma}+\varepsilon_{\beta}}
{\cal F}_0{}^{\gamma}_{\alpha\beta}({\cal A}(\theta), \theta)
\delta(\theta-\theta_1)\delta(\theta-\theta_2)\,,{} &\\{} &
\hat{\cal M}'{}_{\alpha\beta}^{\imath\jmath}({\cal A}(\theta'_1),
\vec{\theta}'_2;\vec{\theta}_2) = (-1)^{\varepsilon_{\jmath}+
\varepsilon_{\beta}}
{\cal M}_0{}^{\imath\jmath}_{\alpha\beta}({\cal A}(\theta'_1),\theta'_1)
\delta(\theta'_2 - \theta'_1)\times {} & \nonumber \\
{} & \frac{1}{2}[\delta(\theta'_2-\theta_1)\delta(\theta'_1-\theta_2)+
\delta(\theta'_1-\theta_1)\delta(\theta'_2-\theta_2)]\,,{} &$$ in a such way that in fulfilling of the corresponding structural equations of the $1$st, $2$nd orders for GThST (5.1), (5.10) taking the properties (5.11), (5.12) for ${\cal F}_0{}^{\gamma}_{\alpha\beta}(
{\cal A}(\theta), \theta)$, ${\cal M}_0{}^{\imath\jmath}_{\alpha\beta}({\cal
A}(\theta),\theta)$ into account the quantities $Z_{0}[{\cal A}]$, $\hat{\cal R}'{}_{\alpha}^{\imath}(\theta;{\theta}')$, $
\hat{\cal F}^{\prime}{}_{\alpha\beta}^{\gamma}(\theta;{\theta}',{\theta}'_1)$ $\hat{\cal M}'{}_{\alpha\beta}^{\imath\jmath}(\theta,\theta_1;{\theta}',
{\theta}'_1)$ satisfy exactly to the $1$st and $2$nd orders structural equations for the gauge algebra of GTGT with $Z_{0}[{\cal A}]$ ($Z[{\cal A}]$ = $Z_{0}[{\cal A}]$ + $
\int d\theta T\bigl({\cal A}(\theta)$, ${\stackrel{\ \circ}{\cal A}}(\theta)
\bigr)$) (3.11), (4.13) with properties (4.14), (4.15) for $\hat{\cal F}'$, $\hat{\cal M}'$. This embedding of the gauge algebra for GThST on $Q(S)$ can be established in the all orders $k>2$ of the gauge algebra.
Derivation of the formulae (5.16), (5.19), (5.20) are based on the rules of connection for superfield derivatives $\displaystyle\frac{\delta\phantom{xxx}}{\delta{
\cal A}^{\imath}(\theta)}$, $\displaystyle\frac{\partial\phantom{xxx}}{\partial{\cal A}^{\imath}(\theta)}$ obtained in \[1\].
$\theta$-Superfield Quantum Electrodynamics
===========================================
As the initial GSTF model in the Lagrangian formulation consider the superfield model of free spinor superfield of spin $\frac{1}{2}$ being by the singular theory of special type \[1\]. The model is described by Dirac bispinor superfield $\Psi(x,\theta)$ = $\bigl(\psi_{\gamma}(x,\theta)$, $\chi^{\dot{\gamma}}(x,
\theta)\bigr)^{T}$ = $\psi(x)$ + $\psi_1(x)\theta$ and by its Dirac conjugate one $\overline{\Psi}(x,\theta)$ = $\Psi^{+}(x,\theta)\Gamma^0$ = $\bigl(\overline{\chi}{}^{\beta}(x,\theta)$, $\overline{\psi}_{\dot{\beta}
}(x,\theta)\bigr)$ = $\overline{\psi}(x)$ + $\overline{\psi}_1(x)\theta$, $\gamma, \beta$=$1,2$, $\dot{\gamma}, \dot{\beta}$=$\dot{1},\dot{2}$ being by elements of $(\frac{1}{2}, 0)$ $\bigoplus$ $(0, \frac{1}{2})$ reducible massive superfield (on $\theta$) representation $T$ of supergroup $J$ = $\Pi(1,3)^{\uparrow}\times P$, ($\Pi(1,3)^{\uparrow}$ = $SO(1,3)^{\uparrow}$ ${\mbox{$\times \hspace{-1em}\supset$}}$ $T(1,3)$) defined on superspace ${\cal M}$ = ${\bf R}^{1,3} \times \tilde{P}$ = $\{(x^{\mu},
\theta)\}$, $\eta_{\mu\nu}$ = ${\rm diag}(1,-1,-1,-1)$.
Let us point out briefly the only condensed contents of index $\imath$ for ${\cal A}^{\imath}(\theta)$ $\mapsto$ $(\Psi(x,\theta), \overline{\Psi}(x,
\theta))$, the Grassmann parities table, the transformation laws of $\Psi(x,\theta),
\overline{\Psi}(x,\theta)$ with respect to $T_{\mid P}$ representation, the superfunction $S_L(\theta)$ defining the GSTF model in question and Euler-Lagrange equations (3.1) in the form of HCLF (3.17) respectively in Ref.\[1\] notations $$\begin{aligned}
{} & {} &
\imath = (\gamma, \dot{\gamma}, \beta, \dot{\beta}, x),\
\begin{array}{lcccc}
{} & \psi(x) & \psi_1(x) & \Psi(x,\theta) & {}\\
\varepsilon_P & 0 & 1 & 0 & ,\ K(x,\theta) \in
\tilde{\Lambda}_{4\vert 0+1}(x^{\mu},\theta;{\bf C}), \\
\varepsilon_{\Pi} & 1 & 1 & 1 & K\in \{\Psi, \overline{\Psi}\},\\
\varepsilon & 1 & 0 & 1 &
\end{array}\\{} & {} &
\delta\Psi(x,\theta) = \Psi'(x,\theta)
- \Psi(x,\theta) = -\mu {\stackrel{\circ}{\Psi}}(x,\theta) = -\mu
\psi_1(x)\,, \nonumber \\
{} & {} &
\delta\overline{\Psi}(x,\theta) = \overline{\Psi}'(x,\theta) -
\overline{\Psi}(x,\theta) = -\mu {\stackrel{\circ}{
\overline{\Psi}}}(x,\theta) = -\mu\overline{\psi}_1(x)\,,\\{} & {} &
\hspace{-1em}S_L^{(1)}(\theta)\equiv
S_L\Bigl(\Psi(\theta), \overline{\Psi}(\theta),
{\stackrel{\circ}{\Psi}}(\theta), {\stackrel{\circ}{\overline{\Psi}}}
(\theta)\Bigr) = T\Bigl({\stackrel{\circ}{\Psi}}(\theta), {\stackrel{\circ}{
\overline{\Psi}}}(\theta)\Bigr) - S_{0}
\bigl(\Psi(\theta), \overline{\Psi}(\theta)\bigr)\,, \\ {} & {} &
\hspace{-1em}T^{(1)}(\theta) \equiv T\Bigl({\stackrel{\circ}{\Psi}}(
\theta),{\stackrel{\circ}{\overline{\Psi}}}(\theta)\Bigr) =
\displaystyle\int d^4 x {\stackrel{\circ}{\overline{\Psi}}}
(x,\theta) {\stackrel{\circ}{\Psi}}(x,\theta) \equiv
\displaystyle\int d^4 x {\cal L}^{(1)}_{\rm kin}(x,\theta)\,, \\ {} & {} & \hspace{-1em}S^{(1)}_0(\theta) \equiv S_{0}\bigl(\Psi(\theta),
\overline{\Psi}(\theta)\bigr)= \displaystyle\int d^4x
\overline{\Psi}(x,\theta)\left(\imath
\Gamma^{\mu}\partial_{\mu}- m\right){\Psi}(x,\theta)
\equiv \displaystyle\int d^4x {\cal L}^{(1)}_0 (x,\theta)\,, \\ {} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l Z[\Psi,\overline{\Psi}]}{\delta
\Psi(x,\theta)} = - \displaystyle\frac{\partial_l S^{(1)}_0(\theta)}{
\partial\Psi(x,\theta)} + \displaystyle\frac{d}{d\theta}
\frac{\partial_l T^{(1)}(\theta)
\phantom{x}}{\partial{\stackrel{\circ}{\Psi}}(x,\theta)} = -\bigl(\imath
\partial_{\mu} \overline{\Psi}(x,\theta)\Gamma^{\mu} +
m\overline{\Psi}(x,\theta)\bigr) = 0\,, \\{} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l
Z[\Psi,\overline{\Psi}]}{\delta \overline{\Psi}(x,\theta)} = -
\displaystyle\frac{\partial_l S^{(1)}_0(\theta)}{\partial
\overline{\Psi}(x,\theta)} +
\displaystyle\frac{d}{d\theta} \displaystyle\frac{\partial_l T^{(1)}(\theta)
\phantom{}}{\partial{\stackrel{\circ}{ \overline{\Psi}}}(x,\theta)} = -
\bigl(\imath\Gamma^{\mu}\partial_{\mu} - m\bigr){\Psi} (x,\theta) = 0\,,
\\{} & {} &
\displaystyle\frac{\partial_l
S_0\bigl({\Psi}(\theta), \overline{\Psi}(\theta)\bigr)}{\partial
\Psi(x,\theta)\phantom{xxxxxx}} = \displaystyle\frac{\partial_{l,\theta,x}
{\cal L}^{(1)}_0(x,\theta)}{\partial \Psi(x,\theta) \phantom{xxxx}} -
\partial_{\nu}\displaystyle\frac{\partial_{l,\theta,x} {\cal L}^{(1)}_0(
x,\theta)}{\partial
\bigl(\partial_{\nu} \Psi(x,\theta)\bigr)\phantom{x}}\,, \\{} & {} &
\displaystyle\frac{\partial_l
T\Bigl({\stackrel{\circ}{\Psi}}(\theta), {\stackrel{\circ}{
\overline{\Psi}}}(\theta)\Bigr)}{ \partial
{\stackrel{\circ}{\Psi}}(x,\theta)\phantom{xxxxxx}} =
\displaystyle\frac{\partial_{l,\theta,x} {\cal L}^{(1)}_{\rm kin}(x,\theta)}{
\partial{\stackrel{\circ}{\Psi}}(x,\theta)\phantom{xxxx}} -
\partial_{\nu}\displaystyle\frac{\partial_{l, \theta,x} {\cal L}^{(1)}_{\rm
kin}(x,\theta)
}{\partial \Bigl(\partial_{\nu}{\stackrel{\circ}{\Psi}}(x,\theta) \Bigr)
\phantom{x}}\;.
$$ Given model appears by nongauge one and is invariant with respect to global $U(1)$ (phase) transformations with constant parameter $\xi$ and elementary electric charge $e$ $$\begin{aligned}
{} & {} &
\Psi(x,\theta) \mapsto \Psi'(x,\theta) = {\rm exp}(-\imath e\xi)
\Psi(x,\theta),\
(\varepsilon_P, \varepsilon_{\Pi}, \varepsilon)\xi = (0,0,0),\ \xi\in{\bf R}
\,, \nonumber \\
{} & {} &
\overline{\Psi}(x,\theta) \mapsto \overline{\Psi}'(x,\theta)=
{\rm exp}(\imath e\xi)\overline{\Psi}(x,\theta)\,. $$ Realizing the Yang-Mills type gauge principle \[8\] let us change the parameter onto arbitrary superfield $\xi(x,\theta)$. In this connection Eqs.(6.6), (6.7) are changed onto $\theta$-superfield generalization of Dirac equations in presence, at least, of external electromagnetic superfield ${\cal
A}^{\mu}(x,\theta)$ and corresponding superfunction $S_{L{}Q}^{(1)}(\theta)$ must be invariant with respect to following from (6.10) GTGT $$\begin{aligned}
{} & {} &
{\cal A}^{\mu}(x,\theta) = A^{\mu}(x) + A^{\mu}_1(x)\theta \mapsto
{\cal A}'^{\mu}(x,\theta) = {\cal A}^{\mu}(x,\theta) + \partial^{\mu}\xi(x,
\theta)\,,\\{} & {} &
C(x,\theta) = C(x) + C_1(x)\theta \mapsto
C'(x,\theta) = C(x,\theta) + {\stackrel{\circ}{\xi}}(x,\theta)
\,,\\{} & {} &
\Psi(x,\theta) \mapsto \Psi'(x,\theta) = {\rm exp}(-\imath e\xi(x,\theta))
\Psi(x,\theta)\,,\\ {} & {} &
\overline{\Psi}(x,\theta) \mapsto \overline{\Psi}'(x,\theta)=
{\rm exp}(\imath e\xi(x,\theta))\overline{\Psi}(x,\theta)\,,\\ {} & {} & \hspace{-2em}
\begin{array}{lccccccc}
{} & {\cal A}^{\mu}(x,\theta) & {A}^{\mu}(x) & {A}_1^{\mu}(x) & C(x,\theta)&
C(x) & C_1(x) & {} \\
\varepsilon_P & 0 & 0 & 1 & 1 & 1 & 0 & \hspace{-0.5em},\
K(x,\theta) \in
\tilde{\Lambda}_{4\vert 0+1}(x^{\mu},\theta;{\bf R}), \\
\varepsilon_{\Pi} & 0 & 0 & 0 & 0 & 0 & 0 & K\in \{{\cal A}^{\mu}, C, \xi\}.
\\
\varepsilon & 0 & 0 & 1 & 1 & 1 & 0 & {}
\end{array} $$ Note the $\varepsilon_P$ Grassmann parity value of superfield ${\cal A}^{\imath}(\theta)$ is not trivial in contrast to corresponding one in Ref.\[1\] because of the ghost superfield $C(x,\theta)$ inclusion into multiplet ${\cal A}^{\imath}(\theta)$ already on the initial level of the model formulation.
Written in the infinitesimal form (3.18) with parameter $\delta\xi(x,\theta)$ the GTGT and GGTGT have the representation respectively under change of superfield ${\cal A}^{\imath}(\theta)$ and index $\imath$ (6.1) contents $$\begin{aligned}
{} & {} &
\delta_g{\cal A}^{\imath}(\theta) = \displaystyle\int d\theta'
\hat{\cal R}^{\imath}({\cal A}(\theta),\theta,\theta')\delta\xi(\theta')
= \displaystyle\int d\theta'd y
\hat{\cal R}^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta;y,\theta')
\delta\xi(y, \theta'), \alpha = ([\xi], y)\,,\nonumber \\
{} & {} &
{\cal A}^{\imath}(\theta) = (\overline{\Psi}(x,\theta), {\Psi}(x,\theta),
{\cal A}^{\mu}(x,\theta),C(x,\theta)),\ \imath = (\gamma,\dot{\gamma},\beta,
\dot{\beta}, \mu,[C],x) = (\tilde{\imath}, x)
\,,\\{} & {} &
\hat{\cal R}^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta;y,\theta') =
\displaystyle\sum\limits_{k\geq 0} \left(\left(\displaystyle\frac{d}{d\theta}
\right)^k\delta(\theta - \theta')\right)
\hat{\cal R}_k^{\tilde{\imath}}({\cal A}(x,\theta),x,y,\theta) =
\nonumber \\
{} & {} & \hspace{2em}
\displaystyle\sum\limits_{k\geq 0} \left(\left(\displaystyle\frac{d}{d\theta}
\right)^k\delta(\theta - \theta')\right)
\hat{\cal R}_k^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta)\delta(x-y)\,,
\\{} & {} &
\hat{\cal R}_0^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta) = \left\{
\begin{array}{ll}
\partial^{\mu},& \hspace{-0.3em}\imath=(\mu, x) \\
- \imath e \overline{\Psi}(x,\theta), & \hspace{-0.3em}
\imath=(\beta,\dot{\beta},x) \\
\imath e {\Psi}(x,\theta), & \hspace{-0.3em}\imath=(\gamma,\dot{\gamma},x)
\end{array}\right.\hspace{-0.7em},
\hat{\cal R}_1^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta)=-1, \imath =
([C], x).$$ GGTGT (6.17), (6.18) forms the Abelian gauge algebra of GTGT in terminology of Sec.4. To construct zero order structural superfunction $S_{LQ}^{
(1)}(\theta)$ for given algebra let us introduce according to Ref.\[8\] the prolonged covariant derivatives in $\theta$-superfield form with respect to representation $T$ of supergroup $J$ (not Lorentz type) $$\begin{aligned}
{\cal D}_A \equiv \partial_A - \imath e {\cal A}_{A}(x,\theta),
\partial_A=(\partial_{\mu}, \frac{d}{d\theta}),\;{\cal D}_A=({\cal D}_{\mu},
{\cal D}_{\theta}),\ {\cal A}_{A}(x,\theta)=({\cal A}_{\mu},
C)(x,\theta)\,.$$ The supercommutator of above derivatives leads to expression for superfield ${\cal A}_{A}(x,\theta)$ strength being invariant with respect to GTGT (6.11)–(6.14) $$\begin{aligned}
{} &
{\cal F}_{AB}(x,\theta)= \displaystyle\frac{\imath}{e}[{\cal D}_A,{\cal
D}_B]_s = \partial_A{\cal A}_{B}(x,\theta) - (-1)^{\varepsilon({\cal A}_A)
\varepsilon({\cal A}_B)}
\partial_B{\cal A}_{A}(x,\theta)\,,{} &\\{} &
{\cal F}_{AB}(x,\theta)=
\left\|
\begin{array}{lr}
F_{\mu\nu} & F_{\mu [C]}\\
F_{[C]\nu} & F_{[C] [C]}
\end{array} \right\|(x,\theta) = \left\|
\begin{array}{cc}
\partial_{[\mu}{\cal A}_{\nu]} &
\partial_{\mu}C- {\stackrel{\ \circ}{\cal A}}_{\mu}\\
{\stackrel{\ \circ}{\cal A}}_{\nu}-\partial_{\nu}C &
2{\stackrel{\,\circ}{C}}
\end{array} \right\|(x,\theta)=
{} &\nonumber \\
{} &
-(-1)^{\varepsilon({\cal A}_A)\varepsilon({
\cal A}_B)}{\cal F}_{BA}(x,\theta),
(A,B)= ((\mu,[C]),(\nu,[C])),
\varepsilon({\cal A}_A)= (0\cdot\delta_{A\mu}, 1\cdot\delta_{A[C]})
\,.{} &$$ The following superfunctions being quadratic on ${\cal F}_{AB}(x,\theta)$ appear by the Poincare and gauge (with respect to GTGT) invariant objects $$\begin{aligned}
{} & {} &
{\cal F}_{AB}(x,\theta){\cal F}^{AB}(x,\theta)= \left(F_{\mu\nu}F^{\mu\nu}
+ 2 F_{\mu [C]}F^{\mu [C]} + 4{\stackrel{\,\circ}{C}}{\stackrel{\,\circ}{C}}
\right)(x,\theta) \equiv
\nonumber \\
{} & {} & \hspace{2em}
-4\left({\cal L}^{(0)}_1(\partial_{\mu}{\cal A}_{\nu}(x,\theta))
+ {\cal L}^{(1)}_1\bigl(\partial_{\mu}C(x,\theta),
{\stackrel{\ \circ}{\cal A}}{}_{\nu}(x,\theta)\bigr) +
{\cal L}^{(2)}_1\bigl({\stackrel{\,\circ}{C}}(x,\theta)\bigr)\right)\,,
\nonumber \\
{} & {} &
{\cal L}^{(1)}_1\bigl(\partial_{\mu}C(x,\theta),{\stackrel{\ \circ}{\cal
A}}{}_{\nu}(x,\theta)\bigr) \equiv 0;\
{\cal L}^{(2)}_1({\stackrel{\,\circ}{C}}(x,\theta)) =- \frac{d}{d\theta}\Bigl(
{C}(x,\theta){\stackrel{\,\circ}{C}}(x,\theta)\Bigr)\,;
\\{} & {} &
\varepsilon_{ABCD}{\cal F}^{AB}(x,\theta){\cal F}^{CD}(x,\theta) =\left(
\varepsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}
+ 4\varepsilon_{\mu\nu\rho [C]}F^{\mu\nu}F^{\rho[C]} +
4\varepsilon_{\mu\nu [C][C]}F^{\mu\nu}{\stackrel{\,\circ}{C}} + \right.
\nonumber \\
{} & {} & \hspace{2em}
\left. 4 \varepsilon_{\mu [C]\nu [C]}F^{\mu [C]}F^{\nu[C]} +
8\varepsilon_{\mu [C][C][C]}F^{\mu [C]}{\stackrel{\,\circ}{C}}\right)(x,
\theta)
- 4\varepsilon_{[C][C][C][C]}{\cal L}^{(2)}_1\bigl({\stackrel{\,\circ}{C}}
(x,\theta)\bigr)\,,\\{} & {} & \hspace{-1.5em}\varepsilon_{ABCD}\hspace{-0.1em} =\hspace{-0.1em}
-(-1)^{\varepsilon({\cal A}_A)
\varepsilon({\cal A}_B)} \varepsilon_{BACD}\hspace{-0.1em}=\hspace{-0.1em}
-(-1)^{\varepsilon({\cal A}_C)\varepsilon({\cal A}_B)}\varepsilon_{ACBD}
\hspace{-0.1em}=\hspace{-0.1em}
-(-1)^{\varepsilon({\cal A}_C)\varepsilon({\cal A}_D)}\varepsilon_{ABDC}
\,.$$ Choosing the elements of superantisymmetric constant tensor $\varepsilon_{ABCD}$ in the form being compatible with even values of its ($\varepsilon_{ABCD}$) $\varepsilon_{P}$, $\varepsilon_{\Pi}$, $\varepsilon$ gradings and with properties (6.24) $$\begin{aligned}
\varepsilon_{0123}=
\varepsilon_{[C][C][C][C]}=1,\;\varepsilon_{\mu\nu\rho [C]}=
\varepsilon_{\mu [C][C][C]}=0,\;
\varepsilon_{\mu [C]\nu [C]}=-\varepsilon_{\mu\nu [C][C]}=
\varepsilon^{(1)}_{\mu\nu}= - \varepsilon^{(1)}_{\nu\mu}\,,
$$ we obtain for (6.23) the result $$\begin{aligned}
\Bigr(\varepsilon_{ABCD}{\cal F}^{AB}{\cal F}^{CD}\Bigl)(x,\theta) =
\Bigl(\varepsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} -
4\varepsilon^{(1)}_{\mu\nu}\Bigl(F^{\mu\nu}
{\stackrel{\,\circ}{C}} - F^{\mu [C]}F^{\nu [C]}\Bigr) +
4{\stackrel{\,\circ}{C}}{}^2\Bigr)(x,\theta)\,. $$ In the first place, note the superfunction ${\cal L}_{1}^{(2)}\bigl({\stackrel{\,\circ}{
C}}(x,\theta)\bigr)$ is the self-dual one and with accuracy up to total derivatives with respect to $x^{\mu}$, $\theta$ the sum of the $2$nd and $3$rd summands in (6.26) is reduced to the form $$\begin{aligned}
4\varepsilon^{(1)}_{\mu\nu}\left(F^{\nu\mu}
{\stackrel{\,\circ}{C}}+ F^{\mu [C]}F^{\nu [C]}\right)(x,\theta) =
4\varepsilon^{(1)}_{\mu\nu}\left({\stackrel{\ \circ}{\cal A}}{}^{\nu}
{\stackrel{\ \circ}{\cal A}}{}^{\mu}+2F^{\nu\mu}{\stackrel{\,\circ}{C}}\right)
(x,\theta)\,.$$ With regard of the last representation the superfunction $S_{L{}Q}^{(1)}(
\theta)$ being invariant with respect to GTGT (6.11)–(6.14), defining the GThGT with nontrivial inclusion of the ghost superfield $C(x,\theta)$ into superfield (on $\theta$) quantum electrodynamics and addition of the “$\tilde{\theta}$-term” (vacuum angle), leading by means of relationships (6.23)–(6.27) to application in the electromagnetic duality theory (see for instance Ref.\[9\]), has the resultant form $$\begin{aligned}
{} & {} &
S_{L{}Q}^{(1)}(\theta) = S_{L{}Q}^{(1)}\bigl({\cal A}_A(\theta),
{\stackrel{\ \circ}{\cal A}}{}_A(\theta), \Psi(\theta), \overline{\Psi}(
\theta),{\stackrel{\circ}{\Psi}}(\theta), {\stackrel{\circ}{\overline{\Psi}}}(
\theta)\bigr) = \nonumber \\
{} & {} &
T_{\rm inv}\bigl({\cal D}_{\theta}\Psi(\theta),{\cal D}^{\ast}_{\theta}
\overline{\Psi}(\theta)\bigr)
- S^{(11)}\bigl(\Psi(\theta),\overline{\Psi}(\theta),{\cal
A}_{\mu}(\theta)\bigr)
- S_0^{(11)}\bigl({\cal A}_{A}(\theta),
{\stackrel{\ \circ}{\cal A}}{}_{A}(\theta)\bigr)\,;\\{} & {} &
T_{\rm inv}(\theta) \equiv
T_{\rm inv}\bigl({\cal D}_{\theta}\Psi(\theta),{\cal D}^{\ast}_{\theta}
\overline{\Psi}(\theta)\bigr)
= \displaystyle\int d^4x{\cal L}_{\rm kin}^{(1)}\bigl({\cal D}_{\theta}
\Psi(x,\theta),{\cal D}^{\ast}_{\theta}\overline{\Psi}(x,\theta)\bigr)
= \nonumber \\
{} & {} &
\displaystyle\int d^4x\Bigl({\cal D}^{\ast}_{\theta}\overline{\Psi}\Bigr)
\Bigl({\cal D}_{\theta}\Psi\Bigr)(x,\theta),\;
{\cal D}^{\ast}_{\theta}\overline{\Psi}(x,\theta) = \left(\frac{d}{d\theta} +
\imath e C(x,\theta)\right)\overline{\Psi}(x,\theta)
\,;\\{} & {} &
S^{(11)}(\theta)\equiv S^{(11)}\bigl(\Psi(\theta),\overline{\Psi}(\theta),
{\cal A}_{\mu}(\theta)\bigr) = \displaystyle\int d^4x
{\cal L}^{(1)}_0\bigl(\Psi(x,
\theta), \overline{\Psi}(x,\theta), {\cal D}_{\mu}\Psi(x,\theta)\bigr)
\,,\nonumber \\
{} & {} &
{\cal L}^{(1)}_0\bigl(\Psi(x,
\theta), \overline{\Psi}(x,\theta), {\cal D}_{\mu}\Psi(x,\theta)\bigr)=
\left(\overline{\Psi}\left(\imath
\Gamma^{\mu}{\cal D}_{\mu}- m\right){\Psi}\right)(x,\theta)\,;\\
{} & {} &
S^{(11)}_0(\theta)\equiv S^{(11)}_0\bigl({\cal A}_{A}(\theta),
{\stackrel{\ \circ}{\cal A}}{}_{A}(\theta)\bigr) =
\displaystyle\int d^4x {\cal L}^{(11)}_0\bigl({\cal A}_{A}(x,\theta),
\partial_B{\cal A}_{A}(x,\theta)\bigr)\,,\nonumber \\
{} & {} & \hspace{-0.5em}
{\cal L}^{(11)}_0(x,\theta)\hspace{-0.1em} = \hspace{-0.1em}\Bigl(
{\cal L}_{\tilde{\theta}} -
\frac{1}{4}{\cal F}_{AB}{\cal F}^{AB}\Bigr)(x,\theta) \hspace{-0.1em} =
\hspace{-0.1em} {\cal L}_{\tilde{\theta}}(x,\theta)
\hspace{-0.1em} + \hspace{-0.1em}
{\cal L}^{(0)}_1(\partial_{\mu}{\cal A}_{\nu}(x,\theta))\hspace{-0.1em} +
\hspace{-0.1em}{\cal L}^{(2)}_1({\stackrel{\,\circ}{C}}(x,\theta)),\\
{} & {} &
{\cal L}_{\tilde{\theta}}(x,\theta)= - \displaystyle\frac{\tilde{\theta}e^2}{
32\pi^2}\varepsilon_{ABCD}{\cal F}^{AB}(x,\theta){\cal F}^{CD}(x,\theta) =
\nonumber \\
{} & {} & \hspace{-1em}
- \displaystyle\frac{\tilde{\theta}e^2}{32\pi^2}\left(
\varepsilon_{\mu\nu\rho\sigma}F^{\mu\nu}(x,\theta)F^{\rho\sigma}(x,\theta)
+ 4\varepsilon^{(1)}_{\mu\nu}\bigl({\stackrel{\ \circ}{\cal A}}{}^{\nu}
{\stackrel{\ \circ}{\cal A}}{}^{\mu}- 2F^{\mu\nu}{\stackrel{\,\circ}{C}}
\bigr)(x,\theta) -
4{\cal L}^{(2)}_1\bigl({\stackrel{\,\circ}{C}}(x,\theta)\bigr)\right).$$ Superfunctions $T_{\rm inv}(\theta)$, $S^{(11)}(\theta)$, $S_{0}^{(11)}(\theta)$ in (6.28) are invariant with respect to GTGT. Euler-Lagrange equations (3.1) for $Z^{(1)}[\Psi,\overline{\Psi}, {\cal A}_A]$ = $\int d\theta
S_{L{}Q}^{(1)}(\theta)$ read as follows $$\begin{aligned}
{} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l Z^{(1)}\phantom{xx}}{\delta
\Psi(x,\theta)} \hspace{-0.1em} = \hspace{-0.1em}
\displaystyle\frac{\partial_l S_{L{}Q}^{(1)}(\theta)}{
\partial\Psi(x,\theta)\phantom{}} + \displaystyle\frac{d}{d\theta}
\frac{\partial_l T_{\rm inv}(\theta)
\phantom{}}{\partial{\stackrel{\circ}{\Psi}}(x,\theta)}\hspace{-0.1em} =
\hspace{-0.1em} -\bigl(\imath
{\cal D}^{\ast}_{\mu}\overline{\Psi}\Gamma^{\mu} +
m\overline{\Psi}\bigr)(x,\theta) + \imath e ({\stackrel{\,\circ}{C}}
\overline{\Psi})(x,\theta)= 0\,, \\{} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l
Z^{(1)}\phantom{xx}}{\delta \overline{\Psi}(x,\theta)} =
\displaystyle\frac{\partial_l S_{L{}Q}^{(1)}(\theta)}{
\partial \overline{\Psi}(x,\theta)\phantom{}} +
\displaystyle\frac{d}{d\theta} \displaystyle\frac{\partial_l T_{\rm
inv}(\theta)\phantom{}}{\partial{\stackrel{\circ}{\overline{\Psi}}}(x,
\theta)} =
- \Bigl(\bigl(\imath\Gamma^{\mu}{\cal D}_{\mu} - m\bigr) +
\imath e {\stackrel{\,\circ}{C}}(x,\theta)\Bigr)\Psi(x,\theta) = 0\,,\\
{} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l
Z^{(1)}\phantom{xx}}{\delta {\cal A}_{\mu}(x,\theta)} =
-\displaystyle\frac{\partial_l \bigl(S^{(11)}(\theta)+S^{(11)}_0(\theta)
\bigr)}{\partial {\cal A}_{\mu}(x,\theta)\phantom{xxxxxxxxx}} +
\displaystyle\frac{d}{d\theta} \displaystyle\frac{\partial_l S^{(11)}_0(
\theta)\phantom{}}{\partial{\stackrel{\ \circ}{\cal A}}{}_{\mu}(x,\theta)}=
-\Bigl(\partial_{\nu}\Bigl[F^{\nu\mu} -
\displaystyle\frac{\tilde{\theta}e^2}{2\pi^2}
\varepsilon^{(1){}\nu\mu}{\stackrel{\,\circ}{C}}\Bigr] +
\nonumber \\
{} & {} &\hspace{-1em}
e \overline{\Psi}\Gamma^{\mu}{\Psi} \Bigr)(x,\theta) = 0,\
\displaystyle\frac{\partial_{l,\theta,x} {\cal L}_{\tilde{\theta}}(x,\theta)}{
\partial {{\cal A}}^{\nu}(x,\theta)\phantom{xx}} - \partial_{\mu}
\displaystyle\frac{\partial_{l,\theta,x}
{\cal L}_{\tilde{\theta}}(x,\theta)}{\partial \bigl(\partial_{\mu}{\cal
A}^{\nu}(x,\theta)\bigr)} = -
\displaystyle\frac{\tilde{\theta}e^2}{2\pi^2}
\varepsilon^{(1){}\nu\mu}\partial_{\nu}{\stackrel{\,\circ}{C}}(x,\theta)
\,,\\ {} & {} &
{} \hspace{-1em}\displaystyle\frac{\delta_l
Z^{(1)}\phantom{xx}}{\delta C(x,\theta)} =
\displaystyle\frac{\partial_l T_{\rm inv}(\theta)}{\partial C(x,\theta)
\phantom{x}} - \displaystyle\frac{d}{d\theta} \displaystyle\frac{\partial_l
S^{(11)}_0(\theta)}{\partial{\stackrel{\,\circ}{C}}(x,\theta)}
= - \displaystyle\frac{d}{d\theta}\left(\imath e\overline{\Psi}
{\Psi}+\displaystyle\frac{\tilde{\theta}e^2}{4\pi^2}
\varepsilon^{(1)}_{\mu\nu}F^{\mu\nu}\right)(x,\theta) = 0\,, $$ appear by DCLF and represent by themselves the 1st (2nd) order with respect to derivatives on $\theta$ and $x^{\mu}$ nonlinear partial differential equations (6.33), (6.34) for spinor superfields ((6.35), (6.36) for electromagnetic and ghost superfields). In view of degeneracy of the model (6.28) the Cauchy problem setting is not trivial in question and remains out the paper’s scope.
From (6.28–6.36) it follows, in particular, the $\theta$-superfield free electrodynamics is described in terms of superfield ${\cal A}_{A}(x,\theta)$ by means of the superfunctional with “$\tilde{\theta}$-term” $$\begin{aligned}
Z_{SED}[{\cal A_A}]\equiv Z^{(1)}[\Psi,\overline{\Psi}, {\cal A}_A]_{\mid
\Psi=\overline{\Psi}=0} = - \int d\theta S^{(11)}_0\bigl({\cal A}_{A}(\theta),
{\stackrel{\ \circ}{\cal A}}{}_{A}(\theta)\bigr)
$$ the such that for $C(x,\theta)=0$ and in absence of the topological summand $\varepsilon_{\mu\nu\rho\sigma}({\cal F}^{\mu\nu}{\cal F}^{\rho
\sigma})(x,\theta)$ in (6.37) it is obtained the GThST described with accuracy up to nonessential number multipliers in $D=4$ by means of the free massless vector superfield ${\cal A}_{\mu}(x,\theta)$ model \[1\]. The model with $Z_{SED}[{\cal A_A}]$ (6.37) itself belongs to the class of GThST as well as it follows from (6.35), (6.36).
To construct the corresponding to Sec.5 Abelian gauge algebra of GTST being by the gauge subalgebra of the gauge algebra of GTGT with structural functions in (6.17), (6.18), (6.28) it is necessary to restrict the model onto hypersurface ${\stackrel{\circ}{\Psi}}(x,\theta)$ = ${\stackrel{\circ}{\overline{\Psi}
}}(x,\theta)$ = ${\stackrel{\ \circ}{\cal A}}{}_A(x,\theta)$ = $0$ and next to determine the structural functions of $0$ and $1$st orders in correspondence with (6.28)–(6.32) $$\begin{aligned}
S(\Psi(\theta),\overline{\Psi}(\theta),{\cal A}_{\mu}(\theta), C(\theta))
\equiv S(\Psi(\theta),\overline{\Psi}(\theta),{\cal A}_{\mu}(\theta)) =
S^{(11)}(\theta) + S^{(11)}_0({\cal A}_{\mu}(\theta))
$$ and with ${\cal R}_0^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta)$ coinciding with $\hat{\cal R}_0^{\tilde{\imath}}({\cal A}(x,\theta),x,\theta)$ (6.18) with the exception of superfield $C(x,\theta)$ not entering in (6.38). Then the formulae (5.17), (5.18) completely establish the embedding of the gauge algebra of GTST into one of GTGT for given models.
At last, in setting $\theta=0$ in (6.11)–(6.18), (6.38) or equivalently using the special involution $\ast$ \[1\] for $\tilde{\theta}=0$ we obtain the ordinary component quantum electrodynamics formulation on classical level being described by ${\cal A}^{\mu}(x)$, $\Psi(x)$, $\overline{\Psi}(x)$ \[10\].
Let us note the unification possibility of the gauge $A_{\mu}(x)$ and ghost $C(x)$ $P_0(\theta)$-component fields into uniform $P_0(\theta)$-component $A_{A}(x)$ of the uniform superfield ${\cal A}_{A}(x,\theta)$ (6.19) in order to realize the BRST symmetry in the superfield form and to construct the action functional(!) by means of the gauge strength of the form (6.20) for the Yang-Mills type theories had been considered in the paper \[11\] (see the references therein). However the form of Abelian superfield ${\cal A}_{A}(x,\theta)$ (6.19) and strength ${\cal F}_{AB
}(x,\theta)$ (6.20) have exhausted the coincidence of the superfield models from the present paper and from Ref.\[11\]. Their difference is traced not only through the whole corresponding formulae spectrum, among them leading to construction of the actions, but is based on the functionally distinct conceptual formulations of the models.
Conclusion
==========
The basic Theorem announced in Ref.\[1\] from which it follows the many properties of GThGTs, GThSTs in the framework of the Lagrangian formulation for GSTF is completely proved and their consequences are studied. Nontrivial differential-algebraic structures, i.e. the gauge algebras of GTGT, GTST have been investigated.
The general results of the Secs.2–5 have obtained the final confirmation on the example of the $\theta$-superfield quantum electrodynamics, being on the classical level by ordinary $\theta$-superfield spinor electrodynamics, realized on the gauge principle basis \[8\] (the so-called minimal inclusion of interaction). The cases of $\theta$-superfield scalar or vector electrodynamics may be deduced from the free massive complex scalar superfields $\varphi(x,\theta)$, $\varphi^{
\ast}(x,\theta)$ model and free massive complex(!) vector superfield ${\cal
A}^{\mu}(x,\theta)$ in $D=4$ model, realized in fact in the Lagrangian formulation of GSTF in Ref.\[1\], by means of the algorithm from Sec.6. All these models representing the GThGTs with Abelian gauge algebra can be generalized in their constructing, in an obvious way, starting from the case of the initial interacting $\theta$-superfield massive spinor, scalar, (complex) vector models.
Specially note the prolongation of the derivative with respect to odd time $\theta$ have led to necessity already on the classical level of the nontrivial inclusion of the ghost superfield $C(x,\theta)$ playing the same role for $\frac{d}{d\theta}$ as the electromagnetic one ${\cal A}^{\mu}(x,\theta)$ for $\partial_{\mu}$.
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2. A.A. Reshetnyak, General Superfield Quantization Method. II. General Superfield Theory of Fields: Hamiltonian Formalism, hep-th/0303262.
3. A.A. Reshetnyak, General Superfield Quantization Method. III. Construction of Quantization Scheme, hep-th/0304142.
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7. I.A. Batalin and G.A. Vilkovisky, J. Math. Phys. 26 (1985) 172.
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10. D.M. Gitman and I.V. Tyutin, Quantization of Fields with Constraints (Springer-Verlag, Berlin and Heidelberg, 1990).
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[^1]: E-mail: reshetnyak@ssti.ru
[^2]: Because the index $i$ possesses by the complicated condensed contents then Eqs.(2.1) are, in general, the system of partial differential equations \[1\]. The only differential operator $\frac{d}{d\theta}$ is specially singled out here
|
---
author:
- '[ Chi Thang Duong Remi Lebret Karl Aberer ]{}'
title: Multimodal Classification for Analysing Social Media
---
[[**Acknowledgement.**]{}]{} We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU used for this research. We would also like to thank the IT support team for their help.
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6.5in -.2in
.2in
.1in
Rouzbeh Allahverdi and Bruce A. Campbell
[*Department of Physics, University of Alberta*]{}
[*Edmonton, Alberta, Canada T6G 2J1*]{}
.1in
.5in
[**Abstract**]{}
We consider inflaton decay to final state bosons with self-interactions of moderate strength. We find that such final state self-interactions qualitatively alter the reheat dynamics. In the case of narrow-band resonance decay, where a quantitative analysis is possible, we show that these final state interactions regulate the decay rate. The phenomenon of parametric amplification is then effectively suppressed, and does not drastically enhance the decay rate and reheat temperature. Detailed applications of our results to realistic classes of inflationary models will be considered elsewhere.
1 \#1[m\_[\#1]{}]{} \#1[${\tilde #1}$]{}
Inflationary cosmology [@1], provides solutions to the flatness, isotropy and stable relic problems of the standard hot big bang. In these models the universe experiences a period of superluminal expansion during which its energy density is dominated by the potential energy of a scalar field (inflaton). Inflation ends when the inflaton enters the oscillatory regime during which we have a matter-dominated FRW-universe, after which the inflaton decays to relativistic particles (reheating). Reheating represents the crucial transition from the epoch of scalar-field dominated dynamics to a hot FRW universe. After reheating, the universe becomes radiation-dominated and its evolution is just that of the standard hot big bang .
In the standard picture of reheating [@1; @2] the effective decay of the inflaton occurs when $\Gamma \simeq H$, where $\Gamma$ is the one particle decay rate, and the reheat temperature of the universe is ${T}_{R} \sim 0.1{(\Gamma)}^{{1\over 2}}$ (from now on ${M}_{pl}=1$ and all dimensionful quantities are expressed in these units unless otherwise indicated). In this picture, which we will refer to as the linear regime, perturbation theory is taken to be valid and the occupation number of the final state bosons is assumed to be smaller than one (for fermions this is assured because of Pauli blocking). It has recently been recognized within different approaches [@3] -[@31] that this picture is incomplete, and nonlinear effects can change it essentially, leading by parametric amplification to an enhanced decay of the inflaton (for a recent review see, e.g. [@3]). There are two different possible regimes of such a parametrically amplified decay. In the first the decay occurs over many oscillation times of the inflaton, but experiences parametric amplification with moderately large occupation numbers for the modes of the decay product field; this “narrow-band resonance” case is amenable to analytic calculation, and we will be able to analyze the modifications to parametric amplification due to final state-self interactions of the decay products quantitatively. In the second case of “broad-band resonance” the amplification of the decay is so strong that there is explosive decay of the inflaton field on a time-scale not hierarchically longer than the oscillation time, with large occupation numbers for the modes of the decay product field; this scenario is more difficult to analyze quantitatively, but the physical mechanisms which we discuss are of sufficient generality that we expect that they alter the decay dynamics in this case also.
The consequences of a parametrically amplified decay could be significant, and pose major difficulties in the construction of viable inflationary models. If the decay is very fast, the final state particle modes have, in general, very large occupation numbers and are far from thermal equilibrium; these fluctuations have certain effects similar to very high temperature thermal corrections [@9; @12] , and after thermalization may themselves lead to high reheat temperatures, with thermal energy densities of order the inflationary energy density. While on the one hand GUT symmetry restoration due to these non-thermal configurations revives GUT scenarios for baryogenesis [@23; @16; @17] , on the other hand it reintroduces the problem of heavy topological defects whose solution was one of the initial motivations for inflation.
Furthermore, realistic models seeking to implement inflation need to be able to stabilize the required flat potential against radiative corrections. The only presently known method to achieve this in the presence of gauge, scalar potential, Yukawa, and gravitational interactions is by using (approximate) supersymmetry to enforce the appropriate non-renormalization. Parametrically amplified decay of the inflaton, and the resulting efficient reheat, poses a mortal danger to realistic, supersymmetric, inflationary models. In supersymmetric theories the reheat temperature is constrained [@32; @33; @34] by the need to avoid thermal overproduction of gravitinos. This leads to an upper bound on the reheat temperature of order $10^{8} - 10^{9} GeV$, which is orders of magnitude below what would be achieved by efficient reheat from amplified inflaton decay. One obvious way to avoid this disaster would be to insure that the inflaton is sufficiently weakly coupled to its bosonic decay products that the decay never experiences parametric amplification. This is typically the case for inflatons in a hidden sector which have only gravitational strength couplings to their (observable sector) decay products. On the other hand, in many models one introduces direct superpotential couplings of the inflaton to the chiral scalars into which it decays, and we wish to consider the nature of the reheat dynamics in such models, and whether they are ruled out by constraints on reheating.
In particular, we will examine the effects that the final state self-interactions of the decay products have on the parametric amplification of the inflaton decay. Because of the large occupation numbers for the modes of the produced decay bosons, we expect that the presence of self-interactions of these bosons will result in large effective masses being induced for these modes. If the bosons are thermalized these may be interpreted as thermal plasma masses from self-interaction; more generally they will occur as induced effective plasma mass terms in the mode equations for the decay field. As the mode occupation numbers increase, so do these induced masses, until they equal the mass of the inflaton, cutting off the decay. Decay resumes with the thermalization of the decay products, and their dilution and redshift by cosmic expansion, such that their induced masses dip below the inflaton mass; the system thus proceeds in a quasi-stationary process of decay and dilution such that the induced mass of the decay products is always of order the inflaton mass. This regulates the parametric amplification of the decay, preventing abrupt and efficient reheat. In our discussion we will analyze the effects of final state self-interactions for inflaton decays that would otherwise be in the regime of narrow-band parametric amplification. However, because the dominant effect is a kinematical cutoff of the decay, due to the self-induced plasma mass of the final state decay products arising from their strong self-interaction, we expect similar effects in the case of broad-band resonance. Indeed, recent studies of the broad-band decay regime [@7; @10; @14] indicate that the explosive decay from the sequence of higher resonance bands can only effectively produce particles whose mass does not exceed that of the inflaton by more than an order of magnitude. So in this case too there will be a kinematical cutoff due to the final-state self-interaction induced plasma mass of the decay products; although it will now be regulated to be not more than of order ten times the inflaton mass the qualitative effects should be otherwise similar. Applications of our present analysis to realistic classes of supersymmetric inflationary models will be considered elsewhere [@31].
As a basis for our subsequent arguments we will consider a chaotic inflation model with the following potential, whose features we take to resemble the generic features of scalar potentials which arise in supersymmetric theories: V=[1 2]{}[m]{}\^[2]{}\^[2]{} + \^[2]{} + [h]{}\^[2]{} \^[2]{}\^[2]{} +[g]{}\^[2]{}\^[4]{} where for schematic simplicity the inflaton $\phi$ and the matter scalar $\chi$ are taken to be real scalar fields and the self-coupling of the $\chi$ field is considered to be of moderate strength ${10}^{-1}< {g}^{2} < 1$. In supersymmetric theories where the decay scalars are standard model chiral scalars which are gauge non-singlets the quartic potential terms in $ \chi$ arise as D-terms and the coupling is of gauge coupling strength $g^{2}$. For inflaton-scalar couplings arising from superpotentials in supersymmetric theories one has $ \sigma = 2 h m $, and the cubic and quartic couplings of the inflaton to $\chi$ are related to each other. In general the superpotential couplings $ h$ may be, and in viable supersymmetric inflationary models are usually chosen to be, much smaller than gauge couplings $ h \ll g$. The inflaton mass $m$ must be bounded by $m{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{10}^{-6}$ in order to be consistent with COBE data on microwave background fluctuations [@35].
First let us review the effects of parametric amplification on inflaton decay, ignoring self-interaction of the decay products. The nonlinear effects that lead to amplified decay act in two different regimes:\
$[1]~~$ ${{m}^{4} \over {\sigma}^{2}}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
\phi {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{{m}^{2} \over \sigma}$ for the cubic coupling and ${{m}^{4} \over {h}^{4}}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{\phi}^{3}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{{m}^{3} \over {h}^{3}}$ for the quartic coupling. This is the narrow-band resonance case which can be analyzed perturbatively, and the dominant effect is the large occupation number for $\chi$’s. This case has been considered in [@7; @8; @10; @11; @18] , where it is shown that parametric amplification occurs and there are narrow-band resonances for $\chi$ production at $k={m \over 2}$ and $k=m$ for the cubic and quartic couplings respectively. This is the case for which we will make quantitative estimates of the effect of inclusion of final state self-interaction of the decay products.\
$[2] ~~$ $\phi {\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{{m}^{2} \over \sigma}$ for the cubic coupling and $\phi{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m\over h}$ for the quartic coupling. Here perturbation theory is not valid and problem is highly nonlinear. In this case there is broad-band resonance for a large domain of momenta [@7; @10] that leads to an explosive decay of the inflaton. As noted above, because of the essentially kinematic nature of the cutoff we expect the final-state self-interaction effects in this case to be qualitatively similar to those in the narrow band case.
Note that if the cubic and quartic couplings of the inflaton to the decay scalar are of the form arising from a superpotential coupling \[$ \sigma\simeq 2 h m $\], then the conditions for parametric amplification are more general for the cubic coupling and it will dominate the decay. Another important observation is that although most inflatons may decay during a stage of amplified decay this does not lead to the decay of the entire energy density of the inflaton. In the case of narrow-band resonance the decay stops no later than the time when $\phi$ has the minimum value in the abovementioned range (${{m}^{4} \over {\sigma}^{2}}$ for the cubic coupling and ${({m \over h})}^{{4 \over 3}}$ for the quartic coupling) [@7; @8], while in the case of broad-band resonance it stops at the time of the transition from broad-band to narrow-band resonance when the decay becomes out of equilibrium [@7; @10; @25]. After that, the remaining energy density of the inflaton is redshifted as ${R}^{-3}$ where energy density of relativistic $\chi$’s is redshifted as ${R}^{-4}$. The decay of the inflaton will then be completed as in the usual picture, and if the energy density of the inflaton dominates at that time there will be significant dilution of relic densities from the first stage of reheat [@25; @16]. In supersymmetric theories this dilution is not, generally, sufficient by itself to solve the problem of overproduction of gravitinos. However, combined with the effect of final state self-interaction, which we consider below, it can successfully resolve the gravitino problem in many models.
Now let us consider generally the changes to the parametric amplification reheat dynamics that arise from the self-interactions of the final state decay products. Since by assumption the inflaton decays to observable sector standard model (s)particles the final state bosons carry gauge quantum numbers. For these fields self-interactions with couplings as strong as the gauge coupling arise at tree-level from D-terms in supersymmetric models [^1],
and at the one-loop level in the non-supersymmetric case. In addition, there are couplings of non-singlet scalars to the gauge fields as well as possible large superpotential couplings. The decay of the inflaton produces quantum fluctuations along the direction of final state particles in field space and drives them to large field values where the effect of self-coupling becomes important. These large field values induce a self-mass for the decay products that subsequently slows down the decay and tends to shut it off .
During the oscillatory regime $\phi \cong{\phi}_{0} \cos{(mt)}$, and immediately after the end of inflation ${\phi}_{0}\sim 10^{-1}$. Subsequently ${\phi}_{0}$ decreases with time because of decay and Hubble expansion. For ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{m \over h}$ the inflaton decays predominantly via the cubic term. As was discussed in [@7; @8; @10], in the range ${({m \over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{m \over h}$ parametric amplification occurs and there is a narrow-band resonance for $\chi$ production at $k={m \over 2}$. In this range for the inflaton field we can use the Feynman diagrams for one-particle decay, but the occupation number of final state particles is non-trivial and must be taken into account in the calcualtions. According to [@7; @8; @10] parametric amplification will effectively convert most of the energy density of the inflaton into radiation once it is in the afore-mentioned range. Assuming rapid subsequent thermalization this leads to a reheat temperature ${T}_{R}\sim m{h}^{-{1 \over 2}}$ that is much higher than that of the usual picture $\sim 0.1{m}^{{1
\over 2}}h$ for reasonable values of $h$. The quartic self-coupling of $\chi$ will induce a finite temperature correction to the mass-squared of $\chi$, if they are thermalized, of order ${g}^{2}{{T}_{R}}^{2}\sim {g}^{2}{m}^{2}{h}^{-1}$ at this time which is much larger than ${m}^{2}$ (as we mentioned above the non-thermal corrections that exist before thermalization are even larger). However at an earlier time ${t}_{d}$ when the thermal (or non-thermal [^2] ) correction is of order ${{m}^{2} \over 4}$ the one-particle decay becomes kinematically forbidden (note that the thermal correction to mass-squared of $\phi$ is of order ${h}^{2}
{{T}_{R}}^{2}$ which is normally smaller than ${m}^{2}$ for $h$ a typical Yukawa type coupling).
The Hubble expansion that subsequently occurs redshifts the correction to the mass-squared of $\chi$ as ${R}^{-2}$ and this causes further decay. As long as ${t}_{d}<{H}^{-1}$ these successive steps of expansion, decay and (perhaps) thermalization continue. Eventually the decay is not effective enough to compensate for expansion and there is a delay before the remaining energy density of the inflaton is converted into relativistic particles as in the usual picture, and the decay of the inflaton is completed.
Now let us perform a detailed mode by mode analysis of the effect of quartic self-coupling of final state particles. Consider the potential
=[1 2]{}[m]{}\^[2]{}\^[2]{}+2hm\^[2]{} where $\phi$ and $\chi$ are both real scalars. By mode expansion of $\chi$ we derive the following equation for each mode
+3[ ]{}+([k]{}\^[2]{}+4hm\_[0]{})\_[k]{}=0 where a dot denotes differentiation with respect to time, and in this equation for the modes associated with comoving wavenumbers, we are using the physical wavenumber $k$, where $R k = k_{comoving}$ with $R$ the scale factor . For the moment we will ignore the effect of Hubble expansion, since it generally occurs over a longer time scale than that of the effects we consider. We will consider issues of thermalization and Hubble expansion below \[for other treatments of the effect of Hubble expansion on the parametric resonance see [@7; @8; @11]\]. By choosing $z={m \over 2}t$, and in the absence of final state self-interactions we derive a Mathieu equation for the modes of the $ \chi$ field \^["]{}+([[k]{}\^[2]{} \^[2]{}]{} + [4hm\_[0]{} \^[2]{}]{})\_[k]{}=0 where prime denotes differentiation with respect to $z$. In the case of narrow-band resonance (in which we perform our calculations) the Mathieu equation has resonance solutions in the first instability band ${({m \over 2})}^{2}-4hm{\phi}_{0}<{k}^{2}<{({m \over
2})}^{2}+4hm{\phi}_{0}$. Modes in this band grow exponentially in time, which one interprets as particle production. The (slow) Hubble expansion will eventually drive the modes out of the instability band, but they spend enough time there to reach a substantial occupation number [@18]. This time ${t}_{b}$ can be approximately calculated from $\delta k \sim k H {t}_{b}$ where $k={m
\over 2}$ and $\delta k=8h{\phi}_{0}$ is the width of the first instability band which gives ${t}_{b} \sim {16 h \over {m}^{2}}$.
The quartic self-coupling ${g}^{2}{\chi}^{4}$ will induce an effective mass-squared ${{m}^{2}}_{eff}$ for the $\chi$ field and the Mathieu equation with the addition of this self-coupling will be modified to become
\^["]{}+([[k]{}\^[2]{} + [[m]{}\^[2]{}]{}\_[eff]{} \^[2]{}]{} + [4hm\_[0]{} \^[2]{}]{})\_[k]{}=0
In a background of isotropically distributed $\chi$’s over a narrow band of momenta (which is the case in $\phi \rightarrow \chi \chi$ decay before thermal distribution is achieved) ${{m}^{2}}_{eff} \sim
{g}^{2} {{n}_{\chi} \over {E}_{\chi}}$ to the leading order, with ${n}_{\chi}$ the number density of $\chi$’s and ${E}_{\chi}$ their energy [@7; @10; @12; @13; @14]. After thermalization, in a thermal background of temperature $T$, we will have the standard result ${{m}^{2}}_{eff} \sim {g}^{2} {T}^{2}$.
At $t=0$ when $\phi$ starts oscillating ${{m}^{2}}_{eff}=0$ and there is resonance in the band ${({m \over
2})}^{2}-4hm{\phi}_{0}<{k}^{2}<{({m \over 2})}^{2}+4hm{\phi}_{0}$. With particle production in this band ${{m}^{2}}_{eff}$ increases and resonance in this band stops when ${{m}^{2}}_{eff} = 8hm{\phi}_{0}$ after which the number density of particles produced in this band remains unchanged. There will be, however, resonance in the band ${({m \over 2})}^{2}-12hm{\phi}_{0}<{k}^{2}<{({m \over
2})}^{2}-4hm{\phi}_{0}$ that also stops when ${{m}^{2}}_{eff}$ increases by a further amount of $8hm{\phi}_{0}$. This incremental change of ${{m}^{2}}_{eff}$ and smooth transition from one resonance band to the next one continues until ${{m}^{2}}_{eff}={({m \over
2})}^{2}$, after that there is no resonance solution for physical states, i.e. states with ${k}^{2}>0$. If we take ${E}_{\chi}\simeq ({m \over 2})$ for particles in all bands a change of $8hm{\phi}_{0}$ in ${{m}^{2}}_{eff}$ correponds to an increase in number density of $\chi$’s by ${\delta n}_{\chi}\sim
{4h{m}^{2}{\phi}_{0} \over {g}^{2} }$. For the first band ${({m
\over 2})}^{2}-4hm{\phi}_{0}<{k}^{2}<{({m \over
2})}^{2}+4hm{\phi}_{0}$ and for $h{\phi}_{0}\ll m$ the occupation number ${f}_{{m \over 2}}$ required to shift the resonance to the next band is calculated to be
\_\~[1 \^[3]{}]{}[f]{}\_[[m 2]{}]{} 4\^[2]{}8h\_[0]{}\~[4 h\_[0]{}[m]{}\^[2]{} \^[2]{}]{}\_[[m 2]{}]{}\~[4\^[2]{} \^[2]{}]{}
It has been shown [@18] that in the small amplitude limit the analytic result of the decay rate in the $n$-th instability band can also be derived from a physical process, $n$ particles with zero momentum that comprise the classical homogeneous inflaton field annihilating into two final state bosons. In particular, for the first instability band and for ${f}_{k}\ll 1$ we can use the one-particle decay rate from the standard perturbation theory $\Gamma
={{h}^{2}m \over 8\pi}$. Narrow-band parametric amplification has been shown to be indeed an induced process [@18] which means for ${f}_{k}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
1$ the enhancement of decay rate by the factor $(1+ {f}_{k})$ must be taken into account in the particle calculation. The time ${t}_{1}$ which is needed to reach ${f}_{{m \over 2}} = 1$ can be approximately calculated as [^3]
\_[1]{}[m]{}[\_[0]{}]{}\^[2]{}\~ [[2h[m]{}\^[2]{}\_[0]{} \^[2]{}]{} ]{} \_[1]{} \~[4 h \_[0]{}]{} and considering the enhancement factor the time $\delta t$ which is needed to reach ${f}_{{m \over 2}}={4{\pi}^{2} \over {g}^{2}}$ is
t\~[4 h \_[0]{}]{}
This is the time for the resonance in the band ${({m \over
2})}^{2}-4hm{\phi}_{0}<{k}^{2}<{({m \over 2})}^{2}+4hm{\phi}_{0}$ to stop. For the next bands ${k}^{2}$ is smaller and this means that less phase space volume is available for decay products or, equivalently, the occupation number for those bands is larger. Therefore, more time will be needed for production of ${\delta
n}_{\chi}\sim {4h{m}^{2}{\phi}_{0} \over {g}^{2} }$ in bands with smaller ${k}^{2}$, but not greatly so, as the larger occupation numbers are obtained rapidly due the the large coherent final state enhancement. A reasonable lower estimate for the decay time ${t}_{d}$ to effectively achieve ${{m}^{2}}_{eff} \sim {{m}^{2} \over
4}$ is
\_[d]{}\~[[[m]{}\^[2]{} 4]{} 8 h m \_[0]{}]{} t \~ [ m8 \^[2]{}[\_[0]{}]{}\^[2]{}]{}
In order to have physically realistic estimates we take ${g}^{2}
\cong {10}^{-1}$ in our calculations. This leads to
t \~[8 h \_[0]{}]{} , [t]{}\_[d]{} \~[m 4[h]{}\^[2]{}[\_[0]{}]{}\^[2]{}]{} So resonance at each band of width $8hm{\phi}_{0}$ ends in a time $\sim{8 \over h{\phi}_{0}}$and in an approximate time of ${m \over 4{h}^{2}{{\phi}_{0}}^{2}}$ decay effectively stops. We notice that ${t}_{d}< {H}^{-1}$ for ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{({m \over h})}^{2}$.
In this analysis the effect of self-coupling of $\chi$ on the solutions of the Mathieu equation was considered only to the first order, which is reasonable for the case of narrow-band resonance $h{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
m$. The key assumption in our treatment of the Mathieu equation in the presence of the nonlinear term was adiabaticity, i.e. that we can use the instantaneous value of ${{m}^{2}}_{eff}$ which is also legitimate since it changes over a time $\delta t\sim {8 \over h{\phi}_{0}}$ which is greater than ${m}^{-1}$, the period of oscillations of $\phi$, again because we are in the narrow-band resonance regime $h{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
m$.
Our results show major differences from the simple parametric amplification case. The effect of large occupation number for final state particles is not that dramatic here because there is a whole range of resonance bands instead of a single one and the effect of the self-interaction of the produced particles drives modes out of the resonance bands much faster than the Hubble expansion. Consequently the leading effect that influences the decay is the self-interaction of the decay products, which stops it very early.
So far we have not considered thermalization of decay products and the Hubble expansion. The temperature of the thermal bath after thermalization of $\chi$’s is calculated from [^4]
([g]{}\_[B]{}+[7 8]{}[g]{}\_[F]{})[T]{}\^[4]{} = \_ = [n]{}\_[E]{}\_ \~[[m]{}\^[4]{} 16 [g]{}\^[2]{}]{} where ${g}_{B}$ , ${g}_{F}$ are the number of bosonic and fermionic degrees of freedom, respectively. We take the number of degrees of freedom to be the one in the minimal supersymmetric standatd model (${g}_{B}={g}_{F}=74$) which leads to a temperature $T\simeq {g}^{-{1
\over 2}}{m \over 5}$. For this temperature, however, the correction to mass-squared of $\chi$ is
\_[eff]{} \~[g]{}\^[2]{} [T]{}\^[2]{} \~g[[m]{}\^[2]{} 25]{} which is much less than ${({m \over 2})}^{2}$ for ${g}^{2} \cong
{10}^{-1}$. This is just what we expected because ${{m}^{2}}_{eff}
\sim {g}^{2}{{n}_{\chi} \over {E}_{\chi}}$ will be smaller after thermalization when the number density decreases and the mean energy of particles increases. Therefore if thermalization is effective, it will lower ${{m}^{2}}_{eff}$ significantly which leads to further decay and thermalization. This sequence of decay and thermalization stops when ${g}^{2} {T}^{2} \sim {{m}^{2} \over 4}$ that is at a temperature $T\sim m$ for ${g}^{2} \cong {10}^{-1}$ if the sequence is completed within a Hubble time.
Considering thermalization of decay products and the Hubble expansion, there are different possibilities depending on the relation among different time scales involved in the problem, ${t}_{osc}\sim {m}^{-1}$, $\delta t\sim {8 \over h {\phi}_{0}}$, ${t}_{d}\sim {m \over 4{h}^{2}{{\phi}_{0}}^{2}}$, ${t}_{th}\sim {32
\pi \over \alpha m}$, [^5] and ${t}_{H}\sim\sqrt{{3 \over 8 \pi}}{1 \over m {\phi}_{0}}$. The most important thing is that ${t}_{osc}$ is considerably smaller than all other time scales as long as we are in the narrow band regime ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{m \over h}$. Therefore changes in ${{m}^{2}}_{eff}$ caused by decay or thermalization (that can be considered as changes in the background) are adiabatic and our analysis is in principle valid, irrespective of the relation among ${t}_{d}$, ${t}_{th}$, ${t}_{H}$. Regarding these time scales there are different cases:
\[1\]- $\delta t{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$. This occurs when the inequalities ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
5\times {10}^{-3}{m \over h}$ (to have ${t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{th}$), ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}3\times {10}^{-5}$ (to have ${t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$) and ${({m \over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m \over h}$ (to have ${t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$) are satisfied which necessarily means ${m \over h}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{10}^{-2}$. In this case the following sequence of events happens: decay of the inflaton to $\chi$’s, end of the decay, and thermalization of decay products to a temperature $T\sim m$, all in a time scale shorter than the Hubble time. Hubble expansion then redshifts $T$ but for ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{({m \over h})}^{2}$ (assuming ${t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{t}_{H}$ all through the way) this sequence keeps $T\sim m$. For ${\phi}_{0}<{({m \over h})}^{2}$ decay is no longer effective to compensate for expansion and we must wait until a later time when ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{h}^{2}$ and decay is completed as in the standard picture. This second stage of decay dilutes the gravitinos that are produced earlier during the first stage.
\[2\]- $\delta t{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$. This occurs for ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}~min[5\times {10}^{-3}{m \over h},3\times {10}^{-5}]$ and ${({m
\over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m \over h}$ which is possible only for ${m \over h}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
5\times {10}^{-3}$. This case is similar to case \[1\], only because of the faster thermalization the temperature is higher and closer to the maximum possible $\sim m$.
\[3\]- $\delta t{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{th}$. This occurs when ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}~Max[5\times {10}^{-3}{m \over h},3\times {10}^{-5}]$ and ${({m
\over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m \over h}$ that is consistent only for ${m \over h}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
3\times {10}^{-5}$. In this case decay stops without effective thermalization in a Hubble time, so the distribution of $\chi$’s is out of equilibrium. Hubble expansion redshifts ${{m}^{2}}_{eff} $ as ${R}^{-2}$ and further decay compensates for this change. A thermal distribution of particles is not achieved, however, unless ${t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$ [^6] . For ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{({m \over h})}^{2}$ the situation is as in case \[1\].
\[4\]- ${t}_{th}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
\delta t{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{d}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{t}_{H}$. This occurs when ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{10}^{-3}{m \over h}$ and ${({m \over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m \over h}$ which is possible only for ${m \over h}{\
\lower-1.2pt\vbox{\hbox{\rlap{$>$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{10}^{-3}$. In this case the decay products thermalize almost instantaneously and there is thermal equilibrium from the very beginning of the decay. Actually both $\delta t$ and ${t}_{d}$ are much greater than $\sim {15 \over h{\phi}_{0}}$ and $\sim {m \over
2{h}^{2}{{\phi}_{0}}^{2}}$ in this case because rapid thermalization of decay products in the thermal bath keeps the occupation number at each resonance band below one. This means that temperature can be much lower than its maximum $\sim m$ and the situation is very similar to the one in the standard picture of reheating.
Cases \[1\] and \[2\] are the most dangerous regarding the problem of gravitino production. In these cases thermal equilibrium can in principle be achieved from the very beginning and can last until the time ${H}^{-1} = \sqrt{{3 \over 8 \pi}}{m}^{-3}{h}^{2}$. Even in these cases the gravitino overproduction is not that serious since at most we have a thermal bath with temperature $T \sim m$ for a time $t \sim \sqrt{{3 \over 8 \pi}}{m}^{-3}{h}^{2}$, much shorter than $t
\sim {m}^{-2}$ (assuming $m\ll {({m \over h})}^{2}$) which is the case for a radiation-dominated universe with temperature $T \sim m$. Case \[3\], on the other hand, is the most secure since the distribution of $\chi$’s is out of equilibrium, a distribution with less mean energy per particle and higher number density compared with a thermal distribution. Depending on the parameters of the model $m,h$ and strength of the self coupling ${g}^{2}$ one or all of these cases can happen for ${({m \over h})}^{2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{m \over h}$ but the temperature is at most of order $m$ as long as the inflaton is in this range, and is redshifted after that. Also the thermal and non-thermal corrections to the mass-squared of $\chi$ are always ${\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{{m}^{2} \over 4}$.
To compare our results with that of parametric amplification without the effect of final state self-interaction let us consider the case $h={10}^{-6}$,$m={10}^{-7}$, where the standard picture predicts ${T}_{R}\sim 0.1{m}^{{1 \over 2}}h={10}^{8}$ GeV. In the simple parametric amplification case almost all the energy density of the inflaton is converted into radiation once ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{m \over h}={10}^{-1}$ and leads to a very high reheat temperature ${T}_{R}\sim{10}^{15}-{10}^{16}$ GeV which, from the point of view of gravitino overproduction, is a disaster. According to our present analysis we are in the case \[3\] in the above all the way from ${\phi}_{0}={m \over h}={10}^{-1}$ to ${\phi}_{0}={({m \over h})}^{2}={10}^{-2}$. This means that thermalization is not effective for ${10}^{-2}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }
{10}^{-1}$ and occurs much later when decay is no longer effective, so the temperature during the first stage of decay is actually lower than $\sim {10}^{12}$ GeV. This results in a gravitino number density after the first epoch of decay which is too large by a factor of at most $10^{12}$. The inflaton decay is then completed via a second stage when ${\phi}_{0}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{{h}^{2} \over 75}\simeq {10}^{-14}$. By this time the temperature of the thermal bath and the momentum of the relativistic particles that might have been produced during the first stage are redshifted by a factor ${({{10}^{-28} \over {10}^{-4}})}^{{1 \over 3}} =
{10}^{-{8}}$. The second stage will determine the effective reheat temperature to be $\sim {10}^{8}$ GeV, and releases a large amount of entropy that dilutes the gravitinos produced during the first stage of decay by a factor of ${({{10}^{12} \over {10}^{{8}}})}^{3} =
{10}^{12}$ which is now sufficient as a dilution factor.
As we noted previously, since the effect which we consider is a kinematical cutoff due to the large self-induced plasma mass of the strongly self-interacting final state decay products, we expect to also have similar effects in the broad-band case. Studies of the broad band resonance case [@7; @10; @14] indicate efficient production of the decay products only for masses up to about an order of magnitude larger than the inflaton mass, so that when the broad-band resonance has built up a sufficient density of the decay products ( typically in a non-thermal “preheat” distribution) that their self-induced plasma mass exceeds this range, the decay will be suppressed. As above the decay will subsequently proceed as thermalization and Hubble expansion reduce the number densities and the induced self-mass for the decay products dips into the accessible range, resulting in a regular, quasi steady-state transfer of energy into the decay products.
Finally, we contrast the effects considered herein with those considered by Khlebnikov and Tkachev [@13], who studied the semi-classical non-linear effects of the inflaton decay coupling in a massless $\lambda {\phi^{4}} $ model. Note that the effects they study have a different and independent origin from those considered here. The effects which we study are [*specifically*]{} due to final state self-interactions of the decay products which are different from, and larger (gauge strength) than, the inflaton decay coupling. If these final-state self-couplings are present, then we saw above that their effects acted to regulate the parametric amplification of inflaton decay.
Returning to the narrow-band case, as treated above, it is useful to ask if there are viable models where we are in the narrow-band regime from the beginning of oscillations, and for which the analysis presented here provides quantitative, as well as qualitative guidance. We note that in a simple chaotic inflation model with potential $V(\phi )={1 \over 2}{m}^{2}{\phi}^{2}$ inflation ends when $V"(\phi )\simeq 24 \pi V(\phi)$ which happens for $\phi {\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{10^{-1}}$. In the case of primordial supersymmetric inflation or new inflationary models the amplitude of post-inflation oscillations around the global minimium is generically substantially smaller than the Planck scale. Depending on the parameters then, some of these models may satisfy the inequalities necessary to be in the narrow-band resonance regime, or even for the nonlinear effects to be negligible. Interestingly enough, it seems that in some viable supersymmetric models this is indeed the case, and the analysis we have here undertaken is quantitatively valid. We will return to these issues elsewhere [@31].
In conclusion, we have seen that the self-interaction of final state bosons of moderate strength, that arises very naturally in supersymmetric models, has an important impact on the decay of observable sector inflatons, besides producing a rapid thermalization rate. As a step towards improved understanding of the reheating process we have considered a simple schematic model representing generic features of supersymmetric theories, with such a final state self-coupling, and have shown that in the case of narrow-band resonance the outcome is qualitatively different from that of simple parametric amplification. Here inflaton decay occurs during two stages: one stage that consists of successive steps of partial decay, thermalization and expansion at early times which ends relatively soon, and a second stage as in the standard picture that completes the decay. In the first stage because of the quasi-adiabaticity we were able to show that the temperature and the amplitude of quantum fluctuations of final state particles are at most of the order of the mass of the inflaton (approximately), and the temperature is several orders of magnitude below the naive predictions of parametric amplification. The second stage of decay then determines the final reheat temperature and releases a substantial amount of entropy; is of particular importance in order to dilute the previously produced gravitinos in realistic supersymmetric models.
[*Note Added*]{}\
After this paper was submitted a preprint appeared from Prokopec and Roos \[hep-ph/9610400\] who do lattice simulations of inflaton decay. In the case where the decay product field has final state self-interactions of moderate strength their simulations exhibit regulation of the broad-band resonance decay, in agreement with the general analytic arguments we have presented above for the broad-band case.
[**Acknowledgements**]{}\
We are deeply indebted to both John Ellis and Andrei Linde for enlightening discussions, thoughtful criticism, and ongoing collaboration. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. In addition, RA wishes to thank the Iranian Ministry of Culture and Higher Education for continuing support.
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hep-ph/9606260. G. Anderson, A. D. Linde, and A. Riotto, preprint FERMILAB-Pub-96/078-A and
hep-ph/9606416. S. Kasuya and M. Kawasaki, preprint ICRR-Report-360-96-11 and hep-ph/9603317. D. T. Son, preprint UW-PT-96-01 and hep-ph/9601377. D. T. Son, preprint UW-PT-96-05 and hep-ph/9604340. M. Yoshimura, Prog. Theor. Phys. [**94**]{}, 873 (1995). M. Yoshimura, preprint TU-96-499, and preprint hep-ph/9603356. M. Yoshimura, preprint TU-96-500, and preprint hep-ph/9605246. H. Fujisaki, K. Kumekawa, M. Yamaguchi, and M. Yoshimura, Phys. Rev. [**D53**]{}, 6805 (1996). H. Fujisaki, K. Kumekawa, M. Yamaguchi, and M. Yoshimura, preprint TU-95-493, and preprint hep-ph/9511381. D. Boyanovsky, H. J. de Vega, R. Holman, D. -S. Lee, and A. Singh, Phys. Rev. [**D51**]{}. 4419 (1995) . D. Boyanovsky, M. D’Attanasio, H. J. de Vega, R. Holman, and D. -S. Lee, Phys. Rev. [**D52**]{}. 6805 (1995) . D. Boyanovsky, M. D’Attanasio, H. J. de Vega, R. Holman, and D. -S. Lee, preprint PITT-95-1755 and hep-ph/9511361. D. Boyanovsky, H. J. de Vega, R. Holman, and J.F.J. Salgado, hep-ph/9608205. D. Boyanovsky, H. J. de Vega, R. Holman, and J.F.J. Salgado, hep-ph/9609007. R. Allahverdi, B.A. Campbell and J. Ellis, in preparation. J. Ellis, J. E. Kim, and D. V. Nanopolous, Phys. Lett. [**145B**]{}, 181 (1984). M. Kawasaki and T. Moroi, Prog. Theor. Phys. [**93**]{}, 879 (1995) for a recent review see: S. Sarkar, preprint OUTP-95-16-P and hep-ph/9602260. B.A. Campbell, S. Davidson, and K.A. Olive, Nucl. Phys. [**B399**]{}, 111 (1993). M.K. Gaillard, H. Murayama, and K.A. Olive, Phys. Lett. [**B355**]{}, 71 (1995).
[^1]: There are, of course, directions in scalar field space which are both D-flat and F-flat in the supersymmetric standard model; in realistic no-scale supergravity models they will in general already have developed Planck scale vevs during inflation [@36]. If the decay coupling of the inflaton happened to align along one of these directions then final state interactions of the type we consider would not affect this particular decay mode. More generally the decay products will themselves, in turn, have decay modes along these directions. We consider these issues elsewhere [@31].
[^2]: Whether the thermal or non-thermal correction should be considered depends on how rapid the thermalization rate is. Details will be discussed shortly.
[^3]: Note that ${e}^{-\Gamma {t}_{1}}\simeq 1-\Gamma {t}_{1}$ for $\Gamma {t}_{1}\ll 1$, which is valid here.
[^4]: In general the number density of particles is not the same before and after thermalization but the energy density is conserved.
[^5]: ${t}_{th}={{\Gamma}_{th}}^{-1}$ with ${\Gamma}_{th}\sim
{\alpha}^{2} {{n}_{\chi} \over {m}^{2}}$ is for an out of equilibrium distribution of $\chi$’s. It is easily seen that for a thermal bath with $T\sim {m \over 2 g}$ (which is the highest temperature that can be achieved) ${m}^{-1}{\
\lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\
}{t}_{th}$ also.
[^6]: In this case Hubble expansion moves particles from one band to another one. This slightly lowers the time $\delta t$ spent in those bands because now there is an initial number of particles in each band. This, however, is negligible since $\delta {n}_{\chi}$ is also redshifted as ${R}^{-3}$.
|
---
author:
- |
\
(for the STAR Collaboration)\
Kent State University, Ohio 44242, USA\
E-mail:
title: Directed flow in Au+Au collisions from the RHIC Beam Energy Scan at the STAR experiment
---
Introduction
============
A goal of research at the Relativistic Heavy Ion Collider (RHIC) is to explore deconfined quark-gluon matter [@STARwhitepaper]. Directed flow ($v_{1}$) is one of the observables that is sensitive to the dynamics of the system in early times during the collision process. Both hydrodynamical and transport model calculations indicate that the $v_{1}$ slope at mid-rapidity, especially for baryons, is sensitive to the Equation of State (EoS) of the system [@Rischke; @Stoecker]. Based on hydrodynamics, a minimum in $v_{1}$-slope is proposed as a signature of a first-order phase transition between hadronic matter and quark-gluon plasma (QGP) [@Rischke; @Stoecker]. STAR has taken data over a wide range of beam energies ($\sqrt{s_{NN}} = $7.7–200 GeV) to explore the QCD phase diagram. One of the goals of this Beam Energy Scan (BES) is to find the softest point of the QCD Equation of State [@BES-I]. The $v_1$-slope of protons from phase-I of BES [@STAR-BESv1] changes sign near $\sqrt{s_{NN}} \sim 10$ GeV with a minimum around $\sqrt{s_{NN}} = $10–20 GeV. Also the net-proton $v_1$-slope shows a double sign change with a more pronounced minimum in the same energy range. Such a behavior shows a qualitative resemblance to a 3-fluid hydrodynamic model calculation with a 1st-order phase transition [@Stoecker]. However, current state-of-the-art models are not able to reproduce the basic trend of energy dependence of proton $v_1$-slope reported by STAR [@hybrid; @phsd; @phsd2; @3FD; @JAM; @JAM2]. Moreover, the $v_{1}$ results from models with nominally similar prescriptions for the equation of state differ by an order of magnitude [@v1review]. More theoretical progress is necessary for a definitive interpretation of the data.
Recently we have measured $v_1$ for $\Lambda$, $\bar{\Lambda}$, $K^{\pm}$, $K_{S}^{0}$ and $\phi$ in Beam Energy Scan Au+Au collisions at $\sqrt{s_{NN}} =
$ 7.7–200 GeV using the STAR detector [@v1ncq]. $\Lambda$ hyperon offers the opportunity to study a second baryon species, and can compliment the proton data. Charged kaons and $K_{S}^{0}$ have been observed to depend on the kaon-nucleon potential at the energy of AGS experiments [@kaon-potn]. One can test whether any such effect can be observed in the low energy domain of BES. Furthermore, the $\phi$ meson is interesting because its mass is close to the mass of baryons, but it is a vector meson. So $\phi$ can serve as a probe to study whether the $v_{1}$ depends on flavor (baryon or meson) or mass of a particle species. The $\phi$ is less affected by hadronic interactions than other species. Thus it can be used as a clean probe to study the contribution of the partonic phase to $v_{1}$. The $v_1$ measurements with ten different hadron species, having different constituent quarks, will help to disentangle the role of produced and transported quarks in heavy-ion collisions.
![(Color online) Rapidity dependence of directed flow ($v_{1}$) for $\Lambda$, $\overline{\Lambda}$, $K^{+}$, $K_s^0$, $K^{-}$ and $\phi$ in 10-40% and 40-80% Au+Au collisions at $\sqrt{s_{NN}} = $ 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV.[]{data-label="fig-v1vsy"}](figures/v1y_snn_all.pdf)
Analysis Details
================
The analysis was done using Au+Au collision data taken by the STAR detector during the years 2010, 2011, and 2014. The STAR detector offers uniform acceptance, full azimuthal coverage and excellent particle identification [@starnim]. The Time Projection Chamber (TPC) [@startpc] is the main detector which performs charged particle tracking near mid-rapidity. The collision centrality is estimated using the charged particle multiplicity in the rapidity region $|\eta| < 0.5$, measured with the TPC. We utilize both the Time Projection Chamber (TPC) and Time-of-Flight (ToF) [@TOF] detectors for particles identification. The first-order reaction plane ($\Psi_1$) is estimated using the two Beam Beam Counters (BBC) [@BBC], covering 3.3 < $|\eta|$ < 5.2, for $\sqrt{s_{NN}} = $ 7.7–39 GeV. The event plane resolution from the BBCs deteriorates at $\sqrt{s_{NN}}$ > 39 GeV. So we utilize the Zero Degree Calorimeters (ZDC) [@GangThesis], which cover $|\eta| >
6.3$, to estimate $\Psi_1$ for $\sqrt{s_{NN}} = $ 62.4 and 200 GeV. The large $\eta$-gap of these forward detectors (BBC and ZDC) relative to the TPC reduces non-flow contribution in our $v_{1}$ measurements.
Results and discussions
=======================
Figure \[fig-v1vsy\] reports the rapidity dependence of $v_{1}$ for $\Lambda$, $\overline{\Lambda}$, $K^{\pm}$, $K_{S}^{0}$ and $\phi$ in $10-40\%$ and $40-80\%$ central Au+Au collisions at $\sqrt{s_{NN}} = $ 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The data points for $\overline{\Lambda}$ at 7.7 GeV are divided by five in order to present the data in the same vertical scale. It is a common practice to present the strength of $v_{1}(y)$ by its slope at mid-rapidity. In Ref. [@STAR-BESv1], the $v_1$ slope parameter ($dv_{1}/dy$) at mid-rapidity was extracted by fitting the data to a cubic function ($F_{1}\,y + F_{3}\,y^{3}$). In this analysis, poor statistics for $\overline{\Lambda}$ and $\phi$ do not allow a stable cubic fit as done in Ref. [@STAR-BESv1]. We extract $dv_{1}/dy$ by using a linear function ($F_{1}\,y$) over the rapidity region $|y|< 0.8$, except for $\phi$ where the fit is performed for $|y|< 0.6$. The energy dependence of the $v_{1}$ slope for $\pi^{\pm}$, $K^{\pm}$ and $K_{S}^{0}$ mesons are shown in the top panel in Fig \[fig-dv1dy\]. It is observed that all the mesons have negative $v_{1}$ slope. The magnitude of the slope increases with decreasing beam energy. The $K^+$ slope lies above $K^-$ at and above $\sqrt{s_{NN}} = $ 11.5 GeV while a reversal of their positions is observed at $\sqrt{s_{NN}} = 7.7$ GeV. The $K_{S}^{0}$ is consistent with the mean of $K^{+}$ and $K^{-}$ at all studied beam energies. Within the present uncertainties, no distinct mass ordering is observed among the mesons.
![(Color online) Top panel: Beam energy dependence of $dv_{1}/dy$ for $\pi^{\pm}$, $K^{\pm}$ and $K_s^0$ in 10-40% Au+Au collisions. Bottom panel: Beam energy dependence of $dv_{1}/dy$ for $\Lambda$, $\overline{\Lambda}$, $p$, $\bar{p}$ and $\phi$ in 10-40% Au+Au collisions.[]{data-label="fig-dv1dy"}](figures/dv1dy_snn_mesons.pdf "fig:") ![(Color online) Top panel: Beam energy dependence of $dv_{1}/dy$ for $\pi^{\pm}$, $K^{\pm}$ and $K_s^0$ in 10-40% Au+Au collisions. Bottom panel: Beam energy dependence of $dv_{1}/dy$ for $\Lambda$, $\overline{\Lambda}$, $p$, $\bar{p}$ and $\phi$ in 10-40% Au+Au collisions.[]{data-label="fig-dv1dy"}](figures/dv1dy_snn_plamphi.pdf "fig:")
The energy dependence of $dv_{1}/dy$ for the baryons and for $\phi$ are presented in the bottom panel of Fig. \[fig-dv1dy\]. We observe that $p$ and $\Lambda$ show a sign change near $\sqrt{s_{NN}} = $11.5 GeV. The $dv_{1}/dy$ of $\bar{p}$ and $\overline{\Lambda}$ remain negative at all beam energies and their magnitude increases with decreasing beam energy. The $\phi$ meson $dv_{1}/dy$ follows the trend of anti-baryons for $\sqrt{s_{NN}} > $ 14.5 GeV, while its slope turns towards zero at lower energies. Current statistical and systematic uncertainties for $\phi$ $v_{1}$ at 7.7 and 11.5 GeV are too large to draw any conclusions.
The $v_1$ can have contributions from both produced and transported quarks. To disentangle the contribution of the two sources, we define a net particle $v_{1}$ as\
$$\begin{aligned}
\label{formula1}
F_{\Lambda} = r_{1}(y) \,F_{\,\overline{\Lambda}} + [1 - r_{1}(y)]\,
F_{{\rm net\,} \Lambda} ,\end{aligned}$$ $$\begin{aligned}
\label{formula2}
F_{p} = r_{2}(y) \,F_{\,\overline{p}} + [1 - r_{2}(y)]\, F_{{\rm
net\,}p} ,\end{aligned}$$ $$\begin{aligned}
\label{formula3}
F_{K^{+}} = r_{3}(y) \,F_{K^{-}} + [1 - r_{3}(y)]\, F_{{\rm net\,}K} ,\end{aligned}$$ where $F$ denotes the $dv_{1}/dy$ for each species and $ r_{i}(y)$ denotes the rapidity dependent ratios of corresponding anti-particle to particle yields. Figure \[eg-netpv1\] presents the energy dependence of directed flow slope for net $\Lambda$, net $p$ and net $K$. It is observed that net baryons (net $p$ and net $\Lambda$) agree within uncertainties at all energies. The net-$K$ $dv_{1}/dy$ is observed to follow net $p$ and net $\Lambda$ for $\sqrt{s_{NN}}>$ 14.5 GeV, while it diverges from the trend below 14.5 GeV.
![(Color online) Beam energy dependence of net-particle $dv_{1}/dy$ in 10-40% Au+Au collisions.[]{data-label="eg-netpv1"}](figures/net_PKL_latest.pdf)
Number of Constituent Quark (NCQ) scaling has been observed in higher flow harmonics ($v_{2}$ and $v_{3}$) at RHIC and LHC energies [@ncq1; @ncq2; @ncq-lhc]. Such scaling is interpreted as evidence of quark degrees of freedom in the early stages of heavy-ion collisions. The $v_{1}$ measurements for ten different particle species allow us to extend tests of a quark coalescence hypothesis using both produced and transported quarks. In a naive quark coalescence picture, we can assume that quarks acquire $v_{1}$ from the medium and then those quarks statistically coalesce to form hadrons. The coalescence sum rule hypothesis can be written as $(v_{n})_{\rm hadron} = \sum (v_{n})_{\rm constituent \, quarks}$. In panel (a) of Fig. \[fig-v1scaling\], we present a test of the quark coalescence sum rule hypothesis utilizing particle species where all the constituent quarks are produced in collisions. The solid black marker presents $dv_{1}/dy$ for $\overline{\Lambda}$. This result is compared with a sum rule hypothesis where we add one third of $\overline{p}$ $dv_{1}/dy$ to $K^{-}$ $dv_{1}/dy$. The factor $\frac{1}{3}$ arises from the assumption that $\bar{u}$ and $\bar{d}$ have the same $v_{1}$. We also assume that the $s$ and $\bar{s}$ have the same $v_{1}$. We observe that this sum rule hypothesis holds for $\sqrt{s_{NN}} = $11.5–200 GeV, while it deviates at 7.7 GeV. The deviation at the lowest energy indicates that the above-mentioned assumptions do not hold at $\sqrt{s_{NN}} = $7.7 GeV.
![(Color online) Directed flow slope ($dv_1/dy|_{y=0}$) versus $\sqrt{s_{NN}}$ for intermediate-centrality (10-40%) Au+Au collisions. Panel (a) compares the observed $\bar{\Lambda}$ slope with the prediction of the coalescence sum rule for produced quarks. The inset shows the same comparison where the vertical scale is zoomed-out; this allows the observed flow for the lowest energy ($\sqrt{s_{NN}} = 7.7$ GeV) to be seen. Panel (b) presents two further sum-rule tests, based on comparisons with net-$\Lambda$ measurements. The expression $K^-(\bar{u}s) - {1 \over 3} \bar{p}\,(\overline{uud})$ represents the $s$ quark flow; there is no corresponding clear-cut expression for transported $u$ and $d$ quarks.[]{data-label="fig-v1scaling"}](figures/dv1dy_scaling.pdf)
Panel (b) in Fig. \[fig-v1scaling\] presents tests of the coalescence sum rule hypothesis using $u$ and $d$ quarks, which could be transported or produced. We utilize net-particle $v_{1}$ to test the coalescence hypothesis in this less straightforward situation. The fraction of transported quarks among the constituent quarks of net particles is larger than in particles roughly in proportion to $N_{\rm particle}/N_{\rm net \, particle}$. Here we assume that all the transported quarks are contained in net particles. In panel (b) of Fig. \[fig-v1scaling\], open black symbols present net-$\Lambda$ $dv_{1}/dy$. The contribution of transported quarks will increase in net $\Lambda$ at low beam energies, where the $u$ and $d$ quarks are more likely to be transported from the colliding nuclei. In the high energy limit, the constituent quarks in net $\Lambda$ are more likely to be produced in the collision. The net-$\Lambda$ result is compared with two different coalescence sum rule calculations. In the first calculation, shown by open red markers, we replace one of the $u$ quarks in net proton with a $\bar{u}$ quark, and also add the $dv_1/dy$ contribution from an $s$ quark. The $s$ quark contribution is obtained by subtracting one third of $\overline{p}$ $dv_1/dy$ from $K^{-}$ $dv_{1}/dy$. The first sum rule calculation is consistent with net $\Lambda$ within errors for $\sqrt{s_{NN}} >$ 19.6 GeV. As the contribution of transported quarks increases in net $\Lambda$ at lower energies, the first sum rule calculation deviates as we scan further below 19.6 GeV. In the second sum rule test (shown by blue open markers), we add one $s$ quark $dv_1/dy$ to two-thirds of net-$p$ $dv_{1}/dy$. Here we assume that the constituent quarks in net $p$ are dominated by transported quarks. We find that this second sum rule calculation agrees with net $\Lambda$ at the lowest beam energy and it deviates with increasing beam energy.
Conclusions and outlook
=======================
In summary, we reported the measurement of directed flow ($v_1$) near midrapidity for $\pi^{\pm}$, $K^{\pm}$, $K_s^0$, $p$, $\bar{p}$, $\Lambda$, $\bar{\Lambda}$ and $\phi$ spanning eight beam energies over the range $\sqrt{s_{NN}} = $7.7 to 200 GeV. We observe that the proton and $\Lambda$ $dv_1/dy$ agree within errors and they change sign near $\sqrt{s_{NN}} = $11.5 GeV. The slopes for $\pi^\pm$, $K^\pm$, $K^0_s$, $\bar{\Lambda}$ and $\bar{p}$ are negative at all available beam energies. The $K^0_s$ is consistent with the mean of $K^+$ and $K^-$ at all beam energies. The $\phi$ $dv_1/dy$ is negative and follows the same trend as $\bar{p}$ and $\bar{\Lambda}$ for $\sqrt{s_{NN}}>14.5$ GeV. Net-particle $dv_1/dy$ for $p$, $\Lambda$ and $K$ agree with each other at and above $\sqrt{s_{NN}} = 14.5$ GeV, but net $K$ diverges at 11.5 and 7.7 GeV. A quark coalescence sum rule hypothesis is tested using $v_1$ measurements. Produced quarks follow coalescence sum-rule behavior at 11.5–200 GeV, but strongly deviate at 7.7 GeV. This offers a new approach to probe the heavy-ion collision process at the quark level.
The STAR collaboration will upgrade detectors in phase-II of the RHIC Beam Energy Scan [@BES-II]. The upgrade of the inner TPC will not only improve particle identification but will also enhance forward rapidity coverage. A new endcap Time-of-Flight detector (eTOF) will further extend particle identification. Furthermore, future measurements will greatly benefit from a new Event Plane Detector (EPD), which will improve the first-order event plane resolution by a factor of two. The EPD will also provide an independent collision centrality determination. These detector upgrades and increased statistics in BES-II will significantly improve the precision and quality of many measurements.
Recently, a model calculation [@hfv1] predicted that the early transient magnetic field can induce a larger $v_{1}$ for heavy quarks than for light quarks. The model calculation also suggests an opposite sign of $dv_{1}/dy$ for charm ($c$) and anti-charm ($\bar{c}$) quarks. This offers hope that heavy quarks will offer a new approach to study the early electromagnetic field. STAR has collected over 2 billion events with the Heavy Flavor Tracker (HFT) detector during the years 2014 and 2016. The HFT has demonstrated excellent performance in reconstructing heavy flavor hadrons. We also look forward to the measurement of $D^{0} (\bar{u}c)$ and $\overline{D^{0}} (u\bar{c})$ directed flow utilizing both the HFT and ZDC detectors in STAR to probe the early transient magnetic field.
Acknowledgments
===============
This work is supported in part by the US Dept. of Energy under grant DE-FG02-89ER40531.
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abstract: 'We present new and accurate radial velocity (RV) measurements of luminous stars of all ages (old horizontal branch, intermediate–age red clump, and young blue plume, as well as red giants of a range of ages; 20.6$\leq$$\leq$22) in the Carina dwarf spheroidal galaxy, based on low-resolution spectra collected with the FORS2 multi-object slit spectrograph at the VLT. This data set was complemented by RV measurements based on medium and high-resolution archive spectra of brighter ($\lesssim$20.6) Carina targets collected with the GIRAFFE multi-object fiber spectrograph at the VLT. The combined sample includes more than 21,340 individual spectra of $\approx$2,000 stars covering the entire body of the galaxy. The mean ($<$RV$>$=220.4$\pm$0.1 ) and the dispersion ($\sigma$=11.7$\pm$0.1 ) of the RV distribution of candidate Carina stars ($\sim$1,210 objects, 180$\le$$RV$$\le$260 , 4$\sigma$) agree quite well with similar measurements available in the literature. To further improve the statistics, the accurate RV measurements recently provided by @walker07 were also added to the current data set. We ended up with a sample of $\sim$1,370 RV measurements of candidate Carina stars that is $\approx$75% larger than any previous Carina RV sample. We found that the hypothesis that the Carina RV distribution is Gaussian can be discarded at 99% confidence level. The mean RV across the body of the galaxy varies from $\sim$220 at a distance of 7 ($\sim$200 pc) from the center to $\sim$223 at 13 ($\sim$400 pc, 6$\sigma$ level) and flattens out to a constant value of $\sim$221 at larger distances (600 pc, 4$\sigma$ level). Moreover and even more importantly, we found that in the Carina regions where the mean RV is smaller the dispersion is also smaller, and the RV distribution is more centrally peaked (i.e. the kurtosis attains larger values). The difference in mean RV is more than 4 (9$\sigma$ level), when moving from E to W, and more than 3 ($\sim$7$\sigma$ level), when moving from N to S. The RV gradient appears to be in the direction of the Carina proper motion. However, this parameter is affected by large uncertainties to reach a firm conclusion. There is evidence of a secondary maximum in RV across the Carina center ($|D|$$\le$200 pc). The occurrence of a secondary feature across the Carina center is also supported by the flat-topped radial distribution based on the photometric catalog. These findings are reminiscent of a substructure with transition properties, already detected in dwarf ellipticals, and call for confirmation by independent investigations.'
author:
- 'M. Fabrizio, M. Nonino, G. Bono, I. Ferraro, P. François, G. Iannicola, M. Monelli, F. Thévenin P. B. Stetson, A. R. Walker, R. Buonanno, F. Caputo, C. E. Corsi, M. Dall’Ora, R. Gilmozzi, C.R. James, T. Merle, L. Pulone, M. Romaniello'
date: 'drafted / Received / Accepted '
title: 'The Carina Project. IV. radial velocity distribution'
---
Introduction
============
Dwarf galaxies are fundamental laboratories to investigate the influence of the environment on star formation and on chemical evolution in stellar systems that are several order of magnitudes smaller than giant galaxies. Empirical evidence indicates that in the Local Group together with the dwarf ellipticals (dE, M32-like) and the dwarf spirals (dS, M33-like) we are facing three different dwarf morphological types. The dwarf spheroidals (dSphs) show either single (Cetus, @monelli10a; Tucana, @monelli10b) or multiple star formation events (Carina, @bono10), are spheroidal in morphology and devoid of neutral hydrogen [@bouchard05]. The dwarf irregulars (dIs) host a mix of old and young stellar populations, but they have recently experienced an intense SF episode [@sanna09; @cole09], moreover, they have irregular morphology and host a significant fraction of neutral hydrogen. The transition dwarfs have properties between the dSphs and the dIs. The observational scenario concerning the dwarfs in the LG was recently enriched by the discovery of ultra-faint dSphs in the SDSS [@zucker06; @tolstoy09; @wyse10], but their properties still need to be investigated in detail. Although the morphological classification appears robust, we still lack firm quantitative constraints concerning the evolution/transition between different dwarf morphological types. @kormendy85 [@kormendy87] found a well defined dichotomy between ellipticals (Es) and spheroidals (Sphs, @kormendy09). Early type galaxies are distributed along a sequence moving from cD to dEs, while spheroidals are distributed along a sequence that overlaps with spirals and dwarf irregulars (see Fig. 1 in @kormendy09). This empirical dichotomy seems to suggest that E and Sph galaxies are stellar systems that underwent different formation and evolution processes. In particular, the Sphs might be either spirals or irregulars that lost their gas or transformed it into stars. However, there is no general consensus concerning the E–Sph dichotomy, since the correlation between the shape of the brightness profile and the galaxy luminosity is continuous when moving from Es to Sphs [@jerjen97; @gavazzi05; @ferrarese06].
Theoretical and empirical uncertainties affect our understanding of several empirical correlations. More than twenty years ago it was suggested by @skillman89 that the luminosity of dwarfs is correlated with their mean metallicity. However, it is not clear whether this correlation is linear over the entire metallicity range or shows a plateau in the metal-poor regime [@helmi06; @mateo08; @kirby08]. More recently, it was suggested by @woo08 that dEs/dSphs in the LG, at fixed total visual luminosity, show larger mass-to-light (M/L) ratios (see their Fig. 1). The same systems are, at fixed total stellar mass, more metal-rich than dIs (see their Fig. 10 and @mateo98araa).
Precise and homogeneous photometric and spectroscopic data are required to address the above open issues concerning the formation and evolution of these systems. The advent of multi-object spectrographs at the 8-10m class telescopes provided the unique opportunity to collect sizable samples of low-, medium- and high-resolution spectra covering the entire body of nearby dwarf galaxies. By using hundreds of radial velocity (RV) measurements it has been suggested by @kleyna04 that Sextans dSph host kinematic substructures, in particular, they found that the RV dispersion of the stellar population located across the center of this system is close to zero and increases outside the core. A different kinematic status has also been suggested by @battaglia08 for the two distinct stellar populations in the Sculptor dSph.
Moreover, evidence of a fall-off in the velocity dispersion at large galactic distances was suggested in Sextans, Draco and Ursa Minor [@kleyna03; @wilkinson04; @wilkinson06]. However, @walker09, using large and homogeneous samples of RV measurements based on high-resolution spectra collected with Mike Multi Fiber Spectrograph (MMFS) at Magellan [@walker07 hereinafter W07], found that the RV profile in several dSphs (Carina, Draco, LeoI, LeoII, Ursa Minor) are flat at large projected radii (R$\ge$1 kpc), while Fornax, Sculptor and Sextans show a gentle decline.
The observational scenario is far from being settled, and indeed, @lokas08 and @lokas09 using the same RV measurements provided by @walker07, but a different algorithm to reject tidally stripped stars, found a significant decline in the RV dispersion profile of Carina, Fornax, LeoI, Sculptor and Sextans. Furthermore, it was also suggested that kinematic status of Carina [@munoz06] and Bootes [@belokurov06] might be disturbed by the Milky Way. In this context, Carina plays a key role because it is relatively close ($\mu$=20.10 mag), shows at least two well separated SF episodes (t=1-6 and 12 Gyr) and it is metal-poor ($<$\[Fe/H\]$>\sim$–1.7 dex).
Our group is involved in a long-term project on the evolutionary properties, variable stars, kinematic and abundances of stellar populations in the Carina dSph. This is the fourth paper of a series and it is focused on the radial velocity distribution of candidate Carina stars. The structure of the paper is as follows. In Section 2, we discuss in detail the different spectroscopic data sets we collected for this experiment, together with the approach we adopted to reduce the data. Section 3 deals with radial velocity measurements and with the observational strategy we adopted to validate radial velocities based on low-, medium- and high resolution spectra. In Section 4 we lay out the selection criteria adopted to identify candidate Carina stars and their radial velocity distribution. In Section 5, we investigate the velocity distribution, the velocity dispersion and the kurtosis of candidate Carina stars as a function of the projected radial distance. In this section, we also address the similarity between spectroscopic and photometric radial distributions. Finally, in Section 6 we summarize the results of this investigation and briefly outline possible future extensions of this photometric and spectroscopic experiment.
Spectroscopic data sets and data reduction
==========================================
To collect low-resolution spectra of old (Horizontal Branch \[HB\], $\sim$20.75 mag), intermediate (Red Clump \[RC\], $\sim$20.5 mag) and young (Blue Plume \[BP\], $\sim$22 mag) stellar tracers (see Fig. 1) we observed five fields with FORS2 at VLT [@appe98] in service mode[^1]. The program was almost completed by one of us (M.N.) during a Visitor mode observations in December 2004. A significant fraction of FORS2 spectra were collected using the 1400V grism, which covers H$_\beta$ ($\lambda$=4861.34 Å) and MgI Triplet (MgIT[^2]) lines, with a nominal resolution of 2100 ($\lambda_c$=5200 Å). The slits used were 06 length, and typically 10 long, with a wavelength coverage $\approx$4560–5860 Å. The exposure times range from 22 minutes ([*short*]{}) to more than one hour ([*long*]{}).
During the December 2004 run, the 1028z grism was also used with a wavelength coverage $\approx$7730–9480 Å. Its nominal resolution is 2560 ($\lambda_c$=8600 Å) and covers the near-infrared (NIR) calcium triplet (CaT[^3]). Using the 1400V grism we secured in 19.12 hrs of exposure time 743 individual spectra of 356 stars with DIMM seeing ranging from 035 to 127, while using the 1028z grism we secured in 1 hr of exposure time 113 spectra of 73 stars with seeing 038–059. Most of the stars with 1028z spectra also have 1400V spectra. We ended up with a total sample of 359 stars either with single or multiple FORS2 spectra. In particular, the LR data set collected with the grism 1400V includes 356 stars and among them eleven have a single spectrum, while the others a number of spectra ranging from two to twelve. The LR data set collected with the grism 1028z includes 73 stars and among them 39 have single spectra, while the others a number of spectra ranging from two to three. In total the LR data set includes 359 stars of which eleven have a single spectrum and the others a number of spectra ranging from two to 14. Table 1 gives the log of the observations of the spectra collected with FORS2. From left to right are listed the date, the pointing, the coordinates, the grism, the exposure time and the seeing.
These data were complemented by archival medium and high-resolution spectra of Red Giant (RG) and RC stars collected with GIRAFFE [@pas02] at VLT[^4] and covering the entire body of the galaxy (see Fig. 1). The GIRAFFE spectra [@koch06] were collected using the grating LR8 with a nominal resolution of 6,500 centered on the NIR CaT. This data set (hereinafter GMR03) includes 10,394 individual fiber spectra of 1,070 stars and were secured with a total exposure time of 95.4 hrs with seeing condition of 04–22. Note that this data set was also adopted by @munoz06 in their analysis of Carina radial velocity distribution. We also use the high-resolution spectra collected with the gratings: HR10 (R=19,800, 5339$\lesssim$$\lambda$$\lesssim$5619 Å), HR13 (R=22,500, 6120$\lesssim$$\lambda$$\lesssim$6405 Å) and HR14A (R=17,740, 6308$\lesssim$$\lambda$$\lesssim$6701 Å). This data set (hereinafter GHR) includes 2,002 individual spectra of 98 stars, secured with a total exposure time of 24 hrs and seeing 057–320. The above data were complemented by a new set of GIRAFFE spectra collected with LR8 and including 8,092 individual spectra of 959 stars. This data set (hereinafter GMR08) was secured with 48.4 hrs of exposure time and seeing of 054–194. We ended up with a total sample of 1,931 stars either with single or multiple GIRAFFE spectra.
The GMR ($<$GMR03$+$GMR08$>$, weighted mean) data set includes 1,887 stars and only one star has a single spectrum, while the others have multiple spectra ranging from two to 35. The HR data set includes 98 stars and all of them have multiple spectra ranging from seven to nine. Moreover, the GHR and the GMR03 spectra have 38 objects in common, while 125 objects are in common between GHR plus GMR03 and GMR08 spectra. The LR resolution data set has 145 objects in common with the GIRAFFE (GHR plus GMR) data set. The objects in common were adopted to calibrate GMR and LR spectra. Data plotted in the top panels of Fig. 1 show the spatial coverage of the different data sets, while the bottom panels show the location of the targets in the , Color-Magnitude Diagram (CMD, @bono10). The numbers in parenthesis give the number of stars for which we estimated the radial velocity and the total number of stars for which we collected at least one spectrum.
Data plotted in the bottom panels of Fig. 1 show several interesting features: [*i)*]{} [*Stellar tracers*]{} – The FORS2 spectra cover old, intermediate and young stellar tracers ($\lesssim$22, 359 RVs). [*ii)*]{} [*Statistics*]{} – Although the exposure time of the GIRAFFE spectra is a factor of 4-8 longer than the FORS2 spectra, their limiting magnitude is $\lesssim$20.6 (fibers vs slits). This means that this data set does not cover young (BP) and truly old (HB) tracers. The RGs are, indeed, a mix of old and intermediate–age stellar tracers. However, the sample of RV measurements based on GIRAFFE spectra (1,985 vs 359) is more than a factor of five larger than FORS2 (FoV $\sim$25 vs $\sim$5 arcmin squared). [*iii)*]{} [*Multiplicity*]{} – A significant fraction of our targets have multiple spectra. This means that we can provide robust estimates of intrinsic errors. [*iv)*]{} – [*A significant overlap of low, medium and high-resolution spectra*]{} – We can constrain the occurrence of possible systematic errors in RV measurements using spectra collected with different instruments and different gratings/grisms.
We reduced our FORS2 data using standard IRAF[^5] tasks, using day time associated bias frames, flat field and calibration lamps. The wavelength calibration was carried out using the task [identify]{}, daytime lamp frames and a set of 8-12 calibration lines. We typically end up with an accuracy better than $\approx$0.02–0.04 Å, $\leq$2.5 . However, we found systematic shifts up to 1 pixel, corresponding to $\approx$30 in radial velocity, for the two adopted grisms. To overcome the problem, we adopted the approach suggested by @kel03. A model of the sky for each slit was created and subtracted from the reduced 2D spectrum. This model was also used to estimate and correct the systematic shifts from the daytime wavelength calibration lamps, using night sky lines, mainly OI 5577.34 Å for the 1400V grism and a set of isolated skylines for the 1028z grism.
Yet another potential source of systematics in RV estimates was the centroiding (see, e.g. @tolstoy01). Thanks to the special care in the mask design, based on previous FORS2 preimaging of selected Carina fields, and to the very precise tracking of VLT, this effect was limited at the level of 5-10 . To estimate the shift between the centroid of the star and the slit we adopted the slit image which is acquired just before the spectrum. For each target the centroid of the slit is evaluated by collapsing the slit along its width and neglecting the pixels belonging to the star. The collapsed slit is then subtracted away from the pixels belonging to the star, whose centroid is then estimated with the task [center]{} in IRAF. It is worth mentioning that we also acquired slit images, for a couple of masks, soon after we collected the spectra. These slit images were adopted to confirm that our estimates of the centroids were minimally affected by tracking problems. The details of the pre-reduction and reduction of FORS2 spectra will be discussed in a forthcoming paper (Nonino et al. 2011, in preparation).
The GIRAFFE raw data were retrieved from the ESO Archive. These data were reduced using IRAF. After the standard bias and flat correction, the spectra were extracted using the traces from flats. Wavelength calibration is based on daily calibration lamps: the formal solution rms was $<$0.02 Å, corresponding to a systematic $<$1 in RV. After the fiber-spectra assignment, performed using the table associated with the raw data, all the spectra were visually inspected in order to remove bad spectra. A master sky for each exposure was created from sky fibers, after the spectra had been cross-correlated to remove shifts and scaled using the intensity of selected night sky lines in the wavelength range 8250–8750 Å, to better match the region with stellar absorption lines. Subsequently the master skies were cross correlated with the targets spectra, scaled in intensity and subtracted. The cross correlation among the stacked skies showed that the relative accuracy in wavelength calibration could give systematics of the order of 500 m s$^{-1}$ in RV. We finally coadded all the individual spectra of the same target using the task [scombine]{}, after correcting for the barycentric motion.
The GIRAFFE spectra taken with the high resolution grisms were reduced following a similar approach.
Radial velocity measurements
============================
To measure the RV of GMR and GHR spectra we fit individual spectral lines in the coadded spectra. The RV of the LR spectra was measured on individual spectra and the final RV was estimated as a weighted mean among the multiple spectra, when available, of the same object. All spectra were normalized to the continuum and the fit to single or multiple lines performed using either a Gaussian or a Moffat function. The interactive code developed to perform the RV measurements will be described in a forthcoming paper (Fabrizio et al. 2011, in preparation). We selected by visual inspection several spectral lines for each data set, namely H$_\beta$ and MgI triplet for FORS2/1400V spectra (see the spectra plotted in the panels d) of Fig. 2), the CaT for FORS2/1028z and GMR spectra (see the spectra plotted in the panels c) and b) of Fig. 2) and several strong FeI lines for GHR spectra (see the spectra plotted in the panels a) of Fig. 2). The quoted lines were typically fit with a Moffat function with $\beta$=2, to properly account for the contribution of the wings in the line fit.
This preliminary estimate of the RV was validated by eye inspection and when judged satisfactory it was adopted to perform an automatic estimate of the RV using a large set of iron and heavy element lines. We adopted the line list for iron, $\alpha$- and heavy-elements recently provided by @roma08 and by @pedi10. The lines were selected according to the wavelength range and the spectral resolution of the different instruments and grisms. We ended up with a sub-sample of 30–70 lines for LR (grisms: 1028z, 1400V) spectra, with $\approx$35 lines for GMR spectra and with $\approx$90 lines for the GHR spectra. For each spectrum we estimate the RVs as a weighted mean of the lines with the highest S/N ratio. This on average means 1-2 dozens of lines for LR and GMR spectra and approximately 40 lines for the GHR spectra.
The error in the RV based on individual lines was assumed equal to the sigma of the fitting function[^6]. The RV of the entire spectrum was estimated as a weighted mean over the different fitted lines. The error in the radial velocity of coadded spectra (GMR,GHR) is the error of the weighted mean. The RV of the LR spectra was estimated following the same approach for stars with a single spectrum and as a weighted mean among the different mean measurements for the stars with multiple spectra.
Data plotted in Fig. 3 show that the intrinsic error of the RV measurements based on GHR spectra is on average (biweight mean)[^7] $<$1 (panels a,b,c) and becomes of the order of 5 for GMR03 spectra (panel d). Note that the intrinsic error of GMR08 spectra is larger and $\sim$6 . The difference is mainly due to the S/N ratio, since the latter sample is on average 0.5 mag fainter and its total exposure time is almost a factor of two shorter.
The intrinsic error of the RVs based on LR spectra is slightly larger and ranges from $\sim$11 (grism 1028z, panel f) to $\sim$10 (grism 1400V, panel g). The difference between the FORS2 data sets is expected. The first two lines of the CaT, at the typical RV of Carina stars, are contaminated by atmospheric OH emission lines. Therefore, the RV measurements based either on low-resolution spectra or on medium-resolution, low S/N ratio spectra might be affected by slightly larger uncertainties in case the sky lines are not precisely subtracted. The reader, interested in a more detailed discussion, is referred to @noni07 and to @walker07. Moreover, the 1028z spectra have a number of repeats that are approximately a factor of two smaller than the number of repeats of the 1400V spectra (see §2). The fraction of objects for which we measured the RV is quite high, and indeed the fraction ranges from $\sim$98% for GHR and GMR03 spectra, to $\approx$90% for LR (1400V) spectra and to more than $\approx$80% for GMR08 and LR (1028z) spectra. Typically the stars with no RV measurements have either low S/N or noisy spectra (see §2) or missing identification in the photometric catalog. Eventually, the RV was measured in 2,165 out of the 2,323 GIRAFFE spectra ($\sim$93%) and in 381 out of the 429 ($\sim$89%) FORS2 spectra (see Fig. 3). In total the RV was measured in 1,812 stars using GIRAFFE spectra and in 324 stars using FORS2 spectra (see Fig. 1). The final catalog includes RV measurements for 1,979 stars. To validate the approach adopted to measure the RVs, we decided to use real spectra. This approach has several indisputable advantages when compared with artificially generated spectra, since the different spectroscopic data sets we are dealing with partially overlap. We can constrain the precision of RVs based on HR spectra, since for each object we have three spectra with similar spectral resolutions and covering three different wavelength regions (HR10, HR13, HR14A, see §2). The panels a) and b) of Fig. 4 show the internal comparison of our RV measurements as a function of the -magnitude. The difference and the RV dispersion of the entire sample and of the candidate Carina stars (red dots, 180$\le$RV$\le$260 , see §4) are minimal.
We can also constrain the precision of the GMR03 spectra, since this subsample has more than three dozens of stars in common with the GHR spectra. The panel c) shows the difference between the RVs based on the $<$GHR$>$ and those based on the GMR03 spectra (weighted mean). The mean (biweight) of the difference is 0.1 for the candidate Carina stars (26) and smaller than 1 for the entire sample (35). The GMR03 RVs were corrected for the difference.
The panel d) shows the difference between the RVs based on $<$GHR$>$ plus GMR03 (weighted mean) spectra and RVs based on GMR08 spectra. We have more than 120 stars in common and all of them are candidate Carina stars. The difference is smaller than –3 and the $\sigma$$\approx$8 . The difference with the RVs based on GMR03 spectra is mainly due to the fact that the GMR08 spectra have a lower S/N ratio and are on average fainter. The GMR08 RVs were corrected for the difference. The panel a) of Fig. 5 shows the difference between the RVs based on the GIRAFFE ($<$GHR+GMR$>$, weighted mean) spectra and on the FORS2-1400V spectra (internal comparison). The mean difference (biweight) over the entire sample of stars in common (140/144) is –3.6 and smaller (–2.7 ) for the candidate Carina stars (122/125), while the dispersion is of the order of 12 . The RVs based on FORS2-1400V were accordingly corrected. The panel b) shows the difference between the RVs based on GIRAFFE and on FORS2 ($<$GHR+GMR$>$ + 1400V, weighted mean) and RVs based on FORS2-1028z spectra. The difference in RV and dispersion are similar to the RVs based on the 1400V grism and they were also corrected. Data plotted in panel c) show the difference between the RVs based on the GIRAFFE ($<$GHR+GMR$>$, weighted mean) and on the FORS2 ($<$1028z+1400V$>$, weighted mean) spectra after the corrections have been applied. The two samples have more than 140 stars in common and the difference is $\sim$1 for the entire sample and vanishing for candidate Carina stars.
The Fig. 6 shows the comparison between our RVs based on GIRAFFE spectra –$<$GHR+GMR$>$– with different RV measurements available in the literature (external comparison). The panel a) shows the comparison with the RVs based on echelle (+2D at LCO2.5m, @mateo93, crosses) and on multi-fiber spectra (HYDRA at CTIO 4m Blanco, @majewski05, filled circles). The difference with the Mateo sample is quite small both in the mean (biweight) and in the velocity dispersion. The difference with the Majewski sample is larger ($\mu$=–3.6, $\sigma$=9.4 ), and it is caused by the lower S/N ratio of the spectra they adopted to estimate the RVs.
The panel b) shows the comparison between our RVs and RVs based either on high-resolution spectra collected with MIKE at Magellan (crosses) or on medium-resolution spectra collected with GIRAFFE at VLT (filled circles) provided by @munoz06. The MIKE spectra were collected in slit mode with a resolution of $\sim$19,000 (red arm) and the RV measurements are based on the NIR calcium triplet. The GIRAFFE spectra adopted by @munoz06 are the same GMR03 spectra we also included in this investigation. The difference with RVs based on MIKE spectra is minimal ($<$1 ), but the sample of stars in common is limited (14). The difference with the RVs based on GIRAFFE spectra is larger $\sim$3 if we account for the entire sample of stars in common (850/889 stars, $\sigma$=5.7 ) and for candidate Carina stars (353/373 stars, $\sigma$=6.7 ). The reasons for this difference are not clear, apart from the fact that we adopted a different approach to prereduce the spectra and to measure the RVs.
The panel c) shows the comparison with the RVs provided by W07. The RV measurements provided by W07 are based on high-resolution (20,000–25,000) spectra collected with MMFS at Magellan. These spectra sample the region across the magnesium triplet (5140–5180Å). We have more than 550 candidate Carina stars in common and the data plotted in this panel show that the two samples of RV measurements agree, within the errors quite well. The difference is –2 (biweight) and the RV dispersion is smaller than 10 . The same outcome applies to the RVs based on FORS2 spectra (see panel d). We have more than 80 candidate Carina stars in common and the difference is $\approx$1 , while the dispersion $\sigma$$\sim$11 .
We ended up with a data set including RV measurements for 1979 stars. For the objects with RVs based on both GIRAFFE and FORS2 spectra we performed a weighted mean, using as weights the inverse square of the radial velocities standard errors of the GMR, GHR and LR spectra. As a final test, we compared our entire RV data set with the RVs provided by W07. Data plotted in panel e) further support the agreement between the two different sets of RV measurements, and indeed the difference ranges from less than 1 for the entire sample (785 objects in common) to 2 for candidate Carina stars (574), while the dispersion is $\sim$9 . The estimates of the mean difference are based, once again, on the biweight and the number of neglected objects is smaller than 9%. The precision of both internal and external validations, and in particular the good agreement with the RVs provided by W07, support the approach we adopted to measure the RVs.
The referee asked us to comment the impact of binaries stars on the current RV measurements. The occurrence of binary stars among RG stars in dSph galaxies is a highly debated topic and their impact on the RV dispersion ranges from a sizable [@queloz95] to a small error [@olszewski96; @hargreaves96] when compared with the statistical error. However, in a recent investigation @minor10 found, by using a detailed statistical approach, that dSph galaxies with RV dispersions ranging from 4 to 10 can be inflated by no more than 20% due to the orbital motion of binary stars. It is worth noting that the current LR spectra cover a time interval of three years (2004-2007) and the number of repeats ranges from two to 14 (only eleven stars with a single spectrum). The GMR spectra cover a time interval of five years (2003-2008) and the number of repeats ranges from two to 35 (only 1 star with a single spectrum). The GHR spectra were collected in 2005 and the number of repeats ranges from seven to nine. The three different data sets have 145 objects in common. This means that a significant fraction of stars in our sample have spectra collected on a time interval of several years. Therefore, the current RV measurements are less prone to significant changes caused by binary stars. Moreover, the conclusions of the current investigation are minimally affected by a possible uncertainty of the order of 20% in the RV dispersion.
Carina radial velocity distribution
===================================
The top panel of Fig. 7 shows the RV distribution of the entire (GIRAFFE+FORS2) data set. The well defined primary peak located at RV$\sim$220 includes candidate Carina stars. The current radial velocity distribution is soundly supported by the radial velocity distribution recently provided by W07, but based on a smaller number of stars (see the middle panel of Fig. 7). This evidence and the minimal difference in the RV of the stars in common allowed us to merge the two RV catalogs. For the objects in common weighted mean RVs were computed using as weights the inverse square standard errors in the individual RV measurements. We ended up with a sample of 2,629 stars (see the bottom panel of Fig. 7).
To further constrain the radial velocity distribution of the candidate Carina stars we ran a Gaussian kernel on each star with a $\sigma$ equal to the RV uncertainty. The solid red line plotted in the top panel of Fig. 8 was computed by summing the individual Gaussians over the entire data set. We estimated the mean and the $\sigma$ by fitting the smoothed RV distribution with a Gaussian (dashed black line). We found that the peak (220.4$\pm$0.1 ) in the RV distribution agrees quite well, within the errors, with similar estimates available in the literature [@koch06; @munoz06; @walker07; @walker09]. The same outcome applies to the RV dispersion. Note that current estimate is based on a homogeneous sample of candidate Carina stars (180$\leq$RV$\leq$260 , $\sim$4$\sigma$, 1208 stars) that is $\sim$55% larger than any previous sample of RVs.
The middle panel of Fig. 8 shows the RV distribution of candidate Carina stars, but based on the RV measurements (780 candidate Carina stars) provided by W07. The two data sets provide, within the errors, very similar kinematic properties. Note that by merging the two data sets of RV measurements we ended up, according to the quoted selection criterion, with a sample of 1378 candidate Carina stars. This sample of Carina RVs is $\approx$75% larger than any previous RV sample. To overcome subtle uncertainties in the estimate of the RV due to the possible presence of outliers in the final data set, we estimated the biweight mean of this sample and we found 220.9$\pm$0.1 . This estimate is based on 1369 stars and this is the sample that we will adopt in the following to estimate the RV distribution (red line in the bottom panel of Fig. 8). The mean based on the Gaussian fit (dashed line) agrees quite well with the biweight mean. However, the tails of the smoothed RV distribution appear shallower than the tails of the fitting Gaussian. There is also evidence that the low-velocity tail might be shallower than the high-velocity one. To provide a more quantitative estimate, we estimated the $\chi^2$ of the two curves, and we found that the hypothesis that the smoothed distribution is Gaussian can be discarded at 99% confidence level.
The referee has expressed concerns about the possible dependence of the Carina RV distribution on the different samples of RV measurements. To constrain this effect we estimated the RV distribution of Carina stars using stars brighter than =20.25 mag (651). We fit the smoothed distribution with a Gaussian and we found a mean RV $\mu$=221.3$\pm$0.1 and a $\sigma$=8.7$\pm$0.1 . We also performed a new test including only stars brighter than =20.50 mag (1050) and we found $\mu$=220.9$\pm$0.1 and a $\sigma$=8.9$\pm$0.1 . The quoted values agree quite well with the estimates based on the entire sample. The two RV distributions based on smaller samples of brighter objects also show asymmetric wings when compared with a Gaussian distribution.
To further constrain the plausibility of the selection criterion to pin point candidate Carina stars, we plotted the kinematic candidates in the ,(left panel of Fig. 9) and in the , (right panel of Fig. 9) CMDs. The symbols of the different spectroscopic data sets are the same as in Fig. 1. Data plotted in this figure show quite clearly the accuracy of the kinematic selection, and indeed the bulk of the candidate Carina stars are located along the expected evolutionary sequences (RG, RC, HB, blue plume). The new sample also provides a clear separation between RG and Asymptotic Giant Branch (AGB, $\sim$20, $\sim$1.6–1.9 mag) stars. Moreover and even more importantly, data plotted in Fig. 9 soundly support the results recently provided by @bono10 concerning the metallicity distribution of Carina stars. The quoted authors found, using a color-color plane (, ) to select candidate Carina stars, that the spread in color of RG stars is significantly smaller than suggested by spectroscopic measurements. The very narrow distribution in color of candidate Carina RGs is further supported by the kinematic selection. There are a few RGs that attains colors redder than typical Carina RGs. They might be either variables stars or misidentified objects.
Discussion
==========
In order to constrain whether the occurrence of asymmetries in the radial velocity distribution is caused by the presence of substructures in Carina, we investigated the change in the RV distribution as a function of the projected radial distance ($\rho$). To avoid spurious fluctuations in the mean RV, we ranked all the candidate Carina stars as a function of $\rho$ and estimated the mean (biweight) and the median using the first 200 objects. The $\rho$ of each subsample was estimated as the mean over the individual distances of the same 200 stars. We estimated the same quantities by moving of one object in the ranked list until we accounted for the most distant 200 stars in our sample located inside the tidal radius ($r_t$$\sim$28.8$\pm$3.6, @mateo98araa). The error on the mean RV for individual bins is smaller than one tenth of . In order to provide robust constraints on the possible uncertainties introduced by the number of stars per bin and by the number of stepping stars we performed a series of MonteCarlo simulations. The estimated mean dispersion is plotted as cyan shaded area across the RV mean in the bottom panel of Fig. 10. Note that changes in the binning criteria can change the extent of the secondary features, but their relative positions are minimally affected.
A glance at the data plotted in the bottom panel of Fig. 10 shows that the mean RV attains a well defined minimum –RV$\sim$219.5$\pm0.4$ – for $\rho\sim$7 ($\sim$200 pc)[^8] from the galaxy center and increases up to $\sim$222.6$\pm$0.4 for $\rho\sim$13 ($\sim$400 pc, 6$\sigma$ level) and a plateau of $\sim$221.4$\pm$0.4 at larger distances (600 pc, 4$\sigma$ level). This evidence is robust, since the mean (biweight, black line) and the median (blue line) attain very similar values and intrinsic errors. To further constrain the change of the kinematic status across the body of the galaxy we also estimated the radial velocity dispersion ($\sigma_{RV}$, solid line middle panel of Fig. 10). Note that to avoid subtle uncertainties caused by asymmetric radial velocity distributions we also estimated for each bin the semi-interquartile range (SIQ, dashed-dotted line in the middle panel of Fig. 10). Data plotted in the middle panel of Fig. 10 show that both the $\sigma_{RV}$ and the SIQ show very similar radial trends, but the former dispersion parameter attains values that are $\approx$50% larger. The difference indicates that the velocity dispersion should be cautiously treated in dealing with asymmetric RV distributions [@strigari10]. We also found that the dispersion is larger in the innermost regions, decreases in coincidence of the minimum in the mean RV and after a mild increase attains an almost constant value in the outermost galaxy regions. Note that the cyan shaded areas across the RV dispersion and the SIQ display the intrinsic error on individual bin according to MonteCarlo simulations. There is a hint that the dispersion could increase in the outskirts, thus supporting the finding by W07. However, more data are required to constrain the change in the projected radial velocity dispersion across the tidal radius.
Interestingly enough, the kurtosis plotted in the top panel of Fig. 10 shows a mirror trend compared with the projected radial velocity dispersion and with the mean RV. This evidence is suggesting that in the Carina regions where the mean RV is smaller, the dispersion is also smaller and the radial velocity distribution more centrally peaked. The opposite trend characterizes the regions in which the mean RV is larger. The outermost regions show flat trends in the quoted parameters. Evidence of kinematic and photometric substructures have already been found in several LG dwarfs [@monelli03; @kleyna04; @walker09]. In particular, @battaglia08 found evidence of a radial velocity gradient in Sculptor, that they interpreted as an indication of internal rotation. The current data indicate that the RV distribution across the body of Carina shows a radial gradient and possible evidence of rotation. However, the RV distribution is far from being uniform, possibly suggestive of the occurrence of well defined spatial substructures.
The referee has expressed concerns about the approach we adopted to estimate the weighted mean between the GIRAFFE and the FORS2 spectra. To clarify this key point, Fig. 11 shows the same data of Fig. 10, but the parameters plotted in the left panels are only based on the GIRAFFE spectra, while those plotted in the right panels are based on the weighted mean between GIRAFFE and FORS2 spectra. The trends of the plotted parameters as a function of the radial distance are the same. They are only slightly more noisy due to the decrease in the sample size.
To further constrain the spatial extent of possible kinematic substructures we investigated the same parameters, but they were estimated as a function of right ascension ($\alpha$) and declination ($\delta$). Data plotted in panel c) of Fig. 12 show the marginal of the radial velocity along the right ascension axis. We found that the mean RV, when moving from E ($\sim$750 pc from the galaxy center) to W ($\sim$–1060 pc from the center), decreases from 221.9$\pm$0.4 to 219.8$\pm$0.4 . The difference is equal to $\sim$2.1 (4$\sigma$ level) and increases by almost a factor of two ($\sim$4.5 , 9$\sigma$ level) if we take into account the absolute maximum ($\sim$223.3$\pm$0.4 , $\sim$320 pc from the center) and the absolute minimum ($\sim$218.8$\pm$0.4 , $\sim$-280 pc from the center) of the mean RV.
It is worth mentioning that the increase in the mean velocity both in the external regions and outside the galactic center appears to be in the same direction of the Carina proper motion (large blue arrow in panel d) of Fig. 12, @piatek03 [@walker09]). However, the uncertainty affecting current estimates of the Carina proper motion is quite large (see small blue arrows). Note that the increase in the mean RV cannot be in the direction of the Galactic center (heavy dashed line). Moreover and even more importantly, we found that in a region of 200 pc across the galactic center the RV shows a secondary maximum that is strongly correlated with the velocity dispersion (panel b) and anticorrelated with the kurtosis (panel a).
The change in the mean RV along the declination axis is similar to the change in the right ascension axis. The mean RV decreases, when moving from N to S (panel e), from 221.9$\pm$0.4 ($\sim$630 pc from the center) to 219.8$\pm$0.4 ($\sim$-650 pc from the center). The difference is at least at 4$\sigma$ level, while the difference between maximum and minimum (222.3$\pm$0.4 , $\sim$238 pc; 219.1$\pm$0.4 , $\sim$41 pc) is more than 3 ($\sim$7$\sigma$ level). The absolute minimum is located close to the center in the IV quadrant and associated to the absolute minimum in RA. The radial trend outside this region shows the quoted secondary maximum and a constant smooth trend at larger distances from the center. The dispersion (panel f) and the kurtosis (panel g) are once again correlated/anticorrelated with the behavior of the mean RV. It is noteworthy, that the quoted findings are minimally affected by the FORS2, GIRAFFE and W07 data sets adopted to estimate the RVs of candidate Carina stars.
To explain the occurrence of a kinematically cold population close to the center of Sextans, @kleyna04 suggested, using also photometric radial distributions, that it could be the aftermath of a merging with a globular cluster. To further constrain the nature of the above kinematic peculiarities we investigated the Carina radial distributions, using the same approach adopted by @monelli03 and the recent photometric catalog by [@bono10]. The isocontour levels (panel b) and the marginal (panel a) along the right ascension axis plotted in Fig. 13 show that the radial distribution is flat-topped across the Carina center. Note that the bright field star HD48652 (=9.14 mag) minimally affects this feature, since it is located at 277 from the Carina center. On the other hand, the radial distribution is symmetric and with a well defined peak along the declination axis. We drew three vertical lines across the center and the two secondary maxima. Interestingly enough, the same lines include the secondary maximum detected along the right ascension axis in the mean RV. These empirical evidence indicates that the regions across the galaxy center might form a substructure probably reminiscent of a transition between a bulge-like and/or a disk-like substructure.
Summary and conclusions
=======================
We presented new radial velocity measurements of Carina dSph stars based on low-resolution spectra (R$\sim$2500) collected with the FORS2 multi-object slit spectrograph at the VLT. The key advantage of the current sample is that for the first time we collected spectra of old (HB, $\sim$20.75 mag), intermediate (RC, $\sim$20.5 mag) and young (BP, $\sim$22 mag) stellar tracers. The bulk of the spectra were collected using the 1400V grism covering the $H_\beta$ and the MgI triplet. For a fraction (20%) of these targets we also collected spectra using the 1028z grism centered on the NIR CaT. We secured 856 spectra of 359 stars with $\sim$20 hours of exposure time.
The above data were complemented by medium (R$\sim$6500) and high-resolution (R$\sim$18000–23000) spectra collected with the GIRAFFE multi-object fiber spectrograph at the VLT. The targets of this spectroscopic data set are RG and RC stars ($\le$20.6 mag). The medium resolution spectra are centered on the NIR CaT, while the high-resolution ones cover a broad wavelength region (5300$\lesssim$$\lambda$$\lesssim$6700 Å). In total this data set includes 20,488 spectra of 1931 stars collected with $\sim$168 hours of exposure time.
The radial velocity distribution based on the above spectra agrees quite well with similar data available in the literature. By assuming candidate Carina stars the objects with RVs ranging from 180 to 260 (4$\sigma$) we ended up with a sample of $\sim$1,210 stars. By fitting these objects with a Gaussian we found a mean RV –$<$RV$>$=220.4$\pm$0.1 – and a velocity dispersion –$\sigma$=11.7$\pm$0.1 – that agree, within the errors, with similar measurements provided by @walker07.
The above kinematic selection was firmly supported by the position of the candidate Carina stars along the expected evolutionary sequences (RG, AGB, RC, HB, Blue Plume) of the CMD. The narrow distribution in color of the candidate Carina RG stars confirms the recent findings by @bono10, based on photometric selection criteria, that the spread in metallicity of Carina stars is smaller than suggested by spectroscopic measurements [@koch06].
The current catalog of Carina RV measurements was complemented by similar measurements recently provided by @walker07. The stars in common were averaged and we ended up with a sample of $\sim$1,370 RV measurements of candidate Carina stars that is $\approx$75% larger than any previous Carina RV sample. We found that the hypothesis that the Carina RV distribution is Gaussian can be discarded at the 99% confidence level. A more detailed investigation of the RV across the body of the galaxy indicates that the mean RV changes from $\sim$220 at a distance of 7 ($\sim$200 pc) from the center to $\sim$223 at 13 (6$\sigma$ level) and attains an almost constant value of $\sim$221–222 at larger distances (3–5$\sigma$ level). Moreover, we found that in the Carina regions where the mean RV is smaller the dispersion also attains smaller values and the RV distribution is more centrally peaked (larger kurtosis).
The mean RV at large distances from the galactic center decreases, when moving from E to W, by more than 2 (4$\sigma$ level). The difference increases by almost a factor of two ($\sim$4 , 9$\sigma$ level) if we account for the difference between the absolute maximum and the absolute minimum of the mean RV. The change in the mean RV, when moving from N to S, is similar. The difference increases from more than 2 (4$\sigma$ level) in the outermost regions to more than 3 (7$\sigma$ level) across the galaxy center. The RV gradient appears to be in the direction of the Carina proper motion. However, this parameter is still affected by large uncertainties to reach a firm conclusion.
Furthermore, there is also evidence of a secondary maximum in RV across the Carina center ($|D|$$\le$200 pc). The occurrence of a secondary feature across the Carina center is also supported by the flat-topped radial distribution (photometric catalog) along the right ascension axis. In particular, the two secondary maxima cover the same galaxy regions of the secondary maximum in the RV radial variation. These results, once independently confirmed, are probably reminiscent of a substructure with transition properties.
Evidence of substructures in dwarf galaxies dates back to @jerjen00 who detected a spiral structure in the Virgo dE IC3328 using deep, ground-based R-band images. This discovery was soundly confirmed by @lisker07 who found that a significant fraction of bright ($M_{\it B}<16$) early-type dwarfs in Virgo are characterized by disk-like features (spiral arms, bars). Furthermore, they also found that the properties of dE in clusters are also correlated with environmental density. This evidence supports numerical simulations of galaxy harassment indicating that late-type galaxies undergo a significant transformation when accreted in a cluster [@mastropietro05; @kormendy09]. It was also suggested that dEs with disk-like features might be the low-luminosity tail of normal disk galaxies [@lisker06]. This working hypothesis was recently supported by new and accurate kinematical data of IC3328 suggesting that the observed velocity dispersion is the aftermath of two distinct substructures, namely a thin stellar disk and a dynamically hot component. However, no robust conclusion could be reached due to limited radial coverage [@lisker09].
Irregular kinematic properties have also been found in the recently discovered ultra-faint dwarf galaxy –Willman I– by @willman10 using both photometric and spectroscopic data. They found that galaxy stars located in innermost regions show a radial velocity offset of 8 when compared with the outermost ones. Moreover, they also found initial hints of asymmetries in the radial velocity distribution, but their spectroscopic sample is too small to reach firm conclusions.
If the complex kinematic properties of Carina will be supported by future more accurate and deep data sets then the occurrence of disk-like features might be considered a wide spread property of gas-poor dwarf galaxies. A detailed comparison between observed and predicted radial velocity distribution and radial velocity dispersion profile is required to constrain the nature of Carina kinematics.
It is a real pleasure to thank an anonymous referee for his/her positive comments on the results of this investigation and for his/her pertinent suggestions and criticisms that helped us to improve the content and the readability of the paper. We would like to thank M. Lombardi, for useful discussions concerning kinematic properties of dwarf galaxies and S. Moehler for several enlightening suggestions concerning the radial velocity measurements of hot stars. We also acknowledge the ESO Cerro Paranal staff for collecting the spectroscopic data in service mode and the ESO User Support Department for useful suggestions in handling the raw spectra. One of us (GB) thanks IAC for support as a science visitor. This publication makes use of data products from VizieR [@ochse00] and from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. We also thank the ESO/ST-ECF Science Archive Facility for its prompt support.
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\[fig1\] {height="0.65\textheight" width="65.00000%"}
\[fig2\] {height="0.65\textheight" width="65.00000%"}
\[fig3\] {height="0.65\textheight" width="65.00000%"}
\[fig4\] {height="0.65\textheight" width="65.00000%"}
\[fig5\] {height="0.65\textheight" width="65.00000%"}
\[fig6\] ![Comparison among RV measurements based on different spectroscopic data sets.The panel a) shows the difference between our GIRAFFE (GHR+GMR) RVs and those provided by @mateo93 and by @majewski05. The panel b) shows the difference between our GIRAFFE (GHR+GMR) RVs and those provided by @munoz06 using two different data sets. The panel c) and d) show the difference between our GIRAFFE (GHR+GMR) and FORS2 RVs with those provided by W07 [@walker07] using spectra collected with MMFS at Magellan. The panel e) shows the difference between our entire sample (GIRAFFE+FORS2) of RVs and those provided by W07. The biweight mean ($\mu$), the standard deviation for the entire sample (black labels) and for the candidate Carina stars (180$\le$RV$\le$260 , red labels) are also labeled. The numbers in square parentheses show the number of objects in common between the two samples before and after the biweight mean. ](f6.ps "fig:"){height="0.65\textheight" width="65.00000%"}
\[fig7\] ![Top: Radial velocity distribution of the entire sample normalized to the maximum as a function of the radial velocity. Middle: Same as the top, but based on RV measurements provided by W07 [@walker07]. Bottom: Same as the top, but based on both our and W07 RV measurements. Note that for the stars in common in the two data sets we computed a weighted mean. See text for details. ](f7.ps "fig:"){height="0.65\textheight" width="65.00000%"}
\[fig8\] ![Logarithmic radial velocity distribution for the candidate Carina stars (180$\le$RV$\le$260 , $\sim$4$\sigma$). The red solid line shows the smoothed radial velocity distribution estimated running a Gaussian kernel on individual RV measurements. The mean ($\mu$) and the $\sigma$ of the Gaussian fit (dashed black line) are labeled together with the total number of candidate Carina stars. Middle: Same as the top, but based on RV measurements provided by W07 [@walker07]. Bottom: Same as the top, but based on both our and W07 RV measurements. ](f8.ps "fig:"){height="0.65\textheight" width="45.00000%"}
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\[fig13\] ![ The panel a)– Normalized radial profile along the right ascension axis of the Carina photometric catalog [@bono10]. The panel b)– Sky distribution of the photometric catalog. Red contours display iso-density levels ranging from 5 to 95% with steps of $\sim$11%. The two thin dashed-dotted lines mark the two secondary peaks, while the thin dotted line marks the center of the galaxy. The blue star shows the position of the bright field star HD48652 (=9.14 mag). The panel c)– Same as the panel a), but along the declination axis. ](f13.ps "fig:"){height="0.55\textheight" width="70.00000%"}
[cccccrc]{} 17-Feb-2004 & s3 & 06:41:41.0 & -50:56:03.6 & 1400V & 1360 & 0.69-0.79\
17-Feb-2004 & l3 & 06:41:39.0 & -50:55:35.7 & 1400V & 4200 & 0.55-0.84\
18-Feb-2004 & s3 & 06:41:41.1 & -50:56:03.3 & 1400V & 1360 & 0.75-0.80\
18-Feb-2004 & l3 & 06:41:38.9 & -50:55:35.0 & 1400V & 4200 & 0.78-0.83\
20-Feb-2004 & l4 & 06:41:27.3 & -50:57:13.0 & 1400V & 4200 & 0.43-0.76\
20-Feb-2004 & s4 & 06:41:26.8 & -50:57:31.4 & 1400V & 1360 & 0.43-0.47\
21-Feb-2004 & l4 & 06:41:27.4 & -50:57:12.6 & 1400V & 4200 & 0.52-0.68\
21-Feb-2004 & l3 & 06:41:38.9 & -50:55:34.9 & 1400V & 4200 & 0.55-1.51\
22-Feb-2004 & l5 & 06:40:51.0 & -51:07:56.4 & 1400V & 4200 & 0.77-0.83\
23-Feb-2004 & s3 & 06:41:41.1 & -50:56:03.3 & 1400V & 1360 & 1.27-1.77\
23-Feb-2004 & s2 & 06:42:00.5 & -50:50:27.3 & 1400V & 1360 & 1.26-1.59\
13-Mar-2004 & s3 & 06:41:41.0 & -50:56:03.5 & 1400V & 1360 & 1.10-1.14\
06-Dec-2004 & l1 & 06:42:04.0 & -50:53:21.2 & 1400V & 1800 & 0.35-0.66\
06-Dec-2004 & l1 & 06:42:04.0 & -50:53:21.2 & 1028z & 1500 & 0.38-0.60\
07-Dec-2004 & l1 & 06:42:04.0 & -50:53:21.2 & 1400V & 2700 & 0.49-0.73\
07-Dec-2004 & l1 & 06:42:04.0 & -50:53:21.2 & 1028z & 900 & 0.51-0.54\
08-Dec-2004 & s1 & 06:42:06.0 & -50:54:00.0 & 1400V & 1800 & 0.80-0.92\
09-Dec-2004 & l2 & 06:42:04.7 & -50:49:57.3 & 1400V & 2400 & 0.42-0.44\
09-Dec-2004 & l2 & 06:42:04.7 & -50:49:57.3 & 1028z & 1200 & 0.53-0.59\
11-Dec-2004 & l2 & 06:42:04.7 & -50:49:57.3 & 1400V & 2400 & 0.55-0.56\
14-Mar-2007 & f3 & 06:41:35.4 & -50:55:55.2 & 1400V & 8130 & 0.48-0.65\
15-Mar-2007 & f3 & 06:41:35.5 & -50:55:55.1 & 1400V & 2710 & 0.70-1.67\
15-Mar-2007 & f3r & 06:41:31.5 & -50:56:35.5 & 1400V & 5420 & 1.02-1.16\
16-Mar-2007 & f3r & 06:41:31.5 & -50:56:35.6 & 1400V & 8130 & 0.51-0.95\
[^1]: ESO programs 072.D-0671(A), P.I.: Bono; 078.B-0567(A), P.I.: Thévenin
[^2]: MgIT: $\lambda$=5167.32, 5172.68, 5183.62 Å
[^3]: CaT: $\lambda$= 8498.02, 8542.09, 8662.14 Å
[^4]: ESO programs 074.B-0415(A), P.I.: Shetrone; 171.B-0520(B), P.I.: Gilmore; 180.B-0806(B), P.I.: Gilmore
[^5]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^6]: In particular, we assume $e_{RV}$=c$\sigma_\lambda$/$\lambda$
[^7]: Following the referee’s suggestion, we adopted the biweight location estimator [@andr72], since it is a robust indicator insensitive to outliers in both Gaussian and non-Gaussian distributions. We adopted the definition given by [@beers90], which includes data up to four standard deviations from the central location. This method is based on an iterative solution, but the process only requires a few steps to convergence.
[^8]: Projected distance on Carina were estimated assuming a true distance modulus of 20.10$\pm$0.12 mag [@dallora03; @pietrynski09] together with the inclination angle and the axes ratio [@mateo98araa].
|
---
abstract: 'We review some of the possible models that are able to describe the current Universe which point out the future singularities that could appear. We show that the study of the dark energy accretion onto black- and worm-holes phenomena in these models could lead to unexpected consequences, allowing even the avoidance of the considered singularities. We also review the debate about the approach used to study the accretion phenomenon which has appeared in literature to demonstrate the advantages and drawbacks of the different points of view. We finally suggest new lines of research to resolve the shortcomings of the different accretion methods. We then discuss future directions for new possible observations that could help choose the most accurate model.'
author:
- |
José A. Jiménez Madrid$^{1,}$[^1] and Prado Martín-Moruno$^{2,}$[^2]\
[$^{1}$Department of Applied Mathematics and Theoretical Physics,]{}\
[Wilberforce Road, Cambridge, CB3 0WA, United Kingdom]{}\
[$^{2}$Colina de los Chopos, Instituto de Física Fundamental,]{}\
[Consejo Superior de Investigaciones Científicas, Serrano 121, 28006 Madrid, Spain]{}
title: 'On accretion of dark energy onto black- and worm-holes'
---
Introduction
============
The discovery of the cosmic acceleration indicated by the observational data [@Mortlock:2000zu; @Riess:1998cb; @Spergel:2003cb] has caused a break in the belief of what could be the matter content of the universe and what might be its possible future evolution. The interpretation of this data in the framework of General Relativity implies that the majority of the content in the universe should be new stuff, which has been called dark energy, possessing anti-gravitational properties, i.e. the equation of state parameter of the dark energy must be $w<-1/3$ ($w=p/\rho$). It even seems to be possible that the equation of state parameter is less than $-1$. In that case the new stuff is known as phantom energy [@Caldwell:1999ew] and the consideration of this fluid could lead the universe to a catastrophic end by the appearance of a future singularity. The most popular of such singularities is the so-called big rip [@Caldwell:2003vq], which is a possible doomsday of the universe where both of its size and its energy density become infinitely larger. However, the big rip is not the only possibility which has been suggested as the catastrophic end to the universe in the new phantom models. It has also been argued that the universe could finish its evolution at a time where its energy density becomes infinitely larger maintaining the scale factor as a finite value, known as the big freeze singularity [@BouhmadiLopez:2006fu; @BouhmadiLopez:2007qb].
It is well known that dark energy should be accreted onto black holes in a different way that ordinary matter does, since that new fluid covers the whole space. Therefore, the study of dark energy accretion onto black holes becomes an interesting field of study which could lead to surprising effects as the possible disappearance of black holes in phantom environments. As we will show in this chapter the mentioned accretion phenomenon was originally studied by Babichev et al. [@Babichev:2005py], although a great number of works have been done to improve the method used by those authors [@MartinMoruno:2006mi; @Gao:2008jv; @MartinMoruno:2008vy].
On the other hand, the accretion process could also imply unexpected consequences in the case that one considers the evolution of cosmological objects even stranger than black holes, wormholes[^3]. Traversable wormholes are short-cuts between two regions of the same universe or between two universes, which could be used to construct time-machines [@Morris:1988tu]. The reason for its strangeness is not related to its bridge character but to that, in order to be traversable and stable, the wormholes must be surrounded by some kind of exotic matter which do not fulfil the null energy condition. Nevertheless, the consideration of phantom models as the possible current description of the universe has caused a revival of interest in traversable wormholes, since this fluid would also violate the null energy condition. Even more, it has been shown that an inhomogeneous version of phantom energy can be the exotic stuff which supports wormholes [@Sushkov:2005kj]. In some cosmological models the accretion of phantom energy onto a wormhole could lead to an enormous growth of its mouth, engulfing the whole universe which would travel through it in a big trip [@GonzalezDiaz:2005yj], avoiding the big rip [@GonzalezDiaz:2004vv] or big freeze [@BouhmadiLopez:2006fu] singularity in the corresponding cases.
In the present chapter we show the method of how to treat the accretion phenomenon onto black- and worm- holes and its cosmological consequences in some models. In Sec. II, we review some candidates that could be responsible for the current cosmological acceleration and which pay special attention to the possible future singularities appearing in some of them. In Sec. III, the procedure of the study of the dark energy accretion onto black holes based on the Babichev et al. method is shown and its application to the models included in Sec. II is presented. The corresponding study in the case of accretion onto wormholes is shown in Sec. IV, where the possible avoidance of the future singularities is highlighted. Since the study of dark energy accretion onto black- and worm- holes is still an open issue, we refer in Sec. V some interesting works which have produced a debate about the used method and we also outline some possible lines for future research to solve the shortcomings. Finally, in Sec. VI, the results are discussed and further comments are added.
Brief review of some candidates to cosmic acceleration.
=======================================================
The origin of the current accelerating expansion of the universe is one of the most interesting challenges in cosmology. Therefore, a plethora of cosmological models have been developed in recent years in order to take into account such acceleration. Although modifications of the Lagrangian of General Relativity or considerations of more than four dimensions could explain the current phase of the universe, the acceleration can also be modelled in the framework of General Relativity theory, which has shown agreement with the observational tests up to now. Nevertheless, the consequences of using the Einstein’s theory is that most part of the universe’s content must be some kind of fluid with anti-gravitational properties, called dark energy.
In order to show the necessity of the inclusion of dark energy as a new component of a universe described by General Relativity, we must consider an homogeneous, isotropic and spatially flat universe, i.e. a Friedmann-Lemaître-Robertson-Walker (FLRW) model with $k=0$. As an approximation, one can consider that this model is only filled with one fluid[^4]. Throughout this chapter we shall use natural units so that G = c = 1. The Friedmann equations can be expressed in the usual way [@Wald] $$\label{uno}
3\left(\frac{\dot{a}}{a}\right)^2=8\pi\rho,$$ $$\label{dos}
3\frac{\ddot{a}}{a}=-4\pi(\rho+3p),$$ where $a(t)$ is the scale factor, which can be used to define the Hubble parameter $H=\dot{a}/a$, $\rho$ and $p$ are the energy density and the pressure of the fluid, respectively. Eq. (\[dos\]) shows that, in order to obtain an accelerating universe, the dark energy must have an energy density and pressure such that $\rho+3p<0$, violating at least the strong energy condition. If one wants to minimise the strange character of the dark energy, then $w>-1$ ($w=p/\rho$) could be imposed, but it must be noted that such restriction is not based in direct observations but in theoretical wishes.
In fact, as we have already mentioned in the introduction, the observational data indicates that $w$ must be around $-1$ and, therefore, values less than $-1$ are not excluded. In that case the fluid is called phantom energy [@Caldwell:1999ew] and violates even the dominant energy condition.
The limiting case, $w=-1$, is also possible and is equivalent to the introduction of a positive cosmological constant. We shall not pay much attention to this case because it is the well-known de Sitter solution and more importantly, the cosmological constant would be accreted neither by black- nor by worm-holes, as it could be expected and we show in the next sections.
In this section we present the simplest dark and phantom energy models, where the equation of state parameter is considered to be closely constant. As we shall see, such a phantom energy model implies the occurrence of a big rip [@Caldwell:2003vq]. Since it could seem that phantom energy implies the occurrence of a big rip singularity, we want to show that such a singularity is not an inherent property of that fluid. So we shall consider Phantom Generalized Chaplygin Gas (PGCG) models, in order to clarify that phantom models could present no future singularities [@Bouhmadi-Lopez:2004me] or future singularities of other kinds [@BouhmadiLopez:2006fu; @BouhmadiLopez:2007qb].
Quintessence with a constant equation of state parameter.
---------------------------------------------------------
The most popular candidate to describe dark energy, allowing a dynamic evolution for this unknown component, is the quintessence model. In this model a spatially homogeneous massless scalar field is considered, which can be interpreted as a perfect fluid with negative pressure, taking the equation of state parameter values on the range $-1<w<-1/3$. Supposing that the equation of state parameter is approximately constant, the conservation law of the fluid in a FLRW spacetime, $\dot{\rho}+3H(p+\rho)=0$, can be integrated to obtain $$\label{tres}
p=w\rho=w\rho_0\left[a(t)/a_0\right]^{-3(1+w)},$$ which can be introduced in Eq. (\[uno\]) leading $$\label{cuatro}
a(t)=a_0\left(1+\frac{3}{2}(1+w)C(t-t_0)\right)^{2/[3(w+1)]},$$ with $C=\left(8\pi\rho_0/3\right)^{1/2}$ and the subscript $0$ denoting the value at the current time $t_0$. Therefore, a universe described with that model would accelerate forever, decreasing the dark energy density in the process.
Phantom quintessence with a constant equation of state parameter.
-----------------------------------------------------------------
Phantom energy can be considered to be a fluid with an equation of state parameter less than $-1$ which would, therefore, violate not only the strong energy condition but also the dominant energy condition[^5]. But, since the observational data suggests that it could be responsible for the current accelerating expansion therefore, such a pathological fluid should be seriously considered as the possible dominating matter content of our universe.
Analogous to the quintessence case, one can easily obtain an expression for the scale factor in this model considering that $w$ is approximately constant. One has $$\label{cinco}
a(t)=a_0\left(1-\frac{3}{2}(|w|-1)C(t-t_0)\right)^{-2/[3(|w|-1)]},$$ with $\rho=\rho_0\left[a(t)/a_0\right]^{3(|w|-1)}$ and $C$ taking the already mentioned value. Therefore, the scale factor (\[cinco\]) increases with time even faster than the scale factor of a de Sitter universe (which has an exponential behaviour) up to $$\label{seis}
t_{br}=t_0+\frac{2}{3(|w|-1)C}>t_0.$$ At this time both the scale factor and the energy density of the fluid blow up in which is known as the big rip singularity [@Caldwell:2003vq]. If our Universe is described by this model, then an observer located on the Earth would see how progressively it could be ripped apart the galaxies, the stars, our solar system and finally, the atoms and nuclei, up until the moment when every component of the universe would be out of the Hubble horizon of the other components.
Phantom Generalized Chaplygin Gas.
----------------------------------
It could seem that the consideration of a universe filled with phantom energy implies the occurrence of a future big rip singularity, but this is not necessarily the case. We support this claim with the example taken from the Phantom Generalized Chaplygin Gas (PGCG) [@BouhmadiLopez:2006fu; @BouhmadiLopez:2007qb; @Bouhmadi-Lopez:2004me].
The Generalized Chaplygin Gas (GCG) is a fluid with an equation of state of the form [@Kamenshchik] $$\label{siete}
p=-\frac{A}{\rho^{\alpha}},$$ where A is a positive constant and $\alpha>-1$ is a parameter. In the particular case $\alpha=1$ we recover the equation of state of a Chaplygin gas. Inserting Eq. (\[siete\]) in the conservation of the energy momentum tensor, one obtains $$\label{ocho}
\rho=\left(A+\frac{B}{a^{3(1+\alpha)}}\right)^{\frac{1}{1+\alpha}},$$ with $B$ a constant parameter. It can be seen that, maintaining $A>0$, such a fluid fulfils the dominant energy condition for $B>0$ and it is violated otherwise. The rather strange equation of state expressed through Eq. (\[siete\]) has been considered firstly in cosmology because the GCG could reproduce a transition from a dust dominated universe at early time to de Sitter behaviour at late time.
On the other hand, PGCG [@BouhmadiLopez:2006fu; @BouhmadiLopez:2007qb; @Bouhmadi-Lopez:2004me; @Khalatnikov:2003im] are fluids with an equation of state of the GCG type, Eq. (\[siete\]), with the parameters $A$, $B$ and $\alpha$ taking values in intervals such that the energy density and pressure of the fluid fulfil the requirements $\rho>0$ and $p+\rho<0$. It can be seen that such requirements lead to four classes of PGCG with
- type I: $A>0$, $B<0$ and $1+\alpha>0$.
- type II: $A>0$, $B<0$ and $1+\alpha<0$.
- type III: $A<0$, $B>0$ and $(1+\alpha)^{-1}=2n>0$.
- type IV: $A<0$, $B>0$ and $(1+\alpha)^{-1}=2n<0$.
We want to emphasise that when PGCG is considered, the sign of the parameters $A$ and $B$ must not to be necessarily positive and also $\alpha$ can be bigger or less than $-1$.
It can be seen that for type I the scale factor is bounded from below by $a_{{\rm min}}=|B/A|^{1/[3(1+\alpha)]}$ and, therefore, it takes values in the interval $a_{{\rm min}}\leq a<\infty$, which corresponds to $0\leq \rho<|A|^{1/(1+\alpha)}$, approaching the energy density a finite constant value when the scale factor tends to infinity. The Friedmann Eq. (\[uno\]) can be analytically integrated for the energy density (\[ocho\]) to lead a functional dependence of the cosmic time and the scale factor in terms of a hypergeometric series [@Bouhmadi-Lopez:2004me]. Since this expression is rather complicated and can be found in the literature, [@Bouhmadi-Lopez:2004me], we consider that it is enough to comment that the resulting expression implies that when the scale factor diverges the cosmic time also blows up, that is, there is no a singularity at a finite time in the future. The future normal behaviour is also indicated by the Hubble parameter which approaches a constant finite non-vanishing value for large scale factors and, therefore, we can conclude that the late time evolution of such a model is asymptotically de Sitter. It can be seen [@BouhmadiLopez:2007qb] that the type III model has a similar evolution for late times than the type I, although both behaviours can differ greatly at early times[^6].
Although we have just discussed that the consideration of phantom models could avoid the occurrence of a future big rip singularity, another kind of doomsday could appear in phantom models. In order to show this fact, let us consider the type II PGCG. In this case the scale factor is bound from above by $a_{{\rm
max}}=|B/A|^{1/[3(1+\alpha)]}$ taking, therefore, values in the interval $0<a\leq a_{{\rm max}}$ which correspond to $A^{1/(1+\alpha)}\leq\rho<\infty$. As in the type I an analytical expression for the scale factor depending on the cosmic time can be found in terms of hypergeometric series [@BouhmadiLopez:2007qb]. Such an expression can be approximated close to the maximum scale factor value and inverted leading to [@BouhmadiLopez:2006fu; @BouhmadiLopez:2007qb] $$\label{nueve}
a\simeq a_{\rm{max}}\left\{1-\left(\frac{8\pi}{3}\right)^{\frac{1+\alpha}{1+2\alpha}}\left[\frac{1+2\alpha}{2(1+\alpha)}\right]^{\frac{2(1+\alpha)}{1+2\alpha}}A^{\frac{1}{1+2\alpha}}
|3(1+\alpha)|^{\frac{1}{1+2\alpha}}(t_{\rm{max}}-t)^{\frac{2(1+\alpha)}{1+2\alpha}}\right\},$$ which implies that the cosmic time elapsed since the universe has a given scale factor $a$ (closed to $a_{{\rm max}}$) until it reaches its maximum value is finite, i.e., $t_{{\rm
max}}-t<\infty$. Therefore, this model would end at a finite singularity where its energy density blows up whereas the scale factor remains finite, called big freeze singularity [@BouhmadiLopez:2006fu]. The type IV model would have a similar behaviour and it exhibits a singularity of the same kind, [@BouhmadiLopez:2007qb]. So, both models describe a universe which would expand accelerating until it freezes its evolution at a finite time where it is infinitely full of phantom energy.
It must be pointed out that from a classical point of view, as such we are considering through this chapter, a singularity would break down the spacetime. Nevertheless, it has been argued that the consideration of quantum effects could avoid the big rip [@Nojiri:2004pf] and the big freeze singularity [@Bouhmadi].
Dark energy accretion onto black holes.
=======================================
As we have pointed out in the previous section, dark energy is filling our Universe and, therefore, it could be expected that it would interact with different cosmological objects like black- and worm-holes. This section is dedicated to the study of the accretion process onto black holes by using the most accepted model dealing with this phenomenon. Nevertheless, it must be pointed out that some controversy has originated around this method and as such other approaches have been proposed, which will be discussed in Sec. V.
The standard method treating the dark energy accretion onto black holes was firstly presented by Babichev et al. [@Babichev:2005py] and is based on the consideration of a black hole described by the Schwarzschild metric, surrounded by a perfect fluid which represents dark energy. In such a framework they considered the zero component of the energy-momentum conservation equation and the projection of this equation along the four-velocity, to derive the dynamical evolution of the black hole mass. We want to summarise a generalisation of this method, presented in Ref. [@MartinMoruno:2006mi], which allows an internal nonzero energy-flow component $\Theta_{0}^{r}$ by the consideration of the simplest non-static generalisation of the Schwarzschild metric, in which the black hole mass can depend generically on time. That is, the metric can be given by
$${\rm d}s^2=\left(1-\frac{2M(t)}{r}\right){\rm d}t^2-\left(1-\frac{2M(t)}{r}\right)^{-1}{\rm d}r^2- r^2\left({\rm d}\theta^2+\sin^2{\rm d}\phi^2\right),$$
where $M(t)$ is the black hole mass.
The zero component of the conservation law for energy-momentum tensor and its projection along the four-velocity can be integrated in the radial coordinate considering a surrounding perfect fluid. On the other hand, it is known that the rate of change of the black hole mass due to accretion of dark energy can be derived by integrating over the surface area the density of momentum $T_0^r$. Taking into account these considerations one gets the following equation [@MartinMoruno:2006mi]
$$\dot{M}=4\pi A_{M} M^2\left(p+\rho\right)e^{-\int_\infty^r f\left(r,t\right){\rm d}r},$$
relating the temporal rate of change of the mass with the pressure and energy density of the perfect fluid which is considered to describe dark energy.
For the relevant physical case of an asymptotic observer, i.e. $r\rightarrow
\infty$, the previous equation simplifies to
$$\label{eq:negrofinal}
\dot{M}=4\pi A_{M} M^2\left(p+\rho\right),$$
where $A_{M}$ is a positive constant of order unity. It must be pointed out that this expression is the same as that obtained by Babichev et al. [@Babichev:2005py] using the usual Schwarzschild metric, with the difference that in the current case it is only valid for asymptotic observers.
It can be noted that Eq. (\[eq:negrofinal\]) shows that the black hole mass, and with it its size, must decrease when the black hole accretes a fluid which violates the dominant energy condition, i.e. a phantom fluid. This would increase when the dominant energy condition is preserved and it remains constant in the cosmological constant case, since a cosmological constant can be modelled by a perfect fluid with $p+\rho=0$. It must be emphasised that Eq. (\[eq:negrofinal\]) has been obtained for a general perfect fluid, therefore it would be valid in a great variety of dark energy models, since a load of them are based on a perfect fluid.
Since accretion of dark energy onto black holes would increase the black hole size in dark energy models fulfilling the dominant energy condition, an interesting question is whether this growth could be large enough to that the black hole might engulf the whole universe. In order to find a possible answer for this question, one can take into account the Friedmann equations to integrate Eq. (\[eq:negrofinal\]), obtaining the temporal evolution of the black hole mass. That is, [@Yo] $$\label{M4D}
M=\frac{M_0}{1+\sqrt{\frac{8\pi}{3}}A_{M}M_0\left(\rho^{1/2}-\rho_0^{1/2}\right)}.$$ This equation can also be expressed in terms of the Hubble parameter in the following way $$\label{M4DH}
M(t)=\frac{M_0}{1+D M_0\left[H(t)-H_0\right]}.$$ Therefore, as mentioned in Ref. [@MartinMoruno:2006mi], a black hole capable of engulfing the whole universe as shown in a model with an equation of state parameter bigger than minus one, should have a current mass which, roughly speaking, is bigger than all the matter content of the current observable universe, making the occurrence of such a phenomenon impossible.
In the following subsections we consider the models reviewed in the previous section to study the evolution of a black hole living in a universe filled with dark energy.
Application to a quintessence model.
------------------------------------
We assume that dark energy is modelled by a quintessence model satisfying the equation of state (\[tres\]) with $w>-1$ constant. As we have already mentioned, Eq. (\[eq:negrofinal\]) implies that the rate of mass change is positive in this model, therefore the black hole mass grows along time due to accretion of quintessence. In order to obtain the dynamical behaviour of the black hole mass, one can introduce the equation of state (\[tres\]) in Eq. (\[M4D\]) which leads to
$$\label{eq:mnegroquintaesencia}
M=\frac{M_0 \left[1+\frac{3}{2}\left(1+w\right)C\left(t-t_0\right)\right]}{1+\frac{3}{2}\left(1+w\right)C\left(t-t_0\right)-4\pi A_{M}\rho_0 M_0\left(1+w\right)\left(t-t_0\right)} .$$
There are something very interesting in this expression for the black hole mass, because it allows the occurrence of a bizarre fate for our universe as we have already pointed out. That is, Eq. (\[eq:mnegroquintaesencia\]) expresses the possibility that our universe might be engulfed by a black hole, since accretion of dark energy could make the mass of the black hole increase so quickly as to yield a black hole size that would eventually exceed the size of the universe in a finite cosmic time. In fact, the time in which the black hole might reach a infinite size would be $$t_{bs}=t_0+\frac{1}{\left(1+w\right)\left(4\pi A_{M}\rho_0 M_0-\sqrt{6\pi\rho_0}\right)}.$$ This time is finite but, although a universe filled with quintessence has no future singularity, present observational data seems to imply that $w$ is not constant and suggests values less than -1, making it unlikely that the occurrence of the considered bizarre phenomenon at any time in the far future would happen, at least in principle. However, if $\dot{w} > 0$ then the black holes would undergo a larger growth due to accretion of dark energy.
Nevertheless, if one studies in deeper detail Eq. (\[eq:mnegroquintaesencia\]), then one obtains (see Ref. [@MartinMoruno:2006mi]) that in order to have a black hole able to reach an infinite mass in an infinite time, this black hole must possess an initial mass such as $(8\pi\rho_0/3)^{1/2}A_{M}M_0=1$ which means, taking into account the observational data, $M\sim
10^{23}M_\odot$ (where $M_\odot$ is the Sun’s mass). Therefore, in order to have an infinitely large black hole in a finite time in the future, the current black hole mass should be bigger than $10^{23}M_\odot$, which is an extremely large value even for black holes in the galaxies centres. Even more, one can estimate the current matter content of the universe assuming that the observable Universe expands at the speed of light, obtaining a total of $10^{23}$ starts [@MartinMoruno:2006mi]. Therefore, it seems that in order to have a black hole able to engulf the whole universe, it should have a current mass equal to the mass of all the observable universe, which would not be possible.
Finally, we want to point out that, even in the case that the accretion phenomenon of dark energy onto black holes could not produce cosmological consequences in terms of a catastrophic end, it could help in the determination of the correct dark energy model. So, if astronomers were able, in practice, to observe a growth bigger than expected of those black holes living in the centre of the galaxies, then this could be an observational measure of the effects of dark energy with $w>-1$.
Application to a phantom quintessence model.
--------------------------------------------
As we have already pointed out, the observational data not only allows that the equation of state parameter takes a value less than $-1$ but even they seem to suggest it, acquiring therefore, special interest in the study of the evolution of a black hole in a universe filled with phantom energy. So, now we consider $w<-1$ and constant, consequently (\[eq:negrofinal\]) shows that the black hole mass decreases with time. More precisely, inserting Eqs. (\[tres\]) and (\[cinco\]) in Eq. (\[M4D\]), we get an accurate expression of the evolution of the black hole mass, i.e.
$$\label{eq:mnegrophantom}
M=\frac{M_0 \left[1-\frac{3}{2}\left(|w|-1\right)C\left(t-t_0\right)\right]}{1-\frac{3}{2}\left(|w|-1\right)C\left(t-t_0\right)+4\pi A_{M}\rho_0 M_0\left(|w|-1\right)\left(t-t_0\right)} .$$
Taking into account that in a universe filled with phantom energy with $w$ constant a big rip singularity will take place in the future, one can introduce the time of occurrence of the big rip, $t_{br}$, in (\[eq:mnegrophantom\]) to get that the black hole mass vanishes at $t_{br}$, independently of the current black hole mass, $M_0$; that is, all black holes disappear at the big rip [@Babichev:2005py]. It can be noted, by inspection of Eq. (\[M4DH\]), that this is not only an interesting property of this phantom model but it can be found in all models which present a singularity with a divergence of the Hubble parameter at a finite time in the future.
Finally, the decrease of black holes due to the phantom energy accretion phenomenon could provide us with a possible observational test of these models. Therefore, if future observations of black holes in the centre of galaxies (or other possible black holes) indicate a growth of those objects less than expected, then it could be associated to accretion of phantom energy, providing us with another measure able to discriminate between different dark energy models, which would complete those that come from GRB [@Wang:2009zzq; @Ghirlanda:2006ax], supernova, or other observational data.
Application to a Generalized Chaplygin model.
---------------------------------------------
Let us now study the evolution of a black hole in a universe filled with a Generalized Chaplygin Gas. Since the consideration of dark energy modelled by some kinds of Phantom Generalized Chaplygin Gas could prevent the occurrence of a big rip [@Bouhmadi-Lopez:2004me], one could expect the avoidance of the weird behaviours that appear in quintessence or phantom models in these frameworks. In order to see if that is the case, one must take into account Eq. (\[M4D\]).
As we have mentioned in Sec. II, a Generalized Chaplygin Gas, with $A>0$ and $\alpha>-1$, preserves dominant energy condition when the parameter[^7] $B>0$ and violated otherwise. So when $B>0$ the black hole mass increases with cosmic time up to a constant value, but there are a set of parameters where if $\alpha$ is close to $-1$ then it would seem that the black hole mass could eventually exceed the size of the universe at finite time in the future [@Jimenez; @Madrid:2005gd]. When dominant energy condition is violated, $B<0$, black hole mass decreases along cosmic time, tending to a nonzero constant value, therefore black holes do not disappear in this model.
On the other hand, it can be seen [@Yo] that performing a deeper study of the mentioned four types of PGCG (where the sign of $A$ and the range on $\alpha$ is not previously fixed) at late times, where the phantom fluid would drive the dynamical evolution of the universe, the results can be summarised as follows:
- Type I and III. Black holes decrease with time, where the mass tend to a nonzero value when the time goes to infinity.
- Type II and IV. Black holes masses decrease, but now all black holes disappear at big freeze, i.e., the mass of all black holes tend to zero when the universe reaches the big freeze singularity with independence of their initial mass (as it should be expected by inspection of equation (\[M4DH\])).
To end this section, we consider that it would be quite interesting to explore the region of parameter space $(\alpha, A , H_0 , \Omega_K ,\Omega_\phi)$ allowed by current observations in order to determine whether there exists any allowed sections leading to a big freeze or a big rip. However, all available analyses [@Bertolami:2004ic; @Biesiada:2004td; @Zhu:2004aq; @Colistete:2005yx; @Lu:2009zzf; @delCampo:2009cz; @Li:2009br; @Wang:2009zzq; @Wu:2007zz; @Lu:2008zzb] are restricted to the physical region where no dominant energy condition is violated. Therefore, the section described by the interval implied by a PGCG necessarily is outside the analysed regions. One has to extend the investigated domains to include values of parameter $A>1$, $A<0$ or $\alpha<-1$ to probe the parameters space where the dominant energy condition is violated.
Consideration to other black holes.
-----------------------------------
Up to now, we have shown the evolution of a Schwarzschild black hole in a universe filled with dark energy. In this subsection, we study the accretion of dark energy onto charged or rotating black holes, to show whether charge or angular momentum have some influence in their evolution.
Let us continue by considering another black hole metric in order to understand better the application of the dark energy accretion mechanism. In [@Babichev:2008jb], Babichev et al. apply a generalisation of the accretion formalism to a Reissner-Nordsröm black hole. In this case, the metric is given by
$${\rm d}s^2=\left(1-\frac{2M}{r}+\frac{e^2}{r^2}\right){\rm d}t^2-\left(1-\frac{2M}{r}+\frac{e^2}{r^2}\right)^{-1}{\rm d}r^2
- r^2\left({\rm d}\theta^2+\sin^2\theta{\rm d}\phi^2\right),$$
where $m^2>e^2$. It can be noted that if $m<|e|$, then the solution would represent a naked singularity, corresponding $m=|e|$ to the extreme case. By integrating the conservations laws for momentum-energy and its projection along four-velocity for the case for a perfect fluid, and taking into account that the rate of change of the black hole mass due to accretion of dark energy can be derived by integration over the surface area the density of momentum $T_0^r$, Babichev et al. get again the same expression (\[eq:negrofinal\]) which relates the temporal rate of change of the black hole mass to the pressure and the energy density of the perfect fluid. So, also in a Reissner-Nordsröm black hole, the black hole mass increases when it is accreting dark energy holding the dominant energy condition and its mass decreases when phantom energy is considered in the accretion process.
At this point the next question arises again, if phantom energy is getting involved then black hole mass decreases, vanishing at big rip singularity. Therefore, since the electric charge $e$ remains constant due to phantom energy accretion, then in a finite time, the black hole must reach the extreme case, transforming the black hole into a naked singularity. Nevertheless, the authors of Ref. [@Babichev:2008jb] perform a more detailed study about this possible transformation in a naked singularity, concluding that there is no accretion of the perfect fluid onto the Reissner-Nordström naked singularity when $m^2<e^2$ and that, in this situation, a static atmosphere of the fluid around the naked singularity would be formed.
It must be emphasised that, although when one is considering cases far from the extreme case, the back reaction can be neglected and the perfect fluid approximation appears to be valid, it seems that this approximation breaks down close to the extremal case, where one has to take into account the back reaction of the perfect fluid onto the background metric. Even more, the same consideration about the avoidance of transformation of a black hole into a naked singularity, can also be applied to a Kerr black hole. Nevertheless, if the back reaction does not prevent the process of phantom accretion onto a charged black hole or rotating black hole, then this process could be a way to violate the cosmic censorship conjecture [@Penrose].
Dark energy accretion onto wormholes.
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The first solution of the Einstein’s equations describing a traversable wormhole was found by Morris and Thorne in their seminal work [@Morris:1988cz]. That solution, obtained under the assumption of staticity and spherical symmetry, describes a throat connecting two asymptotically flat regions of the spacetime without any horizon and can be expressed as $$\label{diez}
{\rm d}s^2=-e^{2\Phi(r)}{\rm d}t^2+\frac{{\rm d}r^2}{1-K(r)/r}+
r^2\left({\rm d}\theta^2+\sin^2{\rm d}\varphi^2\right),$$ where $\Phi(r)$ and $K(r)$ are the shift and shape functions, respectively, both tending to a constant value when the radial coordinate $r\rightarrow\infty$ in order to have asymptotic flatness. It must be noted that, in these coordinates, two coordinate patches are needed to cover the two asymptotically flat regions, each with $r_0\leq r\leq\infty$, with $r_0$ the minimum radius which corresponds to the throat radius, where $K(r_0)=r_0$.
It can be seen [@Morris:1988cz] that solution (\[diez\]) must fulfil some additional requirements in order to describe a traversable wormhole. In particular the outward flaring condition imposes $K'(r_0)<1$ what, through the Einstein’s equations, implies that $p_r(r_0)+\rho(r_0)<0$ (where here $p_r$ denotes the radial component of the pressure). Therefore, the wormhole must be surrounded by some material with unusual characteristics, called exotic matter, which could lead to the neglect of such spacetime.
Nevertheless, as we have already mentioned in the introduction, the discovery of the current accelerated expansion of the Universe and the consideration of phantom energy as a possible candidate for its origin has produced a more natural consideration of the properties of exotic matter, since it seems that phantom energy could be precisely the exotic stuff which supports wormholes. Gonzalez-Diaz [@GonzalezDiaz:2004vv] considered that similarly to black holes accrete dark energy, wormholes could accrete phantom energy producing a great increase of their size, in such a way that the size of a wormhole could be infinitely larger before the universe reaches the big rip singularity. Such a process would produce the moment that the size of the wormhole equals the size of the universe, the universe boards itself in a travel through the wormhole, called big trip. Even more, the notion of phantom energy has been extended to inhomogeneous spherically symmetric spacetimes showing that it can be in fact the exotic material which supports wormholes [@Sushkov:2005kj], which backs up the mentioned idea.
In order to study such a process one can follow a similar method to the one used by Babichev et al. for the black hole case. So, considering the non-static generalisation of Eq. (\[diez\]) obtained by the consideration of an arbitrary dependence of the shape function on the time, $K(r,t)$, and an energy momentum-tensor of a perfect fluid, one can find the equivalent of Eq. (\[eq:negrofinal\]) for the wormhole case, that is the temporal mass rate evolution as measured by an asymptotic observer, which is [@GonzalezDiaz:2007gt] $$\label{once}
\dot{m}=-4\pi Qm^2(p+\rho),$$ with $Q$ a positive constant. That expression shows that the wormhole mass, and with it, its size must increase when the wormhole accretes phantom energy, it decreases by the accretion of dark energy, remaining constant in the cosmological constant case. It must be remarked that in the achieving of Eq. (\[once\]) no assumption about the possible dependence on the energy density or on the pressure of the fluid have been done, allowing an arbitrary dependence with the time and with the radial component, therefore, such an equation is general and take into account the possible back reaction in an asymptotically flat wormhole spacetime[^8].
If one now considers as an approximation that the fluid which surrounds the wormhole is a cosmological one, i.e., an homogeneous and isotropic fluid fulfilling the Friedmann equations (\[uno\]) and (\[dos\]) and the conservation law, then Eq. (\[once\]) can be integrated to obtain [@Yo; @Jimenez; @Madrid:2005gd] $$\label{doce}
m(t)=\frac{m_0}{1-Qm_0\left[H(t)-H_0\right]}.$$
This expression shows that in phantom models, where $H(t)$ is an increasing function, the wormhole throat could become infinitely big if the Hubble parameter reaches the value $H_*=H_0+1/(Qm_0)<\infty$ at some time $t_*$ in the future. It can be seen that this would be the case at least in models which show a future singularity in a finite time in the future characterised by a divergence of the Hubble parameter, because in those models one has $H_0<H_*<H_{{\rm sing}}=\infty$ and, since the Hubble parameter is a strictly increasing and continuous function before the time of the singularity, this implies $t_0<t_*<t_{{\rm sing}}$. Therefore, the size of a wormhole would be bigger than the size of the universe before the occurrence of the future singularity in all models possessing a future singularity where the Hubble parameter blows up, i.e., in such models the universe would travel through a big trip.
In this section we show the implications of the phenomenon of dark and phantom energy accretion onto wormholes in the models presented in Sec. II. That procedure lead, as it is expected, to the decrease of the wormhole size when it accretes dark energy with $w>-1$ and to a growth of the wormhole mouth in phantom cases. Even more, the big rip and big freeze singularities, in the corresponding models, can be avoided by a big trip phenomenon since, although these singularities present a different behaviour of the scale factor, at both singularities the Hubble parameter blows up.
Application to a quintessence model.
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Let us consider that the dark energy is modelled by a quintessence model, satisfying the equation of state (\[tres\]) with $w>-1$ constant. Eq. (\[once\]) tells us that the rate of mass change is negative, so the wormhole mass would decrease with time due to the accretion of quintessence. Furthermore, taking into account the equation of state (\[tres\]), one can solve (\[once\]) getting the following expression which relates the wormhole mass to the cosmic time [@GonzalezDiaz:2005yj],
$$\label{eq:mgusanoquintaesencia}
m=\frac{m_0}{1+\frac{4\pi Q\rho_0 m_0\left(1+w\right)\left(t-t_0\right)}{1+\frac{3}{2}\left(1+w\right)C\left(t-t_0\right)}} .$$
This expression shows us how a wormhole loses mass due to the accretion of quintessence. Even more, if due to any additional hypothetical process this wormhole would have a macroscopic size, then it would be subjected to chronology protection [@Hawking:1991nk]; therefore, vacuum polarisation created particles would catastrophically accumulated on the chronology horizon of the wormhole, letting the corresponding normalised stress-energy tensor to diverge which, at the end of the day, would imply the disappearance of the wormhole.
Application to a phantom quintessence model.
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Now, we are interested in study the evolution of a wormhole in a universe filled with phantom energy with $w<-1$ constant. In order to obtain the temporal evolution of the wormhole, one can introduce the equation of state of phantom energy into the r.h.s. of Eq. (\[once\]), getting [@GonzalezDiaz:2005yj],
$$\label{eq:mgusanophantom}
m=\frac{m_0}{1+\frac{4\pi Q\rho_0 m_0\left(|w|-1\right)\left(t-t_0\right)}{1+\frac{3}{2}\left(|w|-1\right)C\left(t-t_0\right)}}.$$
This expression implies that the exotic mass of the wormhole diverges at the time
$$t_{bt}=t_0+\frac{t_{br}-t_0}{1+\frac{8\pi Qm_0a_0^{3\left(|w|-1\right)/2}}{3C}},$$
where $t_{br}$ is the finite time at which the big rip singularity takes place. Since the wormhole size diverges before that the universe reaches the big rip singularity, it would be a previous time at which the size of the wormhole would be bigger than the universe, being at this time where properly starts the travel of the universe through the wormhole.
The huge growth of the wormhole throat poses the following two problems. On the one hand, since the wormhole spacetimes are usually considered to be asymptotically flat, when the wormhole increases more than the universe it is impossible to place the wormhole on this universe. On the other hand, since the universe is travelling through the wormhole, one can ask where is the universe travelling to? The solution of these equivalent problems requires the consideration of a multiverse scenario. In such a framework the wormhole could be re-infixed in another universe where the wormhole would be asymptotically flat to, giving also a final destination to the universal travel[^9].
Application to a Generalized Chaplygin Gas model.
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Finally, we will study the evolution of a wormhole in the case of a universe filled with a Generalized Chaplygin Gas. Since type I and III of PGCG avoid the occurrence of a future singularity [@BouhmadiLopez:2007qb; @Bouhmadi-Lopez:2004me], it is of special interest to study the possible occurrence of a big trip phenomenon in these models. Following this line of thinking, in Ref. [@Jimenez; @Madrid:2005gd] it is analysed the phantom energy accretion phenomenon onto wormholes when the phantom energy is modelled by a type I PGCG. In order to perform this study, let us temporarily fix $A>0$ and $\alpha>-1$, therefore, solving Eq. (\[doce\]) for the equation of state of a GCG, one obtains
$$\label{eq:mgusanochaplygin}
m=\frac{m_0}{1-Qm_0\sqrt{\frac{8\pi}{3}}\left(\rho^\frac{1}{2}-\rho_0^{1/2}\right)}.$$
For the case where the dominant energy condition is violated, i.e. $B < 0$, we obtain that $m$ increases with time and tends to a maximum, nonzero constant value. If the dominant energy condition is assumed to be hold, i.e. $B > 0$, then $m$ decreases with time, with $m$ tending to nonzero constant values.
It can be seen that, when the cosmic time goes to infinity, then the exotic mass of wormhole approaches to
$$m=\frac{m_0}{1-Qm_0\sqrt{\frac{8\pi}{3}}\left(A^\frac{1}{2\left(1+\alpha\right)}-\rho_0^{1/2}\right)},$$
that is a generally finite value both for $B > 0$ and $B < 0$. Thus, it could be thought that the presence of a Generalized Chaplygin Gas prevents the eventual occurrence of the big trip phenomenon. However, such a conclusion cannot be guaranteed as the size of the wormhole throat could still exceed the size of the universe during its previous evolution. The question is whether the wormhole would grow rapidly enough or not to engulf the universe during the evolution to its final classically stationary state. To avoid a big trip one needs that the radius of the wormhole does not exceed the size of the universe. It can be checked [@Jimenez; @Madrid:2005gd] that GCG generally prevents the occurrence of a big trip when $\alpha$ does not reach values sufficiently close to $-1$, but when $\alpha$ is inside the interval
$$-1<\alpha<\frac{\ln A}{\ln\left(\sqrt{\frac{3}{8\pi}}\frac{1}{m_0 D}+\rho_0^{1/2}\right)^2}-1,$$
a big trip would still take place.
It is worth noticing that, when there is no big trip phenomenon, the wormhole size tends to become constant at the final stages of its evolution being a rather a macroscopic object. So, the wormhole at this stage would be subjected to chronology protection [@Hawking:1991nk] and vacuum polarisation created particles would catastrophically accumulate on the chronology horizon of the wormhole making the corresponding renormalised stress-energy tensor to diverge and hence the wormhole would disappear.
On the other hand, one can study the wormholes’ evolution living in a universe with phantom energy modelled by a type II or IV PGCG [@Yo], where a future big freeze singularity is predicted. Since a big freeze singularity implies the divergence of the Hubble parameter at this singularity, as we have mentioned in the introduction of the present section, this implies that the wormhole size would blow up before the occurrence of the singularity, implying a big trip phenomenon. That can be easily proved taking into account Eqs. (\[uno\]), (\[ocho\]) and (\[doce\]) for type II PGCG, which yields $$\label{masachap}
m(x)=\frac{m_0}{1+\sqrt{\frac{8\pi}{3}}\frac{Q m_0}{A^{\frac{1}{2|1+\alpha|}}}\left[\frac{1}{(1-x_0^{3|1+\alpha|})^{\frac{1}{2|1+\alpha|}}}-\frac{1}{(1-x^{3|1+\alpha|})^{\frac{1}{2|1+\alpha|}}}\right]},$$ where $x=a/a_{{\rm max}}$ ($0\leq x\leq1$) and a similar expression can be obtained for type IV replacing $A$ with $|A|$. In order to study the behaviour of the wormhole mass, one can define the function $F(x)=m_0/m(x)$ which is continuous in the interval $[x_0,1)$. This function takes a value $F(x_0)=1>0$ and tends to minus infinity when $x$ goes to 1 (which corresponds to $a\rightarrow a_{{\rm max}}$), which implies that $F(x)$ vanishes at some $x_*$ with $x_0<x_*<1$. Therefore, $m(x)$ blows up at $x_*$ being the throat size infinitely large before the universe reaches the big freeze singularity (at $x=1$). So the whole universe will travel through the wormhole before the occurrence of the doomsday.
The results can be summarised as follows
- Types I and III. The evolution of the wormhole is the same for GCG.
- Types II and IV. A big trip phenomenon would prevent the expected cosmological doomsday, i.e., the big freeze.
Debate and new lines of research.
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In the present chapter we have used methods based in the Babichev et al. one, in order to consider the dark and phantom energy accretion onto black- and worm-holes. Our intention is not to claim that the study of these processes is a closed issue, on the contrary, it remains open up to now and a lot of discussion has been originated in this way.
In order to point out the shortcomings of the current available methods which deal with the mentioned accretion phenomenon and suggest possible new lines of research. In this section we also include in chronological order some comments which have appeared in the literature supporting, improving or criticising the Babichev et al. method [@Babichev:2004yx]. We alternate works regarding the accretion onto black holes with others dealing with the accretion onto wormholes, because both phenomena can be studied following the same procedure. Nevertheless, as it has been and will be pointed out, the wormhole case is free of some shortcomings which affect the black hole one, since the first one can never be considered as a vacuum solution (at least if one restricts oneself to the traversable wormhole case).
The first work about accretion of dark energy onto black holes was due to E. Babichev, V. Dokuchaev and Yu. Eroshenko [@Babichev:2004yx]. They considered the spherically symmetric accretion of dark energy onto black holes adjusting the analytic relativistic accretion solution onto the Schwarzschild black hole developed by Michel [@Michel], eliminating from the equations the particle number density. So they obtain the expression for the black hole temporal mass rate $$\label{unob}
\dot{M}=4\pi A_{M} M^2\left[\rho_\infty+p\left(\rho_\infty\right)\right],$$ showing that the black hole mass could decrease by the accretion phenomenon. The authors pointed out that such a decrease of the black hole size is due to the violation of the dominant energy condition, since this condition is assumed to be fulfilled in the derivation of the black hole non-decrease area theorem. On the other hand, by integrating (\[unob\]) in a phantom Friedmann universe, they found that the masses of all black holes tend towards zero when the universe approaches the big rip, independently of their initial masses.
Soon after, P. F. Gonzalez-Diaz [@GonzalezDiaz:2004vv] considered the spherically symmetric accretion of dark and phantom energy onto Morris-Thorne wormholes. He assumed that, since the mass of the spherical thin shell of the exotic matter in a Morris-Thorne wormhole, $\mu=-\pi b_0/2$ (where $b_0$ is the radius of the spherical wormhole throat), is approximately just the negative of the amount of the mass required to produce a Schwarzschild wormhole, then the rate of change of the wormhole throat radius should be similar to that obtained by Babichev et al. [@Babichev:2004yx] for the black hole mass but with a minus sign, i.e. $$\label{tresb}
\dot{b_0}=-2\pi^2 Q b_0^2\left(1+w\right)\rho,$$ with $Q\simeq A_{M}$. He concluded, therefore, that the wormhole throat should increase by the accretion of phantom energy. Even more, he showed, by integrating Eq. (\[tresb\]) in a phantom model with constant equation of state parameter, that the wormhole increases even faster than the universe itself, engulfing the whole universe before it reaches the big rip singularity. Therefore, the universe would embarks itself in a big trip.
Later on, P. F. Gonzalez-Diaz and C. L. Siguenza, [@GonzalezDiaz:2004eu], obtained that the phantom energy accretion onto black holes leads to the disappearance of the black holes at the big rip even when Eq. (\[unob\]) is integrated in more complicated models and Babichev et al. recovered their previous result in a more detailed work [@Babichev:2005py] where they also included a deeper study of two dark energy models admitting analytical solutions. On the other hand, works containing relevant implications of the result of Babichev et al. were also published during that time, like the influence of the accretion phenomenon on the black hole and phantom thermodynamics, Ref. [@GonzalezDiaz:2004eu], and the possible survival of black holes at the big rip due to the same phenomenon which could smooth the big rip singularity when quantum effects are taken into account, Ref. [@Nojiri:2004pf].
But the previous mentioned works did not had the final say about this topic. In 2005 V. Faraoni and W. Israel [@Faraoni:2005it] considered the time evolution of a wormhole in a phantom Friedmann universe finding no big trip phenomenon. In that work they commented that the way in which Gonzalez-Diaz applied the Babichev et al. method to the wormhole case in Ref. [@GonzalezDiaz:2004vv] could be wrong, since $\mu=-\pi b_0/2$ must not to be necessarily valid for a time-dependent wormhole embedded in a FLRW universe and, therefore, the wormhole mass time rate due to the accretion phenomenon would not be simply the analogous negative of the black hole mass time rate.
Soon after, Gonzalez-Diaz [@GonzalezDiaz:2005yj] followed a similar procedure as it has been done by Babichev et al. [@Babichev:2004yx], adjusting the Michel theory to the case of Morris-Thorne wormholes, in order to study the dark energy accretion onto wormholes. He obtained Eq. (\[once\]) for the case of an asymptotic observer (which is equivalent to Eq. (\[tresb\]) with a re-definition of the constants). He also claimed that the results obtained by Faraoni and Israel [@Faraoni:2005it] just take into account the inflationary effects of the accelerated expansion of the universe on the wormhole size (also considered by himself years ago in another work [@GonzalezDiaz:2003pb]) and do not include the superposed effects due to the accretion phenomenon, which existence is clarified in that work [@GonzalezDiaz:2005yj].
On the other hand, a number of difficulties related to the big trip process were treated also by Gonzalez-Diaz in Ref. [@GonzalezDiaz:2006na]. First, he showed how the corrections appearing in the expressions of the study of a wormhole metric with a non-static shape function applying the Babichev et al. method [@Babichev:2004yx] should disappear on the asymptotic limit, coinciding with those corresponding calculated expressions in the static case in that regime. Even more, he considered explicitly a metric able to describe a wormhole in a Friedmann universe, arguing that it would ultimately imply the occurrence of a big trip phenomenon. Second, since wormhole spacetimes are usually considered to be asymptotically flat, then when the wormhole increases more than the universe, this object can neither be placed on it nor be asymptotically flat to it. He proposed that in such a situation a multiverse context must be considered, what would allow to re-insert the wormhole in another universe, recovering the meaning of the asymptotic regime where the accretion process has been calculated. In the third place, he considered the possible instability of wormholes due to the quantum creation of vacuum particles on the chronology horizon when the wormhole throat grows at a rate smaller than or nearly the same as the speed of light. However, in a phantom model the accreting wormhole would clearly grow at a rate which exceeds the speed of light asymptotically and so the vacuum particles would never reach the chronology horizon where they have being created, keeping the wormhole stability. Moreover, although it was known that quantum effects could affect the big rip singularity [@Nojiri:2004pf], Gonzalez-Diaz showed that those effects have no influence in the big trip, which would take place before that singularity. In the fourth place, he argued that the big trip phenomenon would not imply any contradiction with the holographic bound, since wormholes able to connect regions after and before the big rip extend the evolution of the universe up to infinite time.
Following that line of thinking, the authors of [@MartinMoruno:2006mi] applied the Babichev et al. method to the simplest non-static generalisation of the Schwarzschild metric, in order to study the dark energy accretion onto black holes with arbitrary accretion rates. Although they are still using a test fluid approach and, therefore, the validity of their result on arbitrary accretion rates is debatable, the non-static metric is enough to take into account internal non-zero energy flow $\Theta_0^r$. As it was suggested by Gonzalez-Diaz in the case of wormholes [@GonzalezDiaz:2006na], a study of the accretion phenomenon using a non-static metric recovers the result obtained in the static case, Eq. (\[unob\]), for asymptotic observers.
Later on, Faraoni published “No “big trips” for the universe”, [@Faraoni:2007kx], where a sceptical attitude about the big trip phenomenon is adopted, based on some shortcomings of the works of Gonzalez-Diaz [@GonzalezDiaz:2004vv; @GonzalezDiaz:2005yj; @GonzalezDiaz:2006na] in particular and the method of Babichev et al. [@Babichev:2004yx] in general. His principal objection regarding the mentioned works of Gonzalez-Diaz was that the use of a static metric can never produce a non-zero radial energy flow onto the hole, i.e. static metrics always imply $\Theta_0^r=0$. Even more, the solution of Gonzalez-Diaz (and the corresponding of Babichev et al. in the black hole case) cannot be adjusted to satisfy the Einstein’s equations, as the used conservation laws would only strictly correspond to vacuum solutions. On the other hand, he also showed that if the phantom fluid is modelled by a perfect fluid, as it is done in the Babichev et al. method and its application to wormholes, the proper radial velocity of the fluid is $v\sim a^{3(1+w)/2}$ which vanishes at the big rip stopping the accretion phenomenon.
Soon after, Gonzalez-Diaz et al., [@GonzalezDiaz:2007gt], applied the method of Babichev et al. to a non-static generalisation of the Morris-Thorne metric, introducing a shape function with an arbitrary dependence on time, $K(r,t)$. They recovered again Eq. (\[once\]) for the temporal mass rate of a wormhole in the asymptotic limit. It must be noted that in their derivation of such expression they allowed an arbitrary dependence of the energy density, pressure and the four velocity of the fluid on both the time and radial coordinates. Therefore, the wormhole mass rate expression Eq. (\[once\]) must be valid in general for asymptotically flat wormholes. It must be emphasised that a wormhole is a non-vacuum solution and that the consideration of a time dependence in the shape function leads also to a non zero $\Theta_0^r$, which take into account the non-zero energy flow onto the hole, therefore taking into account the back reaction. Nevertheless, the authors of Ref. [@GonzalezDiaz:2007gt] considered that the most crucial argument against the big trip included in the paper of Faraoni [@Faraoni:2007kx] is the vanishing of the proper radial velocity at the big rip and its quickly decrease close to it, since it is in the point of introducing an explicitly equations of state for the fluid in order to integrate Eq. (\[once\]) where they were considering an approximation. First of all, the authors noted that, besides the fact that the time where the universe is engulfed by a wormhole is not only before the big rip but even before the divergence of the wormhole mouth, the important quantity which refers to an accretion process is the proper radial flow, which is approximately $\rho v\sim a^{-3(1+w)/2}$ which increases with time for $w<-1$ and diverges at the big rip, what guarantees the process would not be stopped. In the second place, they point out that the accretion of dark and phantom energy onto astronomical objects differs from the accretion of usual energy concentrated in a given region of space onto those objects, because in the first case the energy pervades the whole space being, therefore, a phenomenon not based on any fluid motion, but on increasing more and more space filled with such kind of energy inside the boundary of the considered object.
Later on, C. Gao, X. Chen, V. Faraoni and Y. G. Shen, [@Gao:2008jv], emphasising that the method of Babichev et al. applied to black holes is not taking into account the backreaction of the fluid on the background, claimed that the results obtained by using that method can only be valid in a low matter density background. In that spirit, they used a generalised McVittie metric and inserted a radial heat flux term in the energy-momentum tensor, in order to show that a cosmological black hole (non-asymptotically flat) should increase by the accretion of phantom energy. The authors also pointed out any difficulties to compare their solution with the corresponding of Babichev et al. [@Babichev:2004yx], arguing that this fact could be due to the simplifications taken in both cases.
The work of Gao et al. originated some interesting comments. First, in Ref. [@Zhang:2007yu], X. Zhang pointed out the shortcomings of the Babichev et al. method, in particular the use of a non-cosmological metric and the exclusion of the backreaction of the phantom fluid on the black hole metric. Nevertheless, Zhang found the results of Ref. [@Gao:2008jv] highly speculative, giving the example of the use of a hypothesised metric. Therefore, he decided to use for the moment the method of Babichev et al. in order to extract at least some tentative conclusions, lacking a complete method to the study of the dark energy accretion phenomenon. Second, in [@MartinMoruno:2008vy] where a first attempt to include cosmological effects in the study of the accretion of dark energy onto black holes was done by the consideration of an Schwarzschild-de Sitter spacetime, it was noted that the results achieved in [@Gao:2008jv] were obtained under the assumption of a premise (contradictory with all studies of this problem in the literature) in which their desired result is contained, making circular their whole argument, and their result invalid. Third, in [@Babichev:2008jb], Babichev et al. included some comments expressing their doubts regarding the conclusions of [@Gao:2008jv]. They claimed that the heat flux term is introduced in an unnatural way in the solution of Gao et al. to support their configuration, because the perfect fluid is not accreted in the mentioned solution and that such an introduction could may be lead to instabilities to small perturbations. They also pointed out that the temperature of the fluid blows up at the event horizon.
In summary, regarding the phenomenon of dark and phantom energy accretion onto black holes the method developed by Babichev et al. [@Babichev:2004yx] has been improved to take into account the backreaction on the black hole size [@MartinMoruno:2006mi] in an asymptotically flat space and also in a cosmological one [@MartinMoruno:2008vy] by using a non-static generalisation of the Schwarzschild and Schwarzschild-de Sitter metric, respectively. Nevertheless, the study still lacks the consideration of the complete backreaction of the dark or phantom fluid in an asymptotically dark or phantom universe.
Although the method used to study the dark and phantom energy accretion onto wormholes is similar to the one treating the above mentioned phenomenon, in the case of the phantom energy accretion onto wormholes the backreaction originated by the consideration of the phantom fluid is automatically taken into account by using a non-static generalisation of the Morris-Thorne metric [@GonzalezDiaz:2007gt]. In this case there are no studies, up to our knowledge, considering rigorously the wormhole accretion phenomenon in a cosmological spacetime.
We want however to point out, that the increase (decrease) of the black hole size by the accretion of dark (phantom) energy, and the contrary in the case of wormholes, seems considerably well supported. Such affirmation can be also understood taken into account a different method. It is well known that the formalism developed by Hayward for spherically symmetric spacetimes [@Hayward:2004fz] implies that a dynamical black hole (characterised by a future outer trapping horizon) would increase if it is considering in an environment fulfilling $p+\rho>0$ and decrease if $p+\rho<0$, phenomenon which is due to a flow of such surrounding material into the hole. Therefore, this totally independent study confirms the results presented in this chapter about black holes at least in a qualitative way. The question would be, how large are the quantitative differences which could appear by the consideration of the backreaction and the cosmological space?
On the other hand, regarding wormholes it has been shown [@MartinMoruno:2009iu] that in order to recover using the Hayward formalism the results obtained by the accretion method where the backreaction is included and by the very basis of wormhole physics, a wormhole must be characterised by a past outer trapping horizon. Since it seems that there would be no reason to change the local characterisation of an astronomical object because of the consideration of such an object in a space with a different asymptotically behaviour, the qualitative increase (decrease) of wormholes by the accretion of phantom (dark) energy should be recovered by considering cosmological wormholes spacetimes. Nevertheless, whether a wormhole would suffer a so huge increase to include the whole universe, occurring a big trip, is still an open question.
We want to point out that there are several interesting opened questions concerning accretion onto black holes. More improvement in the accretion theory is needed to take into account the backreaction of the space-time and study the situation where a perfect fluid approximation is not valid, what would clarify whether a black hole can become a naked singularity? A more detailed study about the spin or charge super-radiance is also needed, in order to show whether these processes could concur in such a way that finally the cosmic censorship conjecture would keep hold. Preliminary results indicate that this is the case, but the possibility of violation of the cosmic censorship conjecture produced by the process of dark energy accretion onto a charged or rotating black holes is still open.
Finally, it must be worth noticing that recent papers, [@MersiniHoughton:2008aw; @MersiniHoughton:2009dp], consider the Babichev et al. method to propose a new observational dark energy test. The main idea is based on the black hole mass change induced by the dark energy accretion process which, as we have shown in this chapter, is proportional to $1+w$. Although a direct observation of this change is beyond our current detection possibilities, because the time scale to produce a change in the black hole mass measurable with our present devices would be too long, it would cause observable modifications in the orbital radius of the supermassive black hole binaries, since the black hole binaries would either merge in a more accelerated way than expected if $1+w>0$ or the merging would be progressively stopped if the dominant energy condition is violated, being also possible that the binaries would rip apart in the second case[^10]. At this moment there are two candidates, Galaxy 0402+379 and Radio Galaxy OJ287, for the observation of the mentioned phenomenon what could provide us with more information about the nature of dark energy, and probably more interesting candidates can be expected in the future.
Conclusions and further comments.
=================================
In the present chapter we have shown that if the current accelerating expansion of the universe is explained in the framework of General Relativity, then the consideration of a dynamical dark energy fluid would produce other effects besides the modelling of that acceleration. In this sense, the consideration of dark energy would not be simply the consideration of an ether covering the whole space, since footprints of such a fluid could appear in our Universe by observing the evolution of black- and worm-holes. Whether such effects are measurable in practice, is a question related to the accuracy of the observational data.
The effects in question regarding the dynamical evolution of black- and worm-holes, would be an additional increase (decrease) of the black hole size in the case that the dark energy fulfils (violates) the dominant energy condition, and the contrary in the wormhole case. Even more, these effects would produce changes in the orbital radius of black hole binaries which could be large enough to be detected in practice, helping us to get new constrains to dark energy equation of state parameter.
If one considers the used test fluid approximation to hold in phantom models possessing a singularity at a finite time in the future, then the black holes would tend to disappear at that singularity, which would never be reached since a big trip phenomenon may take place before. On the other hand, in dark energy models with $w>-1$ black holes would not engulf the whole universe, since the current mass of a hole able to exceed the universe size in a finite time should be so huge that it would be bigger than the mass of the observable Universe.
Although, by the arguments presented in this chapter, the qualitative evolution of the considered astronomical objects by the accretion of dark energy seems to be a solid result, the quantitative dynamical behaviour could differ from the mentioned results, since at the final step we are considering the approximation that the surrounding fluid is a cosmological one. In the black hole case this approximation is even stronger, since the non-static generalisation of the Schwarzschild metric, though taking into account the radial flow, lacks of a consideration of the backreaction.
Finally, whether or not the above features studied in this chapter can be taken to imply that certain dark energy models are more consistent than others is a matter that will depend on both the intrinsic consistency of the different models and the current and future observational data.
**Acknowledgements.** We acknowledge La Casa de Aragón to provide us a relaxing and inspiring place where this chapter was designed and discussed. JAJM thanks the financial support provided by Fundación Ramón Areces and the DGICYT Research Project MTM2008-03754. P. M. M. gratefully acknowledges the financial support provided by the I3P framework of CSIC and the European Social Fund and by a Spanish MEC Research Project No.FIS2008-06332/FIS. Special thanks to Jodie Holdway for English revision of this chapter.
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[^1]: Electronic adresses:J.Madrid@damtp.cam.ac.uk, madrid@imaff.cfmac.csic.es
[^2]: Electronic address: pra@imaff.cfmac.csic.es
[^3]: For information about the historical development of wormholes and a deep study of their spacetime, please see Ref.[@Visser]
[^4]: It must be noted that such an approximation is justified because the contribution of the current dark energy density is around $74\%$ of the total energy density of the universe and, from the evolution of the energy density in terms of the scale factor, it is expected that dark energy density decays slower than the ordinary matter energy density when the scale factor increases, being therefore the future dynamic of the universe governed by the dark component. Even more, in the phantom case the phantom energy density would increase with the scale factor, so the same conclusions can be, of course, recovered in this case.
[^5]: It must be noted that if we want to express the phantom fluid by using a scalar field like in the quintessence dark energy case, then it will possess a negative kinetic term.
[^6]: The quantised $\alpha$ parameter in the type IV model eliminates possible past singularities that could appear in type I. A discussion about the evolution of these models at early times is out in the scope of the present chapter, so we refer the interested reader to Ref. [@BouhmadiLopez:2007qb].
[^7]: $B$ is the constant parameter that appears in the energy momentum tensor conservation law (\[ocho\]) for a GCG.
[^8]: It must be emphasised that, whereas in the case of the study of the dark energy accretion onto black hole phenomenon, the solution is not able to take into account the back reaction of the spacetime, in this case we are treating with a non-vacuum solution and allowing arbitrary time dependence on the involved functions and, therefore, up to now we are taking into account any possible back reaction.
[^9]: It is worth noticing that in models showing one big trip, as it is the considered case, the universe would travel through the time of the arrival universe being, in this case, not a proper time travel. On the other hand, in models which present more than one big trip phenomenon the consideration of a multiverse framework would be not more necessary, since the wormhole mouth at the moment that it is bigger than the universe could be connected to the other infinitely large wormhole mouth, travelling in this case the universe along its own time from future to past. The reader interested on this topic is advised to consult Ref. [@Yurov:2006we].
[^10]: The interested reader can look up in [@MersiniHoughton:2008aw; @MersiniHoughton:2009dp] for more details.
|
---
abstract: |
This work presents an implementation of the [*resistive MHD*]{} equations for a generic algebraic Ohm’s law which includes the effects of finite resistivity within full General Relativity. The implementation naturally accounts for magnetic-field-induced anisotropies and, by adopting a phenomenological current, is able to accurately describe electromagnetic fields in the star and in its magnetosphere. We illustrate the application of this approach in interesting systems with astrophysical implications; the aligned rotator solution and the collapse of a magnetized rotating neutron star to a black hole.\
author:
- |
Carlos Palenzuela${}^1$\
${}^1$Canadian Institute for Theoretical Astrophysics, Toronto, Ontario M5S 3H8, Canada\
title: Modeling magnetized neutron stars using resistive MHD
---
\[firstpage\]
MHD – plasmas – gravitation – methods: numerical
Introduction {#sec:intro}
============
Magnetic fields play an important role in the dynamics of many relativistic astrophysical systems such as pulsars, magnetars, gamma-ray burst (GRBs) and active galactic nuclei (AGNs). In many of these scenarios, the Ohmic diffusion timescales of the magnetized plasma is much longer than the characteristic dynamical timescale of the system, so one can formally take the limit of infinite electrical conductivity. This is regarded as the ideal MHD limit, and it is in general a good approximation to describe astrophysical plasmas. Furthermore, such a limit is described by a relatively manageable, but certainly involved hyperbolic system of equations without stiff terms which facilitates its computational implementation. The ideal MHD limit has been extensively used in the last years to study many of the previous systems (i.e., which basically consist of magnetized neutron stars and black hole accretion disks) in the fully non-linear regime.
In spite of its success and convenience, the ideal MHD approximation also has some limitations. At a purely theoretical level, the assumption of vanishing electrical resistivity prevents some important physical phenomena such as dissipation and reconnection of the magnetic field lines. Reconnection efficiently converts magnetic energy into heat and kinetical energy in very short timescales. This process is believed to be the mechanism originating many energetic emissions, such as in soft gamma-ray repeaters (which could be explained by giant magnetar flares), the Y-point of pulsar magnestosphere or even the short Gamma-Ray Bursts [@2011SSRv..160...45U]. In order to describe such processes, schemes going beyond the ideal MHD limit are required.
At the numerical level, all numerical schemes inherit some numerical resistivity which depends strongly on the resolution, making difficult to disentangle physical phenomena from numerical artifacts especially in highly demanding computational scenarios. The presence of magnetic fields demands relatively high resolution to accurately capture all the physical processes involved, many of them occurring at very small scales. This high resolution is particularly important in the case of instabilities which amplify the magnetic field, such as the Kelvin-Helmholtz instability occurring during the merger of binary neutron stars , and the Magneto-Rotational Instability (MRI) occurring in accretion disks [@1991ApJ...376..214B; @1991ApJ...376..223H; @1995ApJ...440..742H; @1998RvMP...70....1B]. Accurate modeling of the rarefied magnetospheres of compact objects similarly requires high resolution. The electromagnetic fields in this region may be easier to model by adopting a different limit of the MHD equations known as the [*force-free limit*]{} [@Goldreich:1969sb]. In this approximation the fluid inertia is neglected, implying that the fluid does not influence directly the dynamics of the electromagnetic fields.
One possibility to overcome these limitations is to consider instead the resistive MHD framework and solve the full Maxwell and hydrodynamic equations. The coupling between these two is provided by the current –by a suitable Ohm’s law–. With a convenient choice of current, including both induction and Ohmic terms, it is possible to recover both the ideal MHD limit, as well as the finite-resistivity scheme required to describe physical dissipation and reconnections. The effect of small-scales-dynamics can also be modeled with moderate resolutions by using a suitable current. Finally, magnetically dominated magnetospheres can be described by a phenomenological current that decouples the fluid from the force-free EM fields.
The numerical evolution of this resistive MHD code is not free of difficulties. The resistive MHD equations can be regarded as an hyperbolic system with relaxation terms that become stiff for some limits of the current. Consequently, numerical evolution of this system represents a numerical challenge, and several works have recently explored different possibilities to implement it [@Komissarov2007; @2009MNRAS.394.1727P; @2009JCoPh.228.6991D; @2010ApJ...716L.214Z; @2011ApJ...735..113T; @2012arXiv1205.2951B; @2012arXiv1208.3487D]. In this work we take a step further in the development of one of these approaches, based on the Implicit-Explicit (IMEX) Runge-Kutta. Our aim is to model both the interior and the exterior of a star with a phenomenological current based on physical arguments. This will be particularly interesting to study the electromagnetic emissions of astrophysical relativistic systems involving magnetized neutron stars.
The capabilities of our approach are tested by considering the force-free aligned rotator solution, a well studied problem in the context of pulsar magnetospheres . These works were restricted to flat spacetime and excluded the interior of the star from the computational domain, thus side-stepped the stiffness problem mentioned above. Recently a hybrid approach, matching both the ideal and force-free system of equations, revisited this problem within a framework capable of studying both star and surrounding magnetosphere within General Relativity [@2011arXiv1112.2622L]. However such a scheme still relies on two different approximations applied in two regions. The approach we present here finally allows for treating the system from a global point of view with a single, general relativistic, framework.
We also revisit another important astrophysical scenario with a much less understood dynamics; the collapse of a magnetized neutron star to a black hole. This system represents even a more challenging problem because of the strong gravity fields, and it has been studied numerically by considering different approximations. An early study matched an analytical solution for the star to an electrovacuum magnetosphere [@Baumgarte:2002b]. More recently, further realism was achieved by adopting the hybrid scheme that matched the numerical solution of the star to force-free magnetosphere [@2011arXiv1112.2622L]. A step towards studying this system within a common, resistive, framework was presented in [@2012arXiv1208.3487D], although the star’s exterior was treated as an electrovacuum magnetosphere. Our approach presented here is able to consistently study the star and its force-free magnetosphere within the general relativistic resistive MHD equations.
The paper is organized as follows. Section \[sec:equations\] summarizes the fully relativistic resistive MHD system, which is slightly different from the one adopted in [@2012arXiv1208.3487D]. In Section \[sec:coupling\_EM\_fluid\] it is discussed a generic family of algebraic Ohm’s law, and how to construct a phenomenological current to recover both the ideal MHD and the force-free limits. Section \[sec:hyperbolic\_relaxation\] summarizes briefly the IMEX Runge-Kutta methods and different techniques to solve generically the implicit step for any algebraic form of the relaxation terms. The application of these methods to the resistive MHD system is performed is section \[sec:rmhd\]. Section \[sec:simulations\] presents our numerical results for the aligned rotator and the collapse of a neutron star to a black hole. We conclude with some remarks.
Throughout this work we adopt geometric units such that $G=c=1$, and the convention where greek indices $\mu,\nu,\alpha,...$ denote spacetime components (ie, from $0$ to $3$), while roman indices $i,j,k,...$ denote spatial ones. Bold letters will represent vectors.
The evolution equations {#sec:equations}
=======================
This section summarizes the general relativistic resistive magnetohydrodynamic equations, that will allow us to model self-gravitating magnetized fluids. The evolution of the spacetime geometry is governed by Einstein equations. The electromagnetic fields and the fluid obey, respectively, the Maxwell and the General Relativistic Hydrodynamic equations. The closure of the system is given by two constitutive equations; the first one is the equation of state, which relates the pressure to the other fluid variables. The second one is Ohm’s law, –defining the coupling between the fluid and the electromagnetic fields– which will be described in the next section.
Einstein Equations {#subsec:BSSN}
------------------
The geometry of the spacetime can be obtained by solving the four-dimensional Einstein equations. These equations can be recast as a standard initial value problem by splitting explicitly the time and the space coordinates through a 3+1 decomposition, such that the line element can be expressed as $$\begin{aligned}
ds^2 &=& g_{\mu\nu}\, dx^{\mu} dx^{\nu} \nonumber \\
&=&-\alpha^2 \, dt^2
+ \gamma_{ij}\left(dx^i + \beta^i\, dt\right)
\left(dx^j + \beta^j\, dt\right),\end{aligned}$$ where $g_{\mu\nu}$ is the spacetime metric, $\gamma_{ij} = g_{ij}$ is the intrinsic metric of the spacelike hypersurfaces, and the lapse function $\alpha$ and the shift vector $\beta^i$ relates how the coordinates change between neighboring hypersurfaces. The normal to the hypersurfaces is given explicitly by $$n^{\mu} = \frac{1}{\alpha}(1 ,-\beta^i) ~~~,~~~
n_{\mu} = (-\alpha, 0) ~~.$$ Indices on spacetime quantities are raised and lowered with the 4-metric and its inverse, while the 3-metric and its inverse are used to raise and lower indices on spatial quantities.
The rate of change of the intrinsic curvature from one hypersurface to another is given by the extrinsic curvature $$K_{ij} = -\frac{1}{2 \alpha} (\partial_t - {\cal L}_{\beta}) \gamma_{ij}$$ where ${\cal L}_{\beta}$ is the Lie derivative along the vector $\beta^i$.
At any given time, the spacetime geometry is then fully defined by the $3+1$ variables $\{ \alpha,\beta^i,\gamma_{ij},K_{ij} \}$. We adopt the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of Einstein’s equations to evolve a suitable combination of these fields, in a form very close to the presented in [@2006PhRvL..96k1101C].
Maxwell equations {#subsec:maxwell}
-----------------
The electromagnetic fields follow Maxwell equations, that in their extended version can be written as [@2010PhRvD..81h4007P] $$\begin{aligned}
\nabla_{\mu} (F^{\mu \nu} + g^{\mu \nu} \psi) &=& - I^{\nu} + \kappa n^{\nu} \psi
\label{Maxwell_ext_eqs2a} \\
\nabla_{\mu} ({}^*F^{\mu \nu} + g^{\mu \nu} \phi) &=& \kappa n^{\nu} \phi~,
\label{Maxwell_ext_eqs2b} \end{aligned}$$ where $\{F^{\mu \nu}, ^*F^{\mu \nu}\}$ are the Maxwell and the Faraday tensors, $I^{\nu}$ is the electric current and $\{ \phi,\psi \}$ are scalars introduced to control dynamically the constraints by exponentially damping them in a characteristic time $1/\kappa$ [@2002JCoPh.175..645D]. When both the electric and magnetic susceptibility of the medium vanish, like in vacuum or in a highly ionized plasma, the Faraday tensor is simply the dual of the Maxwell one, $$\label{Fduals}
{}^*F^{\mu \nu} = \frac{1}{2}\, \epsilon^{\mu \nu \alpha \beta} \,
F_{\alpha\beta}~,\qquad
F^{\mu \nu} = - \frac{1}{2}\, \epsilon^{\mu \nu \alpha \beta}~{}^*F_{\alpha \beta}$$ where $\epsilon^{\mu \nu \alpha \beta}$ is the Levi-Civita pseudotensor of the spacetime, related to the 4-indices Levi-Civita symbol $\eta^{\mu \nu \alpha \beta}$ by $$\label{levicivita}
\epsilon^{\mu \nu \alpha \beta} = \frac{1}{\sqrt{g}}~
\eta^{\mu \nu \alpha \beta} \qquad
\epsilon_{\mu \nu \alpha \beta} = -\sqrt{g}~
\eta_{\mu \nu \alpha \beta}~.$$ In this case, both tensors can be decomposed in terms of the electric and magnetic fields, $$\begin{aligned}
F^{\mu \nu} &=& n^{\mu} E^{\nu} - n^{\nu} E^{\mu}
+ \epsilon^{\mu\nu\alpha\beta}~B_{\alpha}~n_{\beta}
\label{F_em1a} \\
{}^*F^{\mu \nu} &=& n^{\mu} B^{\nu} - n^{\nu} B^{\mu}
- \epsilon^{\mu\nu\alpha\beta}~E_{\alpha}~n_{\beta}
\label{F_em1b}\end{aligned}$$ such that $E^{\mu}$ and $B^{\mu}$ are the electric and magnetic fields measured by a normal observer $n^{\mu}$. Both fields are purely spatial, that is, $E^{\mu} n_{\mu} = B^{\mu} n_{\mu} = 0$.
The covariant Maxwell equations (\[Maxwell\_ext\_eqs2a\], \[Maxwell\_ext\_eqs2b\]) can be written, by performing the 3+1 decomposition, in term of the electromagnetic fields and the divergence-cleaning scalars [@2010PhRvD..81h4007P] as, $$\begin{aligned}
(\partial_t - {\cal L}_{\beta}) E^{i} &-& \epsilon^{ijk} \nabla_j (\alpha B_k)
+ \alpha \gamma^{ij} \nabla_j \psi
\\ &=& \alpha trK E^i - \alpha J^i
\nonumber
\label{maxwellext_3+1_eq1a} \\
(\partial_t - {\cal L}_{\beta}) \psi &+& \alpha \nabla_i E^i =
\alpha q -\alpha \kappa \psi
\label{maxwellext_3+1_eq1b} \\
(\partial_t - {\cal L}_{\beta}) B^{i} &+& \epsilon^{ijk} \nabla_j (\alpha E_k)
+ \alpha \gamma^{ij} \nabla_j \phi
\\ &=& \alpha trK B^i
\nonumber
\label{maxwellext_3+1_eq1c} \\
(\partial_t - {\cal L}_{\beta}) \phi &+& \alpha \nabla_i B^i =
-\alpha \kappa \phi ~~.
\label{maxwellext_3+1_eq1d}\end{aligned}$$ where $\epsilon^{ijk} \equiv \epsilon^{ijk\alpha} n_{\alpha} =
\eta^{ijk}/\sqrt{\gamma}$ is the three-dimensional Levi-Civita pseudotensor. Since $F^{\mu \nu}$ is antisymmetric, the four-divergence of equation (\[Maxwell\_ext\_eqs2a\]) leads to an additional equation for the current conservation of Maxwell solutions, $$\label{conserved_current}
\nabla_{\mu} I^{\,\mu} = 0 ~.$$ The electric current $I^{\nu}$ can be decomposed into components along and perpendicular to the vector $n^{\nu}$, $$\label{current_decomposition}
I^{\nu} = n^{\nu} q + J^{\nu}~~,$$ where $q$ and $J^{\nu}$ are the charge density and the current as observed by a normal observer $n^{\nu}$. Again, $J^{\nu}$ is purely spatial, so $J^{\nu} n_{\nu}=0$. The current conservation (\[conserved\_current\]) can be expressed, with the 3+1 decomposition, as $$\label{consJ_3+1}
(\partial_t - {\cal L}_{\beta}) q + \nabla_i (\alpha J^i) = \alpha trK q$$ Only a prescription for the spatial components $J^i$, which will determine the coupling between the EM fields and the fluid, is required to complete the system of Maxwell equations. This relation, commonly known as Ohm’s law, will be discussed in detail in section \[sec:coupling\_EM\_fluid\].
Hydrodynamic equations {#subsec:hydro}
----------------------
A perfect fluid minimally coupled to an electromagnetic field is described by the total stress-energy tensor $$\begin{aligned}
\label{stress-energy-perfectfluid}
T_{\mu \nu} &=& \left[ \rho (1 + \epsilon) + p \right]
u_{\mu} u_{\nu} + p g_{\mu \nu}
\nonumber \\
&+& {F_{\mu}}^{\lambda} F_{\nu \lambda}
- \frac{1}{4} g_{\mu \nu} ~ F^{\lambda \alpha} F_{\lambda \alpha} \end{aligned}$$ where a factor $1/\sqrt{4\pi}$ has been absorbed in the definition of the electromagnetic fields. Here $\rho$ is the rest mass density, $\epsilon$ the internal energy and $p$ is the pressure, given by a closure relation $p=p(\rho,\epsilon)$ commonly known as the equation of state (EoS). These fluid quantities are measured in the rest frame of the fluid element. However, to describe the system is usually more convenient to adopt an Eulerian perspective where coordinates are not tied to the flow of the fluid. The four-velocity $u^{\mu}$ describes how the fluid moves with respect to the Eulerian observers, and can be decomposed into space and time components, $$\label{velocity_decomposition}
u^{\mu} = W \left( n^{\mu} + v^{\mu} \right)$$ where $v^{\mu}$ corresponds to the familiar three-dimensional velocities as measured by Eulerian observers (i.e., $v^{\mu} n_{\mu} = 0$). The time component is defined by the normalization relation $u^{\mu} u_{\mu} = -1$, such that $$\label{velocity_decomposition2}
W = - n_{\mu} u^{\mu} = (1 - v_i v^i)^{-1/2} ~,~~
$$ where we can now recognize $W$ as the Lorentz factor.
In summary, the magnetized fluid is described by the physical fields (i.e., the fluid variables and the electromagnetic fields) plus the divergence cleaning scalars, which form the set of primitive variables $(\rho, \epsilon, p, v^i, E^i, B^i, q, \phi, \psi)$ The matter evolution must comply with the conservation of the total stress-energy tensor $$\label{Tconserv_eq}
\nabla_{\nu}T^{\mu\nu}= 0,$$ which can be expressed as a system of conservation laws for the energy density $U$ and the momentum density $S_i$, defined from the projections of the stress-energy tensor $$\label{Tmunu_projections}
U = n_{\mu} n_{\nu} T^{\mu \nu} ~~,~~
S_{i} = - n^{\mu} T_{\mu i} ~~,~~
S_{ij} = T_{ij} ~~.$$ In addition to the conservation of energy and momentum, the fluid usually also conserves the total number of particles, $$\label{baryon_conservation}
\nabla_{\mu} ( \rho u^{\mu} ) = 0$$ where $\rho u^{\mu}$ is the baryon number density. This equation is just the relativistic generalization of the conservation of mass.
As mentioned above, it is necessary to specify the EOS to define the pressure and complete the system of hydrodynamic equations. Along this paper we will consider either the polytropic EoS $p = K \rho^{\Gamma}$, which is a good approximation to describe cold stars, and the ideal gas EoS $p = (\Gamma -1) \rho \epsilon$, which allows for shock heating in the fluid.
Resistive MHD system {#subsec:finaleqs}
--------------------
The evolution of the electromagnetic fields follows the Maxwell equations and the conservation of charge, while the fluid fields are governed by the conservation of the total energy, momentum and baryonic number. In order to capture accurately the weak solutions of these non-linear equations in presence of shocks it is important to express them as a set of local conservation laws, namely $$\begin{aligned}
\label{maxwell11}
\partial_t (\sqrt{\gamma} B^i)
&+& \partial_k [\sqrt{\gamma} \left( -\beta^k B^i
+ \alpha ( \epsilon^{ikj} E_j + \gamma^{ik} \phi ) \right) ]
\\
&=&
- \sqrt{\gamma} B^k (\partial_k \beta^i)
+ \sqrt{\gamma} \phi \left( \gamma^{ij} \partial_j \alpha
- \alpha \gamma^{jk} \Gamma^i_{jk} \right)
\nonumber \\
\label{maxwell12}
\partial_t (\sqrt{\gamma} E^i)
&+& \partial_k [\sqrt{\gamma} \left( -\beta^k E^i
- \alpha ( \epsilon^{ikj} B_j - \gamma^{ik} \psi ) \right) ]
\\
&=&
- \sqrt{\gamma} E^k (\partial_k \beta^i)
+ \sqrt{\gamma} \psi \left( \gamma^{ij} \partial_j \alpha
- \alpha \gamma^{jk} \Gamma^i_{jk} \right)
\nonumber \\
&& - \alpha \sqrt{\gamma} J^i
\nonumber \\
\label{maxwell13}
\partial_t (\sqrt{\gamma} \phi) &+& \partial_k [\sqrt{\gamma}
(- \beta^k \phi + \alpha B^k)]
\\
&=&
\sqrt{\gamma}[ - \alpha\, \phi\, trK + B^k (\partial_k \alpha)
-\alpha \kappa \phi]
\nonumber \\
\label{maxwell14}
\partial_t (\sqrt{\gamma} \psi) &+& \partial_k
[\sqrt{\gamma} (- \beta^k \psi + \alpha E^k)]
\\
&=&
\sqrt{\gamma}[- \alpha\, \psi\, trK + E^k (\partial_k \alpha)
+ \alpha q -\alpha \kappa \psi]
\nonumber \\
\label{maxwell15}
\partial_t (\sqrt{\gamma} q)
&+& \partial_k [\sqrt{\gamma} (- \beta^k q + \alpha J^k)] = 0 \\
\label{fluid11}
\partial_t (\sqrt{\gamma} D )
&+& \partial_k [\sqrt{\gamma} (- \beta^k + \alpha v^k) D ] = 0 \\
\label{fluid12}
\partial_t (\sqrt{\gamma} \tau)
&+& \partial_k [\sqrt{\gamma} \left( - \beta^k \tau + \alpha (S^k - v^k D) \right)]
\\
&=& \sqrt{\gamma} [\alpha S^{ij} K_{ij} - S^j \partial_j \alpha] \\
\label{fluid13}
\partial_t (\sqrt{\gamma} S_i)
&+& \partial_k [\sqrt{\gamma} (- \beta^k S_i + \alpha {S^k}_i)]
\\
&=& \sqrt{\gamma} [\frac{\alpha}{2} S^{jk} {\partial_{i}} \gamma_{jk}
+ S_j \partial_i \beta^j - (\tau + D) \partial_i \alpha]
\nonumber\end{aligned}$$ where we have defined $$\begin{aligned}
\label{finalsets}
D &=& \rho W ~,~~~ \\
\tau &=& h W^2 - p + \frac{1}{2} (E^2 + B^2) - \rho W ~,~~~ \\
S_{i} &=& h W^2 v_{i} + \epsilon_{ijk} E^j B^k ~,~~~ \\
S_{ij} &=& h W^2 v_{i} v_{j} + \gamma_{ij} p \\
&& - E_i E_j - B_i B_j
+ \frac{1}{2} \gamma_{ij} (E^2 + B^2) ~,~~
\nonumber\end{aligned}$$ and the enthalpy $h \equiv \rho (1 + \epsilon) + p$. This form of the relativistic resistive MHD equations is basically the same presented already in [@2012arXiv1208.3487D]. Another similar formulation has also been derived recently [@2012arXiv1205.2951B]. Notice also that the energy conservation has been expressed in terms of the quantity $\tau \equiv U - D$ to recover the Newtonian limit of the energy density.
Coupling between the EM fields and the fluid {#sec:coupling_EM_fluid}
============================================
Maxwell and hydrodynamic equations are coupled by means of the current ${I}^{\mu}$, whose explicit form generically depends on the electromagnetic fields and the local fluid properties measured in the comoving frame. Consequently, it is convenient to introduce the electric and magnetic fields measured by an observer comoving with the fluid, namely $e^{\mu} \equiv F^{\mu \nu} u_{\nu}$ and $
b^{\mu} \equiv {}^*F^{\mu \nu} u_{\nu}$. Notice that, since $e^{\mu} u_{\mu} = b^{\mu} u_{\mu} = 0$, there are only three independent components. The Maxwell and Faraday tensors can therefore be expressed as $$\begin{aligned}
F^{\mu \nu} &=& u^{\mu} e^{\nu} - u^{\nu} e^{\mu}
+ \epsilon^{\mu\nu\alpha\beta}~b_{\alpha}~u_{\beta}
\label{F_em2a} \\
{}^*F^{\mu \nu} &=& u^{\mu} b^{\nu} - u^{\nu} b^{\mu}
- \epsilon^{\mu\nu\alpha\beta}~e_{\alpha}~u_{\beta}
\label{F_em2b}\end{aligned}$$ and the electric current can be decomposed into components along and transverse to $u^{\nu}$, $$\label{current_decomposition2}
I^{\mu} = u^{\mu} \tilde{q} + j^{\mu}~~,$$ where $j^{\mu} u_{\mu} = 0$ and $\tilde{q}$ is the charge density measured by the comoving observer. The relation with the Eulerian quantities (\[current\_decomposition\]) can be obtained from $$q = -n_{\mu} I^{\mu} = W \tilde{q} - n_{\mu} j^{\mu}~~.$$ Substituting these results into eq. (\[current\_decomposition2\]) and using the 3+1 decomposition of the four-velocity, one can write the spatial components of the current as $$\label{current_decomposition3}
I_{i} = J_i = (q + j^{\mu} n_{\mu}) v_i + j_i ~~.$$ Since the charge density follows directly from the current conservation (\[consJ\_3+1\]), the prescription for the three-dimensional electrical current $J_i$ is the only missing piece to completely determine Maxwell equations.
Generalized covariant Ohm’s law {#sec:covohmlaw}
-------------------------------
A standard prescription, known as the Ohm’s law, is to consider that the current is proportional to the Lorentz force acting on a charged particle, implying a linear relation between the current and the electric field in the comoving frame. A richer variety of physical phenomena may be described by including also additional terms proportional to the comoving magnetic field, leading to a generalized covariant Ohm’s law of the form, $$\label{ohm_law_covariant}
j^{\mu} = \sigma^{\mu \nu} e_{\nu} + \lambda\, b^{\mu}~,$$ being $\sigma^{\mu \nu}$ the electrical conductivity of the medium [@1978PhRvD..18.1809B] and $\lambda$ a parameter related to the covariant generalization of the mean-field dynamo [@2012arXiv1205.2951B].
The electrical conductivity can be calculated either in the collision-time approximation [@1978PhRvD..18.1809B] or in the framework of relativistic charged multifluids [@2012arXiv1204.2695A], leading to the same main results. The tensorial conductivity can be written as, $$\label{tensorial_conductivity}
\sigma^{\mu \nu} = \frac{\sigma}{1 + \xi^2 b^2} ( g^{\mu \nu}
+ \xi^2 b^{\mu} b^{\nu}
+ \xi \epsilon^{\mu \nu \alpha \beta} u_{\alpha} b_{\beta} ) ~~$$ where the coefficients are given by $$\label{tensorial_conductivity2}
\xi = 1/R = e \tau_r / m_e ~~,~~
\sigma = R / (n_e e) ~~.$$ Here $\tau_r$ is the collision or relaxation time, $n_e$ is the electron density and $e$ and $m_e$ are the electron’s charge and mass. In the framework described in [@2012arXiv1204.2695A], $R$ is introduced as a proportionality constant in the dissipative force between the two components of the fluid. It is easy to check that the first term of the conductivity (\[tensorial\_conductivity\]) leads to the well known isotropic scalar case, while the other two represent the anisotropies due to the presence of a magnetic field, corresponding to the Hall effect.
In order to compute the closure relation (\[current\_decomposition3\]) it is necessary to write the general relativistic Ohm’s law in terms of fields measured by an Eulerian observer. Let us first consider a simplified Ohm’s law neglecting both the dynamo effects and the last term in the tensorial conductivity (\[tensorial\_conductivity\]), $$\label{complete_tensorial_conductivity}
j_{\mu} = \frac{\sigma}{1 + \xi^2 b^2} [ e_{\mu}
+ \xi^2 (e_{\nu} b^{\nu}) b_{\mu} ]~~,$$ as it has also been used in [@2011MNRAS.418.1004Z]. It was pointed out that this current implies an incomplete Hall effect [@2012arXiv1204.2695A], but it will be enough for our later discussion. Within these assumptions, and using that the electric and magnetic fields in the fluid frame can be written as $$\begin{aligned}
\label{em_fluid_frame}
e^{\mu} &=& W n^{\mu} (E^{\nu} v_{\nu}) + W E^{\nu}
+ W \epsilon^{\mu \nu \alpha} v_{\nu} B_{\alpha}
\label{relation_eE} \\
b^{\mu} &=& W n^{\mu} (B^{\nu} v_{\nu}) + W B^{\nu}
- W \epsilon^{\mu \nu \alpha} v_{\nu} E_{\alpha} ~~,
\label{relation_bB}\end{aligned}$$ it is straightforward to obtain the contraction $$\begin{aligned}
\label{contracted_jn}
j_{\mu} n^{\mu} &=& \frac{\sigma}{1 + \xi^2 b^2} [ e_{\mu} n^{\mu}
+ \xi^2 (e_{\nu} b^{\nu}) b_{\mu} n^{\mu} ]
\\
&=& \frac{\sigma}{1 + \xi^2 b^2} [-W (E^k v_k)
- W \xi^2 (E^j B_k) (B^k v_k) ] ~~.
\nonumber\end{aligned}$$ The prescription for the spatial current (\[current\_decomposition3\]) can be now computed, leading to $$\label{ohm_relativistic_spatial}
J_i = q v_i + \frac{\sigma}{1 + \xi^2 b^2}
[{\cal E}_i + \xi^2 (E^k B_k) {\cal B}_i]$$ where we have introduced the shortcuts $$\begin{aligned}
\label{newE}
{\cal E}_i &=& W \left[ E_i + \epsilon_{i j k} v^{j} B^{k}
- (v_{k} E^{k}) v_{i} \right] ~,~~
\\
\label{newB}
{\cal B}_i &=& W \left[ B_i - \epsilon_{i j k} v^{j} E^{k}
- (v_{k} B^{k}) v_{i} \right] ~.\end{aligned}$$ It is important to recall that this current accounts not only for isotropic resistivity but also for some anisotropic effects induced by the magnetic fields.
In the regime of low magnetization (i.e., $p/B^2 \gg 1$) these anisotropic effects are expected to be small, implying $\xi \ll 1$. In this limit the third term in the current (\[ohm\_relativistic\_spatial\]) can be neglected, leading to the well-known isotropic Ohm’s law. The high conductivity of the fluid implies that, in order to get a finite current, the electric field measured by the comoving observers must vanish $$\label{ef_imhd}
e^{\mu} = 0 \longrightarrow E^i = - \epsilon^{ijk} v_j B_k \,.$$ This is the ideal-MHD condition, which states that the electric field is not an independent variable since it can be obtained via a simple algebraic relation from the velocity and the magnetic vector fields.
The anisotropic effects are expected to be important in magnetically dominated fluids (i.e., $p/B^2 \ll 1$). In this limit $\xi \gg 1$, and the second term in the current (\[ohm\_relativistic\_spatial\]) can be neglected. In highly conducting fluids a finite current is recovered only if the electric field is perpendicular to the magnetic field, $$\label{ef_iff}
e^{\mu} b_{\mu} = E^i B_i = 0 ~.$$ since the initial assumption of magnetically dominated fluid prevents the trivial solution $b^i=0$. In the next subsection it will be shown that this relation is one of the constraints of the force-free approximation.
The force-free limit {#subsec:currentforcefree}
--------------------
The magnetospheres of magnetized neutron stars [@Goldreich:1969sb] and black holes immersed in externally sourced magnetic fields [@Blandford1977] are filled with a low-density plasma so rarefied that even moderate magnetic fields stresses can easily dominate over the pressure gradients. In this regime, the main contribution to the stress-energy tensor comes from the electromagnetic part, $T_{\mu \nu} \approx T_{\mu \nu}^{em}$. Allowing by Maxwell equations, the total conservation of energy and momentum can be written as $$\label{div-stress-fluidem2}
0 = \nabla_{\nu} T^{\mu \nu} \approx - F^{\mu \nu} I_{\nu} ~~.$$ The vanishing of the Lorentz force $F^{\mu \nu} I_{\nu}$ leads to an approximation known as force-free limit, which is valid only for magnetically dominated plasmas with negligible inertia. The spatial components of the force-free condition (\[div-stress-fluidem2\]), after performing the 3+1 decomposition, are $$\label{forcefree2}
q E^i + \epsilon^{ijk} J_j B_k = 0 ~~$$ or, after some simple manipulations, $$\label{forcefree3}
J^i = q\, v^i_{d} + (J^k B_k) \frac{B^i}{B^2} ~~~,~~~
E^i B_i = 0 ~~,$$ where we have defined $v^i_{d} \equiv \epsilon^{ijk} E_j B_k/{B^2}$ as the drift velocity. Several options have been proposed to compute the term $J^k B_k$, which is crucial to provide a completely explicit relation for the current. For instance, a closed formed for the current can be calculated by enforcing the constraint $\partial_t (E^i B_i) = 0$ [@2007ApJ...667L..69G]. Another option is to evolve Maxwell equations by considering only the drift term of the current (\[forcefree3\]), and correct the electric field after each timestep to satisfy the other force-free condition $E^i B_i = 0$ [@2004MNRAS.350..427K; @2006ApJ...648L..51S]. This approximation has been used successfully to study numerically pulsar magnetospheres [@2006ApJ...648L..51S] and jets emerging from black holes with an externally sourced magnetic field [@2010Sci...329..927P; @2010PhRvD..82d4045P; @2011PNAS..10812641N].
The force-free limit can also be achieved by considering an effective anisotropic conductivity with a generic form given by [@2004MNRAS.350..427K; @2012ApJ...749L..32M; @2012ApJ...754...36A] $$\label{forcefree4}
J^i = q\, v^i_{d} + \frac{\sigma_{\parallel}}{B^2}
\left[ (E^k B_k) B^i + \chi (E^2-B^2) E^i \right] ~~,~~~$$ where $\sigma_{\parallel}$ is the (anisotropic) conductivity along the magnetic field lines. The additional term proportional to $E^2-B^2$ is introduced in order to enforce the physical constraint $|E|>|B|$. The remarkably close resemblance between the covariant current (\[ohm\_relativistic\_spatial\]) and the force-free one (\[forcefree4\]) suggests that both of them could lead to the same solutions for some limit of the conductivities. However, the force-free current (\[forcefree4\]) attains a particularly interesting feature; due mainly to the assumption of negligible fluid inertia, it does not depend on the fluid fields. This means that the EM fields are decoupled to the fluid variables, an advantage that could be used to model accurately the EM fields in regions where the fluid description is not accurate.
A current for the ideal MHD and the force-free limits {#subsec:current_ideal_ff}
-----------------------------------------------------
The numerical evolution of the ideal MHD equations typically fails in low density regions with high magnetization unless sufficient resolution is available, a situation that arises commonly in the magnetospheres. A standard practice to avoid these failures is to maintain a density floor (i.e., the so called [*atmosphere*]{}) in regions of low density to exploit advanced numerical techniques for relativistic hydrodynamics. The density in the atmosphere is much smaller than that inside the star, so this approach does not affect the star’s dynamics. However, in the magnetosphere the fluid inertia (and pressure) is typically much smaller than that of the electromagnetic field and one generally encounters numerical difficulties. These problems are mitigated by increasing the density in the atmosphere, effectively decreasing the magnetization in the exterior of the star. Although these modifications produce an unphysical modeling of the plasma in the magnetosphere, one could still solve correctly Maxwell equations by using a suitable current that decouples the electromagnetic fields from the fluid variables.
As explained earlier, the covariant current (\[ohm\_relativistic\_spatial\]) reduces to the ideal MHD limit for high isotropic conductivities (i.e., $\sigma \rightarrow \infty$ and $\xi \rightarrow 0$), while that the force-free constraint $E^i B_i = 0$ is enforced for large anisotropic conductivities (i.e., $\sigma,\xi \rightarrow \infty$). This suggests that the solutions for the EM fields in both limits can be achieved just by changing the anisotropic conductivity, independently on the plasma magnetization. Although Ohm’s law (\[ohm\_relativistic\_spatial\]) is quite general, it still couples the EM fields to the velocity. In addition, the parameter $\xi$ is not appropriate to model the fast decay of the magnetic field with the distance to the source. To overcome these difficulties, and in part motivated by the strategy introduced in [@2011arXiv1112.2622L], we introduce the following phenomenological current to include both the ideal MHD and the force-free limits, $$\begin{aligned}
\label{ohm_ideal_forcefree}
J^i &=& q [ (1-H)\, v^i + H\, v^i_{d} ]
\\
&+& \frac{\sigma}{1 + \zeta^2 }
\left[{\cal E}^i
+ \frac{\zeta^2}{B^2} \{ (E^k B_k) B^i + \chi (E^2-B^2) E^i\} \right]~,
\nonumber\end{aligned}$$ where $H$ is a function which vanishes whereas the ideal MHD limit is valid, and tends to $1$ whereas the force-free limit is more appropriate. The anisotropic ratio $\zeta$, which can be reinterpreted from the definition $\xi^2 b^2 \equiv \zeta^2$, can be conveniently set to be a constant in the region where the force-free limit is valid. The physical condition $B^2-E^2>0$ is enforced through a new current term proportional to an anomalous conductivity $\chi$, which only appears whenever $B^2<E^2$. Overdamping of the electric field is avoided by setting this anomalous conductivity to the characteristic decay time $\chi \approx (\alpha \sqrt{\gamma} \sigma \Delta t )^{-1}$, which can be estimated from the time evolution of $B^2-E^2$.
Let us consider the particular astrophysical scenario of magnetized neutron stars. The large fluid conductivity, both inside and outside the star, is modeled by using a large constant $\sigma \approx 10^5$. The anisotropic ratio, which defines the regions described either with the ideal MHD or the force-free limits, is defined as $\zeta = H \sigma$. This choice ensures that the interior of the star (i.e., $H=0$) is dominated by a large isotropic conductivity, reducing the system of equations to the ideal MHD limit. The exterior of the star (i.e., $H=1$) is dominated by the anisotropic terms which enforce the force-free condition. The kernel function $H$ is defined such that vanishes inside the star and its value becomes unity outside. A smooth transition between the inner and the outer region is achieved by using $$\label{eq:kernel}
H(\rho \,,\rho_o)= \frac{2}{1 + e^{2\,K\,(\rho - \rho_o)}}$$ We typically adopt $K \approx 0.001/\rho_{atm}$ and $\rho_o \approx 50-400\, \rho_{atm}$, being $\rho_{atm}$ the value for the density of the magnetosphere.
Hyperbolic systems with relaxation terms {#sec:hyperbolic_relaxation}
========================================
The general system of relativistic resistive MHD equations (\[maxwell11\]-\[fluid13\],\[ohm\_ideal\_forcefree\]) brings about a delicate issue when the conductivity in the plasma undergoes very large spatial variations. In regions with high conductivity, in fact, the system will evolve on timescales which are very different from those in the low-conductivity region. Mathematically, therefore, the problem can be regarded as a hyperbolic system with relaxation terms which requires special care to capture the dynamics in a stable and accurate manner. The prototype of these systems can be written as $$\label{stiff_equation}
\partial_t {\bf U} = F({\bf U}) + \frac{1}{\epsilon} R({\bf U})$$ where $\epsilon >0$ is the relaxation time. In the limit $\epsilon \rightarrow \infty$ the system is hyperbolic with spectral radius $c_h$ (i.e., the absolute value of the maximum eigenvalue). In the other limit $\epsilon \rightarrow 0$ the system is clearly stiff since the time scale of the relaxation (or stiff term) $R({\bf U})$ is much smaller than the maximum speed $c_h$ of the hyperbolic part $F({\bf U})$.
In the stiff limit ($\epsilon \rightarrow 0$) the stability of an explicit time evolution scheme is only achieved with a time step size $\Delta t \leq \epsilon$, a much stronger restriction than the CFL condition $\Delta t \leq \Delta x / c_h $ of the hyperbolic systems. The development of stable and efficient numerical schemes to overcome this restrictive constraint is challenging, since in many applications the relaxation time can vary many orders of magnitude.
Different alternatives to deal with the inherent stiffness of the relativistic resistive MHD equations has been proposed in the last decade; combination of splitting methods and analytical solutions [@Komissarov2007; @2010ApJ...716L.214Z; @2011ApJ...735..113T], discontinuous Galerkin methods [@2011MNRAS.418.1004Z; @2009JCoPh.228.6991D] and Implicit-Explicit (IMEX) Runge-Kutta methods [@2009MNRAS.394.1727P; @2012arXiv1205.2951B; @2012arXiv1208.3487D]. The following subsections summarize the IMEX Runge-Kutta schemes, a family of time integrators which are able to deal with the potentially stiffness issues and are relatively easy to incorporate into an existing relativistic ideal MHD code.
Implicit-Explicit Runge-Kutta methods {#subsec:imex}
-------------------------------------
An efficient way to solve the hyperbolic-relaxation systems is based on the IMEX Runge-Kutta methods. Within this scheme, all the fields are evolved by using a standard explicit time integration except the potentially stiff terms, which are evolved with an implicit time discretization. For the generic system (\[stiff\_equation\]) this scheme takes the form [@ParRus:2005] $$\begin{aligned}
\label{IMEX}
{\bf U}^{(i)} = {\bf U}^n &+& \Delta t \sum_{j=1}^{i-1} {\tilde{a}}_{ij} F({\bf U}^{(j)})
\nonumber \\
&+& \Delta t \sum_{j=1}^{\nu} a_{ij} \frac{1}{\epsilon} R({\bf U}^{(j)}) \\
{\bf U}^{n+1} = {\bf U}^n &+& \Delta t \sum_{i=1}^{\nu} {\tilde{\omega}}_{i} F({\bf U}^{(i)})
+ \Delta t \sum_{i=1}^{\nu} \omega_{i} \frac{1}{\epsilon} R({\bf U}^{(i)})
\nonumber\end{aligned}$$ where ${\bf U}^{(i)}$ are the auxiliary intermediate values of the Runge-Kutta. The coefficients can be represented as $\nu \times \nu$ matrices $\tilde{A}= (\tilde{a}_{ij})$ and $A= (a_{ij})$ such that the resulting scheme is explicit in $F$ (i.e.,$\tilde{a}_{ij}=0$ for $j \ge i$) and implicit in $R$. An IMEX Runge-Kutta is characterized by these two matrices and the coefficient vectors $\tilde{\omega}_i$ and $\omega_i$. Notice that at each substep the auxiliary intermediate values ${\bf U}^{(i)}$ involves solving an implicit equation. Since the simplicity and efficiency of solving the implicit part at each step is of great importance, it is natural to consider diagonally implicit Runge-Kutta (DIRK) schemes ($a_{ij}=0$ for $j > i$) for the stiff terms. A deeper discussion on the IMEX schemes and the detailed form of the schemes considered here are presented in appendix \[appendixA\].
Solving generic systems with IMEX schemes {#subsec:applying}
-----------------------------------------
The vector of evolved fields $\bf{U}$ can be split in two sets of variables $(\bf{V},\bf{W})$, depending on whether or not they contain any relaxation term in their evolution equations. The evolution system can then be generically written as $$\begin{aligned}
\label{split}
\partial_t {\bf W} &=& F_W({\bf V},{\bf W}) \\
\partial_t {\bf V} &=& F_V({\bf V},{\bf W})
+ \frac{1}{\epsilon} R_V({\bf V},{\bf W}) ~,\end{aligned}$$ where we have considered that the relaxation parameter $\epsilon$ can be any function not depending directly on the present value of the ${\bf V}$-fields. The procedure to compute each auxiliary step ${\bf U}^{(i)}$ can be split in two stages:
1. compute first the intermediate values $\{\bf{V^*},\bf{W^*}\}$ which involves information from previous steps, $$\begin{aligned}
\label{first_step}
{\bf W}^{*} = {\bf W}^n &+& \Delta t~ \sum_{j=1}^{i-1}~ {\tilde{a}}_{ij} F_W({\bf U}^{(j)})
\nonumber \\
{\bf V}^{*} = {\bf V}^n &+& \Delta t~ \sum_{j=1}^{i-1}~ {\tilde{a}}_{ij} F_V({\bf U}^{(j)})
\nonumber \\
&+& \Delta t~ \sum_{j=1}^{i-1}~ a_{ij} \frac{1}{\epsilon^{(j)}} R_V({\bf U}^{(j)}) ~~.
\end{aligned}$$
2. include the relaxation term at the present time by solving the implicit equation $$\begin{aligned}
\label{second_step}
{\bf W}^{(i)} &=& {\bf W}^{*}
\nonumber \\
{\bf V}^{(i)} &=& {\bf V^*}
+ a_{ii}~\frac{\Delta t}{\epsilon^{(i)}}~R_V({\bf V}^{(i)},{\bf W}^{(i)})
\end{aligned}$$ which clearly involves only the ${\bf V}$-fields.
The complexity of inverting this implicit equation depends on the particular form of the relaxation terms. From now on we will restrict ourselves to the algebraic case $R_V({\bf U})=f({\bf U})$. Next it is described two different ways to solve this implicit equation; the first one can only be applied when $R_V({\bf U})$ is a linear function, whereas the second one allows $R_V({\bf U})$ to have any non-linear dependence.
### $R_V$ depending linearly on ${\bf V}$ {#subsubsec:linearR}
The simplest case, however enough to cover a broad range of interesting situations, is to consider a linear relaxation term $$\label{stiff_part}
R_V({\bf V},{\bf W}) = A({\bf W}) {\bf V} + S(\bf{W}) ~.$$ The implicit equation (\[second\_step\]) can then be trivially solved $$\begin{aligned}
\label{invert_matrix}
{\bf V}^{(i)} &=& M \left[ {\bf V^*} + a_{ii}~\frac{\Delta t}{\epsilon^{(i)}}~S({\bf W^{(i)}}) \right]
\nonumber \\
M &=& [I - a_{ii}~\frac{\Delta t} {\epsilon^{(i)}} A({\bf W^{(i)}})]^{-1} ~.\end{aligned}$$ The matrix inversion can be performed analytically and written in a compact form for most of the interesting cases, so that the implicit step can be solved in a completely explicit way.
### $R_V$ depending non-linearly on ${\bf V}$ {#subsubsec:nonlinearR}
In the more general case –with an arbitrary non-linear dependence– it is usually not feasible to solve analytically the implicit step, requiring some approximation to find the solution. A convenient approach to solve this problem is to linearize the stiff term around an approximate solution $\{ {\bf \bar{V}} ,{\bf W}^{(i)}\}$, such that $$\begin{aligned}
\label{linearization2}
R_V({\bf V}^{(i)},{\bf W}^{(i)}) &\approx& R_V({\bf \bar{V}},{\bf W}^{(i)})
\\
&+& \left( \frac{\partial R_V}{\partial {\bf V}} \right)_{{\bf \bar{V}},{\bf W}^{(i)}}
({\bf V}^{(i)} - {\bf \bar{V}})~.
\nonumber\end{aligned}$$ Notice that we are linearizing around the solution ${\bf W}^{(i)}$, which is already known at the beginning of the implicit step.
By defining $A \equiv \left( \frac{\partial R_V}{\partial {\bf V}}
\right)_{{\bf \bar{V}},{\bf W}^{(i)}}$, and substituting the previous expansion (\[linearization2\]) in (\[second\_step\]), it is obtained $$\label{second_step_linear}
{\bf V}^{(i)} = {\bf V^*} + a_{ii}~\frac{\Delta t}{\epsilon^{(i)}}
[ R_V({\bf \bar{V}}) + A ({\bf V}^{(i)} - {\bf \bar{V}}) ]$$ This implicit equation can be written, after some manipulations, in the following way $$\begin{aligned}
\label{invert_matrix_linearized_ final}
{\bf V}^{(i)} &=& {\bf \bar{V}} + M [ {\bf V}^* - {\bf \bar{V}} +
a_{ii}~\frac{\Delta t} {\epsilon^{(i)}} ~R_V({\bf \bar{V}},{\bf W}^{(i)}) ]
\nonumber \\
M &\equiv& [I - a_{ii}~\frac{\Delta t} {\epsilon^{(i)}} A({\bf \bar{V}},{\bf W}^{(i)})]^{-1}\end{aligned}$$ The final expression (\[invert\_matrix\_linearized\_ final\]) can be solved through a Newton-Raphson iterative procedure such that, at each iteration $m$, uses an initial guess ${\bf \bar{V}}={\bf V}^{(i)}_{(m-1)}$ to find the next approximate solution ${\bf V}^{(i)}_{(m)}$.
Numerical evolution of the Resistive MagnetoHydroDynamics system {#sec:rmhd}
================================================================
We adopt finite difference techniques on a regular Cartesian grid to solve the problems of interest. To ensure sufficient resolution is achieved in an efficient manner we employ adaptive mesh refinement (AMR) via the HAD computational infrastructure [^1] that provides distributed, Berger-Oliger style AMR [@Liebling] with full sub-cycling in time, together with an improved treatment of artificial boundaries [@Lehner:2005vc]. The refinement regions are determined using truncation error estimation provided by a shadow hierarchy [@Pretoriusphd] which adapts dynamically to ensure the estimated error is bounded within a pre-specified tolerance. The spatial discretization of the geometry is performed using a fourth order accurate scheme, while that High Resolution Shock Capturing methods based on the HLLE flux formula with PPM reconstruction are used to discretize the resistive MHD variables [@Anderson:2006ay; @Anderson:2007kz]. The time-evolution is performed through the method of lines using a third order accurate Implicit-Explicit Runge-Kutta integration scheme described in the previous section. We adopt a Courant parameter of $\lambda = 0.25$ so that $\Delta t_l = 0.25 \Delta x_l$ on each refinement level $l$. On each level, one therefore ensures that the Courant-Friedrichs-Levy (CFL) condition dictated by the principal part of the equations is satisfied.
Evolution of the electric field {#subsec:dtE}
-------------------------------
The relaxation terms of the resistive MHD system are associated to the current, which mainly appears in the time evolution equation of the electric field. The evolved fields can then be split into and non-stiff ${\bf W} = \{ D,\tau, S_i, B^i, \psi,\phi, q \}$ and potentially stiff ${\bf V}= \{ E^i\}$. The evolution of the non-stiff fields is performed by the explicit part of the IMEX Runge-Kutta, and it is very similar to a standard implementation of the ideal MHD equations. The evolution of the electric field contains in addition the relaxation terms, namely $$\begin{aligned}
\label{split_stiff_part}
\partial_t (\sqrt{\gamma} {\bf E}) &=& F_E + (\sqrt{\gamma} R_E) ~~.
\\
F_E &=& - \partial_k [\sqrt{\gamma} \left( -\beta^k E^i
- \alpha ( \epsilon^{ikj} B_j - \gamma^{ik} \psi ) \right) ] ~,
\nonumber \\
& & - \sqrt{\gamma} E^k (\partial_k \beta^i)
+ \sqrt{\gamma} \psi \left( \gamma^{ij} \partial_j \alpha
- \alpha \gamma^{jk} \Gamma^i_{jk} \right)
\nonumber \\
&& - \alpha \sqrt{\gamma} J^i_{e} ~,
\nonumber\\
R_E &=& -\alpha J^i_{s} ~.
\nonumber\end{aligned}$$ where the factor $1/\epsilon$, corresponding to the fluid conductivity, is absorbed in the definition of $R_E$. The current has been split into a potentially stiff part, $J^i_{s}$, and the terms which can be treated explicitly, $J^i_{e}$. For the phenomenological Ohm’s law (\[ohm\_ideal\_forcefree\]) these components can be written explicitly as $$\begin{aligned}
\label{ohm_ideal_forcefree_splitting}
J^i_{e} &=& q [ (1-H)\, v^i + H\, v^i_{d} ] ~,
\\
J^i_{s} &=& \frac{\sigma}{1 + \zeta^2 }
\left[{\cal E}^i
+ \frac{\zeta^2}{B^2} \{ (E^k B_k) B^i + \chi (E^2-B^2) E^i\} \right]~.
\nonumber\end{aligned}$$ Notice that, although the evolution of $q$ is driven by the current, these terms do not become potentially stiff in this equation since they are not proportional to the field itself. However, the delicate balance between the different fields in the current, which allows to get finite values even for very high conductivities, may be broken during the reconstruction of the fields at the interfaces. These unacceptable large errors are prevented in the standard implementations of the force-free equations by computing the charge density from the constraint $q=\nabla_i E^i$ instead of using the charge conservation. The resulting set of equations is still hyperbolic, since the charge density only couples to the EM fields throughout the non-principal term $q v^i$ [@2011CQGra..28m4007P]. Here we prefer to keep the charge density as an evolution field and treat all the fields in the same manner. The errors at the interfaces are avoided by performing directly the reconstruction of the current $J^i$, which is computed just after solving the stiff terms. This ensures that the fluxes of $q$ will remain bounded between the values given by well-defined neighboring points.
Inversion from conserved to primitive variables {#subsec:inversion}
-----------------------------------------------
The numerical evolution of the resistive MHD system (\[maxwell11\]-\[fluid13\]) involves the recovery, after each timestep, of the primitive fields $\{ \rho,~ \epsilon,p,v^i,E^i,B^i,\psi,\phi,q\}$ from the conserved or evolved fields $\sqrt{\gamma} \{ D,\tau, S_i, E^i, B^i, \psi,\phi,q \}$. Although the conserved fields are just algebraic relations of the primitive ones, the opposite is not true; due to the enthalpy and the Lorentz factor these quantities are related by complicated equations that can only be solved numerically, except for particularly simple equations of state.
The solution at time $t=(n+1)\Delta t$ is directly obtained, for most of the conserved quantities, by evolving their (non-stiff) evolution equations. However, the explicit evolution of the potentially stiff fields only provides a partial solution. As explained in the previous section, a complete solution for the electric field involves taking into account the relaxation terms by solving the corresponding implicit equation. For a generic Ohm’s law, these relaxation terms will depend on the velocity and other primitive fields. Nevertheless, the recovery of the primitive variables from the conserved ones involves all the fields, including the electric field. This is a consistency constraint which implies that the recovery process and the implicit step evolution must be solved [*at the same time*]{}. We will next describe an iterative procedure to evolve the stiff part and recover the primitive fields for the phenomenological current (\[ohm\_ideal\_forcefree\]), as described in subsection \[subsubsec:nonlinearR\].
1. To start the iterative process it is required an approximate solution –initial guess– for the electric field ${\bar E}_i$ and the fluid unknowns of the system, that we have chosen to be the single combination $x \equiv h W^2$. The initial guess for this unknown is given simply by the previous time step ${\bar x} = x^{(n)}$. Possible choices for the electric field initial guess are:
- the previous time step ${\bar E}_i = E^{(n)}_i$
- the ideal MHD limit ${\bar E}_i = - \epsilon_{ijk} v^j B^k$, which involves performing first the recovery in the ideal MHD case (see appendix B for details).
- the approximate solution given by the explicit and previous implicit step evolutions ${\bar E}_i = E^*_i$.
- the trivial case ${\bar E}_i = 0$.
It may be difficult to estimate a priori which initial guess is more convenient. For this reason, our scheme starts with the first option and, if no solution is found, tries sequentially the other choices.
2. Subtract the electromagnetic contributions from the energy and momentum densities, $$\begin{aligned}
\label{con2prim_1}
{\tilde \tau} &=& \tau - \frac{1}{2} (E^k E_k + B^k B_k) ~,~~~ \\
{\tilde S}_{i} &=& S_{i} - \epsilon_{ijk} E^j B^k
\end{aligned}$$ such that the Lorentz factor can be computed as, $$\label{con2prim_2}
W^2 = \frac{x^2}{x^2 - {\tilde S}^i {\tilde S}_i} ~~~,~~
c \equiv \frac{1}{W^2} = 1 - \frac{{\tilde S}^2}{x^2}$$
3. Write also the pressure as a function of the conserved variables and the unknown $x$. For the ideal gas EOS $p = (\Gamma -1) \rho \epsilon$ this relation is just $$\label{c2p_pressure}
p = \frac{\Gamma-1}{\Gamma}
\left( \frac{x}{W^2} - \frac{D}{W} \right)$$
4. Obtain an equation $f(x)=0$, written in terms of the unknown $x$ and the conserved fields, such that it is satisfied only for true solutions of $x$. By using the previous expression (\[c2p\_pressure\]) in the definition of ${\tilde \tau}$, we can write $$\label{fx_hybrid}
f(x) = [ 1 - \frac{(\Gamma - 1)}{W^2 \Gamma}] x
+[ \frac{\Gamma-1}{\Gamma W} -1] D
- {\tilde \tau} \,,$$ where $W$ is computed through eq.(\[con2prim\_2\]). The equation $f(x)=0$ can be solved numerically by using an iterative Newton-Raphson solver. The solution in the iteration $m+1$ can be computed as $$\label{newton-raphson}
x_{(m+1)} = x_{(m)} - \frac{f(x_{(m)})}{f'(x_{(m)})} \,.$$ The derivative of the function $f(x)$ can be computed analytically, $$\begin{aligned}
\label{derivativef}
f'(x) &=& 1 - \frac{2 (\Gamma -1) {\tilde S}^2}{\Gamma x^2}
\nonumber \\
&-& \frac{(\Gamma -1) c}{\Gamma}
+ \frac{(\Gamma - 1) D {\tilde S}^2}{\sqrt{c} \Gamma x^3}
\end{aligned}$$
5. Update the primitive fields by using the relations $$\begin{aligned}
\label{con2prim_3}
v_i &=& \frac{{\tilde S}_i}{x} ~~,~~
W^2 = \frac{x^2}{x^2 - {\tilde S}^2} ~~,~~
h = \frac{x}{W^2} ~~~,~~~
\nonumber \\
p &=& \frac{\Gamma-1}{\Gamma} (h - \rho) ~~~,~~~
\rho = \frac{D}{W} ~~~.
\end{aligned}$$
6. Update the electric field –with the updated values of the primitive fields– by solving the implicit equation, corresponding to eq. (\[second\_step\]),
$$E^i = E_{*}^i + a_{ii}~ {\Delta t} ~R_E^i ~~,~~$$
which can be formally solved with the method described in subsection \[subsubsec:nonlinearR\] for ${\bf V}^{(i)}=E^i$, that is, $$\begin{aligned}
\label{invert_matrix_linearized5}
E^i &=& {\bar E}^i
+ M [ {E^*}^i - {\bar E}^i + a_{ii}~ {\Delta t}~R^i_E ]
\\
M &=& [I - a_{ii} \Delta_t A]^{-1}~~,~~
A = \frac{\partial R_E^i}{\partial E^j} ~~.
\end{aligned}$$
For the phenomenological Ohm’s law (\[ohm\_ideal\_forcefree\]), the matrix $M$ to be inverted is $$\begin{aligned}
\label{invert_matrix_linearized6}
&M^{-1}& = \delta^i_j + {\tilde \sigma} \biggl[
W (\delta^i_j - v^i v_j)
\\
&+& \frac{\zeta^2}{B^2} \left.\{ B^i B_j
+ \chi [2 E^i E_j + \delta^i_j (E^2 - B^2)] \right.\}
\biggr]
\nonumber
\end{aligned}$$ with ${\tilde \sigma}
\equiv a_{ii}\, {\Delta t}\, \alpha\, \sigma/(1 + \zeta^2)$.
7. Iterate until the solution $\{x,E^i\}$ satisfies their constitutive equations $f(x),f(E^i) \le 10^{-10}$, being $f(E^i)$ defined by equation (\[invert\_matrix\_linearized5\]).
In occasions the recovery procedure is unable to find a physical state for a given set of conserved variables. In such cases, which usually occur near a star’s surface, failures can be avoided by assuming that the fluid is isentropic in that timestep and therefore satisfying a polytropic EoS $p=K \rho^{\Gamma}$. Since the internal energy is also a function of the density (i.e., $\rho \epsilon = p/(\Gamma-1)$) for isentropic processes, the conserved quantities are overdetermined and the energy equation can be neglected in the recovery procedure, leading to a more robust algorithm.
Notice also that, although our discussion was focused on the phenomenological the Ohm’s law (\[ohm\_ideal\_forcefree\]), the method described in subsection \[subsubsec:nonlinearR\] can be applied to any algebraic form of the current. Even more general cases with derivative terms can be considered, with the condition that those must be evaluated at earlier times. In a similar way, the method for linear relaxation terms described in subsection \[subsubsec:linearR\] can be generically used for non-linear algebraic currents with the condition that the non-linear terms are evaluated at previous time steps, as it was considered in [@2012ApJ...754...36A]. This option does not require an initial guess for the electric field and therefore may be more effective in avoiding unphysical states.
Numerical simulations {#sec:simulations}
=====================
In this section we report our numerical studies of astrophysical scenarios involving the dynamical evolution of a rotating magnetized star and its magnetosphere. The initial data of rigidly rotating neutron stars is provided by the LORENE package [*Magstar*]{} [^2], which adopts a polytropic equation of state $P=K \rho^{\Gamma}$ with $\Gamma=2$, rescaled to $K=100$. Because the fluid pressure in a neutron star is many orders of magnitude larger than the electromagnetic one, moderate magnetic fields will have an insignificant effect on both the geometry and the fluid structure, and so they can be specified freely. For this reason we have chosen an initial poloidal magnetic field inside the star that becomes dipolar in the external region. The electric fields are set by assuming the ideal MHD condition, with an initial zero fluid velocity in the magnetosphere.
During the evolution, which is performed with the methods described in the previous sections, the ideal MHD and the force-free limits are enforced inside/outside the star by using the phenomenological current (\[ohm\_ideal\_forcefree\]). We monitor the electromagnetic luminosity, constructed from the Newman-Penrose scalar $\Phi_2$ [@Newman:1961qr], $$\label{L_em}
L_{\rm em} = \frac{{dE}^{\rm em}}{dt} =
\lim_{r \rightarrow \infty} \int r^2 |\Phi_2|^2 d\Omega ~.$$ that accounts for the energy carried off by outgoing waves to infinity and it is equivalent to the Poynting luminosity at large distances. Additionally we monitor the ratio of particular components of the Maxwell tensor $\Omega_F = F_{tr}/F_{r\phi}$ which, in the stationary, axisymmetric case, can be interpreted as the rotation frequency of the electromagnetic field [@Blandford1977].
The aligned rotator {#subsec:aligned_rotator}
-------------------
We consider first the evolution of an uniformly rotating stable star of mass $M=1.58 M_{\odot}$ and equatorial/polar radius $R=16.1/10.6~{\rm km}$. The star rotates with a period $T=1.3 {\rm ms}$, so that the light cylinder is located at $R_{\rm LC} = c/\Omega_{NS} = 62~{\rm km}$. The strength of the magnetic field at the pole is $B_p = 1.8 \times 10^{14} G$. The numerical domain extends up to $L=300~{\rm km}$ and contains four centered FMR grids with decreasing sizes (and twice better resolved) such that the highest resolution grid has $\Delta x = 0.76~{\rm km}$ and extends up to $76~{\rm km}$ (i.e., beyond the light cylinder).
This initial configuration is evolved until that the solution relaxes to a quasi-stationary state. Different quantities are plotted along the equatorial plane in fig. \[fig:pulsar1d\] and that both the initial and the final magnetic field solutions are displayed in fig. \[fig:pulsar\_bfield\]. The relaxed final state has the characteristic features observed in previous works. The magnetic fields are being dragged by the fluid rotation in the interior of the star (i.e., as in the initial state), producing a tension that forces the magnetic fields in the magnetosphere to co-rotate with the star up to the light cylinder. Beyond this surface, the magnetic field lines open up, creating a current sheet in the equatorial plane where the anomalous resistivity in the current (or bringing back the neglected fluid inertia) is necessary to preserve the physical condition $B^2>E^2$.
We have computed the Poynting-vector luminosity at two surfaces at $R_{ext}=\{76,114\}~{\rm km}$ located outside the light cylinder, where the measures converge to a unique well-defined value. The EM radiation is mainly dipolar (i.e., around $90\%$ of the energy), with a small fraction in higher multipoles. The luminosity can be compared with previous results in flat spacetime geometry where the spherical star is modeled through inner boundary conditions [@2006ApJ...643.1139C; @2006ApJ...648L..51S] $$\label{eq:lsd_dipole}
L_{\rm sd} = {1\over 4}B_{\rm pole}^2 R_{\rm NS}^2 c
\left({\Omega_{\rm NS} R_{\rm NS}\over c}\right)^4~.$$ Our results agrees within a difference of $\approx 20\%$, where we have used $R_{\rm NS}=R_{eq}$. It is unclear where this small disagreement may come from, since there are several possible explanations; the ambiguity in the definition of the radius of oblated stars, an excess of dissipation in the current sheet, or purely strong gravitational effects, which may become important due to the high compactness $M/R = 0.125$ of the star.
We have also monitored both the energy-momentum constraints and the divergence constraints, checking that they remain small and under control during the evolution. In particular, $|\nabla \cdot B|/|B| \leq 0.05$ in all the domain but the current sheet. By comparing the solutions obtained with three different resolutions, each one improving a factor $1.18$ the previous space discretization $\Delta x$, we have observed that the code converges at $1.8$-order. The luminosity for these three resolutions displayed in fig. \[fig:convergence\_luminosity\] shows that, in spite of the spasmodic reconnections happening in the current sheet, the system converges to a quasi-stationary solution with a steady luminosity.
Collapse of a magnetized rotating neutron star {#subsec:rotating_collapse}
----------------------------------------------
\
After assessing the validity of our implementation with the aligned rotator solution, we can consider a more challenging and dynamical case; the collapse of an uniformly rotating magnetized neutron star to a black hole. The initial data is the same as it was considered in [@2011arXiv1112.2622L]; a star lying on the unstable branch with mass $M=1.84 M_{\odot}$ and equatorial/polar radius $R=10.6/7.3~{\rm km}$, rotating with a period $T=0.78 {\rm ms}$ so that the light cylinder is located at $R_{\rm LC} = 37~{\rm km}$. The strength of the magnetic field at the pole is chosen to be $B_p = 1.8 \times 10^{11} G$, although the results may be rescaled to any strength as long as the magnetic pressure is much smaller than the fluid one. The numerical domain extends up to $L=300~{\rm km}$ and contains $6$ centered FMR grids with decreasing sizes such that the highest resolution grid has $\Delta x = 0.19~{\rm km}$ and extends up to $21~{\rm km}$, while that the second highest extends up to $44~{\rm km}$, beyond the initial location of the light cylinder.
Small perturbations arising from numerical truncation errors are enough to trigger the collapse of the unstable star. The horizon appears after around $1 ms$, although the most dynamical part only stands for the last $0.1 ms$, ending when all the matter disappears beyond the horizon and the nearby magnetic fields reconnects in the equatorial plane and escapes to infinity. The conservation of angular momentum implies that the angular velocity of the star increases during the collapse, dragging the magnetic field lines in the magnetosphere and bringing the light cylinder closer to the star. The magnetic fields also grow due to the magnetic flux conservation. Once all the fluid has accreted onto the black hole, the magnetic fields looses their anchorage, reconnects and propagates away from the source. A significant fraction of the energy stored in the magnetosphere is radiated to infinity in this burst. The density of the star, the Poynting vector density $|\Phi_2|^2$ and the magnetic fields are displayed at some representative stages of the collapse in fig. \[fig:collapse\_bfield\].
The growth of the angular velocity and the magnetic field implies that the luminosity of the aligned rotator (\[eq:lsd\_dipole\]) during a quasi-adiabatic collapse will increase as $L_0 (R_{\rm NS}/R)^6$ [@2011PhRvD..83l4035L], being $L_0$ the initial luminosity of the star. However, since the collapse time is shorter than the star’s period, the outer part of the magnetosphere is not able to respond to the changes in the start’s surface, reducing the power of the luminosity to $(R_{\rm NS}/R)^4$ [@2011arXiv1112.2622L]. In addition, strong gravitational effects will soften the growth of both the angular frequency and the radial magnetic field, leading to a much more moderate luminosity growth.
We have computed the electromagnetic luminosity in a sphere located at $R_{ext}=76~{\rm km}$, beyond the light cylinder. The EM radiation is mainly dipolar and grows during the collapse, with a strong burst due to the reconnection when the fluid is completely swallowed by the black hole. The luminosity and the angular velocity – computed inside and outside the star– are displayed in figure \[fig:collapse\_luminosity\]. The energy in the magnetosphere increases by a factor $C_{peak} \approx 2$ during the collapse. The total radiated energy can be expressed as a fraction $\epsilon_{\rm rad}$ of the peak energy $C_{peak} E_{dipole,0}$, namely $$\label{energy_peak_magnetosphere}
E_{\rm rad} \approx
1.4\times 10^{47} \, C_{\rm peak}\, \epsilon_{\rm rad}
\left( \frac{B_{p}}{10^{15} G} \right)^2 \, {\rm erg}.$$ where we have used $E_{\rm dipole,0} = 1.4 \times 10^{47} B_{\rm pole,15}^2 \, {\rm erg}$ for a star of radius $R_{NS} \approx 12~{\rm km}$ [@2011arXiv1112.2622L]. In our simulation we have found $\epsilon_{\rm rad} = 0.6$, implying that the system radiates $E_{\rm rad} \approx 1.6 \times 10^{47}$ergs during the collapse (for a magnetic field of $10^{15}$G). Notice that this value is different from the analytical estimates and indicates the importance of the fast dynamic and strong gravitational effects in this scenario.
Summary {#sec:discussion}
=======
We have presented a formulation of the general relativistic resistive MHD equations. We have discussed different generalizations of the isotropic Ohm’s law, and constructed a phenomenological current such that the system reduces either to the ideal MHD limit or to the force-free approximation just by changing the ratio of isotropic/anisotropic conductivities. We have explained how to deal with the potential stiffness of the equations by using the implicit-explicit Runge-Kutta methods, showing how to perform the implicit evolution of the electric field and the recovery of the primitive from the conserved fields at the same time for any algebraic Ohm’s law. We implemented the formulation within the HAD computational infrastructure and revisited two interesting astrophysical problems; the aligned rotator and the collapse of a rotating neutron star to a black hole. None of these cases has a known analytical solution, although the first case has been studied extensively. We find a reasonable agreement between our results and previous studies of the aligned rotator, recovering the same qualitative features and approximately the same electromagnetic luminosity.
The case of the collapsing star is more challenging and has been only studied previously either assuming an electrovacuum magnetosphere and/or by matching the exterior to the interior solution. Our results are qualitatively similar to those found in [@2011arXiv1112.2622L], although the total radiated energy in our simulations is one order of magnitude larger due to an increase in both the peak energy in the magnetosphere and the fraction of radiated energy. The possible detectability of this burst has been already discussed in detail in [@2011arXiv1112.2622L] and therefore will not be repeated here.
In conclusion, the resistive MHD framework allows to consider a broad range of new phenomena;study reconnections and dissipation with more realistic Ohm’s law - like the resistive solutions of pulsar magnetospheres [@2012ApJ...746...60L]-, model the magnetic growth due to different instabilities by using the mean-field dynamo [@2012arXiv1205.2951B], and compute the magnetosphere interaction of binary systems –like neutron-neutron stars and neutron-black hole–, which may be crucial to study the possible electromagnetic counterparts to the gravitational waves emitted by these systems, among others possibilities. Work on these directions is in progress and it will be reported in the near future.
IMEX {#appendixA}
====
IMEX Runge-Kutta schemes can be represented by a double tableau in the usual Butcher notation [@But:1987; @But:2003]
$$\begin{minipage}{1.2in}
\begin{tabular} {c c c}
${\tilde c}$ & \vline & ${\tilde A}$ \\
\hline
& \vline & ${\tilde \omega}^T$
\end{tabular}
\end{minipage}
\hskip 1.0cm
\begin{minipage}{1.2in}
\begin{tabular} {c c c}
${c}$ & \vline & ${A}$ \\
\hline
& \vline & ${\omega}^T$
\label{butcher_tableau}
\end{tabular}
\end{minipage}$$
where the coefficients $\tilde{c}$ and $c$ used for the treatment of non-autonomous systems are given by the following relation $$\label{definition_cs}
{\tilde c}_{i} = \sum_{j=1}^{i-1}~ {\tilde{a}}_{ij} ~~~,~~~
{c}_{i} = \sum_{j=1}^{i}~ {a}_{ij} ~~~.$$
Solutions of conservation equations have some norm that decreases in time. It would be desirable, in order to avoid spurious numerical oscillations arising near discontinuities of the solution, to maintain such property at a discrete level by the numerical method. The most commonly used norms are the TV-norm and the infinity norm. A scheme is called Strong Stability Preserving (SSP) if maintains a given norm during the evolution [@SpiRuu:2002].
In all these schemes the implicit tableau corresponds to an L-stable scheme (that is, $\omega^T A^{-1} e =1$, being $e$ a vector whose components are all equal to $1$), whereas the explicit tableau is SSP$k$, where $k$ denotes the order of the SSP scheme. We shall use the notation SSP$k(s,\sigma,p)$, where the triplet $(s,\sigma,p)$ characterizes the number of $s$ stages of the implicit scheme, the number $\sigma$ of stages of the explicit scheme and the order $p$ of the IMEX scheme.
There are different IMEX RK schemes available in the literature. We have considered only third order IMEX schemes, some of them found in the literature [@ParRus:2005] and others developed by us. All of them are based on a third order SSP explicit scheme that can be implemented efficiently by using only two levels of fields and one of rhs. It is worth mentioning that these methods are still under development and have few drawbacks. Probably the most serious one is an accuracy degradation for some range of the relaxation time $\epsilon$.
[c c c c c c]{} 0 & & 0 & 0 & 0 & 0\
0 & & 0 & 0 & 0 & 0\
1 & & 0 & 1 & 0 & 0\
1/2 & & 0 & 1/4 & 1/4 & 0\
& & 0 & 1/6 & 1/6 & 2/3\
[c c c c c c]{} $\alpha$ & & $\alpha$ & 0 & 0 & 0\
0 & & -$\alpha$ & $\alpha$ & 0 & 0\
1 & & 0 & $1-\alpha$ & $\alpha$ & 0\
1/2 & & $\beta$ & $\eta$ & $1/2-\beta-\eta-\alpha$ & $\alpha$\
& & 0 & 1/6 & 1/6 & 2/3\
$$\alpha = 0.24169426078821~,~\beta = 0.06042356519705~,~
\eta = 0.12915286960590 \nonumber$$
\[SSP3-433\]
[c c c c c c c]{} 0 & & 0 & 0 & 0 & 0 & 0\
0 & & 0 & 0 & 0 & 0 & 0\
1 & & 0 & 1 & 0 & 0 & 0\
1/2 & & 0 & 1/4 & 1/4 & 0 & 0\
1 & & 0 & 1/6 & 1/6 & 2/3 & 0\
& & 0 & 1/6 & 1/6 & 2/3 & 0\
1.0cm
[c c c c c c c]{} $\alpha$ & & $\alpha$ & 0 & 0 & 0 & 0\
0 & & -$\alpha$ & $\alpha$ & 0 & 0 & 0\
1 & & 0 & $1-\alpha$ & $\alpha$ & 0 & 0\
1/2 & & $a_{41}$ & $a_{42}$ & $a_{43}$ & $\alpha$ & 0\
1 & & 0 & 1/6 & 0 & 2/3 & 1/6\
& & 0 & 1/6 & 0 & 2/3 & 1/6\
$$a_{41} = \frac{1}{8 \alpha} (2 \alpha^2 + 2 \alpha -1) ~~,~~
a_{42} = \frac{1}{8 \alpha} (-4 \alpha^2 + 1) ~~,~~
a_{43} = \frac{1}{4} (-3 \alpha + 1) ~~,~~
\alpha = 1/3~~. \nonumber$$
\[SSP3-533\]
Ideal MHD limit {#appendixB}
===============
The ideal MHD limit can be obtained by requiring the current to be finite even in the limit of infinite isotropic conductivity, leading to the condition $E^i = - \epsilon^{ijk} v_j B_k$. The Ohm’s law current becomes undetermined (i.e., an infinite conductivity multiplying a vanishing electric field in the co-moving frame), but it can still be computed from the redundant Maxwell equation for the electric field evolution (\[maxwellext\_3+1\_eq1a\]). The evolution of the magnetic field can be simplified by substituting the ideal MHD condition in (\[maxwellext\_3+1\_eq1c\]), $$\begin{aligned}
\label{idealmhd_B}
\partial_t (\sqrt{\gamma} B^{i}) &+&
\partial_k [\sqrt{\gamma} \{ (\alpha v^k - \beta^k) B^i
- \alpha v^i B^k + \alpha \gamma^{ki} \phi \}] \nonumber \\
&=& \sqrt{\gamma} [-B^k \partial_k \beta^i
+ \phi \gamma^{ik} (\partial_k \alpha + \Gamma^j_{jk})]\end{aligned}$$
The transformation from conserved to primitive is simplified by eliminating the electric field as an independent variable and may allow us to recover the primitive quantities in a more robust way. Substituting the ideal MHD condition in the definition of the conserved variables $$\begin{aligned}
\tau &=& h W^2 + B^2 - p - D - \frac{1}{2} [(B^k v_k)^2 + \frac{B^2}{W^2}] ~~~,~~~
\label{Tmunu_decomposition2a} \\
S_{i} &=& [h W^2 + B^2] v_{i} - (B^k v_k) B_i ~~~.
\label{Tmunu_decomposition2b}\end{aligned}$$ it is easy to check that $$\label{con2prim_mhd_vB}
v_i B^i = \frac{S_i B^i}{h W^2} ~~.$$ Using this relation, the scalar product $S^i S_i$ can be solved for the Lorentz factor, obtaining $$\label{con2prim_mhd1}
c \equiv \frac{1}{W^2} = 1 - \frac{x^2 S^2 + (2 x + B^2)(S_i B^i)^2}{x^2 (x + B^2)^2}$$
Assuming an ideal gas EoS, and after some manipulations in the definition of $\tau$ (\[Tmunu\_decomposition2a\]), the resulting final equation to solve is $$\begin{aligned}
\label{trascendental_idealMHD}
f(x) &=& [ 1 - \frac{(\Gamma - 1) c}{\Gamma}] x
+ [\frac{(\Gamma-1) \sqrt{c}}{\Gamma} - 1] D
\nonumber \\
&+& [1-\frac{c}{2}] B^2 - \frac{1}{2 x^2} (S_i B^i)^2 - \tau ~~.\end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The author acknowledges his long time collaborators E .Hirschmann, S. Liebling and C .Thompson for useful comments, and particularly to D. Alic for discussions on the matching of the currents, D. Neilsen for his help on implementing the IMEX in HAD, and L. Lehner for carefully reading and discussing this manuscript. This work was supported by the Jeffrey L. Bishop Fellowship. Computations were performed in Scinet.
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\[lastpage\]
[^1]: publicly available at http://had.liu.edu
[^2]: publicly available at http://www.lorene.obspm.fr
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The Auslander-Reiten Translation
in Submodule Categories
Claus Michael Ringel and Markus Schmidmeier
*Dedicated to Idun Reiten*
*on the occasion of her 60$^{\text{\it th}}$ birthday*
[2000 Mathematics Subject Classification:]{} [Primary 16G70, Secondary 18E30]{}
[Keywords:]{} [Auslander-Reiten sequences, approximations, triangulated categories]{}
Let $\Lambda$ be an artin algebra, and $\lamod$ the category of finitely generated $\Lambda$-modules (these are just the $\Lambda$-modules of finite length). The homomorphism category $\Cal H(\Lambda)$ has as objects the maps $f$ in $\lamod$ and morphisms are given by commutative diagrams. In this paper, we draw attention to the full subcategory $\Cal S(\Lambda)$ of $\Cal H(\Lambda)$ of all monomorphisms (or subobjects), but also to the corresponding subcategory $\Cal F(\Lambda)$ of $\Cal H(\Lambda)$ of all epimorphisms (or factor objects). Categories of the form $\Cal S(\Lambda)$ are much more complicated than the underlying module categories $\lamod$; for example, if $\Lambda$ is a uniserial ring, then $\Lambda$ is of finite representation type, whereas the category $\Cal S(\Lambda)$ may have finitely or infinitely many indecomposable objects, or even be of wild representation type, depending on the Loewy length of $\Lambda$. Since Garrett Birkhoff in 1934 proposed the study of such submodule categories, they have proven to provide a rich source for classification problems, and to attract the use of methods from various areas of algebra including representations of finite dimensional algebras, lattices over tiled orders, representations of posets, and matrix classification. In this manuscript we intend to lay the foundation for an Auslander-Reiten type theory of submodule categories. Here is a preview:
If we consider a map $f\:A \to B$ of $\Lambda$-modules as an object of $\Cal H(\Lambda)$, we will write either $$\big(A\sto fB\big) \quad\text{or}\quad \hobject ABf,$$ but often also just $f$, whatever will be convenient and not misleading. The category $\Cal H(\Lambda)$ is an abelian category, in fact it is equivalent to the category of finitely generated modules over the triangular matrix ring $\big({\Lambda\atop0}{\Lambda\atop\Lambda}\big)$, and hence $\Cal S(\Lambda)$ as well as $\Cal F(\Lambda)$ are exact Krull-Schmidt category. We determine the projective and the injective objects in $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$. For example, if $I$ is an indecomposable injective $\Lambda$-module, then $$\big(0\to I\big) \T{and} \big(I\sto{1}I\big)$$ are both indecomposable injective objects in $\Cal S(\Lambda)$, but clearly the first is not injective in $\Cal H(\Lambda)$ (Proposition 1.4).
In order to see that $\Cal S(\Lambda)$ has Auslander-Reiten sequences, we only have to show that $\Cal S(\Lambda)$ is functorially finite in $\Cal H(\Lambda)$, according to Auslander and Smal[ø]{} (Theorem 2.4 in \[1\]). But this is easy: In the abelian category $\lamod$, every map $f\:A\to B$ can be factorized as the composition of an epimorphism $\Epi(f)$ and a monomorphism $\Mono(f)$. The factorization yields a morphism $f\to \Mono(f)$ in $\Cal H(\Lambda)$ and this morphism is a left minimal $\Cal S(\Lambda)$-approximation for the object $f\in \Cal H(\Lambda)$. To obtain a right minimal $\Cal S(\Lambda)$-approximation for $f$, let $e'\:\Ker(f)\to \I\Ker(f)$ be an injective envelope and choose an extension $e\:A\to \I\Ker(f)$ of $e'$. We call the map $$\Mimo(f) = [f\;e]\: \quad A\;\to\;B\oplus\I\Ker(f)$$ a [*minimal monomorphism*]{} for $f$. In this way, we obtain a morphism $\Mimo(f)\to f$ in the category $\Cal H(\Lambda)$ and this morphism turns out to be the desired right minimal $\Cal S(\Lambda)$-approximation for $f\in\Cal H(\Lambda)$ (Proposition 2.4). Thus, the category $\Cal S(\Lambda)$ is functorially finite in $\Cal H(\Lambda)$ and hence has Auslander-Reiten sequences. Note that $\Cal S(\Lambda)$-approximations provide recipes for calculating relative Auslander-Reiten sequences in the subcategory $\Cal S(\Lambda)$ as soon as Auslander-Reiten sequences are known in $\Cal H(\Lambda)$.
In the module category $\Cal H(\Lambda)$, Auslander-Reiten translates are computed via the Nakayama functor, as usual. Using approximations and the equivalence between $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ given by the kernel and cokernel functors, corresponding Auslander-Reiten sequences are obtained for the categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$, see Proposition 3.2. (Surprisingly also the converse is true: Auslander-Reiten sequences in $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ give rise to such sequences in the category $\Cal H(\Lambda)$, see Proposition 3.5.)
Revisiting the construction of minimal monomorphisms, we show that if two morphisms $f,g\:A\to B$ in $\lamod$ differ only by a map which factorizes through an injective $\Lambda$-module, and if $B$ has no nonzero injective direct summands, then $\Mimo(f)$ and $\Mimo(g)$ are isomorphic objects in $\Cal S(\Lambda)$ (Proposition 4.1).
Our key result is this: For an indecomposable nonprojective object $\big(A\sto fB\big)$ in $\Cal S(\Lambda)$, the Auslander-Reiten translate $\tau_{\Cal S}(f)$ can be computed directly within the category $\lamod$. This is done as follows: Let $g\:B\to C$ be the cokernel of $f$. Recall that the Auslander-Reiten translation in $\lamod$ gives rise to a functor $\tau_\Lambda\:\underlamod\to \overlamod$ where $\underlamod$ and $\overlamod$ denote the factor categories of $\lamod$ modulo all maps which factorize through a projective or through an injective $\Lambda$-module, respectively. Next take a representative $h\:D\to E$ for the morphism $\tau_\Lambda(g)$ such that $D$ and $E$ have no nonzero injective direct summands. As $h$ is determined uniquely, up to a map which factorizes through an injective module, we will see that $\Mimo(h)$ is determined uniquely, up to isomorphism, as an object in $\Cal S(\Lambda)$. This is the Auslander-Reiten translate $\tau_{\Cal S}(f)$ for $f$ in the category $\Cal S(\Lambda)$ (Theorem 5.1). Thus we may write: $$\tau_{\Cal S}(f)\;=\;\Mimo \tau_\Lambda \Cok(f)$$
If $\Lambda$ is a self-injective algebra, the stable category $\underlamod = \overlamod$ has the structure of a triangulated category. We observe that if $A\sto{\bar f}B\sto{\bar g}C\sto{\bar h}\Omega^{-1}A$ is a triangle (with $\Omega^{-1}$ the suspension functor) and $f$ is a map representing $\bar f$, then a map $g$ representing the rotate $\bar g$ is obtained as $g=\Cok\Mimo(f)$. The functor $\tau_\Lambda$ commutes with $\Omega^{-1}$ and with the rotation $\bar f\mapsto \bar g$ in a triangle, and as a consequence, the formulae $$\overline{\tau_{\Cal S}^3(f)}\cong -\tau_\Lambda^3\Omega^{-1}(\bar f)
\quad\text{and}\quad
\overline{\tau_{\Cal S}^6(f)}\cong \tau_\Lambda^6\Omega^{-2}(\bar f)$$ hold (Theorem 6.2). In particular, if $\tau_\Lambda$ coincides with $\Omega^2$, it follows that $$\tau_{\Cal S}^3(f) \cong -\Mimo\Omega^{5}(f)$$ for any indecomposable nonprojective object $f$ in $\Cal S(\Lambda)$ (Corollary 6.4). For example, in the special case that $\Lambda$ is a commutative uniserial ring, all the functors $\tau_\Lambda,$ $\Omega^{2}$ and $\Omega^{-2}$ are equivalent to the identity functor on $\underlamod$ and then the formula yields $$\tau_{\Cal S}^6(f)\cong f$$ for $f$ an indecomposable nonprojective object in $\Cal S(\Lambda)$ (Corollary 6.5).
Let $\big(C'\sto cC\big)$ be an indecomposable nonprojective object in $\Cal S(\Lambda)$. The Auslander-Reiten sequence $$\CD 0 @>>> \hobject{A'}A{a} @>f'>f> \hobject{B'}B{b} @>g'>g>
\hobject{C'}C{c} @>>> 0
\endCD$$ in $\Cal S(\Lambda)$ is made up from two short exact sequences in $\lamod$ given by the sequence $0\to A'\sto{f'}B'\sto{g'}C'\to 0$ of the submodules and the sequence $0\to A\sto fB\sto gC\to 0$ of the big modules. By Proposition 7.2, these two sequences are “usually” split exact. We list the exceptions and collect our findings about the structure of the middle term $(b\:B'\to B)$.
Let us stress that the categories $\lamod$, $\Cal H(\Lambda)$ and $\Cal S(\Lambda)$ usually behave very differently. Consider for example the case of $\Lambda = \Lambda_n =
k[T]/T^n.$ These $k$-algebras $\Lambda_n$ are all of finite representation type: indeed, there is (up to isomorphism) precisely one indecomposable $\Lambda$-module of length $i$, for $1 \le i \le n$, and these are all the indecomposables. On the other hand, it is well-known that the matrix ring $\big({\Lambda_n\atop0}{\Lambda_n\atop\Lambda_n}\big)$ is representation finite only for $n \le 3$, thus, for $n \ge 4$ there are infinitely many isomorphism classes of indecomposable objects in $\Cal H(\Lambda_n)$, see for example \[S, Section 2\]. For $n \le 5$, the subcategory $\Cal S(\Lambda_n)$ of $\Cal H(\Lambda_n)$ consists of only finitely many isomorphism classes of indecomposable objects, whereas there are infinitely many isomorphism classes of indecomposable objects in $\Cal S(\Lambda_n)$, for any $n \ge 6,$ see \[RS1\]. One may consult these references and also \[RS2\] for proofs that the categories $\Cal H(\Lambda_4)$ and $\Cal S(\Lambda_6)$ are “tame”, whereas the categories $\Cal H(\Lambda_n)$ for $n \ge 5$ and $\Cal S(\Lambda_n)$ for $n \ge 7$ are “wild”.
[*Notation:*]{} The condition that $\Lambda$ is an artin algebra can be weakened to requiring that $\Lambda$ is a locally bounded associative $k$-algebra or a locally bounded $k$-spectroid \[GR\]; then also coverings of finite dimensional algebras are included. We recall the corresponding definitions:
Let $k$ be a commutative local artinian ring and $\Lambda$ an associative $k$-algebra which need not have a unit element, but it is required that $\Lambda$ equals the $k$-space $\Lambda^2$ of all possible linear combinations of products in $\Lambda$. By $\lamod$ we denote the category of all $\Lambda$-modules $B$ which have finite length when considered as $k$-modules and which are [*unitary*]{} in the sense that $\Lambda B = B$ holds. The algebra $\Lambda$ is said to be [*locally bounded*]{} if there is a complete set $\{e_i:i\in I\}$ of pairwise orthogonal primitive idempotents such that each of the indecomposable projective modules $e_i\Lambda$ and $\Lambda e_i$ has finite length as a $k$-module, for $i\in I$. If $\Lambda$ is locally bounded such that each indecomposable projective $\Lambda$-module is injective and each indecomposable injective $\Lambda$-module is projective, then we say that $\Lambda$ is a [*self-injective algebra.*]{} An artin algebra $\Lambda$ is called [*uniserial*]{} if both modules $\Lambda_\Lambda$ and ${}_\Lambda\Lambda$ have unique composition series. In this case, all one-sided ideals are two-sided, namely the powers of the unique maximal ideal $\m=\Rad\Lambda$.
For the terminology around almost split morphisms we refer the reader to \[ARS\] and \[AS\]. Here we use the term “Auslander-Reiten sequence” for “almost split sequence” and abbreviate “left (right) minimal almost split map” to “source (sink) map”. The Auslander-Reiten translation in a category $\Cal C$ is denoted by $\tau_{\Cal C}$, but in case $\Cal C = \mod \Lambda$, we write $\tau_\Lambda$.
Finally, we want to apologize that our use of brackets when applying functions and functors is not at all consistent. We have inserted brackets whenever we felt that this improves the readability, but we have avoided multiple brackets whenever possible.
This paper was written in 2001 as a general introduction to a proposed volume devoted to the Birkhoff problem (dealing with subgroups of finite abelian groups as well as with invariant subspaces of linear operators). Unfortunately, we had to delay the Birkhoff project. Since the paper seems of interest in its own right, we now have decided to publish it independently. The authors are indebted to the referee, and also to Aslak Bakke Buan and Øyvind Solberg, for helpful remarks concerning the presentation of the results.
**1. Projective and Injective Objects**
In this section we determine the projective and the injective objects in the categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ and their associated sink and source maps. The injective objects in $\Cal S(\Lambda)$ are also called “relatively injective” or “$\Ext$-injective” as they may not be injective in the category $\Cal H(\Lambda)$. Dually, the projective objects in $\Cal F(\Lambda)$ may not be projective when considered as objects in $\Cal H(\Lambda)$.
Let $\Lambda$ be an associative locally bounded $k$-algebra, and let $U(\Lambda)=\big({\Lambda \atop 0}{\Lambda\atop\Lambda}\big)$ be the associative $k$-algebra of upper triangular matrices with coefficients in $\Lambda$. First we recall well-known facts about $U(\Lambda)$ and about the category $\Cal H(\Lambda)$ of morphisms between $\Lambda$-modules of finite $k$-length.
[**Lemma 1.1.**]{} [(Basic facts about $\Cal H(\Lambda)$)]{}
**
[1.]{} The $k$-algebra $U(\Lambda)$ is locally bounded.
[2.]{} The $k$-categories $\mod U(\Lambda)$ and $\Cal H(\Lambda)$ are equivalent.
[3.]{} $\Cal H(\Lambda)$ is an abelian Krull-Schmidt category.
[4.]{} Each object in $\Cal H(\Lambda)$ has a projective cover and an injective envelope.
[5.]{} The category $\Cal H(\Lambda)$ has Auslander-Reiten sequences.
The categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ are defined to be the full subcategories of $\Cal H(\Lambda)$ which consist of all objects $\big(A\sto fB\big)$ in $\Cal H(\Lambda)$ for which $f$ is a monomorphism or an epimorphism, respectively. These three categories are related by the kernel and cokernel functors. $$\matrix
\Cok: & \Cal H(\Lambda)\to\Cal F(\Lambda), & \quad (A\sto fB)
\mapsto (B\lto{\text{can}}\Cok(f)),\cr
\Ker: & \Cal H(\Lambda)\to\Cal S(\Lambda), & \quad(B\sto gC)
\mapsto(\Ker(g)\lto{\text{incl}}B).
\endmatrix$$
[**Lemma 1.2.**]{} [(Basic properties of $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$)]{}
**
[1.]{} With the exact structure given by the category $\Cal H(\Lambda)$, the categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ are exact Krull-Schmidt $k$-categories.
[2.]{} The category $\Cal S(\Lambda)$ is closed under kernels while $\Cal F(\Lambda)$ is closed under cokernels. Both categories are closed under extensions.
[3.]{} The restrictions of the kernel and cokernel functors $$\Ker\: \Cal F(\Lambda)\to \Cal S(\Lambda)\T{and}
\Cok\:\Cal S(\Lambda)\to \Cal F(\Lambda)$$ induce a pair of inverse equivalences.
The equivalence between $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ is useful to deduce the structure of the projective and the injective objects in either of the two categories from the structure of the corresponding modules in $\Cal H(\Lambda)$ which are described in the following
[**Lemma 1.3.**]{} [(Projective and injective modules in $\Cal H(\Lambda)$; the Nakayama functor)]{}
**
[P-1.]{} Let $P$ be an indecomposable projective $\Lambda$-module with radical $\Rad P$. The objects $(0\to P)$ and $(1_P\:P\to P)$ are indecomposable projective objects and have as sink maps the inclusions $$(0\to\Rad P)\to(0\to P)\quad\text{and}\quad
(\Rad P\lto{\text{incl}}P)\to (P\sto1P),$$ respectively.
[P-2.]{} Each indecomposable projective object arises in this way.
[I-1.]{} Let $I$ be an indecomposable injective $\Lambda$-module with socle $\Soc I$. The indecomposable injective objects $(I\to 0)$ and $(1_I\:I\to I)$ have as source maps the canonical maps $$(I\to 0)\to (I/\Soc I\to 0)\quad\text{and}\quad
(I\sto1I)\to(I\lto{\text{can}}I/\Soc I),$$ respectively.
[I-2.]{} Each indecomposable injective object arises in this way.
[N-1.]{} The operation of the Nakayama functor $\nu_{\Cal H}$ on $\Cal H(\Lambda)$ can be expressed in terms of the Nakayama functor $\nu=\nu_\Lambda$ in $\lamod$. For $P$ a projective $\Lambda$-module we have $$\nu_{\Cal H}( 0\to P) \;=\; (\nu P\sto 1\nu P),\qquad
\nu_{\Cal H}( P\sto 1P) \;=\; (\nu P\to 0).$$
[N-2.]{} On morphisms, the Nakayama functor $\nu_{\Cal H}$ is given in the obvious way, for example if $(0,f)\:(0\to P)\to (1_Q\:Q\to Q)$ is a morphism between projective objects, then $\nu_{\Cal H}(0,f)=
(\nu f,0)\:(1_{\nu P}\:\nu P\to \nu P)\to (\nu Q\to 0)$.
We can now describe the projective and the injective objects in $\Cal S(\Lambda)$.
[**Proposition 1.4.**]{} [(Projective and injective objects in $\Cal S(\Lambda)$)]{}
**
[P.]{} The projective objects in $\Cal S(\Lambda)$ and their sink maps are as in Lemma 1.3-P.
[I-1.]{} Let $I$ be an indecomposable injective $\Lambda$-module. Then the map $(1_I\: I \to I)$ is an indecomposable injective object in $\Cal S(\Lambda)$ and has as source map the canonical map $(1_I\: I\to I)\to(1_{I/\Soc I}\:I/\Soc I\to I/\Soc I)$.
[I-2.]{} If $I$ is an indecomposable injective $\Lambda$-module then $(0\to I)$ is indecomposable (relatively) injective in $\Cal S(\Lambda)$ and has as source map the inclusion $(0\to I)\to (\incl\:\Soc I\to I)$.
[I-3.]{} Each indecomposable (relatively) injective object in $\Cal S(\Lambda)$ arises in this way.
[*Proof:*]{} The projective modules in $\Cal H(\Lambda)$ and their sink maps are objects and morphisms in the category $\Cal S(\Lambda)$, hence the statement in Lemma 1.3-P holds for $\Cal S(\Lambda)$. Similarly, the injective modules in $\Cal H(\Lambda)$ and their source maps are in the category $\Cal F(\Lambda)$, so the statement in 1.3-I holds for $\Cal F(\Lambda)$. By applying the kernel functor from 3. in Lemma 1.2 we obtain the objects and morphisms in I-1 and I-2. Since $\Cal H(\Lambda)$ has sufficiently many projective and injective objects, so does $\Cal S(\Lambda)$.
Let us add the dual statement for the category $\Cal F(\Lambda)$. Of course, we know from 1.2 that the categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ are equivalent, thus there is no intrinsic need to deal with both categories separately. On the other hand, it may be useful for further references to have the precise formulations available.
[**Proposition 1.5.**]{} [(Projective and injective objects in $\Cal F(\Lambda)$)]{}
**
[P-1.]{} Let $P$ be an indecomposable projective $\Lambda$-module. Then the map $(1_I\: P \to P)$ is an indecomposable projective object in $\Cal F(\Lambda)$ and has as sink map the inclusion $(1_{\Rad P}\:\Rad P\to\Rad P)\to (1_P\:P\to P)$.
[P-2.]{} If $P$ is an indecomposable projective $\Lambda$-module then $(P\to 0)$ is indecomposable (relatively) projective in $\Cal F(\Lambda)$ with sink map the canonical map $(\can\:P\to P/\Rad P)\to (P\to 0)$.
[P-3.]{} Each indecomposable (relatively) projective object in $\Cal S(\Lambda)$ arises in this way.
[I.]{} The injective objects in $\Cal F(\Lambda)$ and their source maps are as in Lemma 1.3-I.
[*Example.*]{} Consider the case that $\Lambda$ is uniserial with maximal ideal $\m$ and Loewy length $n\geq 2$. In $\Cal S(\Lambda)$, there are two projective indecomposable objects namely $P_1= (\Lambda = \Lambda)$ and $P_2= (0\subseteq \Lambda)$; their sink maps are the inclusions $$(\m\subseteq\Lambda)\quad\to\quad P_1\qquad\text{and}\qquad
(0\subseteq \m)\quad\to\quad P_2,$$ respectively. One of them, the module $P_1=I_1$ is also injective, both in $\Cal H(\Lambda)$ and in $\Cal S(\Lambda)$, and has source map the canonical map $$I_1\quad\to\quad (\Lambda/\m^{n-1}=\Lambda/\m^{n-1});$$ the second projective $P_2=I_2$ is relatively injective in $\Cal S(\Lambda)$, its source map is given by the inclusion $$I_2\quad\to\quad (\m^{n-1}\subseteq\Lambda).$$
**2. Left and Right Approximations**
The categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ are functorially finite in $\Cal H(\Lambda)$ and hence have Auslander-Reiten sequences. To show this, we determine the left and the right approximation for each object in $\Cal H(\Lambda)$, in each of the subcategories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$.
[*Definitions:*]{} Let $\Cal S$ be a subcategory of a module category $\Cal C$, and $C\in\Cal C$. A morphism $f\:C\to S$ with $S\in\add\Cal S$ is a [*left approximation*]{} of $C$ in $\Cal S$ if the map $$\Hom(f,1_{S'})\:\quad \Hom_{\Cal C}(S,S')\to \Hom_{\Cal C}(C,S')$$ is onto for each $S'\in \Cal S$. Moreover, $f$ is [*left minimal*]{} if each endomorphism $s\in\End_{\Cal C}(S)$ which satisfies $fs=f$ is an isomorphism. A [*minimal left approximation*]{} is a left minimal left approximation. Similarly, [*minimal right approximations*]{} are defined.
In the abelian category $\lamod$, a morphism $f\:A\to B$ has a factorization $A\to\Im(f)\to B$ over the image. The two maps $f_1\:A\to \Im(f)$ and $f_2\:\Im(f)\to B$ are determined uniquely as objects in $\Cal H(\Lambda)$, up to isomorphism, and give rise to functors $$\matrix
\Epi: & \Cal H(\Lambda)\to\Cal F(\Lambda),
& \quad\big(A\sto fB\big)\mapsto\big(A\sto{f_1} \Im(f)\big),\cr
\Mono: & \Cal H(\Lambda)\to \Cal S(\Lambda),
& \quad\big(A\sto f B\big)\mapsto\big(\Im(f)\sto{f_2} B\big).
\endmatrix$$ There are functorial isomorphisms $\Epi\cong\Cok\Ker$ and $\Mono\cong\Ker\Cok$.
[**Lemma 2.1.**]{} [(Approximations given by $\Mono$ and $\Epi$)]{}
*Let $(f\:A\to B)$ be an object in $\Cal H(\Lambda)$.*
[1.]{} The map $(f_1,1_B)\:f\to \Mono(f)$ is a left minimal approximation of $f$ in $\Cal S(\Lambda)$.
[2.]{} The map $(1_A,f_2)\:\Epi(f)\to f$ is a right minimal approximation of $f$ in $\Cal F(\Lambda)$.
The [*proof*]{} is immediate from the definitions.
For the object $(f\:A\to B)\in\Cal H(\Lambda)$ there is also a [*right*]{} minimal approximation in $\Cal S(\Lambda)$ and a [*left*]{} minimal approximation in $\Cal F(\Lambda)$, as we are going to show.
First we define $\Mimo(f)$, the [*minimal monomorphism*]{} for $f$, as follows: Let $e'\:\Ker(f)\to \I\Ker(f)$ be an injective envelope and choose an extension $e\:A \to \I\Ker(f)$ of $e'$, thus $e' = f'e$, where $f'\:\Ker(f)\to A$ is the inclusion map. Then $\Mimo (f)$ is the map $$\Mimo( f) = \bmatrix f & e \endbmatrix\: \quad
A \;\to\; B\oplus \I\Ker(f).$$
[**Lemma 2.2.**]{} [($\Mimo$ is well-defined)]{}
[*$\Mimo(f)$ is independent of the choice of $e$, up to isomorphism in $\Cal H(\Lambda)$.*]{}
[*Proof:*]{} Let $e_1, e_2\:A \to \I\Ker(f)$ be two extensions of $e'$, thus $e'=f'e_1=f'e_2$. We see that the difference $e_2-e_1$ vanishes on $\Ker(f)$. Write $f=f_1f_2$ with $f_1$ an epimorphism and $f_2$ a monomorphism. Since $e_2-e_1$ vanishes on $\Ker(f)$, it factorizes through $f_1=\Cok (f')$, thus $e_2-e_1 = f_1\hat e$ for some map $\hat e\:\Im(f)\to \I\Ker(f)$. Since $f_2$ is mono and the target $\I\Ker(f)$ of $\hat e$ is injective, we can extend $\hat e$ to $B$: There is $\tilde e\:B \to \I\Ker(f)$ such that $f_2\tilde e = \hat e$. Altogether we have $f \tilde e = f_1f_2\tilde e = f_1\hat e = e_2-e_1$. It follows that the representations in $\Cal S(\Lambda)$ given by $\bmatrix f & e_1\endbmatrix$ and by $\bmatrix f & e_2\endbmatrix$ are isomorphic. $$\CD
\ssize A @>[f \; e_1]>> \ssize B \oplus \I\Ker(f) \cr
@| @VV{\left[\smallmatrix 1 & \tilde e \cr 0 & 1\endsmallmatrix\right]} V\cr
\ssize A @>>[ f\; e_2]> \ssize B\oplus \I\Ker(f)
\endCD$$
Dually one defines the [*minimal epimorphism*]{}, $\Mepi(f)$, for a map $f\:A\to B$ as follows. The projective cover of the cokernel of $f$, $p'\:\P\Cok(f)\to \Cok(f)$ factorizes over the canonical map $f''\:B\to \Cok(f)$ so there is $p\:\P\Cok(f)\to B$ such that $p'=pf''$. Let $\Mepi(f)$ denote the map $$\Mepi(f)\;=\;\bmatrix f \cr p\endbmatrix\:\quad
A\oplus\P\Cok(f)\longrightarrow B;$$ then the dual version of the above result holds for $\Mepi$.
[**Lemma 2.3.**]{} [($\Mepi$ is well-defined)]{}
There are canonical maps $\Mimo(f)\to f$ and $f\to \Mepi(f)$, $$\CD \hobject A{B\oplus I}{[f\;e]} @>1>\big[{1\atop0}\big]>
\hobject ABf\qquad\text{and}
\qquad\hobject ABf @>[1\;0]>1>
\hobject{A\oplus P}B{[f\;p]^t},
\endCD$$ which give rise to a “dual” version of Lemma 2.1:
[**Proposition 2.4.**]{} [(Approximations defined by $\Mimo$ and $\Mepi$)]{}
[*Proof of the first statement:*]{} Let $(g\:C\to D)$ be an object in $\Cal S(\Lambda)$ and $(u,v)$ a morphism from $g$ to $f$. We need to find $(u',v')\:g\to \Mimo(f)$ such that the composition with the above map $F\:\Mimo(f)\to f$ is just $(u,v)$. Since $g\:C\to D$ is a monomorphism, the composition $C\sto uA\sto eI$ lifts to a map $v_2\:D\to I$ such that $ue=gv_2$. Then the pair $(u',v')=(u,[v,v_2])$ is a morphism $g\to \Mimo(f)$ which satisfies the condition that $(u',v')\,F=(u,v)$. Thus, $F$ is a right approximation. It remains to show that $F$ is right minimal. Let $(u,v)$ be an endomorphism of $\Mimo(f)$ such that $F=(u,v)\,F$ holds and write the map $v\:B\oplus I\to B\oplus I$ as a matrix $v=\big[{v_{11}\atop v_{21}}{v_{12}\atop v_{22}}\big]$. Since $F=(u,v)\,F$ holds, it follows that $u=1_A$, $v_{11}=1_B$, and $v_{21}=0$. Now the condition that $(u,v)$ is a homomorphism amounts to $[f\;e]\big[{1 \atop0} {v_{12}\atop v_{22}}\big]=
1_A[f\;e]$, that is to say, $fv_{12}+ev_{22}=e$ must hold. Restricting both sides to the kernel of $f$ (so that the composition $fv_{12}$ vanishes) yields that the two maps $ev_{22},e\:\Ker(f)\to I$ are equal. Since $I$ is the injective envelope of $\Ker(f)$, minimality implies that $v_{22}$ is an automorphism of $I$. Hence $v=\big[{1_B\atop 0}{v_{12}\atop v_{22}}\big]$ is an automorphism of $B\oplus I$. We have shown that $F$ is right minimal.
As a consequence of Lemma 2.1 and Proposition 2.4 we obtain:
[**Theorem 2.5.**]{} [(Existence of Auslander-Reiten sequences)]{}
[*Proof:*]{} According to the first statements in Lemma 2.1 and Proposition 2.4, each object $f$ in $\Cal H(\Lambda)$ has a left and a right approximation in $\Cal S(\Lambda)$, this is to say, $\Cal S(\Lambda)$ is functorially finite. Similarly it follows from the second statements in Lemma 2.1 and Proposition 2.4 that the category $\Cal F(\Lambda)$ is functorially finite, too. According to \[1, Theorem 2.4\], those categories have Auslander-Reiten sequences.
**3. Transfer of Auslander-Reiten Sequences**
In this section we construct Auslander-Reiten sequences for the categories $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ from corresponding sequences in the module category $\Cal H(\Lambda)$. Surprisingly, also the converse is possible: Auslander-Reiten sequences in $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ give rise to Auslander-Reiten sequences in $\Cal H(\Lambda)$. First we need a lemma.
[**Lemma 3.1.**]{} [(Kernels and Cokernels of Auslander-Reiten sequences)]{}
[*Proof:*]{} We only show the statement about the sequence in $\Cal S(\Lambda)$, and for this sequence we only show that the map $(g',g)\:(b'\:B'\to B)\to (c'\:C'\to C)$ is either a split epimorphism or a right almost split morphism. Note that this implies that $g'$ is onto.
Let $(x'\:X'\to X)$ be an object in $\Cal S(\Lambda)$ and $(t',t)\:x'\to c'$ a morphism which is not a split epimorphism. Let $x\:X\to X_1$ be the cokernel for $x'$, factorize $c=c_1c_2$ as the product of an epimorphism $c_1\:C\to \Im(c)$ and a monomorphism $c_2\:\Im(c)
\to C_1$, and let $t_1\:X_1\to \Im(c)$ be the cokernel map for $(t',t)$, this map satisfies $xt_1=tc_1$. Then $(t,t_1c_2)\:x\to c$ is not a split epimorphism, since its kernel is not, and hence factorizes over the map $(g,g_1)$: There is $(u,u_1)\:(x\:X\to X_1)\to (b\:B\to B_1)$ such that $(t,t_1c_2)=(u,u_1)(g,g_1)$. If $u'\:X'\to B'$ is the kernel map for $(u,u_1)$ then $(t',t)=(u',u)(g',g)$ factorizes over $(g',g)$.
[**Proposition 3.2.**]{} [(Transfer of AR sequences from $\Cal H(\Lambda)$ to $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$)]{}
[*Proof:*]{} We show the first assertion. Note that the sequence $0\to \Epi(a)\to \Epi(b)\to c\to 0$ in $\Cal F(\Lambda)$ is the cokernel sequence for $0\to \Ker(a)\to \Ker (b)\to \Ker (c)\to 0$ in $\Cal S(\Lambda)$ since $c\in\Cal F(\Lambda)$ satisfies $c=\Epi(c)=\Cok\Ker(c)$. By Lemma 3.1, both sequences are either split exact or almost split in their respective categories. The sequences are not split exact since the morphism $\Epi(b)\to c$ is the composition of the right approximation $\Epi(b)\to b$ and the right almost split morphism $b\to c$, and hence is a right almost split morphism.
[**Corollary 3.3.**]{} [(The translation in $\Cal H(\Lambda)$, $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$)]{}
**
[1.]{} If $c\in\Cal F(\Lambda)$ is indecomposable nonprojective then $\tau_{\Cal F}(c)=\Epi\tau_{\Cal H}(c).$
[2.]{} If $c\in\Cal S(\Lambda)$ is indecomposable nonprojective then $\tau_{\Cal S}(c)=\Ker\tau_{\Cal H}\Cok(c).$
[3.]{} If $a\in\Cal S(\Lambda)$ is indecomposable noninjective then $\tau^-_{\Cal S}(a)=\Mono\tau^-_{\Cal H}(a).$
[4.]{} If $a\in\Cal F(\Lambda)$ is indecomposable noninjective then $\tau^-_{\Cal F}(a)=\Cok\tau^-_{\Cal H}\Ker(a).$
In the remainder of this section we describe how Auslander-Reiten sequences in $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ give rise to Auslander-Reiten sequences in $\Cal H(\Lambda)$.
[**Lemma 3.4.**]{} [(The functors $\Epi$ and $\Mono$ reflect some split morphisms)]{}
[*Proof:*]{} The first assertion is clear since $\Epi$ and $\Mono$ are functors; the third assertion is dual to the second, so we only consider the second assertion. Suppose that $(f,h)$ is split monic, so there are maps $u\:C\to A$ and $w\: \Im(c)\to \Im(a)$ such that $ua_1=c_1w$, $fu=1_A$, $hw=1_{\Im (a)}$. Since $B$ is injective, the map $wa_2$ extends to $D$: There is $v\:D\to B$ such that $c_2v=wa_2$. Since $a_2gv=hc_2v=hw a_2=a_2$, the left minimality of the injective envelope $a_2$ implies that $gv$ is an automorphism of $B$. It follows that the morphism $(f,g)$ is split monic.
[**Proposition 3.5.**]{} [(Transfer of AR sequences from $\Cal S(\Lambda)$ and $\Cal F(\Lambda)$ to $\Cal H(\Lambda)$)]{}
**
[1.]{} Let $(c\:C\to C'')$ be an indecomposable nonprojective object in $\Cal F(\Lambda)$. If the first two rows in the diagram $$\CD
\ssize 0 @>>> \ssize A @>f>> \ssize B @>>>\ssize C @>>> \ssize 0 \cr
@. @VaVV @VVbV @VVcV \cr
\ssize 0 @>>> \ssize A'' @>f''>> \ssize B'' @>>> \ssize C'' @>>> \ssize 0 \cr
@. @V\iota VV @VVdV @| \cr
\ssize 0 @>>> \ssize \I(A'') @>g>> \ssize D @>>>\ssize C''
@>>> \ssize 0\cr
\endCD$$ define an Auslander-Reiten sequence in $\Cal F(\Lambda)$ and if the third row is obtained as push-out along the injective envelope $\iota:A''\to \I(A'')$ then the first row and the third row define an Auslander-Reiten sequence in $\Cal H(\Lambda)$.
[2.]{}Let $(a\:A'\to A)$ be an indecomposable noninjective object in $\Cal S(\Lambda)$. If the two lower rows in the diagram $$\CD
\ssize 0 @>>> \ssize A' @>>> \ssize D @>>h> \ssize \P(C') @>>>
\ssize 0 \cr
@. @| @VVdV @VV\pi V \cr
\ssize 0 @>>> \ssize A' @>>> \ssize B' @>>g'> \ssize C' @>>> \ssize 0 \cr
@. @VaVV @VVbV @VVcV \cr
\ssize 0 @>>> \ssize A @>>f> \ssize B @>>>\ssize C @>>> \ssize 0 \cr
\endCD$$ define an Auslander-Reiten sequence in $\Cal S(\Lambda)$ and if the first row is obtained as pull-back along the projective cover $\pi\:\P(C')\to C'$ then the first row and the third row define an Auslander-Reiten sequence in $\Cal H(\Lambda)$.
[*Proof of the first statement:*]{} The map $(f,g)$ in $\Cal H(\Lambda)$ is not a split monomorphism since $\Epi(f,g)=(f,f'')$ is not, by Lemma 3.1. We show that $(f,g)$ is left almost split. Suppose $$(r,s): \quad \big(A\lto{a\iota}\I(A'')\big)\quad\longrightarrow
\quad \big(X\sto xY\big)$$ is a morphism in $\Cal H(\Lambda)$ which is not a split monomorphism. Factorize $x$ over the image as $X\sto{x_1} \Im(x)\sto{x_2}Y$, so $x=x_1x_2$. Then there is $t:A''\to \Im(x)$ such that the two squares commute. $$\CD \ssize A @>r>> \ssize X \cr
@VaVV @VVx_1V \cr
\ssize A'' @>t>> \ssize \Im(x)\cr
@V\iota VV @VVx_2 V \cr
\ssize \I(A'') @>s>> \ssize Y
\endCD$$ By Lemma 3.4, part 2, the morphism $(r,t)$ is not split monic in $\Cal F(\Lambda)$ and hence factorizes over $(f,f'')$: There are $v:B\to X$, $v'':B''\to \Im(x)$ such that $$bv''=vx_1, \quad\text{and}\quad (r,t)=(f,f'')(v,v'').$$ Then $f''(v''x_2)= tx_2 = \iota s$, so we obtain $w:D\to Y$ such that $dw=v''x_2$ and $gw=s$, by the push-out property for $Y$. Thus, $(v,w)$ is a morphism in $\Cal H(\Lambda)$ and our test map $(r,s)$ factorizes over $(f,g)$: $(r,s)=(f,g)(v,w)$. For the proof that $(f,g)$ is a source map in the module category $\Cal H(\Lambda)$, it remains to check that $(f,g)$ is left minimal, and this follows from the indecomposability of the cokernel $(C\to C'')$.
**4. Minimal Monomorphisms and the Stable Category**
Returning to the investigation of minimal monomorphisms, we show that if $f,g\:A\to B$ are two maps which differ by a morphism which factorizes through an injective module, then $\Mimo(f)$ and $\Mimo(g)$ are isomorphic as objects in $\Cal S(\Lambda)$.
First we verify three claims.
[**Claim 1.**]{} [ *Let $f,g\:A \to B$ be maps in $\lamod$ such that $g-f$ factorizes through an injective module. Let $h\:A \to \I(A)$ be an injective envelope. Then the objects in $\Cal S(\Lambda)$ given by the monomorphisms $[f\; h]$ and $[g\; h]$ are isomorphic.*]{}
[*Proof:*]{} The map $g-f$ factorizes through the injective envelope $h$, so there is $u\:\I(A)\to B$ such that $g-f=hu$. The following commutative diagram shows that $[f\;h]$ and $[g\;h]$ are isomorphic objects in $\Cal H(\Lambda)$. $$\CD
\ssize A @>[f \; h]>> \ssize B \oplus \I(A) \cr
@| @VV{\left[\smallmatrix 1 & 0 \cr u & 1\endsmallmatrix\right]} V\cr
\ssize A @>[g \; h]>> \ssize B \oplus \I(A)
\endCD$$
[**Claim 2.**]{} [ *Given $f:A\to B$ and a map $h:A\to I$ with $I$ injective such that $[f\; h]\: A \to B \oplus I$ is a monomorphism, then there is an injective module $J$ and a commutative diagram $$\CD
\ssize A @>[ f \; e \; 0]>> \ssize B\oplus \I\Ker (f) \oplus J \cr
@| @VV d V\cr
\ssize A @>[ f \; h]>> \ssize B\oplus I
\endCD$$ with $d$ an isomorphism and $e\:A\to \I\Ker(f)$ an extension of the injective envelope $\Ker(f) \to \I\Ker(f).$*]{}
The diagram shows that the maps given by the two rows are isomorphic as objects in $\Cal H(\Lambda)$. Note that in $\Cal H(\Lambda)$, the object given by the upper row decomposes as the direct sum of the two objects $([f\; e]\:A \to B\oplus \I\Ker(f))$ and $(0 \to J)$. Up to isomorphism, the first one is just $\Mimo(f)$.
[*Proof:*]{} Since $[f \; h]$ is a monomorphism, the restriction of $h$ to $\Ker(f)$ is injective, thus $I$ contains a submodule isomorphic to $\Ker(f)$ and therefore $I$ decomposes as $I = \I\Ker(f) \oplus J$ for some injective $\Lambda$-module $J$. Then $h$ has the form $h = [ e \; h_2 ]$, where $ e\: A \to \I\Ker(f)$ is an extension of the inclusion $\Ker(f)\to \I\Ker(f)$, and $ h_2\:A \to J$ satisfies $h_2\Ker(f) = 0.$ Write $f = f_1f_2$ with $f_1$ an epimorphism and $f_2$ a monomorphism. Since $h_2$ vanishes on $\Ker(f)$, we can factorize $h_2$ through $f_1$, say $h_2 = f_1v$ for some map $v\:\Im(f)\to J$. Since $J$ is injective and $f_2$ a monomorphism, we obtain a lifting $w\:B\to J$ of $v$ to $B$, thus $v = f_2w$. Altogether we see that $h_2 = f_1v = f_1 f_2 w = fw$. This shows that the following diagram commutes. $$\CD
\ssize A @>[ f \; e \; 0]>> \ssize B\oplus \I\Ker (f) \oplus J \cr
@| @VV{\left[\smallmatrix 1 & 0 &w\cr 0 & 1 & 0\cr 0 & 0 & 1 \endsmallmatrix\right]} V\cr
\ssize A @>[ f \; e \; h_2]>> \ssize B\oplus \I\Ker(f) \oplus J
\endCD$$
[**Claim 3.**]{} [*Suppose that $B$ has no nonzero injective direct summands and that $f,g\:A\to B$ are maps such that $g-f$ factorizes through an injective object. Then $\I\Ker(f)=\I\Ker(g)$.* ]{}
[*Proof:*]{} (a) First we show that if a map $h:A \to B$ factorizes through an injective object, say $h=h_1h_2$ where $h_1\:A\to I$ and $h_2\:I\to B $, then $\Soc\Ker(h)=\Soc (A)$. Indeed, if there were a simple submodule $S$ of $A$ such that the composition $$S\lto{\text{incl}} A \lto{h_1} I \lto{h_2} B$$ is nonzero then we would obtain that $B$ has a nonzero injective direct summand — in contradiction to our assumption on $B$.
\(b) As a consequence we obtain that if $g-f$ factorizes through an injective object, then $\Soc\Ker(f)=\Soc\Ker(g)$ holds. By (a), $\Soc(A)\subseteq \Ker (g-f)$ and hence $$\Ker(f)\cap \Soc(A)\; = \; \Ker(f+(g-f))\cap \Soc(A) \;=\;
\Ker(g) \cap \Soc (A).$$ Thus, $\I\Ker(f)=\I\Soc\Ker(f) = \I\Soc\Ker(g) = \I\Ker(g)$ holds.
[**Proposition 4.1.**]{} [($\Mimo$ independent of maps which factorize through injective)]{}
[ *Suppose that $B$ has no nonzero injective direct summands. Let $f,g\:A\to B$ be maps in $\lamod$ such that $g-f$ factorizes through an injective $\Lambda$-module. Then the objects $\Mimo(f)$, $\Mimo (g)$ are isomorphic in $\Cal S(\Lambda)$.* ]{}
[*Proof:*]{} Let $h\:A\to \I(A)$ be an injective envelope, then the objects $[f \; h]$, $[g \; h]$ are isomorphic by Claim 1. According to Claim 2, there exist injective modules $J_1$, $J_2$, extensions $e_1\:A\to \I\Ker(f)$, $e_2\:A\to \I\Ker(g)$ of the inclusion maps $\Ker(f)\to \I\Ker(f)$, $\Ker(g)\to \I\Ker(g)$, respectively, and isomorphisms $d_1$, $d_2$ such that the diagram below is commutative. $$\CD
\ssize A @>>[f \; e_1 \; 0]> \ssize B\oplus\I\Ker(f)\oplus J_1 \cr
@| @VVd_1V \cr
\ssize A @>>[f \; h ]> \ssize B\oplus \I(A) \cr
@| @VV\cong V \cr
\ssize A @>>[g \; h ]> \ssize B \oplus \I(A) \cr
@| @AAd_2A \cr
\ssize A @>>[g \; e_2 \; 0]> \ssize B \oplus\I\Ker(g)\oplus J_2
\endCD$$ According to Claim 3, the $\Lambda$-modules $\I\Ker(f)$, $\I\Ker(g)$ are isomorphic. By the Krull-Remak-Schmidt theorem for $\Lambda$-modules, $J_1\cong J_2$ follows. Note that the top row in the diagram when considered as an object in $\Cal H(\Lambda)$ is isomorphic to the direct sum $\Mimo(f)\oplus (0\to J_1)$ while the bottom row is isomorphic to $\Mimo(g)\oplus (0\to J_2)$. Applying the Krull-Remak-Schmidt theorem in the category $\Cal H(\Lambda)$ we obtain that $\Mimo(f)\cong \Mimo (g)$.
The following example shows that the condition that $B$ has no nonzero injective direct summands cannot be omitted.
[*Example.*]{} Let $\Lambda$ be a uniserial ring of Loewy length 2 with $m$ a generator of the maximal ideal $\m$. Denote by $\mu_m\:\Lambda\to\Lambda$ the multiplication by $m$. With $f=1\:\Lambda\to\Lambda$ and $g=\mu_m\:\Lambda\to \Lambda$, clearly $g-f$ factorizes through an injective $\Lambda$-module but $\Mimo(f)=f$ and $\Mimo(g)=\big(\Lambda\lto{[\mu_2\;1]}\Lambda
\oplus \Lambda\big)$ are not isomorphic as $\Mimo(g)\cong f\oplus
(0\to\Lambda)$.
The criterion in Proposition 4.1 for $\Mimo(f)\cong\Mimo(g)$ can be refined to obtain an equivalent condition.
[**Theorem 4.2.**]{} [(An equivalent condition for $\Mimo(f)\cong\Mimo(g)$)]{}
[*Proof:*]{} Assume first that $\Mimo(f)\:A\lto{[f\;e_1]}B\oplus I$ and $\Mimo(g)\:A\lto{[g\;e_2]}B\oplus I$ are isomorphic. Then there are maps $a\in\Aut A$ and $h=\Big({b \atop h_{12}}\;{h_{21}\atop h_{22}}\Big)
\in\Aut (B\oplus I)$ such that $[f\;e_1]h=a[g\;e_2]$. Thus, $ag=fb+e_1h_{12}$, so $ag-fb$ factorizes through the injective $\Lambda$-module $I$. Moreover, as $B$ and $I$ have no indecomposable direct summands in common, it follows that the map $h$ is an automorphism if and only if both $b$ and $h_{22}$ are automorphisms.
For the converse, assume that $fb-ag$ factorizes through an injective $\Lambda$-module. By Proposition 4.1, the objects $\Mimo(fb)$ and $\Mimo(ag)$ are isomorphic; assume they are given by maps $$\Mimo(fb)=[fb\;\;e_1], \;
\Mimo(ag)=[ag\;\;e_2]\:
\qquad A\longrightarrow B\oplus I.$$ Thus, $\Mimo(f)\cong\Mimo(g)$, as indicated by the following diagram. $$\CD
\Mimo(f)\:\qquad @. \ssize A @>[f\;e_1]>>
\ssize B\oplus I \cr
@. @| @VV\Big[{b\atop 0}\;{0\atop 1}\Big]V \cr
\Mimo(fb)\:\qquad @. \ssize A @>[fb\;\;e_1]>>
\ssize B\oplus I \cr
@. @| @VV\cong V \cr
\Mimo(ag)\:\qquad @. \ssize A @>[ag\;\;e_2]>>
\ssize B\oplus I \cr
@. @Va VV @| \cr
\Mimo(g)\:\qquad @. \ssize A @>[g\;\;a^{-1}e_2]>>
\ssize B\oplus I \cr
\endCD$$ For the last step note that if $e_2\:A\to I$ is an extension to $A$ of an injective envelope for $\Ker(ag)$, then since $\Ker(ag)=a^{-1}\Ker(g)$, one obtains that $a^{-1}e_2$ is an extension to $A$ of an injective envelope for $\Ker (g)$. By Lemma 2.2, the object $([g\;\;a^{-1}e_2]\:A\to B\oplus I)$ is $\Mimo(g)$, up to isomorphism.
There is the following dual result for minimal epimorphisms.
[**Theorem 4.3.**]{} [(An equivalent condition for $\Mepi(f)\cong\Mepi(g)$)]{}
**5. The Auslander-Reiten Translation**
In Chapter 3 we have seen how the Auslander-Reiten translations in the categories $\Cal H(\Lambda)$, $\Cal S(\Lambda)$, and $\Cal F(\Lambda)$ are related. Using this, we develop a formula to compute the Auslander-Reiten translate $\tau_{\Cal S}(f)$ of an object $(f\:A\to B)$ in $\Cal S(\Lambda)$ directly in the category $\lamod$. Here is the main result:
[**Theorem 5.1.**]{} [(The Auslander-Reiten translation in $\Cal S(\Lambda)$)]{} $$\tau_{\Cal S}(f) = \Mimo\tau_\Lambda\Cok(f).$$ Note that this means the following: We start with the monomorphism $f$ and form its cokernel $g$. We apply $\tau_\Lambda$ to this map. Recall that $\tau_\Lambda(g)$ is only well-defined in the category $\overlamod$ (obtained from $\lamod$ by factoring out all the maps which factorize through an injective object). Represent $\tau_\Lambda(g)$ by a morphism $h\:D\to E$ in $\lamod$ such that $D$ and $E$ have no nonzero injective direct summands. Now apply $\Mimo.$ As we have seen in Chapter 4, this yields a well-defined object in $\Cal S(\Lambda)$, up to isomorphism.
[*Proof:*]{} We proceed as follows. Let $(f\:A\to B)$ be an object in $\Cal S(\Lambda)$ with cokernel $(g\:B\to C)$. In Step 1 we obtain an exact sequence defining $\tau_\Lambda(g)$; in Step 2 we construct an exact sequence involving $\tau_{\Cal H}(g)$, from which $\tau_{\Cal S}(f)$ is computed as $\Ker\tau_{\Cal H}(g)$ by Corollary 3.3. The formula for the computation of a kernel in Step 3 is used in Step 4 to relate the two exact sequences, and finally, in Step 5, we obtain our result.
[*Step 1*]{}. For an epimorphism $g\: B\to C$ in $\lamod$ we determine $\tau_\Lambda(g)$. We start with a minimal projective presentation of $B$, say $$Q\;\lto d \; P \;\lto e \;B\;\longrightarrow\; 0$$ and a projective presentation of $C$ using the map $eg\:P\to C$ and a projective cover $t\: R\to \Ker(eg)$ to obtain the following commutative diagram with exact rows. $$\CD
\ssize Q @>d>> \ssize P @>e>> \ssize B @>>> \ssize 0 \cr
@VsVV @VV1V @VVgV \cr
\ssize R @>>t> \ssize P @>>eg> \ssize C @>>> \ssize 0 \cr
\endCD$$ By applying the Nakayama functor $\nu_\Lambda=\nu=D\Hom_\Lambda(-,{_\Lambda\Lambda})$ in $\lamod$ to the left square we arrive at the diagram defining $\tau_\Lambda(g)$: $$\CD
\ssize 0 @>>> \ssize \tau_\Lambda B @>v>> \ssize \nu Q @>\nu d >>
\ssize \nu P \cr
@. @V\tau_\Lambda(g)VV @VV \nu s V @VV 1V \cr
\ssize 0 @>>> \ssize \tau_\Lambda C @>>w> \ssize \nu R @>>\nu t >
\ssize \nu P
\endCD$$ (Since $eg$ is not necessarily a projective cover, $\nu t$ is not necessarily an injective envelope.)
[*Step 2*]{}. From a projective presentation for $g$ in the category $\Cal H(\Lambda)$ we construct an exact sequence which involves $\tau_{\Cal H}(g)$: Being an epimorphism, the object $g$ has a projective cover given by $(1_P\: P\to P)$ with $P$ as in Step 1. Using the maps $d$ and $t$ we obtain the following projective presentation for $g$ in $\Cal H(\Lambda)$. $$\CD
\hobject QQ1 \oplus \hobject 0R{ }
@>{\left[{d\atop 0}\right]}>{\left[{d\atop t}\right]}> \hobject PP1
@>e>eg> \hobject B{C}{g} @>>> 0
\endCD$$ Note that this differs in general from a minimal projective presentation; in order to obtain a minimal one, we would have to split off a direct summand of the form $(0\to S)$ where $S$ is a direct summand of $R$ (and $S$ is also a direct summand of $Q$). Applying the Nakayama functor $\nu_{\Cal H}$ (see Lemma 1.3-N) to the morphism between the projective modules, we obtain the following sequence $$\CD
0 @>>> \tau_{\Cal H}\Big( \!\!\!\hobject B{C}{g}\!\!\Big)
\oplus \hobject{\nu S}{\nu S}1 @>>> \hobject{\nu Q}0{} \oplus
\hobject{\nu R}{\nu R}1 @>{\left[{\nu d\atop \nu t}\right]}>0>
\hobject{\nu P}0{}
\endCD$$ in which the additional projective summand $(0\to S)$ gives rise to the injective direct summand $(1_{\nu S}\: \nu S\to \nu S)$.
[*Step 3*]{}. For the computation of the kernel of a map $u\: U\to V$ we observe that if $$\CD 0 @>>> \hobject{U'}0{} @>u'>0> \hobject U{V}u @>>>
\hobject {W'}Ww \endCD$$ is an exact sequence in $\Cal H(\Lambda)$ with $(w\:W'\to W)$ in $\Cal S(\Lambda)$, then $\Ker u=u'$.
[*Step 4*]{}. In the following diagram, the right two columns are split exact and form a commutative diagram in $\Cal H(\Lambda)$; the left column is just the kernel sequence. Note that the sequence in Step 2 involving $\tau_{\Cal H}(g)$ occurs as the middle row, while the sequence in Step 1 defining $\tau_\Lambda B$ occurs as the sequence of source modules in the top row. $$\CD
@. 0 @. 0 @. 0 \cr
@. @VVV @VVV @VVV \cr
0 @>>> \hobject{\tau_\Lambda B}0{} @>v>0> \hobject{\nu Q}0{}
@>\nu d>0> \hobject{\nu P}0{} \cr
@. @VV{\gamma \atop 0}V @VV{ [1 \; 0]\atop 0}V @VV{ 1\atop 0 }V \cr
0 @>>> \tau_{\Cal H}\Big( \!\!\!\hobject B{C}{g}\!\!\Big)
\oplus \hobject{\nu S}{\nu S}1
@>>> \hobject{\nu Q}0{}\oplus \hobject{\nu R}{\nu R}1
@>{\left[{\nu d\atop \nu t}\right]}>0>
\hobject{\nu P}0{} \cr
@. @VVV @VV{\left[{0\atop 1}\right]\atop\left[{0\atop 1}\right]}V @VVV \cr
0 @>>> \hobject {\nu R}{\nu R}1 @>1>1> \hobject {\nu R}{\nu R}1 @>>> 0 \cr
@. @. @VVV @VVV \cr
@. @. 0 @. 0 \cr
\endCD$$ Since the object $1_{\nu R}\:\nu R\to \nu R$ is in $\Cal S(\Lambda)$, we can use the left hand column for the computation of $\Ker \tau_{\Cal H}(g)$, using Step 3: $$\Ker\left(\tau_{\Cal H}(g)\oplus (1_{\nu S}\:\nu S\to \nu S)\right)=\gamma.$$ In order to identify $\gamma$, only the first entries in the above diagram play a role, thus we have to deal with the following diagram. $$(*)\qquad \CD
\ssize 0 @>>> \ssize \tau_\Lambda B @>v>> \ssize \nu Q @>\nu d>>
\ssize \nu P \cr
@. @VV\gamma V @VV[1\;0]V @| \cr
\ssize 0 @>>> \ssize X @>>> \ssize \nu Q\oplus \nu R
@>>\left[{\nu d\atop \nu t}\right]>\ssize\nu P
\endCD$$ where $X$ is the first component of $\tau_{\Cal H}(g)\oplus(1_{\nu S}\:\nu S\to \nu S)$. Compare this with the following diagram. $$(**)\qquad\CD
\ssize 0 @>>>\ssize \tau_\Lambda B @>v>> \ssize \nu Q @>\nu d>>\ssize\nu P \cr
@. @VV[\tau_\Lambda(g)\;v]V @VV[1\;0]V @| \cr
\ssize 0 @>>> \ssize \tau_\Lambda C\oplus \nu Q
@>>{\left[{0\atop 1}{w\atop -\nu s}\right]}>
\ssize\nu Q\oplus \nu R @>>{\left[{\nu d\atop \nu t}\right]}> \ssize \nu P
\endCD$$ A glance back to Step 1 shows that the diagram is commutative and has an exact upper row. It only has to be confirmed that the lower row is exact. Clearly, the $2\times 2$-matrix is a monomorphism, and the composition of the last two maps is zero. Now, given $(x,y)\in\nu Q\oplus \nu R$ with $(x)\nu d+(y)\nu t = 0$, then $\nu d=\nu s\,\nu t$ yields that $((x)\nu s+ y)\nu t=0$, therefore $(x)\nu s+y$ is in the image of $w$: There is $z\in\tau_\Lambda C$ such that $(z)w=(x)\nu s+y$ and hence $$(z,x)\left[\smallmatrix 0 & w\cr 1 & -\nu s\endsmallmatrix\right]\;=\;
(x, (z)w-(x)\nu s) \;=\; (x,y).$$ Comparing the two diagrams labelled $(*)$ and $(**)$ we obtain the following isomorphism. $$\hobject {\tau_\Lambda B}X\gamma \quad\cong\quad
\hobject{\tau_\Lambda B}{\tau_\Lambda C\oplus \nu Q}{[\tau_\Lambda (g)
\;v]}$$
[*Step 5*]{}. Write $\tau_{\Cal H}(g)=(h\: E\to F)$, so $\tau_{\Cal S}(f)=\Ker \tau_{\Cal H}(g)=(j\:D\to E)$ where $j=\Ker(h)$. Thus the object given by $\gamma = \Ker (\tau_{\Cal H}(g)\oplus(1_{\nu S}\:\nu S\to \nu S))$ in the previous step has the form $$\hobject{D}E{j}\oplus\hobject 0{\nu S}{}
\quad\cong\quad \hobject{\tau_\Lambda B}X\gamma
\quad\cong
\hobject{\tau_\Lambda B}{\tau_\Lambda C\oplus \nu Q}{[\tau_\Lambda (g)
\;v]},$$ where the second isomorphism has been established in Step 4. By Claim 2 in the previous section, the right hand side is isomorphic to $\Mimo\tau_\Lambda(g)$, up to an injective direct summand. Since neither $j$ nor $\Mimo\tau_\Lambda(g)$ has an injective direct summand — recall that $j$ is a $\tau_{\Cal S}$-translate — the Krull-Remak-Schmidt theorem in $\Cal S(\Lambda)$ implies that $\tau_{\Cal S}(f)=j=\Mimo\tau_\Lambda(g)$.
There is also the following dual version.
[**Theorem 5.2.**]{} [(The Auslander-Reiten translation in $\Cal F(\Lambda)$)]{}
[*Example.*]{} Let $\Lambda$ be a uniserial ring with maximal ideal $\m$, Loewy length $n$ and socle $k=\m^{n-1}$, as in the Example in Chapter 1. Recall that the projective-injective indecomposable $(0\to\Lambda)$ in the category $\Cal S(\Lambda)$ has as sink map the inclusion $(0\to \m)\to(0\to\Lambda)$ and as source map the inclusion $(0\to \Lambda)\to(k\to\Lambda)$, so $\tau_{\Cal S}(k\to\Lambda)=(0\to\m)$. We illustrate the formula in 5.1 by computing the powers of the Auslander-Reiten translation for the module $(k\to\Lambda)$. First, $$\tau_{\Cal S}\Big(\hobject k\Lambda{}\Big)
= \Mimo\tau_\Lambda\Cok\Big(\hobject k\Lambda{}\Big)
= \Mimo\tau_\Lambda\Big(\hobject\Lambda{\Lambda/k}{}\Big)
= \Mimo\Big(\hobject0{\Lambda/k}{}\Big)
= \Big(\hobject0{\Lambda/k}{}\Big)
\cong \Big(\hobject0\m{}\Big)$$ confirming the above result. Further translates are computed easily: $$\eqalign{
\tau_{\Cal S}\Big(\hobject0\m{}\Big)
&=\Mimo\tau_\Lambda\Cok\Big(\hobject0\m{}\Big)
= \Mimo\tau_\Lambda\Big(\hobject\m\m{}\Big)
= \Big(\hobject\m\m{}\Big) \cr
\tau_{\Cal S}\Big(\hobject\m\m{}\Big)
&= \Mimo\tau_\Lambda\Big(\hobject\m0{}\Big)
= \Mimo\Big(\hobject\m0{}\Big)
= \Big(\hobject\m\Lambda{}\Big) \cr
\tau_{\Cal S}\Big(\hobject\m\Lambda{}\Big)
&= \Mimo\tau_\Lambda\Big(\hobject\Lambda{\Lambda/\m}{}\Big)
= \Mimo\Big(\hobject0{\Lambda/\m}{}\Big)
\cong \Big(\hobject0k{}\Big) \cr
\tau_{\Cal S}\Big(\hobject0k{}\Big)
&= \Mimo\tau_\Lambda\Big(\hobject kk{}\Big)
= \Big(\hobject kk{}\Big) \cr
\tau_{\Cal S}\Big(\hobject kk{}\Big)
&= \Mimo\tau_\Lambda\Big(\hobject k0{}\Big)
= \Big(\hobject k\Lambda{}\Big) \cr
}$$ So after six steps we are back at where we started. This is not a coincidence, as we will see in the next chapter.
[*Definition:*]{} Let $\Cal S(\Lambda)_{\Cal I}$ be the full subcategory of $\Cal S(\Lambda)$ consisting of all objects which have no nonzero injective direct summands.
[**Corollary 5.3.**]{} [ *Every object in $\Cal S(\Lambda)_{\Cal I}$ has the form $\Mimo(f)$ for some morphism $f\:A\to B$ where the $\Lambda$-modules $A$ and $B$ have no nonzero injective direct summands.*]{}
[*Definition:*]{} By $\Cal H'(\Lambda)$ we denote the morphism category for $\underlamod$. Thus, the objects are the morphisms $(\bar f\:A\to B)$ in $\underlamod$, and a morphism in $\Cal H'(\Lambda)$ from $(\bar f\:A\to B)$ to $(\bar{f'}\:A'\to B')$ is given by a pair $(\bar a,\bar b)$ where $\bar a\:A\to A'$, $\bar b\:B'\to B$ are morphisms such that $\bar f\bar b=\bar a\bar {f'}$ holds in $\underlamod$.
[**Corollary 5.4.**]{} [ *The functor $$F\:\Cal S(\Lambda)_{\Cal I}\longrightarrow \Cal H'(\Lambda),
\qquad f\mapsto \bar f,$$ is dense, preserves indecomposables, and reflects isomorphisms. Thus, $F$ is a representation equivalence.* ]{}
[*Proof:*]{} Clearly, $F$ is dense: If the object $\bar f$ in $\Cal H'(\Lambda)$ is represented by $f$, then $F\Mimo(f)$ and $\bar f$ are isomorphic in $\Cal H'(\Lambda)$.
The functor $F$ preserves indecomposables: Let $a$ be an indecomposable noninjective object in $\Cal S(\Lambda)$, then $a$ occurs as the first term of an Auslander-Reiten sequence and hence has the form $a=\tau_{\Cal S}(c)$ for some indecomposable nonprojective object $c\in\Cal S(\Lambda)$. In particular, $F(a)=F\Mimo\big(\tau_\Lambda\Cok(c)\big)$ is isomorphic to $\tau_\Lambda\Cok(c)$ in $\Cal H'(\Lambda)$ which is an indecomposable object since $\Mimo$ is additive.
To show that $F$ reflects isomorphisms, let two objects $f,g\in\Cal S(\Lambda)$ be given such that $F(f)$ and $F(g)$ are isomorphic in $\Cal H'(\Lambda)$. By Corollary 5.3, $f=\Mimo(f_1)$ and $g=\Mimo(g_1)$ where $f_1\:A_1\to B_1$ and $g_1\:C_1\to D_1$ are maps between $\Lambda$-modules with no nonzero injective direct summands. Since $\bar{f_1}$ and $\bar{g_1}$ are isomorphic in $\Cal H'(\Lambda)$ there are isomorphisms of $\Lambda$-modules $u\:A_1\to C_1$ and $v\:B_1\to D_1$ such that $f_1v-ug_1$ factorizes through an injective $\Lambda$-module. By Theorem 4.2, $f=\Mimo(f_1)$ and $g=\Mimo(g_1)$ are isomorphic.
One can go a little bit further.
[*Definition:*]{} Let $\Cal I$ be the ideal in the category $\Cal S(\Lambda)$ of all morphisms which factorize through an injective object in $\Cal S(\Lambda)$. By $\Cal S(\Lambda)/\Cal I$ we denote the factor category of $\Cal S(\Lambda)$ modulo $\Cal I$. Clearly, the functor $F\:\Cal S(\Lambda)\to \Cal H'(\Lambda), f\mapsto \bar f$, annihilates every morphism in $\Cal I$ and hence defines a functor $\bar F\:\Cal S(\Lambda)/\Cal I\to \Cal H'(\Lambda)$.
[**Lemma 5.5.**]{} [*The functor $\bar F$ is full and dense.*]{}
[*Proof:*]{} Since the dense functor $F$ from Corollary 5.4 factorizes over $\bar F$, also $\bar F$ is dense. To show that $\bar F$ is full, let $f\:A\to B$ and $g\:C\to D$ be objects in $\Cal S(\Lambda)$, let $e\:A\to I$ be an injective envelope and $e'\:B\to I$ an extension of $e$. If $(\bar u,\bar v)$ is a morphism in $\Hom_{\Cal H'}(\bar f,\bar g)$ then there are maps $u\:A\to C$, $v\:B\to D$, $w\:I\to D$ such that $ug-fv=ew$. The following diagram is commutative $$\CD A @>f>> B \cr @| @VV[1\;e']V \cr A @>[f\;e]>> B\oplus I \cr
@VuVV @VV\big[{v\atop w}\big]V \cr C @>>g> D \endCD$$ and hence represents two morphisms $\CD f@>\big(1_A,[1_B\;e']\big)>\phantom{)}> [f\;e] @>\big(u,\big[{u\atop w}\big]\big)>>g
\endCD$ in the category $\Cal S(\Lambda)$. (Note that the first is a split monomorphism, so that $f$ and $[f\;e]$ become isomorphic objects in the factor category $\Cal S(\Lambda)/\Cal I$.) By applying the functor $F$ to the composition of the two morphisms, we obtain $(\bar u,\bar v)$.
[*Remark:*]{} The following example shows that the functor $\bar F$ is not a categorical equivalence in general. Let $\Lambda$ be a uniserial ring of Loewy length 3 and $m$ a generator of the maximal ideal $\m$. Let $f$ be the object in $\Cal S(\Lambda)$ given by the inclusion $f\:\m/\m^2\to \Lambda/\m^2$. Then the multiplication $\mu_m$ by $m$ is a nonzero nilpotent endomorphism of $f$ which does not factorize through an injective object in $\Cal S(\Lambda)$, but for which $F(\mu_m)=0$ holds. Thus, the functor $\bar F$ is not faithful.
**6. Morphisms in the Stable Category**
In this section we assume that $\Lambda$ is a self-injective algebra. Then the stable category $\underlamod$ modulo all morphisms which factorize through a projective-injective object, together with the suspension functor $\Omega^{-1}$, becomes a triangulated category. We recall the construction of triangles in this category, observe that the assignment $f\mapsto \Cok\Mimo(f)$ for a morphism $f\:A\to B$ is related to the rotation of a triangle in the stable category, and retrieve the formula $\tau_{\Cal S}^6(f)\cong f$ for the case that $\Lambda$ is a uniserial algebra.
We recall from \[H, Chapter I\] that the stable category $\underlamod$ has the following standard triangles. For a morphism $f\:A\to B$ in $\lamod$, take the short exact sequence given by the inclusion of $A$ in its injective envelope, $i\:A\to \I(A)$, and the cokernel map $c\:\I(A)\to \Omega^{-1}A$, and form the push-out diagram along $f$: $$\CD 0 @>>> A @>i>> \I(A) @>c>> \Omega^{-1}A @>>> 0 \cr
@. @VfVV @VVjV @| \cr
0 @>>> B @>>g> C @>>h> \Omega^{-1}A @>>> 0 \endCD$$ Then the standard triangle corresponding to $f$ is $$T(f)\: \qquad A \lto{\bar f} B\lto{\bar g} C \lto{\bar h}
\Omega^{-1}A$$ and each triangle in the stable category is isomorphic to a standard one \[H, Theorem I.2.6\].
A morphism between triangles $T\:A\to B\to C\to \Omega^{-1}A$ and $T'\: A'\to B'\to C'\to \Omega^{-1}A'$ consists of morphisms $\bar a$, $\bar b$, $\bar c$ in the stable category which make the following diagram commutative. $$\CD A @>\bar f>> B @>>> C @>>> \Omega^{-1}A \cr
@V\bar aVV @V\bar bVV @V\bar cVV @VV\Omega^{-1}\bar aV \cr
A' @>>\bar{f'}> B' @>>\bar{g'}> C'@>>> \Omega^{-1}A'
\endCD$$ The two triangles are isomorphic if $\bar a$, $\bar b$, and $\bar c$ are isomorphisms. From \[H, I.1.2 and I.1.6\] we recall that for the existence of a morphism between the triangles it is sufficient to have a map $b\: B\to B'$ such that $\overline{fbg'}=0$. Moreover, a morphism $(\bar a, \bar b, \bar c)$ is an isomorphism if $\bar a$ and $\bar b$ are isomorphisms in $\underlamod$. Thus, isomorphisms between triangles are obtained from pairs $(\bar a,\bar b)$ of isomorphisms which make the left square in the above diagram commutative. This is to say that if $(\bar a,\bar b)\: \big(A\sto{\bar f} B\big)\to
\big(A'\sto{\bar{f'}}B'\big)$ is an isomorphism in the category $\Cal H'(\Lambda)$ of morphisms in the stable category $\underlamod$, then the triangles $T(f)$ and $T(f')$ are isomorphic.
Given a triangle $T\: A\sto{\bar f} B\sto{\bar g}C\sto{\bar h}\Omega^{-1}A$, then the [*rotation*]{} of $T$ below also is a triangle, as required by the axioms of a triangulated category. $$T^\R\:\qquad
B\sto{\bar g}C\sto{\bar h}\Omega^{-1}A\lto{-\Omega^{-1}\bar f}
\Omega^{-1}B$$ This operation $T(f)\mapsto T(f)^\R$ in the triangulated category gives rise to a selfequivalence $\bar f\mapsto {\bar f}^\R$ on the category $\Cal H'(\Lambda)$ of morphisms in the stable category. According to the following Lemma, a map $g$ in $\lamod$ representing ${\bar f}^\R$ in $\underlamod$ is obtained as $g=\Cok\Mimo f$.
[**Lemma 6.1.**]{} [ *For a map $f\:A\to B$ in $\lamod$, the two morphisms ${\bar f}^\R$ and $\overline{\Cok\Mimo f}$ are isomorphic in $\Cal H'(\Lambda)$.*]{}
[*Proof:*]{} In the following diagram with exact rows, the top row defines the cokernel map for $\Mimo(f)=[f\;e]$ while the third row is given by the push-out diagram defining $T(f)$. $$\CD 0 @>>> A @>[f\;e]>> B\oplus \I\Ker (f)@>\Cok>>C' @>>> 0 \cr
@. @| @V{\big[{1\atop0}{0\atop1}{0\atop0}\big]}VV @V[1\;0]VV \cr
0 @>>> A @>[f\;e\;0]>> B\oplus \I\Ker (f)\oplus I' @>>> C'\oplus I' @>>> 0 \cr
@. @| @VdV\cong V @Vd'V\cong V \cr
0@>>> A @>[f\;i]>> B\oplus \I(A)@>\big[{g\atop -j}\big]>> C @>>> 0 \cr
@. @. @V{\big[{1\atop0}\big]}VV @| \cr
@. @. B @>g>> C \endCD$$ All squares with the possible exception of the square at the bottom are commutative: Since the map $[f\;i]$ is a monomorphism it follows from Claim 2 in Chapter 4 that there is an injective module $I'$ and an isomorphism $d$ such that the left square between the second and third row is commutative; if $d'$ is the cokernel map then both squares are commutative. Here the exact sequence in the first row is a direct summand of the sequence in the second row, the complement being the sequence $0\to 0\to I'\sto1I'\to 0$. Finally, the map $g$ which represents the second map in the triangle $T(f)$ occurs as a restriction of the cokernel map in the third row.
Note that all vertical maps become isomorphisms when considered in the stable category. Thus, the two morphisms $\overline\Cok\:B\oplus\I\Ker (f)\to C'$ and $\bar g\:B\to C$ are isomorphic when considered as objects in $\Cal H'(\Lambda)$.
Since $\Lambda$ is a self-injective algebra, there is a third selfequivalence on the morphism category $\Cal H'(\Lambda)$ (besides the suspension and the rotation) given by the Auslander-Reiten translation $\tau_\Lambda$. According to \[ARS, Proposition IV.3.7\], the functors $\tau_\Lambda$ and $\Cal N\Omega^2$ from $\underlamod$ to $\underlamod$ are isomorphic, where $\Cal N=D\Hom_\Lambda(-,\Lambda)$ is the Nakayama automorphism. It follows that the Auslander-Reiten translation $\tau_\Lambda$ preserves triangles. As a consequence, the functor $\tau_\Lambda$ commutes with the rotation $\bar f\mapsto {\bar f}^\R$, and also with the suspension $\Omega^{-1}$ in the sense that for each morphism $f\:A\to B$ there is a commutative diagram in which the vertical maps are isomorphisms. $$\CD \tau_\Lambda\Omega^{-1}A @>\tau_\Lambda\Omega^{-1}f>>
\tau_\Lambda\Omega^{-1}B \cr
@V\eta'(f)V\cong V @V\eta''(f)V\cong V \cr
\Omega^{-1}\tau_\Lambda A @>>\Omega^{-1}\tau_\Lambda f>
\Omega^{-1}\tau_\Lambda B \endCD$$
The selfequivalence on the triangulated category given by the rotation $T\mapsto T^\R$ yields isomorphisms $T^{3\R}\cong -\Omega^{-1}T$ and $T^{6\R}\cong \Omega^{-2}T$. We obtain the following consequence for the Auslander-Reiten translation $\tau_{\Cal S}$ in the submodule category.
[**Theorem 6.2.**]{} [*Suppose $\Lambda$ is a self-injective algebra. If $(f\: A\to B)$ is an indecomposable nonprojective object in $\Cal S(\Lambda)$ then there are the following isomorphisms in the morphism category $\Cal H'(\Lambda)$.*]{} $$\eqalign{(1)& \qquad\overline{\tau_{\Cal S}(f)} \;\cong\;
\tau_\Lambda\big(\overline{\Cok f}\big),\cr
(2)& \qquad\overline{\tau_{\Cal S}^3(f)} \;\cong\;
-\tau_\Lambda^3\Omega^{-1}(\bar f), \;\text{and} \cr
(3)& \qquad\overline{\tau_{\Cal S}^6(f)} \;\cong\;
\tau_\Lambda^6\Omega^{-2}(\bar f).}$$
[**Corollary 6.3.**]{} [*Under the assumptions of the theorem, there are the following isomorphisms in the submodule category $\Cal S(\Lambda)$: $$\tau_S^3(f)\;\cong\;-\Mimo\tau_\Lambda^3\Omega^{-1}(f),\qquad
\tau_S^6(f)\;\cong\;\Mimo\tau_\Lambda^6\Omega^{-2}(f)$$* ]{} [*Proof of Corollary 6.3:*]{} The functor $\Cal S(\Lambda)_I\to \Cal H'(\Lambda)$, $f\mapsto \bar f$ in Corollary 5.3 reflects isomorphisms, so the assertion follows from Theorem 6.2.
[*Proof of Theorem 6.2.:*]{} The first statement (1) follows from Theorem 5.1 since any map $g$ is stably equivalent to $\Mimo (g)$. For the proof of assertion (2) we use Theorem 5.1 to compute $$\tau_{\Cal S}^3\big(A\sto fB\big)\;=\;\Mimo\tau_\Lambda\Cok
\Mimo\tau_\Lambda\Cok\Mimo\tau_\Lambda\Cok(f).$$ Then we obtain the following isomorphisms of objects in $\Cal H'(\Lambda)$: $$\eqalign{ \overline{\tau_{\Cal S}^3(f)}\;
&\cong\; \tau_\Lambda\big(\overline{\Cok\Mimo\tau_\Lambda\Cok \Mimo
\tau_\Lambda\Cok f}\big) \cr
&\cong\; \tau_\Lambda\big(
\overline{\tau_\Lambda\Cok\Mimo\tau_\Lambda\Cok f}^\R\big) \cr
&\cong\; \tau_\Lambda^2 \big(
\overline{\Cok\Mimo\tau_\Lambda\Cok f}^\R\big) \cr
&\cong\; \tau_\Lambda^2 \big(
\overline{\tau_\Lambda\Cok f}^{2\R} \big) \cr
&\cong\; \tau_\Lambda^3 \big( \overline{\Cok f}^{2\R} \big) \cr
&\cong\; \tau_\Lambda^3 \big( {\bar f}^{3\R} \big) \cr
&\cong\; \tau_\Lambda^3 \big(-\Omega^{-1}(\bar f)\;\big)
}$$ where the first isomorphism is justified by (1), the second, fourth, and sixth isomorphisms follow from Lemma 6.1, the third and fifth equalities are come from the commutativity of $\tau_\Lambda$ with the rotation, and the last map is an isomorphism since a threefold rotation of a triangle is obtained by applying the functor $-\Omega^{-1}$.
In order to deduce the third assertion from the second, pick a representative map $g\:\Omega^{-1}A\to\Omega^{-1}B$ in $\lamod$ for the morphism $\overline{\tau_{\Cal S}^3(f)}$ in $\underlamod$ such that $\Omega^{-1}A$ and $\Omega^{-1}B$ have no nonzero injective direct summands. Then $\Mimo(g)$ is an indecomposable nonprojective object in $\Cal S(\Lambda)$ and by Proposition 4.1 its isomorphism class does not depend on the choice of the map $g$. The following morphisms in $\underlamod$ are isomorphic objects in $\Cal H'(\Lambda)$. $$\overline{\tau_{\Cal S}^6(f)}
\;\cong\; \overline{\tau_{\Cal S}^3\Mimo (g)}
\;\cong\; -\tau_\Lambda^3\Omega^{-1}(\bar g)
\;\cong\; -\tau_\Lambda^3\Omega^{-1}
\big(-\tau_\Lambda^3\Omega^{-1}(\bar f)\big)
\;\cong\; \tau_\Lambda^6\Omega^{-2}(\bar f).$$
We conclude this section with three applications.
[**Corollary 6.4.**]{} [*Suppose $\Lambda$ is a self-injective algebra such that $\tau_\Lambda$ coincides with $\Omega^2$. If $(f\: A\to B)$ is an indecomposable nonprojective object in $\Cal S(\Lambda)$, then there is an isomorphism of objects $$\tau_{\Cal S}^3(f) \cong
-\Mimo\Omega^{5} (f)$$ in $\Cal S(\Lambda)$.*]{}
[*Proof:*]{} Since $\tau_\Lambda$ coincides with $\Omega^2$, we can simplify the expression in formula 2 of Theorem 6.2 and see that $$\overline{\tau_{\Cal S}^3(f)} \cong
-\tau_\Lambda^3\Omega^{-1}(\bar f) \cong
-\Omega^6\Omega^{-1}(\bar f) \cong
-\Omega^5(\bar f)$$ in $\Cal H'(\Lambda)$. The functor $\Cal S(\Lambda)_{\Cal I}\to \Cal H'(\Lambda)$, $f\mapsto \bar f$, in Corollary 5.3 reflects isomorphisms, so $\tau_{\Cal S}^3(f)$ and $-\Omega^5(f)$ are isomorphic in $\Cal S(\Lambda)$.
Note that for any symmetric algebra, the functors $\tau_\Lambda$ and $\Omega^2$ coincide (see for example \[ARS, Proposition IV.3.8\]), thus we can apply Corollary 6.4 in this case.
[**Corollary 6.5.**]{} [*Let $\Lambda$ be a commutative uniserial algebra. Then for an indecomposable nonprojective object $(f\:A \to B)$ in $\Cal S(\Lambda)$, there is an isomorphism of objects $$\tau_{\Cal S}^6(f)\;\cong\; f$$ in $\Cal S(\Lambda).$*]{}
[*Proof:*]{} Since $\Lambda$ is a commutative uniserial algebra, all the functors $\tau_\Lambda,$ $\Omega^{2}$ and $\Omega^{-2}$ are equivalent to the identity functor on $\underlamod$, thus Corollary 6.4 shows that $\tau_{\Cal S}^6(f)$ and $\Omega^{10}(f)\; \cong\; f$ are isomorphic objects of $\Cal S(\Lambda)$.
[*Definition:*]{} By $\Bbb A_\infty^\infty$ we denote the doubly infinite linear quiver $$\cdots \quad \lfrom\alpha \bullet^{-1} \lfrom\alpha\bullet^0
\lfrom\alpha\bullet^1\lfrom\alpha \quad \cdots$$ The path algebra $k\Bbb A_\infty^\infty$ of this quiver is the associative $k$-algebra with basis the paths in $\Bbb A_\infty^\infty$. If $\alpha^n$ denotes the ideal spanned by all paths of length at least $n$, then the factor algebra $\Lambda=k\Bbb A_\infty^\infty/\alpha^n$ is a locally bounded associative $k$-algebra. A $\Lambda$-module $A$ consists of a sequence $(A_i)_{i\in\Bbb Z}$ of $k$-modules together with a sequence $(\alpha_i\:A_i\to A_{i-1})_{i\in\Bbb Z}$ of linear maps. By $A[\ell]$ we denote the [*shifted*]{} module given by the spaces $(A_{i-\ell})_i$ and the maps $(\alpha_{i-\ell})_i$.
[**Corollary 6.6.**]{} [ *Let $\Lambda$ be the associative algebra $k\Bbb A_\infty^\infty/\alpha^n$ where $k$ is a field. For an indecomposable nonprojective object $(f\:A\to B)$ in $\Cal S(\Lambda)$, the following formula holds.*]{} $$\tau_{\Cal S}^6(f)\;\cong\; f[n-6]$$
[*Proof:*]{} The Auslander-Reiten translation $\tau_\Lambda$ is given by the shift $A\mapsto A[-1]$ along the arrow $\alpha$, hence the functor $\tau_\Lambda$ on the stable category $\underlamod$ preserves triangles. Also, $\tau_\Lambda$ commutes with $\Omega^{-1}$ and with the rotation in a triangle. Moreover, for a nonprojective indecomposable $\Lambda$-module $A$, the process of taking the cokernel of the injective envelope twice yields the module $A[n]$. With these adjustments, the claim follows from (3) in Theorem 6.2 as in the proof of Corollary 6.4.
**7. Auslander-Reiten Sequences**
In this section we show that “most” Auslander-Reiten sequences in the category $\Cal S(\Lambda)$ become split exact sequences in the category $\lamod$, when restricted to the short exact sequence of the submodules, or to the short exact sequence of the big modules. We describe the exceptions in detail. The remaining sink and source maps are associated with the projective and the injective objects and have been specified in Chapter 1. We only need to assume here that $\Lambda$ is a locally bounded associative algebra. First we deal with the exceptions.
[**Proposition 7.1.**]{} [(Auslander-Reiten sequences with components not split exact)]{}
[*Proof:*]{} 1. We show that the map $(0,g)$ is minimal right almost split. Clearly, this map is right minimal and not a split epimorphism. In order to show that $(0,g)\: \big(A\sto fB\big) \to \big(0\to C\big)$ is right almost split, let $(t',t):(x'\:X'\to X)\to(0\to C)$ be a test map which is not a split epimorphism. Then $t\: X\to C$ is not a split epimorphism in $\lamod$, so there is $u\:X\to B$ such that $t=ug$. Since the composition $X'\sto{x'}X\sto uB\sto gC$ is zero, there is $u': X'\to A$ such that $x'u=u'f$. Thus, $(u',u)$ is a morphism which satisfies $(t',t)=(u',u)(0,g)$.
2\. A straightforward argument shows that the map $(g,\left[{0\atop 1}\right])$ is right minimal. We verify that a test map $(t',t)\: (x'\:X'\to X)\to(1_C\:C\to C)$ which is not a split epimorphism factorizes over $(g,\left[{0\atop 1}\right])$. Since $t'\:X'\to C$ is not a split epimorphism, there is $u'\: X'\to B$ such that $t'=u'g$. Let $u_1\:X\to \I(A)$ be an extension of $u'e'\:X'\to \I(A)$ to $X$ and put $u=[u_1\;t]\:X\to \I(A)\oplus C$. Then $(u',u)$ is a morphism in $\Cal S(\Lambda)$ such that $(t',t)=(u',u)(g,\left[{0\atop 1}\right])$.
The remaining Auslander-Reiten sequences in $\Cal S(\Lambda)$ are made up from two split exact sequences.
[**Proposition 7.2.**]{} [(Auslander-Reiten sequences with components split exact)]{}
[*Proof:*]{} Since the map $c$ is not a split monomorphism, the test maps $$(0,1)\:(0\to C)\to (C'\sto{c} C)\quad\text{and}\quad
(1,c)\:(C'\sto 1 C') \to (C'\sto{c}C)$$ are not split epimorphisms and hence factorize over $(g',g)$. Thus, both $g$ and $g'$ are split epimorphisms.
We combine this result with Theorem 5.1.
[**Corollary 7.3.**]{} [(The middle term of an Auslander-Reiten sequence)]{}
To conclude this chapter, we state the following dual results for the category $\Cal F(\Lambda)$.
[**Proposition 7.4.**]{} [(AR-sequences in $\Cal F(\Lambda)$ with components not split exact)]{}
[**Proposition 7.5.**]{} [(AR-sequences in $\Cal F(\Lambda)$ with split exact components)]{}
[**Corollary 7.6.**]{} [(The middle term of an AR-sequence in $\Cal F(\Lambda)$)]{}
[**References**]{}
Claus Michael Ringel, Fakultät für Mathematik, Universität Bielefeld,
POBox 100131, D-33501 Bielefeld
[ringelmath.uni-bielefeld.de]{} Markus Schmidmeier, Department of Mathematical Sciences, Florida Atlantic University,
Boca Raton, Florida 33431-0991
[markusmath.fau.edu]{}
|
---
abstract: 'In this paper we obtain the estimates on some dynamic integral inequalities in three variables which can be used to study certain dynamic equations. We give some applications to convey the importance of our result.'
address: 'Deepak B. Pachpatte Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra 431004, India'
author:
- 'Deepak B. Pachpatte'
title: Some new dynamic Inequality on time scales in three variables
---
Introduction
============
The study of time scales was initiated in 1989 by Stefan Hilger [@HIG] in his Ph.D dissertation. Since then many authors have studied the dynamic inequalities on time scales. Some analytic inequalities on time scales in one and two variables is studied in [@Hus; @Ozk; @Sun; @Yeh] by various authors. The authors in [@Boh3; @Dbp1; @Dbp2; @Dbp3] have obtained some interesting dynamic integral and iterated inequalities on time scales. Motivated by the results above in this paper we establish new explicit bounds on some dynamic inequalities in three variables which are useful in solving certain dynamic equations.
In what follows $\mathbb{R}$ denotes the set of real numbers, I=\[a,b\] and $\mathbb{T}$ denotes arbitrary time scales. We say that $f:\mathbb{T} \to \mathbb{R}$ is rd-continuous provided $f$ is continuous right dense point of $\mathbb{T}$ and has a finite left sided limit at each left dense point of $\mathbb{T}$ and will be denoted by $C_{rd}$. Let $\mathbb{T}_1$ and $\mathbb{T}_2$ be two time scales with atleast two points and $\Omega = \mathbb{T}_1 \times \mathbb{T}_2$ and $H = \Omega \times I$. The basic information about time scales can be found in [@Boh1; @Boh2]. Now we give the Lemma given in [@Fer] which is required in proving our result.
Lemma \[[@Fer]\] Let $u,a,f \in C'_{rd} \left( {\mathbb{T}_1 \times \mathbb{T}_2 ,\mathbb{R}_ + } \right)$ and $a$ is nondecreasing in each of the variables. If $$u\left( {x,y} \right) \le a\left( {x,y} \right) + \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {f\left( {s,t} \right)u} } \left( {s,t} \right)\Delta t\Delta s,
\tag{1.1}$$ for $\left( {x,y} \right) \in \mathbb{T}_1 \times \mathbb{T}_2 $ then $$u\left( {x,y} \right) \le a\left( {x,y} \right)e_{\int\limits_{y_0 }^y {f\left( {x,t} \right)\Delta t} } \left( {x,x_0 } \right),
\tag{1.2}$$ for $\left( {x,y} \right) \in \mathbb{T}_1 \times \mathbb{T}_2 $.
Main Results
============
Now we give our main result in the following theorem
**Theorem 2.1** Let $u,p,q,f \in C_{rd} \left( {H,\mathbb{R}_ + } \right)$ and $c \ge 0$ be a constant. If $$u\left( {x,y,z} \right) \le p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)\int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,\tau ,q} \right)} } } u\left( {s,\tau ,q} \right)\Delta q\Delta \tau \Delta s,
\tag{2.1}$$ for $(x,y,z) \in H$, then $$u\left( {x,y,z} \right) \le p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)C\left( {x,y} \right)e_{Q\left( {x,y,z} \right)} \left( {x,x_0 } \right),
\tag{2.2}$$ where $$Q\left( {x,y,z} \right) = \int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,\tau ,q} \right)p_2 \left( {s,\tau ,q} \right)\Delta q\Delta \tau \Delta s} },
\tag{2.3}$$ $$C\left( {x,y} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,\tau ,q} \right)p_1 \left( {s,\tau ,q} \right)\Delta q\Delta \tau \Delta s} } }.
\tag{2.4}$$ **Proof.** Now let $$M\left( {s,t} \right) = \int\limits_a^b {f\left( {s,\tau ,q} \right)p_2 \left( {s,\tau ,q} \right)\Delta q}.
\tag{2.5}$$ Then $(2.1)$ becomes $$u\left( {x,y,z} \right) \le p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)\int\limits_{x_0 }^x {\int\limits_{y_0 }^y {M\left( {s,\tau } \right)} } \Delta \tau \Delta s.
\tag{2.6}$$ Now put $$W\left( {x,y} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {M\left( {s,\tau } \right)} } \Delta \tau \Delta s.
\tag{2.7}$$ Then $W(x,y_0)=W(x_0,y)=0$ and $$u\left( {x,y,z} \right) \le p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)W\left( {x,y} \right).
\tag{2.8}$$ From $(2.7),(2.5),(2.8)$ we have $$\begin{aligned}
&W^{\Delta _1 \Delta _2 } \left( {x,y} \right) \\
&= M\left( {x,y} \right) \\
& = \int\limits_a^b {f\left( {x,y,q} \right)u\left( {x,y,q} \right)\Delta q} \\
& \le \int\limits_a^b {f\left( {x,y,q} \right)\left[ {p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)W\left( {x,y} \right)} \right]\Delta q} \\
&= W\left( {x,y} \right)\int\limits_a^b {f\left( {x,y,q} \right)u\left( {x,y,q} \right)\Delta q} + \int\limits_a^b {f\left( {x,y,q} \right)p_1 \left( {x,y,q} \right)\Delta q} \\
& = \int\limits_a^b {f\left( {x,y,q} \right)p_2 \left( {x,y,q} \right)\Delta q} + \int\limits_a^b {f\left( {x,y,q} \right)p_1 \left( {x,y,q} \right)\Delta q}.
\tag{2.9}
\end{aligned}$$ Now from $(2.9)$ above we have by taking delta integral $$\begin{aligned}
W^{\Delta _1 } \left( {x,y} \right)
&\le \int\limits_{y_0 }^y {\int\limits_a^b {W\left( {x,\tau } \right)f\left( {x,\tau ,q} \right)} } p_2 \left( {x,\tau ,q} \right)\Delta q\Delta \tau \\
& + \int\limits_{y_0 }^y {\int\limits_a^b {f\left( {x,\tau ,q} \right)} } p_1 \left( {x,\tau ,q} \right)\Delta q\Delta \tau.
\tag{2.10}
\end{aligned}$$ Again delta integrating above $(2.10)$ we have $$\begin{aligned}
W\left( {x,y} \right)
&\le \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {W\left( {s,\tau } \right)f\left( {x,\tau ,q} \right)} } p_2 \left( {x,\tau ,q} \right)\Delta q\Delta \tau } \\
&+ \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,\tau ,q} \right)} } p_1 \left( {s,\tau ,q} \right)\Delta q\Delta \tau }.
\tag{2.11}
\end{aligned}$$ Put
$B\left( {x,y} \right) = \int\limits_a^b {f\left( {x,y,q} \right)p_2 \left( {x,y,q} \right)\Delta q} $,
and
$C\left( {x,y} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,\tau ,q} \right)} } p_1 \left( {s,\tau ,q} \right)\Delta q\Delta \tau }$.
We get from $(2.11)$ $$W\left( {x,y} \right) \le \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {B\left( {s,\tau } \right)W\left( {s,\tau } \right)} \Delta \tau \Delta s} + C\left( {x,y} \right).
\tag{2.12}$$
Clearly $C(x,y)$ is nondecreasing in $\Omega$ then applying Lemma to $(2.12)$, we get $$W\left( {x,y} \right) \le C\left( {x,y} \right)e_{\overline Q \left( {x,y} \right)} \left( {x,x_0 } \right),
\tag{2.13}$$ where $$\overline Q \left( {x,y} \right) = \int\limits_{y_0 }^y {B\left( {x,\tau } \right)} \Delta \tau.
\tag{2.14}$$ Now using $(2.13)$ in $$u\left( {x,y,z} \right) \le p_1 \left( {x,y,z} \right) + p_2 \left( {x,y,z} \right)W\left( {x,y} \right),$$ we get the result $(2.2)$.
Applications
============
Now in this section we give some applications of our results. Consider the dynamic integral equation of the form $$u\left( {x,y,z} \right) = g\left( {h,y,z} \right) + \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {F\left( {x,y,z,s,t,q} \right)} } } \Delta q\Delta t\Delta s,
\tag{3.1}$$ for $(x,y,z) \in H$ where $g \in C_{rd} \left( {H,\mathbb{R}} \right)$, $F \in C_{rd} \left( {H^2 \times \mathbb{R},\mathbb{R}} \right)$.
Now our next theorem deals with the estimate of solution of $(3.1)$.
**Theorem 3.1** Suppose the function $F$ in $(3.1)$ satisfy the conditions $$\left| {F\left( {x,y,z,s,t,q,u} \right)} \right| \le r\left( {x,y,z} \right)f\left( {s,t,q} \right)\left| u \right|,
\tag{3.2}$$ where $r,f \in C_{rd} \left( {H,\mathbb{R}} \right)$. If $u(x,y,z)$ is a solution of equation $(3.1)$ then $$\left| {u\left( {x,y,z} \right)} \right| \le \left| {g\left( {h,y,z} \right)} \right| + r\left( {x,y,z} \right)C_2 \left( {x,y,z} \right)e_{Q_2 \left( {x,y,z} \right)} \left( {x,x_0 } \right),
\tag{3.3}$$ where $$C_2 \left( {x,y,z} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)} } } \left| {g\left( {s,t,q} \right)} \right|\Delta q\Delta t\Delta s,
\tag{3.4}$$ $$Q_2 \left( {x,y,z} \right) = \int\limits_{y_0 }^y {\int\limits_a^b {f\left( {x,t,q} \right)} } r\left( {x,t,q} \right)\Delta q\Delta t,
\tag{3.5}$$ for $(x,y,z) \in H$.
**Proof.** Let $u \in C_{rd} \left( {H,\mathbb{R}} \right)$ be a solution of $(3.1)$, we have $$\begin{aligned}
\left| {u\left( {x,y,z} \right)} \right|
&\le \left| {g\left( {h,y,z} \right)} \right| + \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q} \right)} \right|} } } \Delta q\Delta t\Delta s \\
&\le \left| {g\left( {h,y,z} \right)} \right| \\
&+ r\left( {x,y,z} \right)\int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)} } } \left| {u\left( {s,t,q} \right)} \right|\Delta q\Delta t\Delta s.
\tag{3.6}\end{aligned}$$ Now applying the Theorem $(2.1)$ gives the estimate $(3.3)$.
Now for obtaining estimates in our next theorem we suppose that $F$ satisfies Lipschitz type conditions.
**Theorem 3.2** Suppose that the function $F$ in $(3.1)$ satisfies the condition $$\left| {F\left( {x,y,z,s,t,q,u} \right) - F\left( {x,y,z,s,t,q,v} \right)} \right| \le r\left( {x,y,z} \right)f\left( {s,t,q} \right)\left| {u - v} \right|,
\tag{3.7}$$ where $r,f \in C_{rd} \left( {H,\mathbb{R}} \right)$. If $u(x,y,z)$ is a solution of $(3.1)$ then $$\begin{aligned}
&\left| {u\left( {x,y,z} \right) - g\left( {x,y,z} \right)} \right| \\
&\le k\left( {x,y,z} \right) + r(x,y,z)C_3 (x,y,z)e_{Q_2 \left( {x,y,z} \right)} \left( {x,x_0 } \right),
\tag{3.8}\end{aligned}$$ for $(x,y,z) \in H$ where $$C_3 \left( {x,y,z} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)\left| {k\left( {s,t,q} \right)} \right|\Delta q\Delta t\Delta s} } },
\tag{3.9}$$
and $$k\left( {x,y,z} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,g(s,t,q)} \right)} \right|} } } \Delta q\Delta t\Delta s,
\tag{3.10}$$ for $(x,y,z) \in H$.
**Proof.** Let $u \in C_{rd} \left( {H,\mathbb{R}} \right)$ be a solution of equation $(3.1)$. Then we have
$$\begin{aligned}
&\left| {u\left( {x,y,z} \right) - g\left( {x,y,z} \right)} \right| \\
&\le \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,u(s,t,q)} \right)} \right|} } } \Delta q\Delta t\Delta s \\
& \le \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,u(s,t,q)} \right)} \right.} } } \\
&\left. { - F\left( {x,y,z,s,t,q,u(s,t,q)} \right)} \right|\Delta q\Delta t\Delta s \\
& + \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,g(s,t,q)} \right)} \right|} } } \Delta q\Delta t\Delta s \\
& \le k\left( {x,y,z} \right) \\
& + r\left( {x,y,z} \right)\int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)} } } \left| {u\left( {s,t,q} \right) - h\left( {s,t,q} \right)} \right|\Delta q\Delta t\Delta s,
\tag{3.11}\end{aligned}$$
for $(x,y,z) \in H$.
Now an application of theorem $(2.1)$ to $(3.11)$ gives the estimate $(3.8)$.
Now we consider equation $(3.1)$ and the integral equation $$h\left( {x,y,z} \right) = v\left( {x,y,z} \right) + \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {G\left( {x,y,z,s,t,q,h(x,y,z)} \right)\Delta q\Delta t\Delta s} } },
\tag{3.12}$$ for $v \in C_{rd} \left( {H,\mathbb{R}} \right)$,$G \in C_{rd} \left( {H^2 \times \mathbb{R},\mathbb{R}} \right)$.
Now we give the following theorem.
**Theorem 3.3** Suppose the function $F$ in $(3.1)$ satisfies the condition $(3.7)$ then for every solution $h \in C_{rd} \left( {H,\mathbb{R}} \right)$ of $(3.11)$ and $u \in C_{rd} \left( {H,\mathbb{R}} \right)$ solution of $(3.1)$ we have the estimates $$\begin{aligned}
\left| {u\left( {x,y,z} \right) - h\left( {x,y,z} \right)} \right|
&\le \left[ {\overline g \left( {x,y,z} \right) + \overline k \left( {x,y,z} \right)} \right] \\
& + r\left( {x,y,z} \right)C_4 \left( {x,y,z} \right)e_{Q_2 \left( {x,y,z} \right)} \left( {x,x_0 } \right).
\tag{3.13}\end{aligned}$$ for $(x,y,z) \in H$ in which $$C_4 \left( {x,y,z} \right) = \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)\left[ {\overline g \left( {s,t,q} \right) + \overline k \left( {s,t,q} \right)} \right]\Delta q\Delta t\Delta s} } }.
\tag{3.14}$$ $$\overline g \left( {x,y,z} \right) = \left| {g\left( {x,y,z} \right) - v\left( {x,y,z} \right)} \right|.
\tag{3.15}$$ $$\begin{aligned}
\overline k \left( {x,y,z} \right)
&= \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,h(s,t,q)} \right)} \right.} } } \\
&\left. { - G\left( {x,y,z,s,t,q,h(s,t,q)} \right)} \right|\Delta q\Delta t\Delta s.
\tag{3.16}\end{aligned}$$ for $(x,y,z) \in H$.
**Proof.** Since $u(x,y,z)$ and $v(x,y,z)$ are respectively solutions of $(3.1)$ and $(3.12)$ we have $$\begin{aligned}
&\left| {u\left( {x,y,z} \right) - h\left( {x,y,z} \right)} \right| \\
&\le \left[ {\overline g \left( {x,y,z} \right) + \overline k \left( {x,y,z} \right)} \right] \\
&+ \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,u(s,t,q)} \right)} \right.} } } \\
&\left. { - F\left( {x,y,z,s,t,q,h(s,t,q)} \right)} \right|\Delta q\Delta t\Delta s \\
&+ \int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {\left| {F\left( {x,y,z,s,t,q,u(s,t,q)} \right)} \right.} } } \\
&\left. { - G\left( {x,y,z,s,t,q,h(s,t,q)} \right)} \right|\Delta q\Delta t\Delta s \\
&\le \overline g \left( {x,y,z} \right) + \overline k \left( {x,y,z} \right) \\
&+ r\left( {x,y,z} \right)\int\limits_{x_0 }^x {\int\limits_{y_0 }^y {\int\limits_a^b {f\left( {s,t,q} \right)\left| {u\left( {s,t,q} \right) - h\left( {s,t,q} \right)} \right|\Delta q\Delta t\Delta s} } }.
\tag{3.17}\end{aligned}$$ Now an application of Theorem $2.1$ to $(3.17)$ yields $(3.13)$.
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---
abstract: 'In theoretical analyses of the moving contact line, an infinite force along the solid wall has been reported based off the non-integrable stress along a [*single*]{} interface. In this investigation we demonstrate that the stress singularity is integrable and results in a [*finite*]{} force at the moving contact line if the contact line is treated as a one-dimensional manifold and all [*three*]{} interfaces that make up the moving contact line are taken into consideration. This is due to the dipole nature of the vorticity and pressure distribution around the moving contact line. Mathematically, this finite force is determined by summing all the forces that act over an infinitesimally small cylindrical control volume that encloses the entire moving contact line. With this finite force, we propose a new dynamic Young’s equation for microscopic dynamic contact angle that is a function of known parameters only, specifically the interface velocity, surface tension, and fluid viscosity. We combine our model with Cox’s model for apparent dynamic contact angle and find good agreement with published dynamic contact angle measurements.'
address:
- 'Department of mechanical and aerospace, University of Florida, Gainesville, FL.'
- 'Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL.'
author:
- Peter Zhang
- Kamran Mohseni
bibliography:
- '/Users/Peter/GitHub/REF/RefA2.bib'
- '/Users/Peter/GitHub/REF/ref\_zhang.bib'
title: Theoretical Model of a Finite Force at the Moving Contact Line
---
moving contact line, dynamic contact angle, multiphase flows
Introduction
============
The moving contact line (MCL) is a unique and challenging problem that influences many natural and industrial processes such as drop impact [@YarinAL:06a], boiling [@DhirVK:98a], industrial coatings [@WeinsteinS:04a], and inkjet printing [@DerbyB:10a], among others. Some of the aspects that make the MCL such a complex problem include its multiscale nature [@SnoeijerJH:13b], the apparent breakdown of the no-slip assumption [@Dussan:76a], hysteresis [@EralH:13a], and dependence on surface properties [@QuereD:08a]. In recent years industrial and medical applications have spurred MCL research towards even more challenging problems such as wetting failure [@deGennesPG:85a; @LandauL:88a], air entrainment [@KumarS:14a], electrowetting [@MugeleF:05a], and phase change at the contact line [@SnoeijerJH:12a; @SnoeijerJH:13a]. While these problems are important for advancing multiphase technology, there is still no consensus on the physics that govern contact line movement over a smooth surface [@SnoeijerJH:13b; @SpeltP:14a; @BlakeTD:06a; @BonnD:09a]. Understanding the MCL dynamics over a smooth surface is essential to developing models for more complex wetting phenomena and thus it has remained a topic of significant interest for several decades.
![Schematic of a droplet sliding down an inclined plate. Near the moving contact line, the interface has minimal curvature and intersects the solid with a dynamic contact angle $\phi$.[]{data-label="fig:schematic"}](./fig1.pdf){width="0.49\linewidth"}
Two early investigations of this fundamental problem were conducted by Moffatt [@MoffattHK:64a] and Huh & Scriven [@HuhC:71a]. In these works, the MCL geometry was modeled by three planar interfaces intersecting with a contact angle $\phi$ and characterized by an interface velocity $U$, as shown in figure \[fig:schematic\]. Moffatt examined a viscous fluid displacing an inviscid gas and reported a solution with an “infinite stress and pressure (both of order $r^{-1}$) on the the plate at the corner”. Huh & Scriven extended this analysis to two viscous fluids and reported that “the total force exerted on the solid surface is logarithmically infinite" due to the singular stress at the MCL. These investigations, among others [@HockingLM:77a; @deGennesPG:85a; @CoxRG:86a; @ShikhmurzaevYD:97a; @EggersJ:04a; @BlakeTD:06a; @BonnD:09a; @SnoeijerJH:13b; @SpeltP:14a], have concluded that the hydrodynamic solution does not accurately model the MCL because an infinite force is not physical and this result has come to be known as the MCL problem, or “Huh & Scriven’s paradox” [@BonnD:09a].
Since these early works, several MCL theories have been developed and include the molecular kinetic theory (MKT) [@BlakeTD:69a], interface formation theory (IFT) [@ShikhmurzaevYD:97a; @ShikhmurzaevYD:07a], and hydrodynamic theory [@CoxRG:86a; @VoinovOV:76a; @Dussan:76a] among others [@PetrovP:92a; @PismenL:02a; @SeppecherP:96a]. In many of these models, microscopic physical mechanisms relieve the stress singularity and allows the interface to move relative to the solid without inducing a singular force. MKT describes the motion of the contact line as a molecular process where molecules attach and detach from the solid surface with some characteristic frequency and displacement. The rate at which these molecules are displaced induces changes in the local surface tension which subsequently affects the dynamic contact angle. Typically the length scale of the molecular displacement is on the order of nanometers while the frequency of this displacement is inversely proportional to the viscosity of the fluid. Through experiment the molecular frequency and displacement have been determined for specific fluid-solid pairs however, it is often difficult to predict these parameters for general cases [@BlakeTD:06a].
Interface formation theory is built on the premise that near a MCL, fluid elements move from one interface to another in finite time so that the fluid exhibits a rolling type motion rather than a sliding type motion associated with fluid slip. As fluid elements transition from one interface to another, IFT posits that the fluid properties will gradually change such that a surface tension gradient will be generated near the MCL. Shikhmurzaev [@ShikhmurzaevYD:97a] proposes an equation of state to relate the surface tension to the interfacial density multiplied by a phenomenological coefficient. Assuming that the Young’s equation is valid for dynamic contact lines, the predicted surface tension gradient results in a change in contact angle. Proponents of IFT report that the model contains no singularities, preserves the rolling kinematics of MCLs, and captures the effects of the local flow field on the dynamic contact angle [@ShikhmurzaevYD:06a]. Based on these characteristics, and the possibility of simulating high Capillary number flows without slip, IFT has been praised as a potentially far-reaching approach [@BlakeTD:06a].
The hydrodynamic theory of the MCL is based on classical continuum fluid mechanics but relaxes the no-slip condition. In many cases the Navier-slip boundary condition [@NavierCL:23a] and a constant slip length is used to determine the flow near the MCL. However, recent molecular dynamics simulations have indicated that the slip length is dependent on the fluid stress [@Troian:97a] and that fluid slip near the MCL may require a more generalized slip boundary condition [@Mohseni:16b]. In general, hydrodynamic models find that the interface shape varies logarithmically with respect to distance from the contact line and is a function of the microscopic contact angle and inner to outer length scale ratio [@CoxRG:86a; @VoinovOV:76a; @HockingLM:82a]. In many applications, a constant microscopic contact angle and length scale ratio allows the hydrodynamic model to capture the shape of the interface [@RameE:96a; @SuiY:13a]. However, there is no physical reason why these parameters should be constant and there have been a growing number of publications that have reported that these parameters should vary as a function of contact line velocity [@RameE:04a; @ShenC:98a; @ZhouM:92a].
In each of the theories reviewed above, molecular scale physics are used to remove the stress singularity at the MCL. As a consequence, the MCL becomes a multiscale problem that couples macroscopic dynamics to microscopic physical parameters like the molecular equilibrium frequency of MKT, the interfacial density of IFT, or the slip length in hydrodynamic models. Currently these microscopic parameters are difficult to determine theoretically and are most often obtained by fitting the theory to experimental measurements. Through the fitting process, most of these theories have reported agreement with experimental measurements [@HoffmanRL:75a; @CoxRG:86a; @ShikhmurzaevYD:02a; @SevenoD:09a; @BlakeT:11a; @ItoT:15a] and it has been difficult to judge which theory captures the true physics of the contact line. As a result, the physics of the MCL are still debated to this day and the field continues to grow larger and more diverse. For additional information regarding the MCL, we refer the reader to the following references [@BlakeTD:06a; @Dussan:79a; @BonnD:09a; @SnoeijerJH:13b; @SpeltP:14a; @deGennesPG:85a].
Looking back on the evolution of the MCL problem, it appears that many investigations were motivated by the conclusions of Moffatt and Huh & Scriven who determined that a stress that scales as $1/r$ is not integrable and that the hydrodynamic solution of the MCL subject to the no-slip boundary condition incorrectly predicts an infinite force. Interestingly we have observed that a continuum field that scales as $1/r$ is treated as an [*integrable*]{} singularity in other fields, and even in continuum fluid flows. In electromagnetism, an electric field that scales as $1/r$ is integrable and correlated to the total charge [@GriffithsDJ:72a]. In potential flow theory, a velocity field that scales with $1/r$ is integrable and directly related to the total mass flux of a line source [@Batchelor:67a]. Even in Stokes flows, the two-dimensional Stokeslet contains an stress singularity that scales as $1/r$ and is considered an integrable singularity that is correlated to a finite force [@CrowdyDG:10a]. Given these numerous examples of integrable $1/r$ singularities, one begins to wonder why is the MCL different?
Motivated by these observations, this work will revisit the classic hydrodynamic solution and present an alternative perspective of the stress singularity. We will begin with a brief review of the Stokes solution to the MCL problem in §\[sect:geo\_sol\]. In §\[sect:f\_MCL\], we will show that classical hydrodynamics models the contact line region as a mathematical line, i.e. a one-dimensional manifold. By treating the MCL as a one-dimensional manifold, we find that the total force exerted by the fluid is finite and a function of the surface tension, interface velocity, and fluid viscosity. §\[sect:discussion\] presents a discussion of this finite force and a comparison with previous works to show that the logarithmically infinite force only arises if the MCL is treated as a two-dimensional manifold. Based on this finite force result, we propose a model for the microscopic dynamic contact angle in §\[sect:DCA\_model\] and provide supporting experimental comparisons. Similarities between the MCL, Stokeslet, and cusped fluid interface are discussed in §\[sect:force\_comp\] as they all exhibit singular stresses and finite forces. Concluding remarks are found in §\[sect:conclusion\].
Stokes flow solution to the moving contact line problem {#sect:geo_sol}
=======================================================
![Geometry of a moving contact line in a cylindrical coordinate system ($r,\theta$) whose origin is fixed at the contact line. In this moving reference frame, the solid boundary moves with a velocity $U$ relative to the fluid-fluid interface whose shape is given by $\phi(r)$.[]{data-label="fig:gen_corner"}](./fig2.pdf){width="0.6\linewidth"}
The primary analysis and results of this paper are based off the well-known Stokes solution to the MCL problem originally presented by Cox [@CoxRG:86a]. To establish a foundation for the following discussion, this section will provide a brief review of Cox’s solution. At a top level, Cox’s solution to the MCL flow, shown in figure \[fig:gen\_corner\], is obtained using perturbation theory and expanding about the zeroth order solution in the small Capillary number limit. As shown in the past [@CoxRG:86a; @SnoeijerJ:06a], the zeroth order flow corresponds to the boundary driven planar wedge whose solution is known analytically from the works of Moffatt [@MoffattHK:64a] and Huh & Scriven [@HuhC:71a] and valid in the intermediate and outer regions where the no-slip condition is valid. From this zeroth order solution, Cox and others have iteratively solved for higher order terms. Below, we present Cox’s derivation of the zeroth and first order solution.
The analysis performed by Cox begins with assuming that the Reynolds number is significantly smaller than one (${\mbox{\textit{Re}}}= \rho \ell U/\mu \ll 1$) so that the dynamics of fluids A and B near the MCL are governed by the Stokes and continuity equations given by $$\begin{gathered}
\label{eq:stokes1}
\nabla^2 \bm{u}^*_A= {\boldsymbol{\nabla}}p^*_A, \hspace{2cm} {\boldsymbol{\nabla}}{\boldsymbol{\cdot}}\bm{u}^*_A= 0,\\
\label{eq:stokes2}
\lambda \nabla^2 \bm{u}^*_B= {\boldsymbol{\nabla}}p^*_B, \hspace{2cm} {\boldsymbol{\nabla}}{\boldsymbol{\cdot}}\bm{u}^*_B= 0.\end{gathered}$$ $\rho$ denotes the density, $\ell$ the characteristic length scale, $U$ the interface velocity, $\mu$ the viscosity, $\lambda = \mu_B/\mu_A$ the viscosity ratio, $\bm{u}^*$ the dimensionless velocity, and $p^*$ the dimensionless pressure. If fluid A or B is not specified in the subscript, then the variable is representative of either fluid. In this MCL problem, the fluid is subject to the no-slip and zero penetration boundary condition along all interfaces in addition to the continuity of tangential stress and balance of normal stress at the fluid-fluid interface. In order to make the problem tractable, Cox assumed that Capillary number is significantly smaller than one (${\mbox{\textit{Ca}}}=\mu U /\sigma \ll 1$) and expanded the velocity, pressure, and fluid-fluid interface shape as $$\begin{gathered}
\label{eq:u_expan} \bm{u}^* = \bm{u}^*_{0} + {\mbox{\textit{Ca}}}\bm{u}^*_{1} + \dotsc,\\
\label{eq:p_expan} p^* = p^*_0 + {\mbox{\textit{Ca}}}p^*_1 + \dotsc , \\
\label{eq:phi_expan} \phi = \phi_0 + {\mbox{\textit{Ca}}}\phi_1 + \dotsc .\end{gathered}$$ Here, $\sigma$ denotes the surface tension. Given the expansions above, other quantities of interest such as the stress tensor, $\bm{T}$, or interface curvature, $\kappa$, can be written using similar expansions. Substituting equations (\[eq:u\_expan\])-(\[eq:p\_expan\]) into equation (\[eq:stokes1\])-(\[eq:stokes2\]) and collecting terms of order ${\mbox{\textit{Ca}}}^0$ yields the zeroth order governing equations which are given by $$\begin{gathered}
\nabla^2 \bm{u}^*_{A0}= {\boldsymbol{\nabla}}p^*_{A0}, \hspace{2cm} {\boldsymbol{\nabla}}{\boldsymbol{\cdot}}\bm{u}^*_{A0}= 0,\\
\lambda \nabla^2 \bm{u}^*_{B0}= {\boldsymbol{\nabla}}p^*_{B0}, \hspace{2cm} {\boldsymbol{\nabla}}{\boldsymbol{\cdot}}\bm{u}^*_{B0}= 0.\end{gathered}$$ As noted by Cox, Capillary number only appears in the normal stress boundary condition at the fluid-fluid interface, i.e. $$\begin{gathered}
{\mbox{\textit{Ca}}}\bm{\hat{n}} {\boldsymbol{\cdot}}[\![ \bm{T}_0 + {\mbox{\textit{Ca}}}\bm{T}_1 + \dotsc ]\!] {\boldsymbol{\cdot}}\bm{\hat{n}'} = \kappa_0 + {\mbox{\textit{Ca}}}\kappa_1 + \dotsc,\end{gathered}$$ where $[\![ {\boldsymbol{\cdot}}]\!]$ denotes the jump of a quantity across an interface whose normal vector is denoted by $\bm{\hat{n}'}$. Collecting terms of order ${\mbox{\textit{Ca}}}^0$, one obtains $\kappa_0 = 0$ so that the zeroth order solution is the flow in a planar wedge with angle $\phi_0$. To obtain the zeroth order solution, Cox rewrites the Stokes equation as the biharmonic stream function equation, $\nabla^4 \psi = 0$, whose general solution is known and given in \[app:bih\_sol\]. Applying the aforementioned boundary conditions, Cox identifies the zeroth order stream function as $$\begin{gathered}
\psi_0 =rU[A\cos(\theta) + B \sin(\theta) + C\theta\cos(\theta)+D\theta \sin(\theta)],\end{gathered}$$ where ($r,\theta$) is the local cylindrical coordinate system. The coefficients $A$, $B$, $C$, and $D$ are analytically known and presented in \[app:coeffs\], in addition to the zeroth order velocity, pressure, and vorticity. The normal stress jump that appears in the zeroth order solution, i.e. $$\begin{gathered}
-(r/\ell)^{-1}m(\phi_0,\lambda) = -\dfrac{2}{r/\ell} [ \lambda (C_B \cos \phi_0 +D_B \sin \phi_0) - (C_A \cos \phi_0 +D_A \sin \phi_0) ],\end{gathered}$$ is accounted for by the curvature of the first order interface shape, that is $$\begin{gathered}
\dfrac{\partial \phi_1}{\partial r} = (r/\ell)^{-1}m(\phi_0,\lambda).\end{gathered}$$ Integrating the equation above yields $\phi_1 = m(\phi_0,\lambda) \ln (r/\ell) + Q$ and the interface shape $$\begin{gathered}
\label{eq:int_shape}
\phi = \phi_0+{\mbox{\textit{Ca}}}[m(\phi_0,\lambda) \ln (r/\ell) + Q]+ \dotsc,\end{gathered}$$ where $Q$ is an unspecified constant of integration. The equation above is the widely recognized general form of the fluid-fluid interface under steady motion [@CoxRG:86a; @SibleyD:15a]. In the paper by Cox, higher order terms were not reported as they had a negligible effect. In the next section, we revisit this classic solution to the MCL flow and demonstrate that the singular stress of the dominant zeroth order solution exerts a finite force at the moving contact line.
Forces acting at the contact line {#sect:f_MCL}
=================================
![Schematic of the cylindrical control volume $V$ with radius $\epsilon$ that is bounded by the surface $\mathcal{S}$ and centered at the MCL. The moving contact line has a contact angle of $\phi_0$. $\bm{\hat{n}}$ and $\bm{\hat{t}}$ denote the unit normal and tangential vectors of the contour $\mathcal{C}$. $\mathcal{I}$ denotes the interfacial surfaces and $\sigma$ denotes the surface tension force between fluid A, B, and the solid.[]{data-label="fig:CL_volume"}](./fig3.pdf){width="0.89\linewidth"}
In the past, moving contact line analyses have often utilized integral equations that were derived for fluid interfaces. However, the moving contact [*line*]{} is a one-dimensional manifold unlike fluid interfaces, which are two-dimensional manifolds. A fluid interface is uniquely defined by a single normal vector while a contact line has multiple normal vectors. This multivaluedness subsequently appears in the hydrodynamic solution which exhibits a multivalued velocity, stress, vorticity, and pressure along the contact line. In light of these contact line attributes, let us take a step back and consider a summation of forces acting on a finite sized cylindrical control volume, $V$, with radius $\epsilon$ centered around the contact line, as shown in figure \[fig:CL\_volume\]. This volume is bounded by the surface $\mathcal{S}$ and intersects the interfacial surfaces denoted by $\mathcal{I}$. As we will demonstrate later, this is the control volume necessary to derive the Young’s equation and is the one-dimensional analogue to the rectangular control volume used to derive the balance of forces on a fluid interface which is a two-dimensional manifold. For a steady problem, the sum of all the forces acting on the control volume $V$ is given by $$\begin{aligned}
\label{eq:sum_forces}
\Sigma \bm{f} = \underbrace{\iint_\mathcal{S} \bm{\hat{n}} {\boldsymbol{\cdot}}\bm{T} dA}_\text{surface force} +\underbrace{ \iiint_V \rho \bm{g} dV}_\text{body force} +\underbrace{\iint_\mathcal{I} {\boldsymbol{\nabla}}_\pi {\boldsymbol{\cdot}}\bm{T}_\pi dA}_{\substack{\text{surface tension} \\ \text{gradient}}} + \underbrace{ \int_{C\!L} [\![ \bm{T_\pi} {\boldsymbol{\cdot}}\bm{\hat{t}'} ]\!] ds}_{\substack{\text{surface tension} \\ \text{force}}} = 0,\end{aligned}$$ assuming massless interfaces [@SlatteryJC:07a]. $\bm{g}$ denotes the body force, the subscript $\pi$ denotes surface quantities, $C\!L$ denotes the contact line, and $\bm{\hat{t}'}$ denotes the unit tangential vectors of the interfaces. In the limit as $\epsilon \to \ell_i = 0$, where $\ell_i$ denotes the distance from the contact line over which fluid slip occurs, the equation above represents the sum of all forces acting exactly at the MCL. Note that in some works $\ell_i$ is sometimes denoted by the slip length, $\ell_s$, because they are often considered to be the same order of magnitude. However they are not equivalent and $\ell_i$ and $\ell_s$ are treated as separate length scales in this work. For this analysis, we will assume that there are zero body forces ($\bm{g} = 0$) and zero surface tension gradients (${\boldsymbol{\nabla}}_\pi {\boldsymbol{\cdot}}\bm{T}_\pi = 0$). As a result, the $x$ and $y$ component of the forces (per unit contact line length) acting on the control volume are reduced to $$\begin{gathered}
\label{eq:MCL_fx}
\Sigma f_x = \lim_{\epsilon \to 0}\oint_\mathcal{C} \bm{\hat{n}}{\boldsymbol{\cdot}}\bm{T} {\boldsymbol{\cdot}}\bm{\hat{e}_x} ds + \sigma_{AS} - \sigma_{BS} + \sigma_{AB} \cos(\phi_0) = 0,\\
\label{eq:MCL_fy}
\Sigma f_y = \lim_{\epsilon \to 0}\oint_\mathcal{C} \bm{\hat{n}}{\boldsymbol{\cdot}}\bm{T} {\boldsymbol{\cdot}}\bm{\hat{e}_y} ds + \sigma_{AB} \sin(\phi_0) = 0,\end{gathered}$$ where $\mathcal{C}$ denotes the contour path. The effects of surface tension gradients are considered in a related work by Thalakkottor & Mohseni [@mohseni:17r].
For the forces acting in the $x$ direction, we decompose the stress integral into three segments that lie inside each material so that the integral above is rewritten as $$\begin{aligned}
\notag
\Sigma f_x = &\int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}(\bm{T}_{A0} + {\mbox{\textit{Ca}}}\bm{T}_{A1} + \dotsc) {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta + \int_{\phi_0}^\pi \bm{\hat{e}_r} {\boldsymbol{\cdot}}(\bm{T}_{B0} + {\mbox{\textit{Ca}}}\bm{T}_{B1} + \dotsc) {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta \\
\label{eq:fx_split}
&+\int_\pi^{2\pi} \bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{T}_S {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta+ \sigma_{AS} - \sigma_{BS} + \sigma_{AB}\cos(\phi_0) = 0.\end{aligned}$$ Here, the integral of $\bm{T}_S$ represents the force induced by the stress of the enclosed solid, $\bm{f}_\text{S}$. To evaluate the fluid stress tensor integrals, we begin with the zeroth order solution and use the stress tensor decomposition $\bm{T} = \bm{\hat{T}} -2\mu \bm{B}$, where $\bm{\hat{T}} = -p\bm{I} + 2\mu\bm{\Omega}$ is the reduced stress tensor, $\bm{\Omega}$ is the vorticity tensor, and $\bm{B} = (\nabla {\boldsymbol{\cdot}}\bm{u})\bm{I} - (\nabla\bm{u})^T$ is the surface deformation rate tensor. With this decomposition, the first stress tensor integral in equation (\[eq:fx\_split\]) is rewritten as $$\begin{gathered}
\label{eq:stress_decomp}
\int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{T}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta = \int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta - 2\mu_A\int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{B}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta,\end{gathered}$$ for the zeroth order solution. The force contribution of the surface deformation rate tensor is identically zero, as one can show that $$\begin{gathered}
\bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{B}_{A0} = -\dfrac{\partial u_{0r}}{\partial r} \bm{\hat{e}_r} - \dfrac{\partial u_{0\theta}}{\partial r} \bm{\hat{e}_\theta} = 0,\end{gathered}$$ since both the radial and azimuthal components of the zeroth order velocity are independent of $r$. This is consistent with the findings of Wu et al [@WuJZ:06a], who reported that the surface deformation rate tensor does not contribute to the total surface force over a closed boundary if viscosity is constant. By substituting the zeroth order solution for pressure and vorticity into the first term on the right hand side of equation (\[eq:stress\_decomp\]), we obtain $$\begin{aligned}
\notag
f_{A0,x} &= \int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta = \int_0^{\phi_0} (-p_{A0}\bm{\hat{e}_r} +\mu_A\omega_{A0} \bm{\hat{e}_\theta}) {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta\\ \notag
&=-\mu_A U\int_0^{\phi_0} \left\{ \dfrac{2}{r} [C_A\cos(\theta) + D_A\sin(\theta)]\cos(\theta) + \dfrac{2}{r} [C_A\sin(\theta)-D_A\cos(\theta)]\sin(\theta) \right\} r d\theta\\ \notag
&= -2\phi_0 \mu_A U C_A.\end{aligned}$$
The result above shows that the zeroth order viscous force contribution from fluid A in the $x$ direction ($f_{A0,x}$) is independent of $r$ and a function of contact angle, viscosity, and interface velocity only. The same analysis performed on the segment that lays in fluid B and for the zeroth order fluid stress integrals in the $y$ direction allows us to rewrite equations (\[eq:MCL\_fx\]) and (\[eq:MCL\_fy\]) as $$\begin{gathered}
\label{eq:fx_final}
2\phi_0 \mu_A U C_A + 2(\pi-\phi_0)\mu_B U C_B - \sigma_{AB}\cos(\phi_0) = \sigma_{AS} - \sigma_{BS} + f_{\text{S},x},\\
\label{eq:fy_final}
2\phi_0 \mu_A U D_A + 2(\pi-\phi_0)\mu_B U D_B -\sigma_{AB}\sin(\phi_0) = f_{\text{S},y},\end{gathered}$$ and represents the balance of forces in the $x$ and $y$ direction when the sum of all forces is zero and the moving contact line is steady. The equations above are zeroth order accurate and have been organized so that the left hand side contains the forces that the fluid exerts on the solid, and the right hand side contains the forces that the solid exerts on the fluid. Note that the expressions above are independent of $r$, so that even in the limit as $\epsilon \to 0$, the total MCL force of the zeroth order solution remains [*finite*]{} despite a singular stress. Conceptually, one can think of this as an infinite stress acting on an infinitesimally small area, resulting in a finite force.
The analysis above, which has been performed for the zeroth order solution, can also be applied to the first order solution. Given the first order interface shape in equation (\[eq:int\_shape\]), and assuming that $\bm{u}_1$ is finite at the contact line, the first order stream function will have the form given by $$\begin{gathered}
\psi_{1} = (r/\ell)^{2} \ln (r/\ell) q_{2L,1}(\theta)+ \sum_{n = 1}^\infty (r/\ell)^{n} q_{n,1}(\theta).\end{gathered}$$ From this stream function, the velocity field and stress tensor can be analytically determined so that first order equivalent of equation (\[eq:stress\_decomp\]), taken in the limit as $r \to 0$, becomes $$\begin{gathered}
\label{eq:stress_first}
\lim_{r \to 0} \int_0^{\phi_0} \bm{\hat{e}_r} {\boldsymbol{\cdot}}({\mbox{\textit{Ca}}}\bm{T}_{A1}) {\boldsymbol{\cdot}}\bm{\hat{e}_x} r d\theta = \lim_{r \to 0} \int_0^{\phi_0} {\mbox{\textit{Ca}}}\left[\ln(r/\ell) h_{2L}(\theta)+ \sum_{n = 1}^\infty (r/\ell)^{n-2} h_n(\theta) \right] r d\theta.\end{gathered}$$ Here, $h$ simply denotes the function that collects the terms of $\bm{\hat{e}_r} {\boldsymbol{\cdot}}({\mbox{\textit{Ca}}}\bm{T}_{A1}) {\boldsymbol{\cdot}}\bm{\hat{e}_x}$ that scale with the same order of $r$. In this limit where $r \to 0$, all terms of the integrand vanish except the $n = 1$ term. Thus the first order viscous force of fluid A acting at the MCL is given by $$\begin{gathered}
{\mbox{\textit{Ca}}}f_{A1,x} = {\mbox{\textit{Ca}}}\int_0^{\phi_0} h_1(\theta) d\theta.\end{gathered}$$ As $f_{A0,x}$ and $f_{A1,x}$ are of the same order of magnitude and because ${\mbox{\textit{Ca}}}\ll 1$, we conclude that the first order force is significantly smaller than the zeroth order force. Similarly, we find that the first order forces due to fluid B and in the $y$ direction are also negligible so that the balance of forces at the MCL is given by equation (\[eq:fx\_final\]) and (\[eq:fy\_final\]). In the work by Cox [@CoxRG:86a], the first order velocity field and corresponding viscous forces are also neglected as they have a negligible effect on the interface shape when compared to the zeroth order forces.
Based on the finite force results above, we remark that the zeroth order MCL solution is a prime example of the singular mathematical models described by Dussan & Davis [@Dussan:74a], much like the well-known Stokeslet [@Guazzelli:11a]. In both the Stokeslet and the zeroth order MCL solution, the force acting over a small area is modeled as mathematical line with infinite stress and finite force. Therefore, they can be considered physically realistic, at least in regard to conservation of mass and momentum. In the context of moving contact lines, it may appear unusual for a model to contain a singular stress and finite force. However, we emphasize that the Stokeslet, potential line source, and potential line vortex all exhibit the same characteristics and have all been successfully used in modeling a wide variety of physical phenomena. In the following discussion, we present a complex formulation of the MCL problem that yields the same finite force at the MCL, but avoids some of the algebra through the use of Cauchy’s residue theorem.
Complex formulation of the moving contact line force {#complex-formulation-of-the-moving-contact-line-force .unnumbered}
----------------------------------------------------
In the analysis above, the steps required to find the MCL force can be some what cumbersome and therefore, we introduce a relatively simpler complex formulation of the problem in this section. The advantages of this formulation will become clear in §\[sect:force\_comp\], when the MCL problem is compared with the Stokeslet and cusped fluid interface.
As demonstrated by Langlois & Deville [@LangloisWE:64a], any flow satisfying the Stokes equation will ensure that the pressure and vorticity are harmonic conjugates. Thus, we can define the function $G = \mu \omega + ip$ representing the shear and normal stresses. For the zeroth order solution, $G_0$ is given by $$\begin{gathered}
\label{eq:G}
G_0 = \mu \omega_0 + ip_0 = 2\mu U\dfrac{-D+iC}{z}.\end{gathered}$$ It is immediately apparent that $G_0$ has a simple pole at the location of the contact line where $z = 0$ and represents a dipole distribution. Furthermore, we observe that the complex function $G$ and the reduced stress tensor $\bm{\hat{T}}$ are composed of pressure and vorticity only. Thus, it is not all that surprising that the contour integral of $\bm{\hat{T}}$ can be written in terms of a complex contour integral of $G$, that is $$\begin{aligned}
\notag\oint G dz &= \oint ( \mu\omega + ip)(dx + i dy)\\
\notag &= \oint (-p\boldsymbol{\hat{e}_x}- \mu\omega \boldsymbol{\hat{e}_y}) {\boldsymbol{\cdot}}\left(\dfrac{dy}{ds} \boldsymbol{\hat{e}_x}- \dfrac{dx}{ds} \boldsymbol{\hat{e}_y} \right) ds + i\oint ( \mu\omega \boldsymbol{\hat{e}_x} -p \boldsymbol{\hat{e}_y}) {\boldsymbol{\cdot}}\left(\dfrac{dy}{ds} \boldsymbol{\hat{e}_x}- \dfrac{dx}{ds} \boldsymbol{\hat{e}_y} \right) ds\\
\label{eq:f_stress} & = {\oint \boldsymbol{\hat{n}} {\boldsymbol{\cdot}}\mathbf{\hat{T}} {\boldsymbol{\cdot}}\boldsymbol{ \hat{e}_x} ds} + i{\oint \boldsymbol{\hat{n}} {\boldsymbol{\cdot}}\mathbf{\hat{T}} {\boldsymbol{\cdot}}\boldsymbol{ \hat{e}_y} ds}.\end{aligned}$$ Here, the real and imaginary components are exactly equal to the stress integral terms of equations (\[eq:MCL\_fx\]) and (\[eq:MCL\_fy\]) and correspond to the viscous force exerted by the fluid in the $x$ and $y$ directions. In contrast to the previous analysis, this formulation reveals that we can avoid the tedious algebra of integrating $\bm{\hat{T}}$, and instead evaluate the left hand side of equation (\[eq:f\_stress\]) using Cauchy’s method of residues [@MitrinovicD:84a]. However, the standard residue theorem cannot be applied over the contour $\mathcal{C}$, as the function $G$ is piecewise holormophic within the contour. Therefore, it is necessary to decompose the contour $\mathcal{C}$ into the three subcontours $\mathcal{C}_A$, $\mathcal{C}_B$, and $\mathcal{C}_S$ that enclose each phase, see figure \[fig:contour\_decomp\]. Inside each subcontour, $G$ is entirely holomorphic and the pole resides at the MCL ($z = 0$). Thus, we can treat the contour integral of $G_{A}$ and $G_{B}$ using Cauchy’s residue theorem and the Sokhotski-Plemelj formulas [@EstradaR:12a]. For the zeroth order solution, we obtain $$\begin{gathered}
\oint_\mathcal{C} G_0 dz =\oint_{\mathcal{C}_A} G_{A0} dz + \oint_{\mathcal{C}_B} G_{B0} dz + \oint_{\mathcal{C}_S} G_S dz,\\
\oint_{\mathcal{C}_A} G_{A0} dz = \phi_0 i \operatorname{Res}(G_{A0}) = -2\phi_0 \mu_A U C_A -i2\phi_0 \mu_A U D_A,\\
\oint_{\mathcal{C}_B} G_{B0} dz = (\pi-\phi_0) i \operatorname{Res}(G_{B0}) = -2(\pi-\phi_0) \mu_B U C_B -i2(\pi-\phi_0) \mu_B U D_B.\end{gathered}$$ Additional details regarding the Sokhotski-Plemelj formulas and the treatment of singularities residing on the contour can be found in \[app:SP\_theorem\]. As before, the integral of $G_S$ is the force exerted by the stress of the solid.
![Decomposition of the contour $\mathcal{C}$ into $\mathcal{C}_A$, $\mathcal{C}_B$, and $\mathcal{C}_S$.[]{data-label="fig:contour_decomp"}](./fig4.pdf){width="0.99\linewidth"}
The real component of the complex contour integrals above can be substituted into equation (\[eq:fx\_split\]), and we once again obtain equation (\[eq:fx\_final\]), representing the balance of viscous and surface tension forces at the MCL. Similarly the imaginary components can be used to obtain equation (\[eq:fy\_final\]). Interestingly, one can use the complex contour integral of $G$ to find the total surface force for any Stokes flow. As we will show in §\[sect:force\_comp\], this complex analysis correctly captures the viscous force exerted by two other singular Stokes flows that have similar pressure and vorticity fields. While we have only presented a relatively simple complex formulation relevant to the MCL force, complex variables can also be used to define the stream function and velocity to analytically solve for viscous flows in a variety of problems [@CrowdyDG:10a; @CrowdyD:17a]. In the following section, we compare our result to previous works and discuss the physical implications and limitations of this solution.
Discussion of the MCL force {#sect:discussion}
===========================
The analysis of the previous section shows that the force at the moving contact line predicted by the Stokes solution is finite. However, previous works [@Batchelor:67a; @HuhC:71a] have reported a logarithmically infinite total force on the solid. So why does the analysis above predict a finite force when others report an infinite force? To understand the distinction, we first replicate the result of Huh & Scriven, by integrating the stress of fluid A along [*only*]{} the fluid-solid boundary, i.e. $$\begin{gathered}
\label{eq:surf_f_int}
f_{AS,x} = \int_0^R \bm{\hat{e}_\theta} {\boldsymbol{\cdot}}\bm{T}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} dr \quad \text{at} \quad \theta = 0^\circ,\end{gathered}$$ where $R$ is some finite length. From the solution provided in \[app:coeffs\], one can show that $\bm{\hat{e}_\theta} {\boldsymbol{\cdot}}\bm{T}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} = -\mu_A \omega_{A0}$, and that the integral above is improper, as $\omega_{A0}$ is singular at $r = 0$. Therefore, this integral can only be evaluated in the limit, that is $$\begin{aligned}
f_{AS,x} &= \lim_{\epsilon \to 0}\int_\epsilon^R -\dfrac{2\mu_A U}{r} [C_A\sin(0) - D_A\cos(0)] dr\\
&= \lim_{\epsilon \to 0} 2\mu_A D_A U[\ln R - \ln \epsilon] \\
& = \infty.
\end{aligned}$$ Thus, the viscous force exerted by fluid A along the fluid-solid interface is logarithmically infinite. The same analysis performed for fluid B at $\theta = 180^\circ$ yields another infinite force. [*Individually*]{}, fluid A and B exert infinite forces along the fluid-solid interface, however, the sum of these two infinite forces is undefined or infinite depending on the sign of $D$. Note that in this approach, only the forces along the fluid-solid interface are considered. This would contradict Young’s equation, which clearly includes the surface tension force of the fluid-fluid interface that only exists at $r = 0$ on the solid boundary, and nowhere else.
Based on the discussion above, we find that the distinction between our analysis and previous results, is the control volume that is used to derive the force integral. In the analysis of Huh & Scriven, the force integral given by equation (\[eq:surf\_f\_int\]) is derived for rectangular control volumes containing discontinuities across a two-dimensional manifold or surface, i.e. a fluid interface [@LealLG:07a]. In contrast, the force integral presented in equation (\[eq:stress\_decomp\]) is specifically derived for volumes containing line discontinuities like the contact line and thus includes the forces acting along the fluid-fluid interface. In works that report an infinite force, it appears that the MCL was viewed as an extension of the interface between a single fluid and solid. Thus, the force was determined by integrating the fluid stress along the fluid-solid interface only. However, the MCL is [*not*]{} an extension of a fluid-solid interface, but rather a line defined by the intersection of three immiscible materials. From this perspective, it is natural to define a control volume that encloses all three materials and the MCL. In related analyses of the contact line Slattery et al [@SlatteryJC:07a] and Andreotti & Snoeijer [@SnoeijerJH:16a] have also chosen the same cylindrical control volume. By defining the control volume in this fashion, we include the forces that act on the fluid-fluid interface and capture the multivalued nature of the MCL.
![Annular contour $\mathcal{C}'_A$ (shown in red) with inner radius $R_i$ and outer radius $R_o$ residing in fluid A. []{data-label="fig:annul_cont"}](./fig5.pdf){width="0.49\linewidth"}
To physically understand why the total force at the MCL remains finite, consider the annular contour $\mathcal{C}'_A$ with inner radius $R_i$ and outer radius $R_o$ that lies inside fluid A, as shown in figure \[fig:annul\_cont\]. In the zeroth order solution, the stress along the two radial segments and the stress along the two azimuthal segments are exactly equal and opposite, that is $$\begin{gathered}
\underbrace{-\left[\bm{\hat{e}_\theta} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} \right]_{\theta = 0}}_\text{fluid-solid interface} = \underbrace{\left[\bm{\hat{e}_\theta} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x} \right]_{\theta = \phi_0}}_\text{fluid-fluid interface} = -\dfrac{2 \mu_A UD_A}{r},\\
\underbrace{-\left[(\bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x})r \right]_{r = R_i}}_\text{inner arc} = \underbrace{\left[(\bm{\hat{e}_r} {\boldsymbol{\cdot}}\bm{\hat{T}}_{A0} {\boldsymbol{\cdot}}\bm{\hat{e}_x})r \right]_{r = R_o}}_\text{outer arc} = -2 \mu_A UC_A.\end{gathered}$$ Thus, the force acting along the fluid-fluid interface is balanced by the force acting along the fluid-solid interface. Similarly, the forces along the two azimuthal arcs balance each other and the total surface force acting on the contour $\mathcal{C}'_A$ is zero. In fact, any contour path that does not enclose or pass through the singularity will yield zero total surface force. If we shrink the radius of the inner arc to zero, the contour $\mathcal{C}'_A$ is reduced to $\mathcal{C}_A$ and passes through the singularity, as shown in figure \[fig:contour\_decomp\]. Here, the logarithmically infinite force along the fluid-solid interface is balanced by the force along the fluid-fluid interface, as the stress along each boundary approaches positive and negative infinity at the same rate. A similar cancellation of logarithmic singularities is reported in the work of Jones [@JonesMA:03a] when the free vortex sheet is shed tangentially from the plate edge. In the end, the total force exerted by the fluid is given by the azimuthal arc and remains finite no matter how small $\mathcal{C}_A$ becomes.
While the analysis above demonstrates that there is a finite force at the MCL, the model is not without its limitations. One such limitation is the viscous dissipation per unit volume, which scales as $r^{-2}$. This dissipation is non-integrable for both fluids and results in a singular total energy. However, this does not affect the balance of momentum so long as density and viscosity are constant. This singular energy is not unique to the MCL problem and appears in several other two-dimensional singular continuum models. For example, the potential line source/sink and line vortex have infinite kinetic energy at the singularity [@Batchelor:67a]. From electromagnetism, the energy per unit volume for an infinite line charge or infinitely long current carrying wire is also singular [@GriffithsDJ:72a]. These singular magnitudes in energy are a consequence of modeling some finite sized physical feature as a mathematical line, where some desired integral quantity is preserved. For example, the potential line vortex is the limiting case of a Rankine vortex where the radius of the central core is reduced to zero while preserving the total circulation. In this limit, circulation can only be preserved if angular velocity approaches infinity, therefore, the potential line vortex exhibits infinite kinetic energy. Despite this non-physical kinetic energy, potential flow theory has successfully modeled a wide range of high Reynolds number flows. Similarly, the MCL model presented in this manuscript is the limit where the fluid slip region has been reduced to an infinitely small point while preserving the total force. We recognize that these singular continuum models are idealized representations of the true physical phenomena. For the MCL problem, this relatively simple model will require additional development in order to capture the transfer of energy. However, it is still valid when considering forces and momentum transfer near the MCL. In the following section, we explore the impact of this model on the prediction of dynamic contact angle.
Dynamic contact angle model {#sect:DCA_model}
===========================
The force balance presented above is essentially a dynamic Young’s equation that can be used to model the dynamic contact angle. To do so, we assume that the solid is relatively rigid such that any deformation of the solid near the MCL is extremely small [@SlatteryJC:07a; @LesterG:61a]. As a result, ${f_{\text{S},x}}$ is small relative to the other terms of equation (\[eq:fx\_final\]) and the force balance at the contact line can be written as $$\begin{gathered}
\label{eq:DCA}
2\phi_0 \mu_A U C_A + 2(\pi-\phi_0)\mu_B U C_B - \sigma_{AB}\cos(\phi_0) = \sigma_{AS} - \sigma_{BS}.\end{gathered}$$ In the equation above, the viscosity, interface velocity, and surface tension are known and therefore one can solve for the only remaining unknown variable, namely the dynamic contact angle $\phi_0$. This angle corresponds to the microscopic contact angle since equation (\[eq:DCA\]) represents the force balance at $r = \ell_i$. For the idealized hydrodynamic solution, $\ell_i = 0$ and $\phi_0$ is the microscopic angle measured at the solid surface. In problems where slip occurs over a finite but significantly smaller length than the capillary length ($\ell_i \ll \ell_c$), $\phi_0$ corresponds to the microscopic angle measured just outside the slip region. In the limit $U \to 0$, equation (\[eq:DCA\]) is simplified to the static Young’s equation. The static Young’s equation can be used to replace the right hand side of equation (\[eq:DCA\]) with $-\sigma_{AB}\cos(\phi_\text{static})$ so that the dimensionless dynamic Young’s equation is rewritten as $$\begin{gathered}
\label{eq:nond_DCA}
\cos(\phi_0) - \cos(\phi_\text{static}) = {\mbox{\textit{Ca}}}_A [2\phi_0 C_A + 2(\pi-\phi_0)C_B\lambda].\end{gathered}$$
This non-dimensional form reveals that the change in microscopic contact angle scales with Capillary number and is consistent with diffuse interface and molecular kinetic models. In its simplest form, the diffuse interface model predicts that microscopic contact angle will scale as $$\begin{gathered}
\label{eq:DI_DCA}
\cos(\phi_0) - \cos(\phi_\text{static}) \sim {\mbox{\textit{Ca}}}\dfrac{\zeta}{\ell_i},\end{gathered}$$ where $\zeta$ is the interface width [@EW:07a; @SnoeijerJH:13b]. Molecular kinetic theory proposes the relation $$\begin{gathered}
\label{eq:MKT_DCA}
\cos(\phi_0) - \cos(\phi_\text{static}) = {\mbox{\textit{Ca}}}F_B(\nu_0,\xi),\end{gathered}$$ where $F_B$ is a Boltzman factor that is a function of the molecular equilibrium frequency and molecular displacement, $\nu_0$ and $\xi$ [@BlakeTD:06a; @BlakeTD:11a; @SnoeijerJH:13b]. The right hand side of equations (\[eq:nond\_DCA\]-\[eq:MKT\_DCA\]) show that the change in microscopic contact angle scales with Capillary number multiplied by a factor representing the proposed physical mechanisms of each model. In our proposed model, the factor on the right hand side represents the total viscous force acting on the contact line region and only contains macroscopic parameters that are known a priori, e.g. viscosity. In contrast, microscopic parameters such as the interface thickness, molecular displacement, and molecular equilibrium frequency need to be empirically determined from experimental data. Interestingly, we have not specified the specific physical mechanism which regularizes the stress singularity at microscopic scales, whether it be slip or molecular attachment and detachment. Our analysis essentially finds that regardless of the microscopic physical mechanisms, the total force exerted must equal the net change in fluid momentum. As conservation of momentum should be satisfied for any physical model, it is somewhat expected that equations (\[eq:nond\_DCA\]-\[eq:MKT\_DCA\]) are similar in form. Note that this perspective on the MCL is conceptually similar to vortex sheets which are used to model boundary layers in potential flow theory. While the exact velocity profile within the boundary layer is not resolved by the vortex sheet, the total integrated circulation is captured and potential flow theory is able to accurately model the forces acting on solid bodies.
To demonstrate the utility of this model, we apply it to the Brookfield std. viscosity fluid and 70% glycerol solution, whose dynamic contact angles were experimentally measured by Hoffman [@HoffmanRL:75a] and Blake & Shikhmurzaev [@ShikhmurzaevYD:02a], respectively. In this example, these two fluids and experiments are chosen for their contrasting fluid properties and experimental set ups. However, the same analysis can be applied to other fluids and geometries as well. As shown in table \[tab:fluid\_prop\], the Brookfield fluid has a high viscosity and perfectly wets the solid while the glycerol solution has a significantly lower viscosity and only partially wets the solid. With respect to the experimental set up, Hoffman measured the contact angle of a liquid slug as it was pushed through a 1.95 mm precision bore tube while Blake & Shikhmurzaev measured the dynamic contact angle created by plunging a smooth tape into a bath of fluid. Despite these two very different fluids and experiments, we will see that the theoretical model is in good agreement with the experimental measurements.
Given the fluid properties in table \[tab:fluid\_prop\], we use equation (\[eq:DCA\]) to obtain a theoretical prediction of microscopic dynamic contact angle as a function of Capillary number, as shown in figure \[fig:dynamic\_CA\](a). To account for the roughness of the solid surface we have used $\phi({\mbox{\textit{Ca}}}\to 0^+)$ in place of $\phi_\text{static}$ in equation (\[eq:DCA\]). At first glance, we observe that the glycerol solution exhibits a relatively smaller change in contact angle as ${\mbox{\textit{Ca}}}$ increases. This is due to the significantly larger viscosity ratio that allows the Brookfield fluid to more easily reduce the contact angle of the receding air phase. To compare our results with those reported by Hoffman and Blake & Shikhmurzaev, we combine our model for microscopic contact angle ($\phi_0$) with Cox’s model for apparent contact angle ($\phi_D$) [@CoxRG:86a]. Cox’s zeroth order model is given $$\begin{gathered}
\label{eq:cox_g}
g(\lambda,\phi_D) = g(\lambda,\phi_0) + {\mbox{\textit{Ca}}}\ln (1/\varepsilon),\end{gathered}$$ where all terms are of order 1 and $\varepsilon = \ell_i/\ell_o$ is the ratio of the inner length scale to the outer length scale. $g(\lambda,\phi)$ is given by $$\begin{gathered}
g(\lambda,\phi) = \int_0^\phi \dfrac{\lambda(\beta^2-\sin^2\beta)[(\pi-\beta)+\sin\beta\cos\beta]+[(\pi-\beta)^2-\sin^2\beta](\beta-\sin\beta\cos\beta)}{2\sin\beta[\lambda^2(\beta^2\sin^2\beta)+2\lambda\{\beta(\pi-\beta)+\sin^2\beta\}+\{ (\pi-\beta)^2-\sin^2\beta\}]} d\beta.\end{gathered}$$ The apparent contact angle predicted by the combination of these two models is shown in figure \[fig:dynamic\_CA\](b) with $\varepsilon = 10^{-4}$. Note that the outer length scale is the distance at which the apparent contact angle is measured and is often interpreted as the capillary length scale. Hoffman and Blake & Shikhmurzaev did not report the length scale of their contact angle measurements, however it was likely smaller than the capillary length scale due to their use of microscopes. Consequently, experimentally obtained values of $\varepsilon$ are slightly larger than one might expect if one were to use the capillary length as the outer length scale.
$\mu$ \[N s/m$^2]$ $\rho$ \[kg/m$^3]$ $\sigma_{AB}$ \[N/m\] $\phi_\text{static}$
--------------------------------- -------------------- -------------------- ----------------------- ----------------------
Brookfield std. viscosity fluid 98.8 974 0.0217 $0^\circ$
70% glycerol solution 0.023 1181 0.0635 $67^\circ$
: Fluid properties of Brookfield std. viscosity fluid and 70% glycerol solution investigated by Hoffman [@HoffmanRL:75a] and Blake & Shikhmurzaev [@ShikhmurzaevYD:02a], respectively.[]{data-label="tab:fluid_prop"}
![(a) Microscopic dynamic contact angle predicted for the Brookfield fluid and 70% glycerol solution investigated by Hoffman [@HoffmanRL:75a] and Blake & Shikhmurzaev [@ShikhmurzaevYD:02a], respectively. Microscopic dynamic contact angle is obtained using equation (\[eq:DCA\]) and the MCL solution presented in §\[sect:geo\_sol\]. (b) Apparent dynamic contact angle comparison between experimental measurements and current theoretical model. The current model uses Cox’s model [@CoxRG:86a] where the microscopic contact angle, $\phi_0$, is theoretically predicted by equation (\[eq:DCA\]). []{data-label="fig:dynamic_CA"}](./fig6a.pdf){width="0.99\linewidth"}
![(a) Microscopic dynamic contact angle predicted for the Brookfield fluid and 70% glycerol solution investigated by Hoffman [@HoffmanRL:75a] and Blake & Shikhmurzaev [@ShikhmurzaevYD:02a], respectively. Microscopic dynamic contact angle is obtained using equation (\[eq:DCA\]) and the MCL solution presented in §\[sect:geo\_sol\]. (b) Apparent dynamic contact angle comparison between experimental measurements and current theoretical model. The current model uses Cox’s model [@CoxRG:86a] where the microscopic contact angle, $\phi_0$, is theoretically predicted by equation (\[eq:DCA\]). []{data-label="fig:dynamic_CA"}](./fig6b.pdf){width="0.99\linewidth"}
(a)
(b)
Overall, there is good agreement between the theoretical apparent dynamic contact angle and the experimental data of both Hoffman and Blake & Shikhmurzaev. At low ${\mbox{\textit{Ca}}}$, the current model diverges slightly from the experimental data of the glycerol solution. This difference could be created by errors in contact angle measurement or by differences in the way apparent contact angle is defined. In figure \[fig:collapsed\_comp\] we provide additional comparisons for the remaining fluids that were tested by Hoffman and Blake & Shikhmurzaev. In this figure the magnitude of $g(\lambda,\phi_D)-g(\lambda,\phi_0)$ is plotted against ${\mbox{\textit{Ca}}}\ln(1/\varepsilon)$ so that the model can be represented by a single curve regardless of static contact angle or viscosity ratio. As before, the microscopic contact angle is predicted by equation (\[eq:DCA\]) and the combined model captures the dynamic contact angle behavior. There is some deviation at high ${\mbox{\textit{Ca}}}$, however this is to be expected as the current model is derived by assuming ${\mbox{\textit{Ca}}}\ll 1$. The results presented here can be extended to receding contact lines by combining our model with the model proposed by Eggers [@EggersJ:04c; @EggersJ:05a]. One only needs to substitute negative values of $U$ into equation (\[eq:DCA\]) so that the advancing and receding fluids are reversed.
![ Comparison of the current model to experimental data measured by Hoffman [@HoffmanRL:75a] and Blake & Shikhmurzaev [@ShikhmurzaevYD:02a]. All glycerol solution data points were adapted from [@ShikhmurzaevYD:02a] while the remaining data points were adapted from [@HoffmanRL:75a]. To collapse the model regardless of static contact angle or viscosity ratio, the data is presented as $g(\lambda,\phi_D)-g(\lambda,\phi_0)$ vs. ${\mbox{\textit{Ca}}}\ln(1/\varepsilon)$ where $\varepsilon = 10^{-4}$. []{data-label="fig:collapsed_comp"}](./fig7.pdf){width="0.99\linewidth"}
In this current model, $\phi_0$ is a function of ${\mbox{\textit{Ca}}}$ and is in agreement with the experimental observations of Ramé et al. [@RameE:04a]. However, the finite force analysis does not provide a theoretical means of determining the length scale ratio, $\varepsilon$. Regardless, we found that the results are not particularly sensitive to $\varepsilon$ (in agreement with Bonn et al. [@BonnD:09a]) and therefore a constant value of $10^{-4}$ was used for all experimental comparisons despite the different fluid properties. This suggests that, at least for these test cases, $\varepsilon$ could be treated as a constant that is independent of the fluid properties and experimental set up.
In addition to providing a theoretical model of the dynamic contact angle, the results above will impact other aspects of MCL dynamics. For example, microscopic contact angle determines the sign and strength of vorticity near the MCL [@Mohseni:18f] as well as the surface tension force which is given by $\sigma \cos \phi_0$. These effects are important in microfluidics and heat/mass transfer applications where vorticity and surface tension forces influence mixing and contact line pinning. Furthermore, this result may be useful in numerical simulations where physical phenomena of interest typically span several orders of magnitude. Convergence of numerical solutions are extremely sensitive to the prescribed contact angle and grid size [@ZaleskiS:09a]. In general, grid convergence is achieved when the grid resolution of the simulation is the same order of magnitude as the slip length, typically $10^{-7}$m to $10^{-9}$m for water. Thus, the required number of grid points for most simulations is extremely large. In order to reduce the computational cost and improve accuracy, the Stokes flow solution could be used as a subgrid model. In such a scheme, the minimum grid size would be determined by the validity of the small ${\mbox{\textit{Re}}}$ assumption. The implementation of a Stokes flow subgrid model is outside the scope of this publication, but will be investigated in future works.
Comparison with similar singular Stokes flows {#sect:force_comp}
=============================================
While unfamiliar in the context of a moving contact line, a finite force corresponding to a singular stress is not unprecedented. In fact, there exist two other Stokes flows that contain stress singularities and that have known finite forces, namely the cusped interface flow and Stokeslet. The cusped interface flow, investigated by Richardson [@RichardsonS:68a] and Joseph et al [@JosephDD:90a], is created by two submerged cylinders rotating with constant angular velocity. Under the right conditions, the fluid interface will develop a cusp singularity. The stream function near a cusped interface is reported by Richardson as $$\begin{gathered}
\psi = \dfrac{\sigma}{2\pi\mu} r \ln (r) \sin(\theta).\end{gathered}$$ It is easily shown that the complex formulation of this flow is given by $$\begin{gathered}
G = \mu\omega + ip = -\dfrac{\sigma i}{\pi z}.\end{gathered}$$ Evaluating equation (\[eq:f\_stress\]) using Cauchy’s residue theorem for a singularity that lies on the contour and at a cusp yields a force per unit length of $f = 2\sigma$, in agreement with Richardson.
A similar analysis can be performed for a two-dimensional Stokeslet, i.e. the flow that is created by an infinitely small cylinder moving through a quiescent fluid [@JothiramB:87a; @Guazzelli:11a]. For a Stokeslet that is aligned with the $y$-axis, the stream function and complex function $G$ are given by $$\begin{gathered}
\psi = \dfrac{\alpha}{4\pi} r \sin(\theta)[1-\ln (r)],\\
G = \mu\omega + ip = \dfrac{\mu \alpha i}{2\pi z},\end{gathered}$$ where $\alpha$ is the strength of the Stokeslet. Evaluating equation (\[eq:f\_stress\]) for a contour path that encloses the Stokeslet yields a force per unit length of $f = -\mu \alpha$. This result is consistent with the force reported by Avudainayagam & Jothiram [@JothiramB:87a]. As discussed previously, the viscous surface force in a Stokes flow is solely determined by the pressure and vorticity. From the complex function $G$, we see that despite the different velocity fields, both the Stokeslet and the cusped fluid interface have vorticity and pressure fields that take the form of a dipole. In addition to a singular stress and finite force, we also note that the Stokeslet, cusped interface, and MCL singularity all predict infinite viscous dissipation per unit volume at $r = 0$. While this dissipation is singular, numerous applications of the Stokeslet singularity [@CrowdyDG:10a; @Pozrikidis:90a] have demonstrated that the finite force predicted by these singular models can still be used to model physical problems.
Concluding remarks {#sect:conclusion}
==================
In this publication, the force at a moving contact line was theoretically investigated using the hydrodynamic solution of the MCL. By defining a cylindrical control volume around the MCL, we were able to show using both real and complex analysis that the total viscous force exerted by the fluids on the solid is finite despite a singular stress. Unlike previous treatments of the contact line, this control volume accounts for the viscous forces that act on the fluid-fluid interface in addition to the forces that act on the fluid-solid interface, much like the Young’s equation. With this finite force, we proposed a model for microscopic dynamic contact angle that is a function of the interface velocity, fluid viscositiy, and surface tension. As validation, we combined our model for microscopic contact angle with Cox’s model for apparent contact angle and achieved a good match with experimental measurements. Interestingly, the results reported in this work have been alluded to in previous publications. Cox recognized the possibility of a velocity dependent microscopic contact angle and stated “it is uncertain whether such an angle \[microscopic angle\] would depend on the spreading velocity” [@CoxRG:86a]. In a more direct observation, Voinov [@VoinovOV:76a] stated “In this case $\alpha_m$ \[microscopic angle\] can be a function of the velocity”. Dussan & Davis [@Dussan:74a] recognized the similarities between the MCL problem and other singular models which led them state: “There exist physical situations where the force distributed over a small area is replaced by a force acting at a point or a line. (This implies an unbounded stress tensor)". In the end, we would like to emphasize that the analysis and conclusions made in this work are for a MCL [*model*]{} much like the Stokeslet and cusped fluid interface. The finite force result does not imply that phenomena like slip or thermal activation do not occur, but merely that the net effect of these microscopic phenomena should yield the same total change in momentum within a microscopic control volume enclosing the contact line.
In the field of wetting and dewetting, the concept of a finite force despite a singular stress is somewhat unusual. However, if we look to other fields, we find that there are many two-dimensional singular continuum models with similar characteristics. In electromagnetism, electric fields are singular at the locations of line charges. Through Gauss’ law, we know that the strength of the point charge is finite despite the fact that the electric field is singular. In potential flow theory, line sources and line vortices are regularly used to model flows at high Reynolds numbers. Despite the fact that the velocity or shear stress approaches infinity at these singularities, they still conserve physical quantities such as mass flux and circulation. These singular models do not resolve the exact physics that occur at the singularity and are capable of correctly capturing the global features of the problem. In this sense, there may be fluid slip extremely close to the moving contact line, however we do not need to resolve it as we can already obtain the contact line force and nearby velocity field. Essentially the finite force at the moving contact line is merely a different physical application of the same mathematical concepts that are applied in other fields. At present, the MCL model retains a singular stress and infinite viscous dissipation at the corner singularity and are known limitations of this model. Motivated by the results of this work, future works will seek to extend this model to accurately capture the transfer of energy at the MCL.
General solution to the biharmonic equation {#app:bih_sol}
===========================================
The biharmonic stream function equation is given by $$\begin{gathered}
\nabla^4 \psi = 0.\end{gathered}$$ In polar coordinates, the solution is found using the technique of separation of variables with $\psi = R(r)\Theta(\theta)$. The solution that was initially reported by Michell [@MichellJH:89a], and later extended by Filonenko-Borodich [@FilonenkoBM:58a], is given by $$\begin{gathered}
\label{eq:psi_sol} \psi = (r/\ell)^2\ln (r/\ell) q_{2L}(\theta) + (r/\ell) \ln (r/\ell) q_{1L}(\theta) + \ln (r/\ell) q_{0L}(\theta) + \sum_{n = -\infty}^\infty (r/\ell)^n q_n(\theta).\end{gathered}$$ The functions $q$ are given by $$\begin{aligned}
q_{2L} &= P_{2L}[A_{2L}+B_{2L}\theta],\\
q_{1L} &= P_{1L}[A_{1L} \cos(\theta)+ B_{1L} \sin(\theta) +C_{1L} \theta \cos(\theta) +D_{1L} \theta \sin(\theta)],\\
q_{0L} &= P_{0L}[A_{0L} + B_{0L}\theta],\\
q_0 &= P_0[A_{0} + B_{0} \theta + C_{0} \cos(2\theta) + D_{0} \sin(2\theta)],\\
q_1 &= P_1[A_{1} \cos(\theta) + B_{1} \sin(\theta) + C_{1} \theta \cos(\theta) + D_{1} \theta \sin(\theta)],\\
q_2 &= P_2[A_{2} + B_{2} \theta + C_{2} \cos(2\theta) + D_{2} \sin(2\theta)],\\
q_n & = P_n[A_{n} \cos((n-2)\theta) +B_{n} \cos(n\theta) +C_{n}\sin((n-2)\theta) + D_{n} \sin(n\theta)] \quad \text{for} \quad n \geq 3,\\
q_n &= P_n[A_{n} \cos ((n+2)\theta) +B_{n} \cos(n\theta) + C_{n} \sin((n+2)\theta) + D_{n} \sin(n\theta)] \quad \text{for} \quad n\leq -1,\end{aligned}$$ where $P$ is a dimensional coefficient and $A$, $B$, $C$, and $D$ are coefficients determined by the boundary conditions. The velocity, vorticity, and pressure can all be determined using the equations given by $$\begin{gathered}
\bm{u} = {\boldsymbol{\nabla}}\times \psi \bm{\hat{e}_z},\\
\omega = -{\boldsymbol{\nabla}}^2 \psi,\\
{\boldsymbol{\nabla}}p = \mu\nabla^2 \bm{u}.\end{gathered}$$
In the past, stream functions of various order $r$ have been investigated individually and correlated to unique classes of flows. The paint scraper problem investigated by Taylor [@Taylor:62a] is described terms of order $(r/\ell)^{-1}$. Similarly, the MCL problem investigated by Huh & Scriven [@HuhC:71a] is described by terms of order $(r/\ell)^{-1}$. Moffatt eddies and the hinged plate flow correspond to terms with $n \geq 1$ [@MoffattHK:64a]. The cusped fluid interface and Stokeslet solution described by Richardson [@RichardsonS:68a] and Joseph et al [@JosephDD:90a] is described by terms of order $(r/\ell) \ln (r/\ell)$. Solutions that combine multiple orders of $n$ have been shown to represent more complex flows such as evaporation near the contact line [@SnoeijerJH:12a; @SnoeijerJH:13a].
Zeroth order solution to the moving contact line flow {#app:coeffs}
=====================================================
The zeroth order solution to the moving contact line problem described in §\[sect:geo\_sol\] was first reported by Moffatt [@MoffattHK:64a] and Huh & Scriven [@HuhC:71a]. In a planar wedge geometry, where fluid A has a contact angle of $\phi_0$, the flow is governed by the biharmonic stream function equations given by $$\begin{gathered}
\nabla^4 \psi_{A0} = 0,\\
\nabla^4 \psi_{B0} = 0.\end{gathered}$$ The boundary conditions of the zeroth order solution include no-slip at the fluid-solid interface, zero mass flux through all interfaces, and continuity of tangential velocity and shear stress across the fluid-fluid interface. The resulting boundary conditions on $\psi_{A0}$ and $\psi_{B0}$ are therefore given by $$\begin{aligned}[c]
\psi_{A0}(\theta = 0) &= 0, \\
\psi_{B0}(\theta = \phi_0) &= 0, \\
\left[ \dfrac{1}{r} \dfrac{\partial \psi_{A0}}{\partial \theta} \right]_{\theta = 0} &= U,\\
\left[ \dfrac{\partial \psi_{A0}}{\partial \theta} \right]_{\theta = \phi_0} &= \left[ \dfrac{\partial \psi_{B0}}{\partial \theta} \right]_{\theta = \phi_0},
\end{aligned}
\hspace{3 cm}
\begin{aligned}[c]
\psi_{A0}(\theta = \phi_0) &= 0,\\
\psi_{B0}(\theta = \pi) &= 0,\\
\left[ \dfrac{1}{r} \dfrac{\partial \psi_{B0}}{\partial \theta} \right]_{\theta = \pi} &= -U,\\
\left[ \dfrac{\partial^2 \psi_{A0}}{\partial \theta^2} \right]_{\theta = \phi_0} &= \left[ \lambda \dfrac{\partial^2 \psi_{B0}}{\partial \theta^2} \right]_{\theta = \phi_0},
\end{aligned}$$ where $\lambda = \mu_B/\mu_A$ denotes the viscosity ratio. Solving the system of equations created by the boundary conditions above yields the stream function, velocity, pressure, and vorticity is given by $$\begin{gathered}
\psi_0 =rU[A\cos(\theta) + B \sin(\theta) + C\theta\cos(\theta)+D\theta \sin(\theta)],\\
u_{0r} = \dfrac{1}{r} \dfrac{\partial \psi}{\partial \theta} =U\left[-A\sin(\theta) + B\cos(\theta) + C[\cos(\theta)-\theta\sin(\theta)] + D[\sin(\theta) + \theta\cos(\theta)] \dfrac{}{} \right],\\
u_{0\theta} = -\dfrac{\partial \psi}{\partial r} = -U[A\cos(\theta)+B\sin(\theta)+C\theta\cos(\theta)+D\theta\sin(\theta)],\\
p_0 = \dfrac{2\mu U}{r} [C\cos(\theta)+D\sin(\theta)],\\
\omega_0 = \dfrac{2U}{r} [C\sin(\theta) - D\cos(\theta)].\end{gathered}$$ For each fluid $A$, $B$, $C$, and $D$ are constant coefficients given by $$\begin{aligned}
A_A &= 0,\\
B_A &= \dfrac{(8\phi_0(\lambda - 1))\sin(\phi_0)^2 + 4\pi\lambda\phi_0\sin(2\phi_0) + 8\phi_0(\pi^2 - \lambda\phi_0^2 - 2\pi\phi_0 + \phi_0^2 + \pi\lambda\phi_0)}{\Delta},\\
C_A &= [-8\pi\lambda\sin(\phi_0)^2 + (8\pi\phi_0 - 2\lambda + 4\lambda\phi_0^2 - 4\pi^2 - 4\phi_0^2 - 4\pi\lambda\phi_0 + 2)\sin(2\phi_0) \\
& \quad+ \lambda\sin(4\phi_0) - \sin(4\phi_0)]/\Delta,\\
D_A &= \dfrac{(8 - 8\lambda)\sin(\phi_0)^4 + (16\pi\phi_0 + 8\lambda\phi_0^2 - 8\pi^2 - 8\phi_0^2 - 8\pi\lambda\phi_0)\sin(\phi_0)^2}{\Delta}, \end{aligned}$$ $$\begin{aligned}
A_B &= \dfrac{8\pi^2\sin(\phi_0)^2 + (-\pi(4\pi\phi_0 - 2\lambda + 4\lambda\phi_0^2 - 4\phi_0^2 + 2))\sin(2\phi_0) + \pi(\sin(4\phi_0) - \lambda\sin(4\phi_0))}{\Delta},\\
B_B &= [(8\lambda\phi_0 - 8\phi_0 - 8\phi_0\pi^2 + 8\pi\phi_0^2 - 8\pi\lambda\phi_0^2)\sin(\phi_0)^2 + (2\pi - 2\pi\lambda)\sin(2\phi_0)^2 \\
& \quad + (4\pi\phi_0 - 4\pi^2)\sin(2\phi_0) + 8\phi_0\pi^2 - 16\pi\phi_0^2 - 8\lambda\phi_0^3 + 8\phi_0^3 + 8\pi\lambda\phi_0^2]/\Delta,\\
C_B &= \dfrac{-8\pi\sin(\phi_0)^2 + (4\pi\phi_0 - 2\lambda + 4\lambda\phi_0^2 - 4\phi_0^2 + 2)\sin(2\phi_0) + \lambda\sin(4\phi_0) - \sin(4\phi_0)}{\Delta},\\
D_B &= \dfrac{(8 - 8\lambda)\sin(\phi_0)^4 + (8\pi\phi_0 + 8\lambda\phi_0^2 - 8\phi_0^2)\sin(\phi_0)^2}{\Delta},\\
\Delta &= (\phi_0(8\lambda - 8) - 8\pi\lambda)\sin(\phi_0)^2 + ((4\lambda - 4)\phi_0^2 + 8\pi\phi_0 - 2\lambda - 4\pi^2 + 2)\sin(2\phi_0)\\
& \quad + (\lambda - 1)\sin(4\phi_0) + (8 - 8\lambda)\phi_0^3 + (8\lambda\pi - 16\pi)\phi_0^2 + 8\pi^2\phi_0.\end{aligned}$$ Note that the coefficients are a function of the contact angle, $\phi_0$, and viscosity ratio, $\lambda$, only.
Complex contour integrals with singularities on the contour {#app:SP_theorem}
===========================================================
The contour integrals considered in this paper typically contain singularities that are simultaneously on the contour and at a corner. In this section, we demonstrate how Cauchy’s residue theorem is affected by singularities on the boundary based on §1.5 of [@EstradaR:12a].
Consider the analytic function $W = 1/(z-\xi)$ and the closed contour $C$ that defines an interior region, $S_I$, and an exterior region, $S_O$. The solution to the complex contour integral of $W$ along the path $C$ is given by $$\begin{gathered}
\oint_C W dz=\begin{cases}
2\pi i, & \xi \in S_I,\\
0, & \xi \in S_O,\\
\pi i, & \xi \in C \text{ along a smooth segment of C},\\
\phi i, & \xi \in C \text{ and at a corner},
\end{cases}
\end{gathered}$$ where the four cases correspond to the location of $\xi$, as shown in figure \[fig:sokhotski\]. In the first two cases, the singularity lies inside the contour ($\xi \in S_I$) or outside the contour ($\xi \in S_O$) and Cauchy’s residue theorem yields a solution of $2\pi i\operatorname{Res}(W)$ and 0, respectively. If $\xi$ lies on a smooth segment of $C$, then the Sokhotski-Plemelj theorem states that the integral will equal $\pi i\operatorname{Res}(W)$, which in this case is $\pi i$. If $\xi$ lies on $C$ and at a corner, then the Sokhotski-Plemelj theorem states that the integral will equal $\phi i\operatorname{Res}(W) = \phi i$, where $\phi$ is the angle of the corner.
![(a) $\xi$ lies inside the contour in the region $S_I$ (b) $\xi$ lies outside the contour in the region $S_O$ (c) $\xi$ lies on the path $C$ along a smooth segment (d) $\xi$ lies on the path $C$ at a corner point.[]{data-label="fig:sokhotski"}](./fig8a.pdf){width="0.69\linewidth"}
![(a) $\xi$ lies inside the contour in the region $S_I$ (b) $\xi$ lies outside the contour in the region $S_O$ (c) $\xi$ lies on the path $C$ along a smooth segment (d) $\xi$ lies on the path $C$ at a corner point.[]{data-label="fig:sokhotski"}](./fig8b.pdf){width="0.69\linewidth"}
![(a) $\xi$ lies inside the contour in the region $S_I$ (b) $\xi$ lies outside the contour in the region $S_O$ (c) $\xi$ lies on the path $C$ along a smooth segment (d) $\xi$ lies on the path $C$ at a corner point.[]{data-label="fig:sokhotski"}](./fig8c.pdf){width="0.69\linewidth"}
![(a) $\xi$ lies inside the contour in the region $S_I$ (b) $\xi$ lies outside the contour in the region $S_O$ (c) $\xi$ lies on the path $C$ along a smooth segment (d) $\xi$ lies on the path $C$ at a corner point.[]{data-label="fig:sokhotski"}](./fig8d.pdf){width="0.69\linewidth"}
\(a) $\xi \in S_I$
\(b) $\xi \in S_O$
\(c) $\xi \in C$
\(d) $\xi \in C$ and at a corner
References {#references .unnumbered}
==========
|
---
abstract: 'Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. , a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation. The package provides an interface to , making it possible to access the core run modes of in . With these run modes, users can find approximate solutions to zero-dimensional systems and positive-dimensional systems, test numerically whether a point lies on a variety, sample numerically from a variety, and perform parameter homotopy runs.'
author:
- 'Daniel J. Bates[^1], Elizabeth Gross[^2], Anton Leykin[^3], Jose Israel Rodriguez[^4]'
title: Bertini for Macaulay2
---
Numerical algebraic geometry
=============================
[*Numerical algebraic geometry (numerical AG)*]{} refers to a set of methods for finding and manipulating the solution sets of systems of polynomial equations. Said differently, given $f:\mathbb C^N\to\mathbb C^n$, numerical algebraic geometry provides facilities for computing numerical approximations to isolated solutions of $V(f)=\left\{z\in\mathbb C^N | f(z)=0\right\}$, as well as numerical approximations to generic points on positive-dimensional components. The book [@SWbook] provides a good introduction to the field, while the newer book [@BertiniBook] provides a simpler introduction as well as a complete manual for the software package [@Bertini].
is a free, open source software package for computations in numerical algebraic geometry. The purpose of this article is to present a [@M2] package that provides an interface to . This package uses basic datatypes and service routines for computations in numerical AG provided by the package . It also fits the framework of the package [@NAG4M2], a native implementation of a collection of numerical AG algorithms: most of the core functions of have an option of using instead of the native solver.
In the remainder of this section, we very briefly describe a few fundamental concepts of the field. In the subsequent sections, we describe the various run modes of that have been implemented in this interface. We conclude with Section 5, which describes how to use within .
Finding isolated solutions
--------------------------
The core computational engine within is [*homotopy continuation*]{}. This is a three-stage process for finding a superset of all isolated solutions in $V(f)$. Given a polynomial system $f(z)$, the three steps are as follows:
1. Choose an easily-solved polynomial system $g(z)$ that reflects the structure of $f(z)$, and solve it. Call this set of solutions $S$.
2. Form the homotopy $$H(z,t)=(1-t)f(z) + \gamma tg(z),$$ with $\gamma\in\mathbb C$ a random complex number. Notice that $H(z,1)=\gamma g(z)$, the solutions of which are known, and $H(z,0)=f(z)$, for which we seek the solutions.
3. There is a real curve extending from each solution $z\in S$. Use predictor-corrector methods, adaptive precision, and endgames to track along all of these paths as $t$ goes from 1 to 0.
Assuming $g(z)$ is constructed in one of several canonical ways [@SWbook], there is a probability one guarantee that this procedure will result in a superset of all isolated solutions of $f(z)=0$.
There are many variations on this general technique, and there are many minor issues to consider when implementing this method. However, due to space limitations, we leave the reader to explore the references for more information on this powerful method.
Finding irreducible components
------------------------------
Given an irreducible algebraic set $X$ of dimension $k$, it is well known that $X$ will intersect almost any linear space of codimension $k$ in a finite set of points. In fact, there is a Zariski open subset of the set of all linear spaces of codimension $k$ for which intersection with $X$ yields some fixed number of points, called the [*degree*]{} of $X$, $\deg X$.
This fundamental fact underlies the computation of positive-dimensional irreducible components in numerical algebraic geometry. Suppose algebraic set $Z$ decomposes into irreducible components $Z_{i,j}$, $$Z=\cup_{i=0}^{\dim Z}\cup_{j\in \Lambda_i} Z_{i,j},$$ where $i$ is the dimension of $Z_{i,j}$ and $j$ is just the index of component $Z_{i,j}$ in dimension $i$, stored in finite indexing set $\Lambda_i$.
In numerical algebraic geometry, the representation $W$ of an algebraic set $Z$ consists of a representation $W_{i,j}$ for each irreducible component $Z_{i,j}$ of $Z$. In particular, [*witness set*]{} $W_{i,j}$ is a triple $(f, L_{i}, \widehat{W}_{i,j})$, consisting of polynomial system $f$, linear functions $L_{i}$ corresponding to a linear space of codimension $i$, and [*witness point set*]{} $\widehat{W}_{i,j} = Z_{i,j}\cap V(L_{i})$.
There are a variety of ways to compute $W$, many of which are described in detail in [@BertiniBook]. Most of these methods can be accessed through the package by using optional inputs to specify the desired algorithm.
Solving zero-dimensional systems
================================
In the following sections we outline and give examples of the different run modes implemented in the interface package .
Finding solutions of zero-dimenstional systems
----------------------------------------------
The method [bertiniZeroDimSolve]{} calls to solve a polynomial system and returns solutions as a list of [Points]{} using the data types from . Diagnostic information, such as the residuals and the condition number, are stored with the coordinates of the solution and can be viewed using [peek]{}.
i1 : R=CC[x,y];
i2 : f = {x^2+y^2-1,(x-1)^2+y^2-1};
i3 : solutions=bertiniZeroDimSolve(f)
o3 = {{.5, .866025}, {.5, -.866025}}
i4 : peek solutions_0
o4 = Point{ConditionNumber => 88.2015 }
Coordinates => {.5, .866025}
CycleNumber => 1
FunctionResidual => 3.66205e-15
LastT => .000390625
MaximumPrecision => 52
NewtonResidual => 4.27908e-15
SolutionNumber => 3
Users can specify to use regeneration, an equation-by-equation solving method, by setting the option [USEREGENERATION]{} to 1.
i5 : solutions=bertiniZeroDimSolve(f, USEREGENERATION=>1);
In common applications, one would like to classify solutions, e.g., separate real solutions from non-real solutions, and, thus, recomputing solutions to a higher accuracy becomes important. The method [bertiniRefineSols]{} calls the sharpening module of and sharpens a list of solutions to a desired number of digits using Newton’s method.
i6 : refinedSols=bertiniRefineSols(f, solutions, 20);
(coordinates refinedSols_0)_1
o6 = .86602540378443859659+3.5796948761134507351e-83*ii
Parameter homotopies
--------------------
Many fields, such as statistics, physics, biochemistry, and engineering, have applications that require solving a large number of systems from a parameterized family of polynomial systems. In such situations, computational time can be decreased by using parameter homotopies. For an example illustrating how parameter homotopies can be used in statistics see [@HRS12].
The method [bertiniParameterHomotopy]{} calls to run both stages of a parameter homotopy. First, assigns a random complex number to each specified parameter and solves the resulting system, then, after this initial phase, computes solutions for every given choice of parameters using a number of paths equal to the exact root count.
i7 : R=CC[a,b,c][x,y];
i8 : f={a*x^2+b*y^2-c, y};
i9 : bertiniParameterHomotopy(f,{a,b,c},{{1,1,1},{2,3,4}})
o9 = {{{-1, 0}, {1, 0}}, {{-1.41421, 0}, {1.41421, 0}}}
Solving positive-dimensional systems
====================================
Given a positive-dimensional system $f$, the method [bertiniPosDimSolve]{} calls to compute a numerical irreducible decomposition. This decomposition is assigned the type [NumericalVariety]{} in . In the default settings, uses a classical cascade homotopy to find witness supersets in each dimension, removes extra points using a membership test or local dimension test, deflates singular witness points, then factors using a combination of monodromy and a linear trace test.
i10 : R = CC[x,y,z];
i11 : f = {(y^2+x^2+z^2-1)*x, (y^2+x^2+z^2-1)*y};
i12 : NV = bertiniPosDimSolve f
o12 = A variety of dimension 2 with components in
dim 1: [dim=1,deg=1]
dim 2: [dim=2,deg=2].
o12 : NumericalVariety
Once the solution set to a system, i.e., the variety $V$, is computed and stored as a [NumericalVariety]{}, [bertiniComponentMemberTest]{} can be used to test numerically whether a set of points $p$ lie on the variety $V$. For every point in $p$, [bertiniComponentMemberTest]{} returns the components to which that point belongs. As for sampling, [bertiniSample]{} will sample from a witness set $W$. These methods call the membership testing and sampling options in respectively.
i13 : p={{0,0,0}};
i14 : bertiniComponentMemberTest (NV, p)
o14 = {{[dim=1,deg=1]}}
i15 : component=NV#1_0
i16 : bertiniSample(component,1)
o16 = {{0, -8.49385e-20+7.48874e-20*ii, -.148227-.269849*ii}}
Solving homogeneous systems
===========================
The package includes functionality to solve a homogenous system that defines a projective variety. In , the numerical computations are performed on a generic affine chart to compute representatives of projective points. To solve homogeneous equations, set the option $\tt ISPROJECTIVE$ to $1$. If the user inputs a square system of $n$ homogeneous equations in $n+1$ unknowns, then the method [bertiniZeroDimSolve]{} outputs a list of points in projective space.
i17 : R = CC[x,y,z];
i18 : f = {y^2-4*z^2,16*x^2-y^2};
i19 : bertiniZeroDimSolve(f,ISPROJECTIVE=>1);
o19 = {{.251411+.456072*ii, 1.00564+1.82429*ii, .502821+.912143*ii},
{.106019+.160896*ii, .424078+.643585*ii, -.212039-.321792*ii},
{-.15916-.12286*ii, .636639+.49144*ii, -.318319-.24572*ii},
{-.48005-.092532*ii, 1.9202+.370128*ii, .960101+.185064*ii}}
If $f$ is a positive-dimensional homogeneous system of equations, then the method calls to compute a numerical irreducible decomposition of the projective variety defined by $f$.
i20 : R = CC[x,y,z];
i21 : f = {(x^2+y^2-z^2)*(z-x),(x^2+y^2-z^2)*(z+y)};
i22 : NV = bertiniPosDimSolve(f,ISPROJECTIVE=>1)
o22 : = A projective variety with components in projective dimension:
dim 0: [dim=0,deg=1]
dim 1: [dim=1,deg=2]
Using from
============
The package depends on the package, a collection of basic datatypes and service routines common to all packages for numerical AG: e.g., an interface package [@PHCpackM2] to another polynomial homotopy continuation solver, [@PHC], also has this dependence. The dependencies between the metioned packages are depicted on the diagram below: dashed arrows stand for a dependency that is optional and is engaged if an executable for the corresponding software is installed.
(NAG) [[*NumericalAlgebraicGeometry*]{}]{}; (NAGtypes) [[*NAGtypes*]{}]{}; (Bertini) [[*Bertini*]{}]{}; (PHCpack) [[*PHCpack*]{}]{}; (NAG) – (NAGtypes); (Bertini) – (NAGtypes); (PHCpack) – (NAGtypes); (NAG) – (Bertini); (NAG) – (PHCpack);
While independent from the package, our interface provides a valuable option for this package: the user can set as a default solver for homotopy continuation tasks.
i23 : needsPackage "NumericalAlgebraicGeometry";
i24 : setDefault(Software=>BERTINI)
An alternative way is to specify the [Software]{} option in a particular command:
i25 : CC[x,y]; system = {x^2+y^2-1,x-y};
i26 : sols = solveSystem(system, Software=>M2engine)
o26 = {{-.707107, -.707107}, {.707107, .707107}}
i27 : refsols = refine(system, sols, Bits=>99, Software=>BERTINI);
i28 : first coordinates first refsols
o28 = -.707106781186547524400844362105-2.13764004134262114934289955768e-140*ii
o28 : CC (of precision 100)
The unified framework for various implementations of numerical AG algorithms should be particularly convenient to a user doing numerical computations with tools from many packages.
[99]{}
D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler. Bertini: Software for numerical algebraic geometry. Available at [http://www.nd.edu/$\sim$sommese/bertini]{}.
D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler. [*Numerically solving polynomial systems with Bertini.*]{} To be published by SIAM, 2013.
D.R. Grayson and M.E. Stillman. Macaulay2, a software system for research in algebraic geometry, available at [www.math.uiuc.edu/Macaulay2/.]{}
E. Gross, S. Petrović, and J. Verschelde. [*Interfacing with PHCpack.*]{} J. Softw. Algebra Geom. 5, 20–25, 2013.
J. Hauenstein, J. Rodriguez, and B. Sturmfels. *Maximum likelihood for matrices with rank constraints*, preprint, arXiv:1210.0198.
A. Leykin. [*Numerical algebraic geometry.*]{} J. Softw. Algebra Geom. 3, 5–10, 2011.
A.J. Sommese and C.W. Wampler. [*The numerical solution to systems of polynomials arising in engineering and science.*]{} World Scientific, Singapore, 2005.
J. Verschelde. [*Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation*]{} ACM Trans. Math. Softw. 25(2):251–276, 1999. Software available at [http://www.math.uic.edu/$\sim$jan]{}.
[^1]: Department of Mathematics, Colorado State University, Fort Collins, CO 80523. [bates@math.colostate.edu]{}. This work was partially supported by NSF grant DMS–1115668.
[^2]: Department of Mathematics, North Carolina State University, Raleigh, NC 27695; [eagross@ncsu.edu]{}. This work was partially supported by NSF award DMS–1304167.
[^3]: School of Mathematics, Georgia Tech, Atlanta, GA; [leykin@gatech.math.edu]{}. This work was partially supported by NSF grant DMS–1151297.
[^4]: Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720 [jo.ro@berkeley.edu]{}. This work was partially supported by NSF grant DMS–0943745.
|
---
abstract: 'Transactional memory (TM) has emerged as a promising abstraction for concurrent programming alternative to lock-based synchronizations. However, most TM models admit only *isolated* transactions, which are not adequate in multi-threaded programming where transactions have to interact via shared data *before* committing. In this paper, we present *Open Transactional Memory* (OTM), a programming abstraction supporting *safe, data-driven* interactions between *composable* memory transactions. This is achieved by relaxing isolation between transactions, still ensuring atomicity: threads of different transactions can interact by accessing shared variables, but then their transactions have to commit together—actually, these transactions are transparently *merged*. This model allows for *loosely-coupled* interactions since transaction merging is driven only by accesses to shared data, with no need to specify participants beforehand. In this paper we provide a specification of the OTM in the setting of Concurrent Haskell, showing that it is a conservative extension of current STM abstraction. In particular, we provide a formal semantics, which allows us to prove that OTM satisfies the *opacity* criterion.'
title: A Specification of Open Transactional Memory for Haskell
---
Introduction {#sec:introduction}
============
=-1 The advent of multicore architectures has emphasized the importance of abstractions supporting correct and scalable multi-threaded programming. In this model, threads can collaborate by interacting on data structures (such as tables, message queues, buffers, etc.) kept in shared memory. Traditional lock-based mechanisms (like semaphores and monitors) used to regulate access to these shared data are notoriously difficult and error-prone, as they easily lead to deadlocks, race conditions and priority inversions; moreover, they are not composable and hinder parallelism, thus reducing efficiency and scalability. *Transactional memory* (TM) has emerged as a promising abstraction to replace locks [@moss:transactionalmemorybook; @st:dc1997]. The basic idea is to mark blocks of code as *atomic*; then, execution of each block will appear either as if it was executed sequentially and instantaneously at some unique point in time, or, if aborted, as if it did not execute at all. This is obtained by means of *optimistic* executions: the blocks are allowed to run concurrently, and eventually if an interference is detected a transaction is restarted and its effects are rolled back. Thus, each transaction can be viewed in isolation as a *single-threaded* computation, significantly reducing the programmer’s burden. Moreover, transactions are composable and ensure absence of deadlocks and priority inversions, automatic roll-back on exceptions, and increased concurrency.
However, in multi-threaded programming different transactions may need to interact and exchange data *before* committing. In this situation, transaction isolation is a severe shortcoming. A simple example is a request-response interaction between two transactions via a shared buffer, like in a master/worker situation. We could try to synchronize the threads accessing the buffer [[[]{.nodecor}]{}]{} by means of two semaphores `c1`, `c2` as follows:\
``` {baseline="t"}
// Party1 (Master)
atomically {
<put request in b>
up(c1);
<some other code; may abort>
down(c2); // wait for answer
<get answer from b; may abort>
}
```
``` {baseline="t"}
// Party2 (Worker)
atomically {
down(c1); // wait for data
<get request from b>
<compute answer; may abort>
<put answer in b>
up(c2);
}
```
\
Unfortunately, this solution does not work: any admissible execution requires an interleaved scheduling between the two transactions, thus violating isolation; hence, the transactions deadlock as none of them can progress. It is important to notice that this deadlock arises because interaction occurs between threads of *different* transactions; in fact, the solution above is perfectly fine for threads outside transactions or within the same transaction.
To overcome this limitation, in this paper we propose a programming model for *safe, data-driven* interactions between memory transactions. The key observation is that *atomicity* and *isolation* are two disjoint computational aspects:
- an *atomic non-isolated* block is executed “all-or-nothing”, but its execution can overlap others’ and *uncontrolled* access to shared data is allowed;
- a *non-atomic isolated* block is executed “as it were the only one” (i.e., in mutual exclusion with others), but no rollback on errors is provided.
Thus, a “normal” block of code is neither atomic nor isolated; a mutex block (like Java *synchronized* methods) is isolated but not atomic; and a usual TM transaction is a block which is both atomic and isolated. Our claim is that *atomic non-isolated blocks can be fruitfully used for implementing safe composable interacting memory transactions*—henceforth called *open transactions*.
In this model, a transaction is composed by several threads, called *participants*, which can cooperate on shared data. A transaction commits when all its participants commit, and aborts if any thread aborts. Threads participating to different transactions can access to shared data, but when this happens the transactions are *transparently merged* into a single one. For instance, the two transactions of the synchronization example above would automatically merge becoming the same transaction, so that the two threads can synchronize and proceed. Thus, this model relaxes the isolation requirement still guaranteeing atomicity and consistency; moreover, it allows for *loosely-coupled* interactions since transaction merging is driven only by run-time accesses to shared data, without any explicit coordination among the participants beforehand.
In summary, the contributions of this paper are the following:
- We present *Open Transactional Memory*, a transactional memory model where multi-threaded transactions can interact by non-isolated access to shared data. Consistency and atomicity are ensured by transparently *merging* transactions at runtime.
- We describe this model in the context of Concurrent Haskell (Section \[sec:cot\]). Namely, we define two monads [[[]{.nodecor}]{}]{} and [[[]{.nodecor}]{}]{}, representing the computational aspects of atomic *multi-threaded open* (i.e., non-isolated) transactions and atomic *single-threaded isolated* transactions, respectively. Using the construct [[[]{.nodecor}]{}]{}, programs in the [[[]{.nodecor}]{}]{} monad are executed “all-or-nothing” but without isolation; hence these transactions can merge at runtime. When needed, blocks inside transactions can be executed in isolation by using the construct [[]{.nodecor}]{}. Both OTM and ITM transactions are *composable*, and we exploit Haskell type system to forbid irreversible effects inside these two monads.[^1]
- We provide a formal operational semantics of our system (Section \[sec:semantics\]). This semantics defines clearly the behaviour also in less intuitive situations, and serves as a reference for implementations. Using this semantics we prove that OTM satisfies the *opacity* correctness criterion for transactions [@gk:ppopp08].
Some concluding remarks and directions for future work are in Section \[sec:conclusions\].
Concurrency in Haskell {#sec:background}
======================
Haskell was born as pure lazy functional language; side effects are handled by means of monads [@pw:popl1993]. For instance, I/O actions have type [[[a]{.nodecor}]{}]{} and can be combined together by the monadic bind combinator [[>>=]{.nodecor}]{}. Therefore, the function [[:: Char -> IO ()]{.nodecor}]{} takes a character and delivers an I/O action that, when performed (even multiple times), prints the given character. Besides external inputs/outputs, values of [[[]{.nodecor}]{}]{} include operations with side effects on mutable (typed) cells. A cell holding values of type [[[]{.nodecor}]{}]{} has type [[[a]{.nodecor}]{}]{} and may be dealt with only via the following operations:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
newIORef :: a -> IO (IORef a)
readIORef :: IORef a -> IO a
writeIORef :: IORef a -> a -> IO ()
```
Concurrent Haskell [@pgf:popl1996] adds support to threads which independently perform a given I/O action as explained by the type of the thread creation function:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
forkIO :: IO () -> IO ThreadId
```
The main mechanism for safe thread communication and synchronisation are *MVars*. A value of type [[[a]{.nodecor}]{}]{} is mutable location (as for [[[a]{.nodecor}]{}]{}) that is either empty or full with a value of type [[[]{.nodecor}]{}]{}. There are two fundamental primitives to interact with MVars:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
takeMVar :: Mvar a -> IO a
putMvar :: Mvar a -> a -> IO ()
```
The first empties a full location and blocks otherwise whereas the second fills an empty location and blocks otherwise. Therefore, MVars can be seen as one-place channels and the particular case of [[[()]{.nodecor}]{}]{} corresponds to binary semaphores.
We refer the reader to [@jones:2010awkward-squad] for an introduction to concurrency, I/O, exceptions, and cross language interfacing (the “awkward squad” of pure, lazy, functional programming).
STM Haskell [@hmpm:ppopp2005] builds on Concurrent Haskell adding *transactional actions* and a transactional memory for safe thread communication, called *transactional variables* or *TVars* for short.
Transactional actions have type [[[a]{.nodecor}]{}]{} and are concatenated using [[[]{.nodecor}]{}]{} monadic “bind” combinator, akin I/O actions. A transactional action remains tentative during its execution and (its effect) is exposed to the rest of the system by
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
atomically :: STM a -> IO a
```
which takes an STM action and delivers an I/O action that, when performed, runs the transaction guaranteeing atomicity and isolation with respect to the rest of the system.
Transactional variables have type [[[a]{.nodecor}]{}]{} where [[[]{.nodecor}]{}]{} is the type of the value held and, like IOrefs, are manipulated via the interface:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
newTVar :: a -> STM (TVar a)
readTVar :: TVar a -> STM a
writeTVar :: TVar a -> a -> STM ()
```
For instance, the following code uses monadic bind to combine a read and write operation on a transactional variable and define a “transactional update”:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
modifyTVar :: TVar a -> (a -> a) -> STM ()
modifyTVar var f = do
x <- readTVar var
writeOTVar var (f x)
```
Then, [[(modifyTVarxf)]{.nodecor}]{} delivers an I/O action that applies [[]{.nodecor}]{} to the value held by [[]{.nodecor}]{} and updates [[]{.nodecor}]{} accordingly—the two steps being executed as a single atomic isolated operation.
The primitives recalled so far cover memory interaction, but STM allows also for *composable blocking*. In STM Haskell, blocking translates in “this thread has been scheduled too early, i.e., the right conditions are not fulfilled (yet)”. The programmer can tell the scheduler about this fact by means of the primitive:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
retry :: STM a
```
The semantics of [[]{.nodecor}]{} is to abort the transaction and re-run it after at least one of the transactional variables it has read from has been updated—there is no point in blindly restarting a transaction.
Finally, transactions can be composed as alternatives by means of
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
orElse :: STM a -> STM a -> STM a
```
which evaluates its first argument, and if this results is a [[]{.nodecor}]{} the second argument is evaluated discarding any effect of the first.
Composable open transactions {#sec:cot}
============================
In this section we present the key ideas of the paper by gradually introducing the primitives from the [[]{.nodecor}]{}library, summarised in Figure \[fig:base-interface\].
Although the OTM model can be implemented in any language, we consider Haskell because its expressive type system offers a perfect environment for studying the ideas of transactional memory. In [@hmpm:ppopp2005] this has been used to single out computations which can be executed in transactions, i.e. terms which can perform memory effects, from those which can perform irreversible input/output effects. In this paper we refine further this approach by using the type system to separate *isolated* transactions from those which can interact, and hence merged.
``` {tabsize="3" gobble="2"}
data ITM a
data OTM a
-- henceforth, t is a placeholder for ITM or OTM --
-- Sequencing, do notation ------------------------
(>>=) :: t a -> (a -> t b) -> t b
return :: a -> t a
-- Running isolated and atomic computations -------
atomic :: OTM a -> IO a
isolated :: ITM a -> OTM a
retry :: ITM a
orElse :: ITM a -> ITM a -> ITM a
-- Exceptions -------------------------------------
throw :: Exception e => e -> t a
catch :: Exception e => t a -> (e -> t a) -> t a
-- Threading --------------------------------------
fork :: OTM () -> OTM ThreadId
-- Transactional memory ---------------------------
data OTVar a
newOTVar :: a -> ITM (OTVar a)
readOTVar :: OTVar a -> ITM a
writeOTVar :: OTVar a -> a -> ITM ()
```
The key point is to separate isolation from atomicity. In fact, isolation is a computational aspect which can be *added* to atomic transactions. From this perspective, we distinguish between isolated atomic actions and (non isolated) atomic actions. The former are values of type [[[a]{.nodecor}]{}]{} and the latter of [[[a]{.nodecor}]{}]{}. Each type of actions can be sequentially composed (by the corresponding monadic binders) preserving atomicity and, for the former, isolation.
The function [[]{.nodecor}]{} takes an isolated atomic action and delivers an atomic action whose effects are guaranteed to be executed in isolation with respect to other actions. Then, [[]{.nodecor}]{} takes an atomic action and delivers an I/O action that when performed runs a transaction whose effects are kept tentative until it commits. Tentative effects are shared among all non-isolated transactions. Therefore, any value of type [[[a]{.nodecor}]{}]{} can be seen as a value of [[[a]{.nodecor}]{}]{} for the I/O they deliver is the same:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
atomically = atomic . isolated
```
#### Isolation
[[]{.nodecor}]{}supports composable blocking via the primitive [[]{.nodecor}]{}, under [[]{.nodecor}]{}slogan “a thread that has to be blocked because it has been scheduled too soon”. As for [[]{.nodecor}]{}, retrying a transactional action actually corresponds to block the threads on some condition. Note that [[::OTMa]{.nodecor}]{} is not a primitive since it can be defined from that of [[]{.nodecor}]{} as [[retry]{.nodecor}]{}.
Checks may be declared as follows:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
check :: Bool -> ITM ()
check b = if b then return () else retry
```
although similar primitives may be implemented at the runtime level in order to use this information in thread scheduling.
[[]{.nodecor}]{}provides a mechanism for safe thread communication by means of transactional variables called *OTVars*, similar to [[]{.nodecor}]{}’s TVars but supporting *open* transactions. These variables are values of type [[[a]{.nodecor}]{}]{} where [[[]{.nodecor}]{}]{} is the type of value held. Creating, reading and writing OTVars is done via the interface shown in Figure \[fig:base-interface\]. All these actions are both atomic and isolated as ensured by their type. Therefore, when it comes to actions of type [[[a]{.nodecor}]{}]{}, OTVars are basically TVars; e.g. [[]{.nodecor}]{} from [[]{.nodecor}]{}corresponds to:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
modifyOTVar :: OTVar a -> (a -> a) -> ITM ()
modifyOTVar var f = do
x <- readOTVar var
writeOTVar var (f x)
```
From its type it is immediate to see that the update is both atomic and isolated. In fact, read and write operations are glued together by the [[>>=]{.nodecor}]{} combinator, preserving both properties.
Likewise, invariants on transactional variables can be easily checked by composing reads and checks as follows:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
assertOTVar :: OTVar a -> (a -> Bool) -> ITM ()
assertOTVar var p = do
x <- readOTVar var
check (p x)
```
#### Blocking
A semaphore is a counter with two fundamental operation: [[]{.nodecor}]{} which increments the counter and [[]{.nodecor}]{} which decrements the counter if it is not zero and blocks otherwise. Semaphores are implemented using [[]{.nodecor}]{}as OTVars holding a counter:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
type Semaphore = OTVar Int
```
Then, [[]{.nodecor}]{} and [[]{.nodecor}]{} are two trivial atomic and isolated updates, with the latter being guarded by a pre-condition:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
up :: Semaphore -> ITM ()
up s = modifyOTvar s (1+)
down :: Semaphore -> ITM ()
down s = do
assertOTVar s (> 0)
modifyOTVar s (-1+)
```
Actions can also be composed as alternatives by means of the primitive [[]{.nodecor}]{}. For instance, the following takes a family of semaphores and delivers an action that decrements one of them, blocking only if none can be decremented:
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
downAny :: [Sempahore] -> ITM ()
downAny (x:xs) = down x `orElse` downAny xs
downAny [] = retry
```
#### Interaction
The interchangeability of [[]{.nodecor}]{}and [[]{.nodecor}]{}ends when isolation is dropped. In fact, [[]{.nodecor}]{}offers shared OTVars as a mechanism for safe *transaction interaction*. This means that non-isolated transactional actions see the effects on shared variables of any other non-isolated transactional action, as they are performed concurrently on the same object. This flow of information introduces dependencies between concurrent tentative actions tying together their fate: an action cannot make its effects permanent, if it depends on informations produced by another action which fails to complete. [[]{.nodecor}]{}guarantees coherence of transactional actions in presence of interaction through shared transactional variables. Thus, OTVars enables loosely-coupled interaction right inside atomic actions taking the programming style of [[]{.nodecor}]{}a step further. For instance, communication, rendezvous, brokering, and in general, multi-party interactions can all be atomic (non-isolated) actions.
In order to substantiate these claims, let us see open transactions in action by implementing a synchronisation scenario as described in Section \[sec:introduction\]. In this example a master process outsources part of an atomic computation to some thread chosen from a worker pool; data is exchanged via some shared variable, whose access is coordinated by a pair of semaphores. Notably, both the master and the worker can abort the computation at any time, leading the other party to abort as well. This can be achieved straightforwardly using [[]{.nodecor}]{}:\
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
master c1 c2 = do
-- put request
isolated (up c1)
-- do something else
isolated (down c2)
-- get answer
```
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
worker c1 c2 = do
-- do something
isolated (down c1)
-- get request
-- put answer
isolated (up c2)
```
\
Both functions deliver atomic actions in [[]{.nodecor}]{}, and hence are not isolated. We used semaphores for the sake of exposition but we could synchronize by means of more abstract mechanisms, like barriers, channels or futures, which can be implemented using [[]{.nodecor}]{}.
#### Concurrency
Differently from [[]{.nodecor}]{}, [[]{.nodecor}]{}supports parallelism inside non-isolated transactions. We can easily fork new threads without leaving [[[]{.nodecor}]{}]{} but, like any effect of a transactional action, thread creation and execution remain tentative until the whole transaction commits. Forked threads participate to their transaction and impact its life-cycle (e.g. issuing aborts) as any other participant. This means that before committing, all forked threads have to complete their transactional action, i.e. terminate. Therefore, although the whole effect delivered by the transaction has happened concurrently, forked threads never leave a transaction alive.
Because of their transactional nature, threads forked inside a transaction do not have compensations nor continuations (i.e. I/O actions to be executed after an abort or after a commit). Compensations are pointless since aborts revert all effects including thread creation. It is indeed possible to replace the primitive [[]{.nodecor}]{} with one supporting I/O actions as continuations like
``` {tabsize="3" xleftmargin="2ex" gobble="1"}
forkCont :: OTM a -> (a -> IO ()) -> OTM ThreadID
```
In fact, this mechanism can be implemented by means of the primitives already offered [[]{.nodecor}]{}: since commits are synchronisation points, the above corresponds to the parent thread forking a thread for each continuation, after the atomic action is successfully completed.
On the other hand, by definition isolated atomic actions have to appear as being executed in a single-threaded setting; hence [[[]{.nodecor}]{}]{}, like [[[]{.nodecor}]{}]{}, does not support thread creation.
Formal specification of [[]{.nodecor}]{} {#sec:semantics}
========================================
------- ----------------- -------------------------------------------------------------
Value $V\Coloneqq$ $r \mid {\textnormal{\ttfamily\textbackslash $x$\;->\;$M$}}
\mid {\textnormal{\ttfamilyreturn\;$M$}}
\mid {\textnormal{\ttfamily$M$\;>>=\;$N$}}
\mid {\textnormal{\ttfamilythrow\;$M$}}
\mid {\textnormal{\ttfamilycatch\;$M$\;$N$}}
\mid {\textnormal{\ttfamilyputChar\;$c$}}
\mid $
$
{\textnormal{\ttfamilygetChar}}
\mid {\textnormal{\ttfamilyfork\;$M$}}
\mid {\textnormal{\ttfamilyatomic\;$M$\;$N$}}
\mid {\textnormal{\ttfamilyisolated\;$M$}}
\mid {\textnormal{\ttfamilyretry}}
\mid {\textnormal{\ttfamily$M$\;`orElse`\;$N$}}
\mid $
$
{\textnormal{\ttfamilynewOTVar\;$M$}}
\mid {\textnormal{\ttfamilyreadOTVar\;$r$}}
\mid {\textnormal{\ttfamilywriteOTVar\;$r$\;$M$}}
$
Term $M, N\Coloneqq$ $x \mid V \mid M\,N \mid \dots$
------- ----------------- -------------------------------------------------------------
--------------- ----------------------------- -------------------------------------------------------------------------------------------------------
Thread $T_t \Coloneqq$ ${\lthrparenthesisM{]\mspace{-3.7mu})}_{t}} \mid {{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,k}}$
Thread family $P \Coloneqq$ $T_{t_1} \parallel \dots \parallel T_{t_n}\qquad \forall i,j\ t_i \neq t_j $
Expression $\mathbb{E}\Coloneqq$ ${[-]}\mid {\textnormal{\ttfamily$\mathbb{E}$ >>= $M$}}$
Plain process $\mathbb P_t \Coloneqq$ ${{(\mspace{-3.7mu}[}\mathbb E{]\mspace{-3.7mu})}_{t}} \parallel P \hfill t \notin P$
Transaction $\mathbb T_{t,k} \Coloneqq$ ${{(\mspace{-3.7mu}[}{\mathbb E;M}{]\mspace{-3.7mu})}_{t,k}} \parallel P \hfill t \notin P$
Any process $\mathbb A_t \Coloneqq$ $\mathbb{P}_t \mid \mathbb{T}_{t,k}$
--------------- ----------------------------- -------------------------------------------------------------------------------------------------------
Syntax and abstract machine states
----------------------------------
We fix an Haskell-like language extended with the [[]{.nodecor}]{}primitives of Figure \[fig:base-interface\]. The syntax is summarised in Figure \[fig:syntax\] where the meta-variables $x$ and $r$ range over a given countable set of variables and of location names , respectively. We assume Haskell typing conventions and denote the set of all well-typed terms by ${\textsf{Term}}$.
Terms of this language are evaluated by an abstract state machine whose states are pairs ${\langle P; \Sigma\rangle}$ formed by:
- a *thread family* (process) $P = T_{t_1} \parallel \dots \parallel T_{t_n}$,
- a *memory* $\Sigma = \langle \Theta, \Delta ,\Psi \rangle$, where $\Theta : {\textsf{Loc}}\rightharpoonup {\textsf{Term}}$ is the *heap* and $\Delta : {\textsf{Loc}}\rightharpoonup {\textsf{Term}}\times {\textsf{TrName}}$ is the *working memory*; ${\textsf{TrName}}$ is a set of names used to identify active transactions; $\Psi$ is a forest of threads identifiers.
#### Threads
Threads are the smaller unit of execution the machine scheduler operates on; they execute [[]{.nodecor}]{}terms and do not have any private transactional memory. A thread outside transactions is represented by ${\lthrparenthesisM{]\mspace{-3.7mu})}_{t}}$ where $M$ is the term being evaluated and $t$ is a unique *thread identifier* (Figure \[fig:ctm-contextes\]). A thread inside a transaction $k$ is represented by ${{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,k}}$ where $M$ is the term being evaluated inside the transaction $k$ and $N$ is the term being evaluated as *continuation* after $k$ commits or aborts.
At any time, all thread identifiers are stored in the auxiliary structure $\Psi$, which is a forest reflecting how threads are forked: if $t'$ has been forked by $t$ while inside $k$ then $t'$ belongs to $k$ too and occurs in $\Psi$ as a child of $t$.
We shall present thread families borrowing the parallel operator $\parallel$ from process algebra (Figure \[fig:ctm-contextes\]). The operator is associative, commutative and defined only on threads whose thread identifiers are distinct. The notation is extended to thread families (i.e. processes) with $\mathbf{0}$ denoting the empty family.
#### Memory
The memory $\Sigma$ is divided in the heap $\Theta$ and in the distributed working memory $\Delta$ (plus the auxiliary structure $\Psi$ recording thread fork hierarchy). As for traditional closed (ACID) transactions (e.g. [@hmpm:ppopp2005]), operations inside a transaction are evaluated against $\Delta$ and effects are propagated to $\Theta$ only on commits. When a thread inside a transaction $k$ accesses a location outside $\Delta$ the location is *claimed by transaction $k$* and remains claimed until $k$ commits, aborts or restarts. Threads in $k$ can interact only with locations claimed by $k$, but active transactions can be merged to share their claimed locations.
We shall denote the set of all possible states as ${\textsf{State}}$, and reference to each projected component of $\Sigma$ by a subscript, i.e. $\Sigma_\Theta$ for the heap and $\Sigma_\Delta$ for the working memory. When describing updates to the memory $\Sigma$, we adopt the convention that $\Sigma'$ has to be intended equals to $\Sigma$ except if stated otherwise, i.e. by statements like $\Sigma'_\Theta = \Sigma_\Theta[r \mapsto M]$. Finally, the completely undefined partial function $\varnothing$ denotes the empty heap and working memory.
Operational semantics
---------------------
$$\begin{gathered}
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Eval\;\begin{array}{c}
rule:admn-eval
\\[0pt}}{\thepage}{\textsc{Eval\;\begin{array}{c}
rule:admn-eval
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Eval\;\begin{array}{c}
rule:admn-eval
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
M \not\equiv V \quad \mathcal{V}[M] = V
}{
M \to V
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{BindVal\;\begin{array}{c}
rule:admn-bindv
\\[0pt}}{\thepage}{\textsc{BindVal\;\begin{array}{c}
rule:admn-bindv
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{BindVal\;\begin{array}{c}
rule:admn-bindv
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
}{
{\textnormal{\ttfamilyreturn\;$M$\;>>=\;$N$}} \to {\textnormal{\ttfamily$N$\,$M$}}
}
\qquad
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{BindEx\;\begin{array}{c}
rule:admn-binde
\\[0pt}}{\thepage}{\textsc{BindEx\;\begin{array}{c}
rule:admn-binde
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{BindEx\;\begin{array}{c}
rule:admn-binde
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\textnormal{\ttfamilye}} \in \{{\textnormal{\ttfamilyretry}},{\textnormal{\ttfamilythrow\;$N$}}\}
}{
{\textnormal{\ttfamilye\;>>=\;$M$}} \to {\textnormal{\ttfamilye}}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{CatchVal\;\begin{array}{c}
rule:admn-catchv
\\[0pt}}{\thepage}{\textsc{CatchVal\;\begin{array}{c}
rule:admn-catchv
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{CatchVal\;\begin{array}{c}
rule:admn-catchv
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\textnormal{\ttfamilyr}} \in \{{\textnormal{\ttfamilyretry}},{\textnormal{\ttfamilyreturn\;$N$}}\}
}{
{\textnormal{\ttfamilyr\;`catch`\;$M$}} \to {\textnormal{\ttfamilyr}}
}
\quad
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{CatchEx\;\begin{array}{c}
rule:admn-catche
\\[0pt}}{\thepage}{\textsc{CatchEx\;\begin{array}{c}
rule:admn-catche
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{CatchEx\;\begin{array}{c}
rule:admn-catche
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
}{
{\textnormal{\ttfamilythrow\;$M$\;`catch`\;$N$}} \to {\textnormal{\ttfamily$N$\,$M$}}
}
\end{gathered}$$
$$\begin{gathered}
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{InChar\;\begin{array}{c}
rule:input-char
\\[0pt}}{\thepage}{\textsc{InChar\;\begin{array}{c}
rule:input-char
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{InChar\;\begin{array}{c}
rule:input-char
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
}{
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilygetChar}}]}}; \Sigma\rangle}
\xrightarrow{?c}
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilyreturn\;$c$}}]}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{OutChar\;\begin{array}{c}
rule:output-char
\\[0pt}}{\thepage}{\textsc{OutChar\;\begin{array}{c}
rule:output-char
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{OutChar\;\begin{array}{c}
rule:output-char
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
}{
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilyputChar\;$c$}}]}}; \Sigma\rangle}
\xrightarrow{!c}
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilyreturn\;()}}]}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{TermIO\;\begin{array}{c}
rule:termio
\\[0pt}}{\thepage}{\textsc{TermIO\;\begin{array}{c}
rule:termio
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{TermIO\;\begin{array}{c}
rule:termio
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
M \to N
}{
{\langle {\mathbb{P}_{t}{[M]}}; \Sigma\rangle}
\xrightarrow{}
{\langle {\mathbb{P}_{t}{[N]}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{ForkIO\;\begin{array}{c}
rule:forkio
\\[0pt}}{\thepage}{\textsc{ForkIO\;\begin{array}{c}
rule:forkio
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{ForkIO\;\begin{array}{c}
rule:forkio
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
t'\notin \mathsf{threads}{{\mathbb{P}_{t}{[{\textnormal{\ttfamilyfork\;$M$}}]}}}
}{
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilyfork\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{}
{\langle {\mathbb{P}_{t}{[{\textnormal{\ttfamilyreturn\;$t'$}}]}} \parallel
{{(\mspace{-3.7mu}[}{M;{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t',k}}; \Sigma\rangle}
}
\end{gathered}$$
$$\begin{gathered}
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{TermT\;\begin{array}{c}
rule:term
\\[0pt}}{\thepage}{\textsc{TermT\;\begin{array}{c}
rule:term
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{TermT\;\begin{array}{c}
rule:term
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
M \to N
}{
{\langle {\mathbb{T}_{t,k}{[M]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[N]}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{ForkT\;\begin{array}{c}
rule:fork
\\[0pt}}{\thepage}{\textsc{ForkT\;\begin{array}{c}
rule:fork
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{ForkT\;\begin{array}{c}
rule:fork
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
t'\notin \mathsf{threads}({\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyfork\;$M$}}]}})
\qquad
\Sigma'_\Psi=
\mathsf{add\_child}(t,t',\Sigma_\Psi)
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyfork\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreturn\;$t'$}}]}} \parallel
{{(\mspace{-3.7mu}[}{M;{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t',k}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{NewVar\;\begin{array}{c}
rule:newvar
\\[0pt}}{\thepage}{\textsc{NewVar\;\begin{array}{c}
rule:newvar
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{NewVar\;\begin{array}{c}
rule:newvar
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
r \notin \operatorname{dom}(\Sigma_\Theta)\cup\operatorname{dom}(\Sigma_\Delta)\qquad
\Sigma'_\Delta = \Sigma_\Delta[r\mapsto (M,k)]
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilynewOTVar\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreturn\;$r$}}]}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Read1\;\begin{array}{c}
rule:read-miss
\\[0pt}}{\thepage}{\textsc{Read1\;\begin{array}{c}
rule:read-miss
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Read1\;\begin{array}{c}
rule:read-miss
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
r \notin \operatorname{dom}(\Sigma_\Delta) \quad
\Sigma_\Theta(r) = M \quad
\Sigma'_\Delta = \Sigma_\Delta[r \mapsto (M,k)]
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreadOTVar\;$r$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreturn\;$M$}}]}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Read2\;\begin{array}{c}
rule:read-hit
\\[0pt}}{\thepage}{\textsc{Read2\;\begin{array}{c}
rule:read-hit
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Read2\;\begin{array}{c}
rule:read-hit
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma_\Delta(r) = (M,j) \qquad
\Sigma'_\Delta = \Sigma_\Delta[k \mapsto j]
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreadOTVar\;$r$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,j}{[{\textnormal{\ttfamilyreturn\;$M$}}]}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Write1\;\begin{array}{c}
rule:write-miss
\\[0pt}}{\thepage}{\textsc{Write1\;\begin{array}{c}
rule:write-miss
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Write1\;\begin{array}{c}
rule:write-miss
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
r \notin \operatorname{dom}(\Sigma_\Delta) \qquad
\Sigma'_\Delta = \Sigma_\Delta[r \mapsto (M,k)]
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilywriteOTVar\;$r$\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreturn\;()}}]}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Write2\;\begin{array}{c}
rule:write-hit
\\[0pt}}{\thepage}{\textsc{Write2\;\begin{array}{c}
rule:write-hit
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Write2\;\begin{array}{c}
rule:write-hit
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma_\Delta(r) = (N,j) \qquad
\Sigma'_\Delta = \Sigma_\Delta[k \mapsto j][r \mapsto (M,j)]
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilywriteOTVar\;$r$\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyreturn\;()}}]}}[k \mapsto j]; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Or1\;\begin{array}{c}
rule:orfirst
\\[0pt}}{\thepage}{\textsc{Or1\;\begin{array}{c}
rule:orfirst
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Or1\;\begin{array}{c}
rule:orfirst
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\textnormal{\ttfamilyop}} \in \{{\textnormal{\ttfamilythrow}}, {\textnormal{\ttfamilyreturn}}\}
\qquad
{\langle {{(\mspace{-3.7mu}[}{M;{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
\xrightarrow{\tau}^*
{\langle {{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilyop\;$N$}};{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,j}}; \Sigma'\rangle}
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamily$M$\;`orElse`\;$M'$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle \mathbb{T}'_{t,j}[{\textnormal{\ttfamilyop\;$N$}}]; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Or2\;\begin{array}{c}
rule:orsecond
\\[0pt}}{\thepage}{\textsc{Or2\;\begin{array}{c}
rule:orsecond
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Or2\;\begin{array}{c}
rule:orsecond
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\langle {{(\mspace{-3.7mu}[}{M;{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
\xrightarrow{\tau}^*
{\langle {{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilyretry}};{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,j}}; \Sigma'\rangle}
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamily$M$\;`orElse`\;$M'$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,k}{[M']}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Isolated\;\begin{array}{c}
rule:isolated
\\[0pt}}{\thepage}{\textsc{Isolated\;\begin{array}{c}
rule:isolated
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Isolated\;\begin{array}{c}
rule:isolated
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\textnormal{\ttfamilyop}} \in \{{\textnormal{\ttfamilythrow}}, {\textnormal{\ttfamilyreturn}}\}
\qquad
{\langle {{(\mspace{-3.7mu}[}{M;{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
\xrightarrow{\tau}^*
{\langle {{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilyop\;$N$}};{\textnormal{\ttfamilyreturn}}}{]\mspace{-3.7mu})}_{t,j}}; \Sigma'\rangle}
}{
{\langle {\mathbb{T}_{t,k}{[{\textnormal{\ttfamilyisolated\;$M$}}]}}; \Sigma\rangle}
\xrightarrow{\tau}
{\langle {\mathbb{T}_{t,j}{[{\textnormal{\ttfamilyop\;$N$}}]}}; \Sigma'\rangle}
}
\end{gathered}$$
$$\begin{gathered}
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{New\;\begin{array}{c}
rule:tr-new
\\[0pt}}{\thepage}{\textsc{New\;\begin{array}{c}
rule:tr-new
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{New\;\begin{array}{c}
rule:tr-new
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
}{
{\langle {{(\mspace{-3.7mu}[}{\textnormal{\ttfamilyatomic\;$M$\;>>=\;$N$}}{]\mspace{-3.7mu})}_{t}}; \Sigma\rangle}
\xrightarrow{new\langle k\rangle}
{\langle {{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Commit\;\begin{array}{c}
rule:tr-commit
\\[0pt}}{\thepage}{\textsc{Commit\;\begin{array}{c}
rule:tr-commit
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Commit\;\begin{array}{c}
rule:tr-commit
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma'_\Theta = \mathsf{commit}(k,\Sigma) \qquad
\Sigma'_\Delta = \mathsf{cleanup}(k,\Sigma)
}{
{\langle {{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilyreturn\;$M$}};N}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
\xrightarrow{co\langle k\rangle}
{\langle {{(\mspace{-3.7mu}[}{\textnormal{\ttfamilyreturn\;$M$\;>>=\;$N$}}{]\mspace{-3.7mu})}_{t}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Abort1\;\begin{array}{c}
rule:tr-abort-1
\\[0pt}}{\thepage}{\textsc{Abort1\;\begin{array}{c}
rule:tr-abort-1
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Abort1\;\begin{array}{c}
rule:tr-abort-1
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma'_\Theta = \mathsf{leak}(k,\Sigma) \qquad
\Sigma'_\Delta = \mathsf{cleanup}(k,\Sigma) \\
\Sigma'_\Psi = \mathsf{remove}(r,\Sigma_\Psi) \qquad
r = \mathsf{root}(t,\Sigma_\Psi)
}{
{\langle {{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilythrow\;$M$}};N}{]\mspace{-3.7mu})}_{t,k}}; \Sigma\rangle}
\xrightarrow{ab\langle k, t, M\rangle}
{\langle {{(\mspace{-3.7mu}[}{\textnormal{\ttfamilythrow\;$M$\;>>=\;$N$}}{]\mspace{-3.7mu})}_{t}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Abort2\;\begin{array}{c}
rule:tr-abort-2
\\[0pt}}{\thepage}{\textsc{Abort2\;\begin{array}{c}
rule:tr-abort-2
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Abort2\;\begin{array}{c}
rule:tr-abort-2
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma'_\Theta = \mathsf{leak}(k,\Sigma) \qquad
\Sigma'_\Delta = \mathsf{cleanup}(k,\Sigma) \\
\Sigma'_\Psi = \mathsf{remove}(r,\Sigma_\Psi) \qquad
r = \mathsf{root}(t,\Sigma_\Psi) \quad
r = \mathsf{root}(t',\Sigma_\Psi)
}{
{\langle {{(\mspace{-3.7mu}[}{M';N}{]\mspace{-3.7mu})}_{t',k}}; \Sigma\rangle}
\xrightarrow{\overline{ab}\langle k, t, M\rangle}
{\langle {{(\mspace{-3.7mu}[}{\textnormal{\ttfamilythrow\;$M$\;>>=\;$N$}}{]\mspace{-3.7mu})}_{t'}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{Abort3\;\begin{array}{c}
rule:tr-abort-3
\\[0pt}}{\thepage}{\textsc{Abort3\;\begin{array}{c}
rule:tr-abort-3
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{Abort3\;\begin{array}{c}
rule:tr-abort-3
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
\Sigma'_\Theta = \mathsf{leak}(k,\Sigma) \qquad
\Sigma'_\Delta = \mathsf{cleanup}(k,\Sigma)
\\
\Sigma'_\Psi = \mathsf{remove}(r,\Sigma_\Psi) \qquad
r = \mathsf{root}(t,\Sigma_\Psi) \qquad
r \neq \mathsf{root}(t',\Sigma_\Psi)
}{
{\langle {{(\mspace{-3.7mu}[}{M';N}{]\mspace{-3.7mu})}_{t',k}}; \Sigma\rangle}
\xrightarrow{\overline{ab}\langle k, t, M\rangle}
{\langle {{(\mspace{-3.7mu}[}{\textnormal{\ttfamilyretry}}{]\mspace{-3.7mu})}_{t'}}; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{MCastAb\;\begin{array}{c}
rule:tr-multicast-abort
\\[0pt}}{\thepage}{\textsc{MCastAb\;\begin{array}{c}
rule:tr-multicast-abort
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{MCastAb\;\begin{array}{c}
rule:tr-multicast-abort
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\langle P; \Sigma\rangle} \xrightarrow{ab\langle k, t, M\rangle} {\langle P'; \Sigma'\rangle}
\quad
{\langle Q; \Sigma\rangle} \xrightarrow{\overline{ab}\langle k, t, M\rangle} {\langle Q'; \Sigma'\rangle}
\hfill
}{
{\langle P\parallel Q; \Sigma\rangle} \xrightarrow{ab\langle k, t, M\rangle} {\langle P'\parallel Q'; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{MCastCo\;\begin{array}{c}
rule:tr-multicast-commit
\\[0pt}}{\thepage}{\textsc{MCastCo\;\begin{array}{c}
rule:tr-multicast-commit
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{MCastCo\;\begin{array}{c}
rule:tr-multicast-commit
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\langle P; \Sigma\rangle} \xrightarrow{co\langle k\rangle} {\langle P'; \Sigma'\rangle}
\quad
{\langle Q; \Sigma\rangle} \xrightarrow{co\langle k\rangle} {\langle Q'; \Sigma'\rangle}
\hfill
}{
{\langle P \parallel Q; \Sigma\rangle} \xrightarrow{co\langle k\rangle} {\langle P' \parallel Q'; \Sigma'\rangle}
}
\\
{{{
{\text{\scriptsize[{ \protected@write \@auxout {}{\string \newlabel {\hline}{{\textsc{MCastGroup\;\begin{array}{c}
rule:tr-multicast-context
\\[0pt}}{\thepage}{\textsc{MCastGroup\;\begin{array}{c}
rule:tr-multicast-context
\\[0pt}}{\hline}{}} } \hypertarget{\hline}{\textsc{MCastGroup\;\begin{array}{c}
rule:tr-multicast-context
\\[0pt}}
}\!\!]}}\hfill
} \\[-10pt]
}
\end{array}}]{
{\langle P; \Sigma\rangle} \xrightarrow{\beta} {\langle P'; \Sigma'\rangle}
\quad
\beta \neq \tau
\quad
\mathsf{transaction}(\beta) \notin \mathsf{transactions}(Q)
\hfill
}{
{\langle P \parallel Q; \Sigma\rangle} \xrightarrow{\beta} {\langle P' \parallel Q; \Sigma'\rangle}
}
\end{gathered}$$
$$\begin{aligned}
\mathsf{threads}(T_{t_1} \parallel \dots \parallel T_{t_n}) &{\triangleq}\{t_1, \dots t_n\}
\\
\mathsf{transaction}(\beta) &{\triangleq}k \text{ for }
\beta\in\{new\langle k\rangle, co\langle k\rangle, ab\langle k, t, M\rangle, \overline{ab}\langle k, t, M\rangle\}
\\
(\Delta[k \mapsto j])(r) &{\triangleq}\begin{cases}
\Delta(r) &\!\text{if}\ \Delta(r)= (M,l),l\neq k\\
(M,j) &\!\text{if}\ \Delta(r) = (M, k)
\end{cases}
\\
\mathsf{transactions}(P) &{\triangleq}\begin{cases}
\mathsf{transactions}(P_1) \cup \mathsf{transactions}(P_2) &
\!\text{if } P = P_1 \parallel P_2\\
\{k\}&\!\text{if } P = {{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,k}}\\
\emptyset &\!\text{otherwise}
\end{cases}
\\
P[k \mapsto j] &{\triangleq}\begin{cases}
P_1[k \mapsto j] \parallel P_2[k \mapsto j] &\!\text{if } P = P_1 \parallel P_2\\
{{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,j}} &\!\text{if } P = {{(\mspace{-3.7mu}[}{M;N}{]\mspace{-3.7mu})}_{t,k}}\\
P &\!\text{otherwise}
\end{cases}
\\
\Theta[r \mapsto M](s) &{\triangleq}\begin{cases}
M &\!\text{if } r = s\\
\Theta(s) &\!\text{otherwise}
\end{cases}
\\
\Delta[r \mapsto (M,k)](s) &{\triangleq}\begin{cases}
(M,k) &\!\text{if } r = s\\
\Delta(s) &\!\text{otherwise}
\end{cases}
\\
\mathsf{cleanup}(k,\Sigma)(r) &{\triangleq}\begin{cases}
\perp &\!\!\text{if } \Sigma_\Delta(r) = (M,k)\\
\Sigma_\Delta(r) &\!\!\text{otherwise}
\end{cases}
\\
\mathsf{commit}(k,\Sigma)(r) &{\triangleq}\begin{cases}
M &\!\text{if } \Sigma_\Delta(r) = (M,k)\\
\Sigma_\Theta(r) & \!\text{otherwise}
\end{cases}
\\
\mathsf{leak}(k,\Sigma)(r) &{\triangleq}M \!\text{ if } \Sigma_\Theta(r) = M \text{ or } \Sigma_\Theta(r) = {\perp} \text{ and } \Sigma_\Delta(r) = (M,k)
\end{aligned}$$
The dynamics of the machine is defined by two transition relations presented in . The first relation $M \to N$ is defined on terms only and models pure computations (Figure \[fig:semantics-term\]). Rule allows a term $M$ that is not a value to be evaluated by means of an auxiliary (partial) function $\mathcal{V}[M]$ yielding the value $V$; the other rules define the semantics of the monadic [[[]{.nodecor}]{}]{} and exception handling in a standard way. We remark the symmetry between [[[]{.nodecor}]{}]{} and [[[]{.nodecor}]{}]{} and how [[]{.nodecor}]{} is treated as an exception by and as a result value by .
Relation $\to$ can be thought as accessory to the second relation ${\langle P; \Sigma\rangle} \xrightarrow{\beta} {\langle P'; \Sigma'\rangle}$, which describes state transitions. Since several rules can apply to a given state according to different evaluation contexts as per Figure \[fig:ctm-contextes\], this relation is non-deterministic; this models the fact that the scheduler can choose which thread to execute next among various possibilities. Labels $\beta$ describe the kind of transition, and are defined as follows: $$\beta ::= \tau
\mid new\langle k\rangle
\mid co\langle k\rangle
\mid ab\langle k, t, M\rangle
\mid \overline{ab}\langle k, t, M\rangle
\qquad
\text{for $k\in{\textsf{TrName}}$, $M\in{\textsf{Term}}$}$$
Transitions labelled by $\tau$ represent *internal* steps of transitions, i.e., steps which do not need a coordination among transactions: reduction of pure terms, thread creation and memory operations. These transitions are defined by the rules in Figure \[fig:semantics-tau\]. Reading a location falls into two cases: rule models the reading of an unclaimed location and its memory effect is to record the claim in $\Delta$, while rule models the reading of a claimed location and its effect is to merge the transactions of the current thread with that claiming the location. Writes behave similarly. Rules and describe the semantics of alternative sub-transactions: if the first one [[]{.nodecor}]{}-es the second is executed discarding any effect of the first. Rule spawns a new thread for the current transaction; a term [[[M]{.nodecor}]{}]{} can appear inside [[]{.nodecor}]{}, thus allowing multi-threaded open transactions, but its use inside [[[]{.nodecor}]{}]{} is prevented by the type system and by the shape of as well.
The remaining labels describe state transitions concerning the life-cycle of transactions: creation, commit, abort, and restart (Figure \[fig:semantics-trs-mgr\]). These operations require a coordination among threads; for instance, an abort from a thread has to be propagated to every thread participating to the same transaction. This is captured in the semantics by labelling the transition with the operation and the name of the transaction involved; this information is used by the derivation rules to force synchronisation of all participants of that transaction. To illustrate this mechanism, we describe the commit of a transaction $k$, namely ${\langle P; \Sigma\rangle} \xrightarrow{co\langle k\rangle} {\langle P'; \Sigma'\rangle}$. First, by means of we split $P$ into two subprocesses, one of which contains all threads participating in $k$ (those not in $k$ cannot do a transition whose label contains $k$). Secondly, using recursively we single out every thread in $k$. Finally, we apply provided that every thread is ready to commit, i.e., it is of the form ${{(\mspace{-3.7mu}[}{{\textnormal{\ttfamilyreturn\;$M$}};N}{]\mspace{-3.7mu})}_{t,k}}$.
Aborting a transaction works similarly, but it based on vetoes instead of an unanimous vote. Aborts are triggered by unhandled exceptions raised by some thread, but threads react to this situation in different ways:
- threads in the same tree of the thread rasing the exception have been forked within the transaction; hence, the root thread is aborted and all other threads in the tree are killed because their creation, as for any transactional side-effect, have to be discarded;
- threads in different trees joined the transaction after it was created, due to a merging; hence, these threads just retry their transaction, since aborting would require them to handle exceptions raised by “foreign” threads.
Notice that there are no derivation rules for ${\textnormal{\ttfamilyretry}}$, since its meaning is to inform the scheduler that the execution is stuck; hence the machine has to re-execute the transaction from the beginning (or a suitable check-point), following a different execution order, if and when possible.
Opacity
-------
In this section we use the formalisation of [[]{.nodecor}]{}to prove that it meets the *opacity* criterion.
The opacity correctness criterion for transactional memory [@gk:ppopp08] is an extension of the classical *serialisability property* for databases with the additional requirement that even non-committed transactions must access consistent states. Intuitively, this property ensures that:
effects of any committed transaction appear performed at a single, indivisible point during the transaction lifetime,
effects of any aborted transaction cannot be seen by any other transaction, and
transactions always access consistent states of the system.
In order to formally capture these intuitive requirements let us recall some notions from [@gk:ppopp08]. A *history* is a sequence of `read`, `write`, `commit`, and `abort` operations[^2] ordered according to the time at which they were issued (simultaneous events are arbitrarily ordered) and such that no operation can be issued by a transaction that has already performed a `commit` or an `abort`. A transaction $k$ is said to be in a history $H$ if the latter contains at least one operation issued by $k$. Any history $H$ defines a *happens-before* partial order $\prec_H$ where $k \prec_H k'$ iff the transaction $k$ becomes committed or aborted in $H$ before $k'$ issues its first operation. If $\prec_H$ is total then $H$ is called *sequential*. For a history $H$, let $\mathit{complete}(H)$ be the set of histories obtained by adding either a commit or an abort for every live transaction in $H$.
We are now able to recall Guerraoui and Kapałka’s definition[^3] of opacity.
A history $H$ is said to be *opaque* if there is a sequential history $S$ equivalent to some history in the set $\mathit{complete}(H)$ such that ${\prec_S} \subseteq {\prec_H}$.
As shown in [@gk:ppopp08], opacity corresponds to the absence of mutual dependencies between live transactions, where a dependency is created whenever a transaction reads an information written by another or depends from its outcome.
For a history $H$ let $\ll$ be a total order on the set $T$ of all transactions in $H$. An *opacity graph* for $H$ and $\ll$, $OPG(H,\ll)$, is a bi-coloured directed graph on $T$ such that a vertex is *red* if the corresponding transaction is either running or aborted, it is *black* otherwise, and such that there is an edge from $k$ to $k'$ whenever any of the following holds:
1. $k'$ happens-before $k$;
2. $k$ reads something written by $k'$;
3. $k'$ reads some location written by $k$ and $k' \ll k$;
4. $k'$ is neither running nor aborted and there are a location $r$ and a transaction $k''$ such that $k' \ll k''$, $k'$ writes to $r$, and $k''$ reads $r$ from $k$.
The edge is red if the second case applies otherwise it is black. If all edges from red nodes in $OPG(H,\ll)$ are also red then the graph is said to be *well-formed*.
Let $H$ be a history and let $k$ be a transaction appearing in it. A `read` operation by $k$ is said to be *local* (to $k$) whenever the previous operation by $k$ on the same location was a `write`. A `write` operation by $k$ is said to be *local* (to $k$) whenever the next operation by $k$ on the same location is a `write`. We denote by $\mathit{nonlocal}(H)$ the longest sub-history of $H$ without any local operations. A history $H$ is said *locally-consistent* if every local `read` is preceded by a `write` operation that writes the red value; it is said *consistent* if, additionally, whenever some $k$ reads $v$ from $r$ in $\mathit{nonlocal}(H)$ then some $k'$ writes $v$ to $r$ in $\mathit{nonlocal}(H)$.
A history $H$ is opaque if and only if
$H$ is consistent and
there exists a total order $\ll$ on the set of transactions in $H$ such that $OPG(\mathit{nonlocal}(H),\ll)$ is well-formed and acyclic.
In [@gk:ppopp08] transactions may encapsulate several threads but cannot be merged. Therefore, in order to study opacity of [[]{.nodecor}]{}we extend the set of operations considered in loc. cit. with explicit merges. Let $k,k'$ be two running transactions in the given history; when they merge, they share their threads, locations, and effects. From this perspective, $k$ is commit-pending and depends from $k'$ and hence in the opacity graph, $k$ is a red node connected to $k'$ by a red edge. Hence, merges can be equivalently expressed at the history level by sequences like:\
new $x$;
$k'$ writes on $x$;
$k$ reads from $x$;
$k$ prepares to commit.
\
These are the only dependencies found in histories generated by [[]{.nodecor}]{}.
\[th:noloops\] For $H$ a history describing an execution of a [[]{.nodecor}]{}program and a total order $\ll$, $OPG(\mathit{nonlocal}(H),\ll)$ is a forest of red edges where only roots may be white.
By inspection of the rules it is easy to see that
transactions may access only locations they claimed;
claimed locations are released only on `commit`s, `abort`s and retries;
transactions have to merge with any transaction holding a location they need.
Therefore, at any time there is at most one running transaction issuing operations on a given location, hence `read`s and `write`s do not create edges. Thus edges are created only during the execution of merges and, by inspecting the above implementation, it easy to see that
any transaction can issue at most one merge;
a transaction issuing a merge is a red node;
the edge created by a merge is red.
Therefore, transactions form a forest made of red edges where any non-root node is red.
[[]{.nodecor}]{}meets the opacity criterion.
A forest formed by red edges whose sources are always red is acyclic and well-formed.
Conclusions {#sec:conclusions}
===========
In this paper we have presented OTM, a programming model supporting interactions between composable memory transactions. This model separates isolated transactions from non-isolated ones, still guaranteeing atomicity; the latter can interact by accessing to shared variables. Consistency is ensured by transparently *merging* interacting transactions at runtime. We have showed the versatility and simplicity of OTM by implementing some examples which are incompatible with isolation, and we have given a formal semantics for OTM, which allowed us to prove that this model satisfies the opacity criterion.
There are two main directions for future work each posing its own challenges. First, like [[]{.nodecor}]{}, this model supports nesting (via [[]{.nodecor}]{}); however, this feature is currently limited to isolated (sub)transactions. Supporting nesting of open transaction requires additional care in the handling of side-effects: is merging transactions at different level of nesting feasible and meaningful or are we breaking the intuition behind the programming model? Secondly, an implementation is due in order to validate experimentally the model. A possible approach is to implement [[]{.nodecor}]{}completely in Haskell on top of [[]{.nodecor}]{}. This solution does not need any specific support from the Haskell RunTime (HRT) but cannot benefit of the performance gains offered by a deeper integration, thus hindering any fair comparison with existing TM models, like [[]{.nodecor}]{}. On the other hand, integrating [[]{.nodecor}]{}with the HRT and the Glasgow Haskell Compiler, akin [[]{.nodecor}]{}, would be more efficient but also more complex and invasive.
=-1 We have presented OTM within Haskell (especially to leverage its type system), but this model is general and can be applied to other languages. A possible future work is to port this model to an imperative object oriented language, such as Java or C++; however, like other TM implementations, we expect that this extension will require some changes in the compiler and/or the runtime.
This work builds on the ideas in [@mpt:coord15] where we described an abstract calculus with shared memory and open transactions. In *loc. cit.* we showed how this model is expressive enough to represent $TCCS^m$ [@ksh:fossacs2014], a variant of the Calculus of Communicating Systems with transactional synchronization. Being based on CCS, communication in $TCCS^m$ is synchronous; however, nowadays asynchronous models play an important rôle (see e.g. actors, event-driven programming, etc.). It may be interesting to generalize the discussion so as to consider also this case, e.g. by defining an actor-based calculus with open transactions. Such a calculus can be quite useful also for modelling speculative reasoning for cooperating systems [@ma2010:speculative; @mmp:dais14; @mmp:eceast2014; @mpm:gcm14w; @mp:memo14]. A local version of actor-based open transactions can be implemented in [[]{.nodecor}]{}using lock-free data structures (e.g., message queues) in shared transactional memory.
#### Acknowledgements
We thank Nicola Gigante and Valentino Picotti for their valuable feedback about the [[]{.nodecor}]{}programming model.
[10]{}
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V. Koutavas, C. Spaccasassi, and M. Hennessy. Bisimulations for communicating transactions. In A. Muscholl, editor, [*Proc. [FOSSACS]{}*]{}, volume 8412 of [*LNCS*]{}, pages 320–334. Springer, 2014.
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M. Miculan, M. Peressotti, and A. Toneguzzo. Open transactions on shared memory. In [*[COORDINATION]{}*]{}, volume 9037 of [*LNCS*]{}, pages 213–229. Springer, 2015.
S. L. Peyton Jones. Tackling the awkward squad: monadic input/output, concurrency, exceptions, and foreign-language calls in Haskell. , 180:47, 2001.
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[^1]: In fact, OTM model can be implemented in any programming language, provided we have some means, either static or dynamic, to forbid irreversible effects inside transactions.
[^2]: The definition given in [@gk:ppopp08] considers finer-grained events; in particular, `read` and `write` operations are formed by `request`, `execution`, and `response` events. However in *loc. cit.* the authors restrict to histories where `request`-`execution`-`response` sequences are not interleaved, hence we can consider the simpler `read`/`write`s events in the first place.
[^3]: The original definition requires the history $H$ to be also “legal”, but this notion is relevant only in presence of non-transactional operations which both [[]{.nodecor}]{}and [[]{.nodecor}]{}prevent by design.
|
---
abstract: 'We demonstrate, by considering the triangular lattice spin-$1/2$ Heisenberg model, that Monte Carlo sampling of skeleton Feynman diagrams within the fermionization framework offers a universal first-principles tool for strongly correlated lattice quantum systems. We observe the fermionic sign blessing—cancellation of higher order diagrams leading to a finite convergence radius of the series. We calculate the magnetic susceptibility of the triangular-lattice quantum antiferromagnet in the correlated paramagnet regime and reveal a surprisingly accurate microscopic correspondence with its classical counterpart at all accessible temperatures. The extrapolation of the observed relation to zero temperature suggests the absence of the magnetic order in the ground state. We critically examine the implications of this unusual scenario.'
author:
- 'S. A. Kulagin$^{1,2}$, N. Prokof’ev$^{1,3}$, O. A. Starykh$^{4}$, B. Svistunov$^{1,3}$, and C. N. Varney$^{1}$'
bibliography:
- 'references.bib'
title: Bold Diagrammatic Monte Carlo Method Applied to Fermionized Frustrated Spins
---
The method of bold diagrammatic Monte Carlo simulation (BDMC) [@bold1] allows one to sample contributions from millions of skeleton Feynman diagrams and extrapolate the results to the infinite diagram order, provided the series is convergent (or subject to resummation beyond the convergence radius). Recent experimentally certified application of BDMC to unitary fermions down to the point of the superfluid transition [@NatureP] makes a strong case for BDMC method as a generic method for dealing with correlated fermions described by Hamiltonians without small parameters. One intriguing avenue to explore is to apply it to frustrated lattice spin systems, where, on one hand, standard Monte Carlo (MC) simulation fails because of the sign problem [@Loh1990], and, on the other hand, the system’s Hamiltonian can be always written in the fermionic representation [@PopovFedotov1; @PopovFedotov2; @Fermionization] which contains no large parameters—exactly what is needed for the anticipated convergence of BDMC series with the diagram order.
The BDMC approach is based on the [*sign blessing*]{} phenomenon, when, despite the factorial increase in the number of diagrams with expansion order, the series features a finite convergence radius because of dramatic (sign alternation induced) compensation between the diagrams. With the finite convergence radius, the series can be summed either directly, or with resummation techniques that can be potentially applied down to the critical temperature of the phase transition, if any. (At the critical temperature thermodynamic functions become nonanalytic, and the diagrammatic expansion involving explicit symmetry breaking by the finite order parameter is necessary to treat the critical region and the phase with broken symmetry.) In the absence of the sign blessing, the resummation protocols become questionable in view of the known mathematical theorems regarding asymptotic series. At the moment, there is no theory allowing one to prove the existence of a finite convergence radius analytically. The absence of Dyson’s collapse [@Dyson] in a given fermionic system is merely providing hope that the corresponding diagrammatic series is not asymptotic and cannot be [*a priori*]{} taken as a sufficient condition for the sign blessing. Hence, the applicability of BDMC method to a given system can be established only on the basis of a direct numerical evidence for series convergence and comparison with either experiment or alternative controllable techniques, such as high-temperature series [@singh05] and numerical linked cluster (NLC) expansions [@Rigol]. In this Letter, we report the first successful application of BDMC method to fermionized quantum spin systems by simulating the canonical model of frustrated quantum magnetism—the triangular lattice antiferromagnetic spin-1/2 Heisenberg model (TLHA). We demonstrate that BDMC method for this frustrated magnet [*is indeed*]{} subject to the sign blessing phenomenon which allows us to obtain basic static and dynamic correlation functions with controllable (about one percent or better) accuracy. The agreement with extrapolated high-temperature expansions is excellent.
In addition, we report a very surprising finding of extreme similarity between short-distance static spin correlations of the quantum and classical spin models, evaluated at different but uniquely related to each other temperatures. This accurate (within the error bars) quantum-to-classical correspondence holds at all temperatures accessible to us, $T\geq 0.375 $ (here and below temperature is measured in the units of the exchange constant $J$). Specifically, the [ *entire*]{} static correlation function of the quantum model at a given temperature $T$—having quite nontrivial pattern of sign-alternating spatial dependence and temperature evolution, thus forming a system’s fingerprint—turns out to be equal, up to a global temperature dependent normalization factor, to its classical counterpart at a certain temperature $T_{\rm cl}\equiv T_{\rm cl}(T)$. Extrapolation of the obtained $T_{\rm cl}(T)$ curve to the $T=0$ limit results in a finite value of $T_{\rm cl}(0)>0$, suggesting a quantum-disordered ground state of the quantum model.
The Hamiltonian of the TLHA is given by $$H = J \sum_{<i,j>} \, {\vec S}_i \cdot {\vec S}_j \, .
%\qquad (J>0) \, .
\label{HM}$$ Here ${\vec S}_i$ is the spin-1/2 operator on the $i$th site of the triangular lattice and the sum is over the nearest neighbor pairs coupled by the positive exchange integral, $J>0$. As found by Popov and Fedotov [@PopovFedotov1; @PopovFedotov2], the grand canonical Gibbs distribution of the model can be reformulated identically in terms of purely fermionic operators using $${\vec S}_i\, =\, \frac{1}{2}\sum_{\alpha, \beta}\, f_{i \alpha}^{\dagger }
\vec{\sigma }_{\alpha \beta} f_{i \beta}^{\,} \, ,
\label{SF}$$ where $f_{i \beta}$ is the second quantized operator annihilating a fermion with spin projection $\alpha,\beta =\pm 1$ on site $i$, and $\vec{\sigma }$ are the Pauli matrices. The representation leads to a flat-band fermionic Hamiltonian, $H_{\rm F}$, with two-body interactions and amenable to direct diagrammatic treatment. To eliminate statistical contributions from nonphysical states having either zero or two fermions, Ref. introduced an imaginary chemical potential to $H_{\rm F}$: $$H_{\rm F} \to H_{\rm F} - i (\pi/2) T \sum_{i} (n_{i} -1) \, , \quad n_{i} =\sum_\alpha f_{i \alpha}^{\dagger }f_{i \alpha}^{\,}\, .
\label{HF}$$ The added term commutes with the original Hamiltonian and has no effect on properties of the physical subspace $\{ n_i=1 \}$ whatsoever. Moreover, the grand canonical partition functions and spin-spin correlation functions of the original spin model and its fermionic version are also identical because (i) physical and nonphysical sites decouple in the trace and (ii) the trace over nonphysical states yields identical zero on every site. As a result, one arrives at a rather standard Hamiltonian for fermions interacting through two-body terms. A complex value of the chemical potential, which can also be viewed as a peculiar shift of the fermonic Matsubara frequency $\omega_n = 2\pi (n+1/2) T \to 2\pi (n+1/4) T$, is a small price to pay for the luxury of having the diagrammatic technique.
We perform BDMC simulations using the standard $G^2W$-skeleton diagrammatic expansion of the fermionic model - in the real space–imaginary time representation [@kpssv2013], see also [@Kris08]. The first and most important question to answer is whether the sign blessing phenomenon indeed takes place. In Fig. \[fig:1\] we show comparison between the calculated answer for the static uniform magnetic susceptibility, $\chi_u$, and the NLC expansion result [@Rigol] at $T=2$. This temperature is low enough to ensure that we are in the regime of strong correlations because $\chi_u$ is nearly a factor of two smaller than the free-spin answer $\chi_u^{(0)} = 1/4T$. On the other hand, this temperature is high enough to be sure that the high-temperature series can be described by Padé approximants without significant systematic deviations from the exact answer [@singh05; @Rigol] (at slightly lower temperature the bare NLC series starts to diverge). We clearly see that the BDMC series converges to the correct result with an accuracy of about three meaningful digits and there is no statistically significant change when more than a hundred thousand of 7th order diagrams [@remark2] are accounted for. The error bar for the 7th order point is significantly increased due to factorial growth in computational complexity. Feynman diagrams are usually formulated for the system in the thermodynamic limit. In practice, for reasons of convenient data handling, our code works with finite system sizes $L$ with periodic boundary conditions (its performance does not depend on $L$). In all cases we choose $L$ to be much larger than the correlation length $\xi$ and check that doubling the system size makes no detectable changes in the final answer. The 4th order result can be obtained after several hours of CPU time on a single processor.
Interestingly enough, when temperature is lowered down to $T=1$ which is significantly below the point where the bare NLC series start to diverge, see Fig. \[fig:2\], the BDMC series continue to converge exponentially. This underlines the importance of performing simulations within the self-consistent skeleton formulation.
![\[fig:1\] (Color online) Uniform susceptibility calculated within the $G^2W$-skeleton expansion as a function of the maximum diagram order retained in the BDMC simulation (black dots) for $T/J=2$. The result of the high-temperature expansion (with Padé approximant extrapolation) [@singh05; @Rigol] is shown by the red square and horizontal line. ](fig1.eps){width="0.8\columnwidth"}
In Fig. \[fig:2\] we show results of the BDMC simulation performed at temperatures significantly below the mean-field transition temperature. We observe excellent agreement (within our error bars) with the Padé approximants used to extrapolate the high-temperature series data to lower temperature [@singh05]. Within the current protocol of dealing with skeleton diagrams we were not able to go to a lower temperature due to the development of near singularity in the response function (and thus in the effective-interaction propagator) at the classical ordering wave vector ${\mathbf
Q}=(4\pi/3,0)$, in units of inverse lattice constant. In future work, we plan to apply pole-regularization schemes to overcome this technical problem.
![\[fig:2\] (Color online) Uniform susceptibility as a function of temperature (red dots) for the triangular Heisenberg antiferromagnet calculated within the BDMC approach. NLC expansion results based on triangles (labeled as 7T and 8T) and sites (labeled as 12S and 13S) [@Rigol] are shown along with two different Padé approximant extrapolations [@singh05]. ](fig2.eps){width="0.8\columnwidth"}
We now turn to the static susceptibility $$\chi({\bf r}) \, =\, \int_0^{1/T} d\tau \, \langle\, S^z_{\bf 0}(0)
\, S^z_{\bf r} (\tau) \, \rangle \, .
\label{K}$$ Here $S^z_{\bf r}(\tau)$ is the Matsubara spin operator on the lattice site labeled by the integer index vector ${\bf r}$. For the simplicity of comparing susceptibility with its classical counterpart, we normalize it to unity at the origin, $\chi({\bf r}) \to \chi({\bf r})/
\chi({\bf 0})$, doing the same with the classical $\chi_{\rm cl}({\bf r})$. The latter is obtained by Metropolis simulation of the classical Heisenberg model in which quantum spin operators are replaced with classical unit vectors ${\bf n}_{\bf r}$. For every accessible temperature $T$ we observe a perfect match (within the error bars, which are about $1\%$) between quantum correlator $\chi({\bf r})$ and its classical counterpart, for $r$ ranging from 1 to 5 (which includes 10 different sites), calculated at a certain temperature $T_{\rm cl}(T)$. A typical example of the match is presented in Fig. \[fig:fingerprint\]. We note in passing that the equal-time correlation function, $ \langle\, S^z_{\bf
0}(0) \, S^z_{\bf r} (0) \, \rangle$, while having qualitatively similar shape to that of , does not match the classical correlator $\chi_{\rm cl}({\bf r}) = \langle\, n^z_{\bf 0} \, n^z_{\bf r} \, \rangle $, especially so for sites at which the sign of the correlation changes with temperature (such as sites 3 and 7 in Fig. \[fig:fingerprint\] for which the sign of $\chi({\bf r})$ changes from ferromagnetic at high $T$ to antiferromagnetic one below $T \approx 0.5$).
Mapping long-range correlations in quantum models onto the renormalized classical behavior is rather standard on approach to the ordered state [@sachdev_book]. What we observe is fundamentally different: quantum-to-classical correspondence, or QCC, is valid in the intermediate temperature regime at all distances, including nearest-neighbor sites, and when the correlation length $\xi$ is still very short, of the order of the lattice spacing, $\xi \sim 1$. It is worth noting that this short-distance correspondence is also very different from the high-$T$ quasiclassical wave regime of Ref. which allows for the classical description at distances $r \sim \xi \gg 1$.
We find that QCC also takes place for the $s=1/2$ square lattice Heisenberg antiferromagnet where, thanks to the absence of a sign problem for the path-integral Monte Carlo simulation, it has a relative accuracy of $\sim 0.3\%$ at all temperatures (down to the ground state in a finite-size system). These facts suggest that QCC in 2D is extremely accurate and thus may take place for other lattices (however, it does not hold for 1D chains).
{width="0.75\columnwidth"} {width="0.63\columnwidth"} {width="0.7\columnwidth"}
The quality of the matching procedure allows us to establish the $T_{\rm cl}(T)$ correspondence with an accuracy of about one percent. In the right panel of Fig. \[fig:fingerprint\] we plot the final result along with the asymptotic high-temperature relation $T_{\rm cl}=(4/3)T$ reflecting the difference between the $(S^z)^2=1/4$ and $\langle (n^z)^2 \rangle =
1/3$. An immediate consequence of observed QCC in Fig. \[fig:fingerprint\] is that the entire $\bf{q}$ dependence of the static susceptibility $\chi({\bf q}, \omega_n=0)$ of the quantum model is given by the susceptibility of the classical model at temperature $T_{\rm cl}(T)$, which is readily available from classical Monte Carlo simulations.
Due to the limited low-temperature range of the $T_{\rm cl}(T)$ curve for the TLHA it is perhaps too early to make any definite conclusion regarding its extrapolation down to the $T=0$ limit. One possibility is that it smoothly extrapolates to a finite value $T_{\rm cl}(0)=0.28$, implying that the ground state is some kind of a spin liquid. This possibility was discussed by Anderson [@Anderson73] almost forty years ago but was subsequently rejected on the basis of numerous investigations which include exact diagonalization [@bernu92; @sindzingre; @yunoki], Green’s function MC calculations [@capriotti], series expansion [@zheng1999], density matrix renormalization group [@white07] studies, as well as large-$S$ (spin wave) [@miyake1992; @chubukov1994; @chernyshev2009], large-N [@sachdev1992], and functional renormalization group analysis [@thomale2011]. Note, however, that the spin correlation length for the classical model at $T_{\rm cl} \approx 0.28$ is above $10^3$ lattice periods [@kawamura10] and thus simulations of small system sizes $L\sim 10$ would be severely affected by finite-size effects. The value of $T_{\rm cl}(0) \approx 0.28 $ is surprisingly close (essentially within the error bars) to the temperature obtained by extrapolating transition temperatures for the $q=3$ Potts transition in finite magnetic fields $h$ to the $h=0$ limit [@misha; @seabra]. Large-scale MC simulations performed in zero magnetic field also identify $T_{\rm cl}=0.285(5)$ as the critical point of the chiral transition [@kawamura84; @kawamura10; @xu95]. However, the debate with regards to the existence of the chiral transition is not settled yet—an alternative scenario [@azaria92; @southern93] predicts a sharp crossover to a more standard nonlinear sigma-model type behavior around $T_{\rm cl}=0.28$.
![\[fig:6\] (Color online) Blue (square) symbols: local static susceptibility $\chi(r=0,\omega_m=0)$, multiplied by $4 T$, as a function of $T/J$. Black circles show $T$ dependence of the ($4$ times) local spin correlation function $\chi(r=0,\tau=\beta/2)$. ](fig6.eps){width="0.8\columnwidth"}
The other possibility is that the QCC curve $T_{\rm cl}(T)$ will cross over to the standard renormalized classical behavior in the long wavelength limit and will arrive at $T_{\rm cl}(0) = 0$, implying the ordered quantum ground state. This is exactly what happens for the square lattice antiferomagnet, see inset in the right panel in Fig. \[fig:fingerprint\]. In fact, it is also known (and can be readily deduced from the correspondence plot and Fig.2 of Ref. ) that in the TLHA the renormalized classical regime with large correlation length emerges only below temperature $T \approx 0.25$ [@singh; @zheng2006], which is well below our lowest data point $T=0.375$. Clearly, more data at lower temperatures are required in order to resolve this fascinating question.
The normalization factor $\chi(0) \equiv \chi(r=0,\omega_m=0)$ in Fig. \[fig:fingerprint\] is given by the local static susceptibility which of course is different for the classical and quantum system. For the classical Heisenberg model $T\chi(0)$ is simply $1/3$, independent of temperature, while in the quantum system this quantity is $T$ dependent, as Fig. \[fig:6\] shows. The same figure also shows the local spin correlation function at $\tau=\beta/2$, $\chi(r=0,\tau=\beta/2) = T \sum_m e^{i \pi m} \chi(r=0,\omega_m)$. This too probes quantum fluctuations, i.e., contributions to the sum from terms with $\omega_m \neq 0$. As expected, both curves deviate from unity with the lowering of $T$, reflecting the increasing role of quantum fluctuations.
Perhaps the most striking feature of QCC is its predictive power in the search for spin-liquid states. Indeed, if QCC is confirmed for a given model of quantum magnetism and the classical ground state is not ordered due to macroscopic degeneracy then the quantum ground state is not ordered as well; i.e., it is a spin liquid. Moreover, even if the classical ground state is ordered but the correspondence curve $T_{\rm cl}(T)$ is such that $T_{\rm cl}(0)\ne 0$, the quantum ground state is still not ordered similarly to its finite-temperature classical counterpart. While the final outcome for the TLHA remains to be seen, our data convincingly show an unusual classical-to-quantum correspondence with regards to the static spin correlations.
We thank M. Rigol for comments and the original NLC data and R.R.P. Singh for comments. This work was supported by the National Science Foundation under Grants No. PHY-1005543 (S.K., N.P., B.S., and C.N.V.) and No. DMR-1206774 (O.A.S.), and by a grant from the Army Research Office with funding from the DARPA.
|
---
abstract: 'We find prominent similarities in the features of the time series for the (model earthquakes or) overlap of two Cantor sets when one set moves with uniform relative velocity over the other and time series of stock prices. An anticipation method for some of the crashes have been proposed here, based on these observations.'
author:
- 'Bikas K. Chakrabarti'
- Arnab Chatterjee
- Pratip Bhattacharyya
title: 'Two Fractal Overlap Time Series: Earthquakes and Market Crashes'
---
Introduction {#intro}
============
Capturing dynamical patterns of stock prices are major challenges both for epistemologists as well as for financial analysts [@CCB:book]. The statistical properties of their (time) variations or fluctuations [@CCB:book] are now well studied and characterized (with established fractal properties), but are not very useful for studying and anticipating their dynamics in the market. Noting that a single fractal gives essentially a time averaged picture, a minimal two-fractal overlap time series model was introduced [@CCB:Chakrabarti:1999; @CCB:Pradhan:2003; @CCB:Pradhan:2004] to capture the time series of earthquake magnitudes. We find that the same model can be used to mimic and study the essential features of the time series of stock prices.
The two fractal-overlap model of earthquake {#sec:overlapmodel}
===========================================
Let us consider first a geometric model [@CCB:Chakrabarti:1999; @CCB:Pradhan:2003; @CCB:Pradhan:2004; @CCB:Bhattacharyya:2005] of the fault dynamics occurring in overlapping tectonic plates that form the earth’s lithosphere. A geological fault is created by a fracture in the earth’s rock layers followed by a displacement of one part relative to the other. The two surfaces of the fault are known to be self-similar fractals. In the model considered here [@CCB:Chakrabarti:1999; @CCB:Pradhan:2003; @CCB:Pradhan:2004; @CCB:Bhattacharyya:2005], a fault is represented by a pair of overlapping identical fractals and the fault dynamics arising out of the relative motion of the associated tectonic plates is represented by sliding one of the fractals over the other; the overlap $O$ between the two fractals represents the energy released in an earthquake whereas $\log O$ represents the magnitude of the earthquake. In the simplest form of the model each of the two identical fractals is represented by a regular Cantor set of fractal dimension $\log 2 / \log 3$ (see Fig. \[ccb:fig:generatio\]). This is the only exactly solvable model for earthquakes known so far. The exact analysis of this model [@CCB:Bhattacharyya:2005] for a finite generation $n$ of the Cantor sets with periodic boundary conditions showed that the probability of the overlap $O$, which assumes the values $O=2^{n-k} (k=0, \ldots , n)$, follows the binomial distribution $F$ of $\log_2 O = n-k$ [@CCB:Bhattacharyya:2006]: $$\begin{aligned}
\lefteqn{\Pr \left ( O=2^{n-k} \right )
\equiv \Pr \left ( \log_2 O = n-k \right )} \nonumber \\
& &= \left ( \begin{array}{c} n\\ n-k \end{array} \right )
\left ( {1 \over 3} \right )^{n-k} \left ( {2 \over 3} \right )^k
\equiv F(n-k).
\label{eq:binomial-regular}\end{aligned}$$
Since the index of the central term (i.e., the term for the most probable event) of the above distribution is $n/3 + \delta$, $-2/3 < \delta < 1/3$, for large values of $n$ Eq. (\[eq:binomial-regular\]) may be written as $$F \left ( {n \over 3} \pm r \right ) \approx \left ( \begin{array}{c}
n\\ n \pm r \end{array} \right )
\left ( {1 \over 3} \right )^{{n \over 3} \pm r}
\left ( {2 \over 3} \right )^{{2n \over 3} \mp r}
\label{eq:binomial-central}$$
by replacing $n-k$ with $n/3 \pm r$. For $r \ll n$, we can write the normal approximation to the above binomial distribution as $$F \left ( {n \over 3} \pm r \right ) \sim {3 \over \sqrt{2 \pi n}}
\exp{ \left ( -{9r^2 \over 2n} \right )}
\label{eq:normal-approx}$$
Since $\log_2 O = n-k = {n \over 3} \pm r$, we have $$F \left ( \log_2 O \right ) \sim {1 \over \sqrt{n}}
\exp{\left [ - {\left ( \log_2 O \right )^2 \over n} \right ]},
\label{eq:normal-approx'}$$
not mentioning the factors that do not depend on $O$. Now $$F \left ( \log_2 O \right ) \mathrm{d} \left ( \log_2 O \right )
\equiv G(O) \mathrm{d} O
\label{eq:equivalence}$$ where $$G(O) \sim {1 \over O}
\exp \left [ - {\left ( \log_2 O \right )^2 \over n} \right ]
\label{eq:log-normal2}$$
is the log-normal distribution of $O$. As the generation index $n \to \infty$, the normal factor spreads indefinitely (since its width is proportional to $\sqrt{n}$) and becomes a very weak function of $O$ so that it may be considered to be almost constant; thus $G(O)$ asymptotically assumes the form of a simple power law with an exponent that is independent of the fractal dimension of the overlapping Cantor sets [@CCB:Bhattacharyya:2006]: $$G(O) \sim {1 \over O} \ \mathrm{for} \ n \to \infty.
\label{eq:power-law2}$$
The Cantor set overlap time series {#sec:timeseries}
==================================
We now consider the time series $O(t)$ of the overlap set (of two identical fractals [@CCB:Pradhan:2004; @CCB:Bhattacharyya:2005]), as one slides over the other with uniform velocity. Let us again consider two regular cantor sets at finite generation $n$. As one set slides over the other, the overlap set changes. The total overlap $O(t)$ at any instant $t$ changes with time (see Fig. \[ccb:figu1\](a)). In Fig. \[ccb:figu1\](b) we show the behavior of the cumulative overlap [@CCB:Pradhan:2004] $Q^o(t) = \int_0^t O(\tilde{t}) d\tilde{t}$. This curve, for sets with generation $n=4$, is approximately a straight line [@CCB:Pradhan:2004] with slope $(16/5)^4$. In general, this curve approaches a strict straight line in the limit $a \rightarrow \infty$, asymptotically, where the overlap set comes from the Cantor sets formed of $a-1$ blocks, taking away the central block, giving dimension of the Cantor sets equal to $\mathrm{ln}(a-1)/\mathrm{ln}a$. The cumulative curve is then almost a straight line and has then a slope $\left[(a-1)^2/a\right]^n$ for sets of generation $n$. If one defines a ‘crash’ occurring at time $t_i$ when $O(t_i)-O(t_{i+1}) \ge \Delta$ (a preassigned large value) and one redefines the zero of the scale at each $t_i$, then the behavior of the cumulative overlap $Q^o_i(t) = \int_{t_{i-1}}^t O(\tilde t) d \tilde{t},\; \tilde{t} \le t_i$, has got the peak value ‘quantization’ as shown in Fig. \[ccb:figu1\](c). The reason is obvious. This justifies the simple thumb rule: one can simply count the cumulative $Q^o_i(t)$ of the overlaps since the last ‘crash’ or ‘shock’ at $t_{i-1}$ and if the value exceeds the minimum value ($q_o$), one can safely extrapolate linearly and expect growth upto $\alpha q_o$ here and face a ‘crash’ or overlap greater than $\Delta$ ($=150$ in Fig. \[ccb:figu1\]). If nothing happens there, one can again wait upto a time until which the cumulative grows upto $\alpha^{2}q_o$ and feel a ‘crash’ and so on ($\alpha=5$ in the set considered in Fig. \[ccb:figu1\]).
The stock price time series {#sec:stocktimeseries}
===========================
We now consider some typical stock price time-series data, available in the internet. The data analyzed here are for the New York Stock Exchange (NYSE) Indices [@CCB:NYSE]. In Fig. \[ccb:figu2\](a), we show that the daily stock price $S(t)$ variations for about $10$ years (daily closing price of the ‘industrial index’) from January 1966 to December 1979 (3505 trading days). The cumulative $Q^s(t) = \int_0^t S(t) dt$ has again a straight line variation with time $t$ (Fig. \[ccb:figu2\](b)). Similar to the Cantor set analogy, we then define the major shock by identifying those variations when $\delta S(t)$ of the prices in successive days exceeded a preassigned value $\Delta$ (Fig. \[ccb:figu2\](c)). The variation of $Q_i^s(t) = \int_{t_{i-1}}^{t_i} S(\tilde{t}) d\tilde{t}$ where $t_i$ are the times when $\delta S(t_i) \le -1$ show similar geometric series like peak values (see Fig. \[ccb:figu2\](d)); see [@CCB:PFE].
We observed striking similarity between the ‘crash’ patterns in the Cantor set overlap model and that derived from the data set of the stock market index. For both cases, the magnitude of crashes follow a similar pattern — the crahes occur in a geometric series.
A simple ‘anticipation strategy’ for some of the crashes may be as follows: If the cumulative $Q_i^s(t)$ since the last crash has grown beyond $q_0 \simeq 8000$ here, wait until it grows (linearly with time) until about $17,500$ ($\simeq 2.2q_0$) and expect a crash there. If nothing happens, then wait until $Q_i^s(t)$ grows (again linearly with time) to a value of the order of $39,000$ ($\simeq (2.2)^2 q_0$) and expect a crash, and so on.
The same kind of analysis for the NYSE ‘utility index’, for the same period, is shown in Figs. \[ccb:figu3\].
Earthquake magnitude time series
================================
Unlike in the case of stock price time series where accurate data are easily available, the time series for earthquake magnitudes $M(t)$ at any fault involves considerably coordinated measurements and comparable accuracies are not easily achievable. Still from the available data, as in the case of stock market (where the integrated stock price $Q^s (t)$ shows clear linear variations with time and this fits well with that for the cumulative overlap $Q^o (t)$ for the fractal overlap model; see also [@CCB:EQbook]), the integrated earthquake magnitude $Q^m (t) = \int_0^t M(t) dt$ of the aftershocks does also show such prominent linear variations (see Fig. \[ccb:figu4\]). We believe, the slopes of these linear $Q^m (t)$ vs. $t$ curves for different faults would give us the signature of the corresponding fractal structure of the underlying fault. It may be noted in this context, in our model, the slope becomes $[(a-1)^2/a]^n$ for an $n$th generation Cantor set, formed out of the remaining $a-1$ blocks having the central block removed.
Summary
=======
Based on the formal similarity between the two-fractal overlap model of earthquake time series and of the stock market, we considered here a detailed comparison. We find, the features of the time series for the overlap of two Cantor sets when one set moves with uniform relative velocity over the other looks somewhat similar to the time series of stock prices. We analyze both and explore the possibilities of anticipating a large (change in Cantor set) overlap or a large change in stock price. An anticipation method for some of the crashes has been proposed here, based on these observations.
[99.]{} Sornette D (2003) Why Stock Markets Crash? Princeton Univ. Press, Princeton; Mantegna RN, Stanley HE (1999) Introduction to Econophysics. Cambridge Univ. Press, Cambridge
Chakrabarti BK, Stinchcombe RB (1999) Physica A 270:27-34
Pradhan S, Chakrabarti BK, Ray P, Dey MK (2003) Phys. Scr. T106:77-81
Pradhan S, Chaudhuri P, Chakrabarti BK (2004) in Continuum Models and Discrete Systems, Ed. Bergman DJ, Inan E, Nato Sc. Series, Kluwer Academic Publishers, Dordrecht, pp.245-250; cond-mat/0307735
Bhattacharyya P (2005) Physica A 348:199-215
Bhattacharyya P, Chatterjee A, Chakrabarti BK (2007) Physica A, 381:377-382 NYSE Daily Index Closes from http://www.unifr.ch/econophysics
Chakrabarti BK, Chatterjee A, Bhattacharyya P (2006) in Takayasu H (Ed) Practical Fruits of Econophysics, Springer, Tokyo, pp. 107-110; arxiv:physics/0510047.
Bhattacharyya P, Chakrabarti BK (Eds) (2006) Modelling Critical and Catastrophic Phenomena in Geoscience, Lecture Notes in Physics, vol. 705, Springer-Verlag, Heidelberg
U S Geological Survey, Southern California Catalogs, www.data.scec.org.
|
[ **Conformal Metrics\
**]{} [Daniela Kraus and Oliver Roth]{}\
[University of Würzburg\
Department of Mathematics\
D–97074 Würzburg\
Germany]{}\
Introduction
============
Conformal metrics connect complex analysis, differential geometry and partial differential equations. They were used by Schwarz [@Sch1891], Poincaré [@Poi1898], Picard [@Pic1890; @Pic1893; @Pic1905] and Bieberbach [@Bie12; @Bie16], but it was recognized by Ahlfors [@Ahl38] and Heins [@Hei62] that they are [*ubiquitous*]{} in complex analysis and geometric function theory. They have been instrumental in important recent results such as the determination of the spherical Bloch constant by Bonk and Eremenko [@BE].
The present notes grew out of a series of lectures given at the CMFT Workshop 2008 in Guwahati, India. The goal of the lectures was to give a self–contained introduction to the rich field of conformal metrics assuming only some basic knowledge in complex analysis.
Our point of departure is the fundamental concept of [*conformal invariance*]{}. This leads in a natural way to the notion of curvature in Section \[sec:curvature\] and in particular to a consideration of metrics with constant curvature and their governing equation, the Liouville equation. The study of this equation and its numerous ramifications in complex analysis occupies large parts of our discussion. We focus on the case of negative curvature and first find in an elementary and constructive way all [*radially symmetric*]{} solutions and thereby explicit formulas for the hyperbolic metric of all circularly symmetric domains like disks, annuli and punctured disks.
In Section 3, we discuss Ahlfors’ Lemma [@Ahl38] including a simplified proof of the case of equality using only Green’s Theorem and a number of standard applications such as Pick’s Theorem and Liouville’s Theorem. Following a suggestion of A. Beardon and D. Minda Ahlfors’ Lemma is called the [*Fundamental Theorem*]{} throughout this paper – a grandiose title but, as we shall see, one that is fully justified. In a next step, we describe the elementary and elegant construction of a conformal metric on the twice–punctured plane with curvature bounded above by a negative constant due to Minda and Schober [@MinScho3], which is based on earlier work of Robinson [@Rob39]. The power of this tool is illustrated by proving the little Picard Theorem, Huber’s Theorem and Picard’s Big Theorem.
The topic of Section 4 is M. Heins’ [@Hei62] celebrated theory of [ *SK–metrics*]{}, which we develop for simplicity only for [*continuous*]{} metrics. Following Heins’ point of view, we emphasize the analogy between SK–metrics and subharmonic functions and prove for instance a Gluing Lemma for SK–metrics. A crucial step is the solution of the Dirichlet problem for conformal metrics of constant negative curvature on [*disks*]{}. As there is no analog of the Poisson formula, the construction is a little more involved and is carried out in detail in the Appendix using only some basic facts from classical potential theory and a fixed point argument. After this preparation it is quite easy to carry over a number of fundamental properties and concepts from the theory of harmonic and subharmonic functions such as Harnack’s monotone convergence theorem and Perron families to SK–metrics. As an application the existence of the hyperbolic metric on any domain with at least two boundary points is established. The usefulness of the Gluing Lemma is illustrated by deriving a precise asymptotic estimate for the hyperbolic metric close to an isolated boundary point. This then implies the [*completeness*]{} of the hyperbolic metric and leads to a simple proof of Montel’s Big Theorem.
In Section 5, we describe the general (local) solution of Liouville’s equation in terms of bounded analytic functions. This exhibits an intimate relation between constantly curved conformal metrics and analytic function theory by associating to every metric with curvature $-1$ its [*developing map*]{}, which in general is a multivalued analytic function. The connection is realized via the [*Schwarzian derivative*]{} of a conformal metric, which makes it possible to reconstruct a constantly curved metric from its Schwarzian using the Schwarzian differential equation. We give two applications to the hyperbolic metric. First, a quick proof that the Schwarzian of the hyperbolic metric of any domain with an isolated boundary point has always a pole of order $2$ at this specific point is obtained. Second, it is shown that the inverse of the developing map of the hyperbolic metric is always single–valued. This combined with the results of Section 4 immediately proves the Uniformization Theorem for planar domains. We close the paper by showing how the techniques of the present paper can be used to derive an explicit formula for the hyperbolic metric on the twice–punctured plane originally due to Agard. This involves some simple facts from special function theory, in particular about hypergeometric functions.
This paper contains no new results at all. Perhaps some of the proofs might be considered as novel, which is more or less a byproduct of our attempts to try to present the material in a way as simple as possible. In order not to interrupt the presentation, we have not included references in the main text. Some references can be found in notes at the end of every section, which also contain some suggestions for further reading. We apologize for any omission. We have included a number of exercises and we intend to provide solutions for them at [*www.mathematik.uni-wuerzburg.de/$\sim$kraus*]{}. There is no attempt to present a comprehensive treatment of “conformal metrics” in this paper; we rather focus on some selected topics and strongly recommend the monograph [@KL] of L. Keen and N. Lakic and the forthcoming book of A. Beardon and D. Minda on the same topic for extensive discussions of conformal metrics. This is required reading for anybody interested in conformal metrics! The adventurous reader is directed to Heins’ paper [@Hei62].
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank all the participants of the CMFT Workshop for their valuable questions and comments. We owe a particular debt of gratitude to the other “Resource persons” of the workshop Lisa Lorentzen, Frode R[ø]{}nning, Richard Fournier, S. Ponnusamy and Rasa and Jörn Steuding. We are very grateful to Alan Beardon, Edward Crane, David Minda, Eric Schippers and Toshi Sugawa for many helpful discussions and conversations, and in particular for being tireless advocates of the theory of conformal metrics. And finally, we wish to say a big thank–you to Meenaxi Bhattacharjee and Stephan Ruscheweyh.
> [*“Pardon me for writing such a long letter; I had not the time to write a short one.”*]{}
[**Glossary of Notation**]{}
-------------------------------- -------------------------------------------------------------------------------------------------- --------------
[**Symbol**]{} [**Meaning**]{} [**Page**]{}
\[2mm\] ${{\mathbb{N}}}$ set of non–negative integers
\[1mm\] ${{\mathbb{R}}}$ set of real numbers
\[1mm\] ${{\mathbb{C}}}$ complex plane
\[1mm\] ${{\mathbb{C}}}'$ punctured complex plane ${{\mathbb{C}}}\backslash\{ 0 \}$
\[1mm\] ${{\mathbb{C}}}''$ twice–punctured plane ${{\mathbb{C}}}\backslash \{0,1\}$
\[1mm\] $D$, $G$ domains in the complex plane
\[1mm\] $\overline{M}$ closure of $M \subseteq {{\mathbb{C}}}$ relative to ${{\mathbb{C}}}$
\[1mm\] $\partial M$ boundary of $M \subseteq {{\mathbb{C}}}$ relative to ${{\mathbb{C}}}$
\[1mm\] $K_R(z_0)$ open disk in ${{\mathbb{C}}}$, center $z_0 \in {{\mathbb{C}}}$, radius $R>0$
\[1mm\] ${{\mathbb{D}}}$ unit disk in ${{\mathbb{C}}}$, i.e., ${{\mathbb{D}}}=K_1(0)$
\[1mm\] ${{\mathbb{D}}}'$ punctured unit disk in ${{\mathbb{C}}}$, i.e., ${{\mathbb{D}}}'={{\mathbb{D}}}\backslash \{ 0\}$
\[1mm\] ${{\mathbb{D}}}_R$ open disk in ${{\mathbb{C}}}$, center $0$, radius $R>0$
\[1mm\] ${{\mathbb{D}}}'_R$ punctured disk $\{z \in {{\mathbb{C}}}\, : \, 0<|z|<R\}$
\[1mm\] $\Delta_R$ complement of $\overline{{{\mathbb{D}}}}_R$, i.e., $\{z \in {{\mathbb{C}}}\, : |z|>R\}$
\[1mm\] $A_{r,R}$ annulus $\{z \in {{\mathbb{C}}}\, : \, r<|z|<R\}$
\[1mm\] $ \Delta$ Laplace Operator $\Delta:=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$
or generalized Laplace Operator
\[1mm\] $\kappa_{\lambda}$ (Gauss) curvature of $\lambda(z) \, |dz|$
\[1mm\] $f^*\lambda$ pullback of $\lambda(z) \, |dz|$ under $f$
\[1mm\] $\lambda_G(z) \, |dz|$ hyperbolic metric of $G$
\[1mm\] $L_{\lambda}(\gamma)$ $\lambda$–length of path $\gamma$
\[1mm\] $\d$ distance function associated to $\lambda(z) \, |dz|$
\[1mm\] $C(M)$ set of continuous functions on $M\subseteq {{\mathbb{C}}}$
\[1mm\] $C^k(M)$ set of functions having all derivatives of order $\le k$ continuous
in $M \subseteq {{\mathbb{C}}}$
\[1mm\] $m{_\zeta}$ two–dimensional Lebesgue measure w.r.t. $\zeta \in {{\mathbb{C}}}$
-------------------------------- -------------------------------------------------------------------------------------------------- --------------
Curvature {#sec:curvature}
=========
In the sequel $D$ and $G$ \[def:G\] always denote domains in the complex plane ${{\mathbb{C}}}$.
Conformal metrics
-----------------
Conformal maps preserve angles between intersecting paths[^1], but the euclidean length $$\int \limits_{\gamma} \, |dz|:=\int \limits_{a}^{b} |\gamma'(t)| \, dt$$ of a path $\gamma : [a,b] \to {{\mathbb{C}}}$ is in general not conformally invariant. It is therefore advisable to allow more flexible ways to measure the length of paths.
A continuous function $\lambda : G \to [0,+\infty)$, $\lambda \not\equiv 0$, is called conformal density on $G$.
Let $\gamma : [a,b] \to G$ be a path in $G$ and $\lambda$ a conformal density on $G$. Then $$L_{\lambda}(\gamma):=\int \limits_{\gamma} \lambda(z) \, |dz|:=\int
\limits_{a}^{b} \lambda(\gamma(t)) \, |\gamma'(t)| \, dt$$ is called the $\lambda$–length of the path $\gamma$ and the quantity \[def:leng\] $\lambda(z) \, |dz|$ is called conformal pseudo–metric on $G$. If $\lambda(z)>0$ throughout $G$, we say $\lambda(z) \, |dz|$ is a conformal metric on $G$. We call a conformal pseudo–metric $\lambda(z) \, |dz|$ on $G$ regular, if $\lambda$ is of class $C^2$ in $\{z \in G \, : \, \lambda(z)>0\}$.
The $\lambda$–length of $\gamma$ only depends on the trace of $\gamma$, but not on its parameterization.
Let $\lambda(w) \, |dw|$ be a conformal pseudo–metric on $D$ and $w=f(z)$ a non–constant analytic map from $G$ to $D$. We want to find a conformal pseudo–metric $\mu(z) \, |dz|$ on $G$ such that $L_{\mu}(\gamma)=L_{\lambda}(f \circ \gamma)$ for every path $\gamma$ in $G$. There is clearly only one possible choice for $\mu(z) \, |dz|$ since by the change–of–variables formula $$L_{\lambda}(f\circ \gamma)=\int \limits_{f \circ \gamma} \lambda(w) \,
|dw|=
\int \limits_{\gamma} \lambda(f(z)) \, |f'(z)| \, |dz|
=L_{\lambda\circ f \cdot |f'|}(\gamma) \, .$$
Let $\lambda(w) \, |dw|$ be a conformal pseudo–metric on $D$ and $w=f(z)$ a non–constant analytic map from $G$ to $D$. Then the conformal pseudo–metric $$\label{eq:pullback}
(f^*\lambda)(z) \, |dz|:=\lambda(f(z)) \, |f'(z)| \, |dz|$$ is called the pullback of $\lambda(w) \, |dw|$ under $w=f(z)$.
Thus $$\label{eq:length}
L_{f^*\lambda}(\gamma)=L_{\lambda}(f \circ\gamma)\, .$$
Gauss curvature
---------------
Let $\lambda(w) \, |dw|$ be a conformal pseudo–metric. We wish to introduce a quantity $T_{\lambda}$ which is conformally invariant in the sense that $$T_{f^*\lambda}(z)=T_{\lambda}(f(z)) \,$$ for all conformal maps $w=f(z)$.
Consider (\[eq:pullback\]). We need to eliminate the conformal factor $|f'(z)|$. For this we note that $\log |f'|$ is harmonic, so $\Delta (\log |f'|)=0$ and therefore $$\Delta (\log f^*\lambda)(z)=\Delta (\log \lambda \circ f)(z)+\Delta (\log
|f'|)(z)=\Delta (\log \lambda \circ f)(z)=\Delta (\log \lambda) (f(z)) \, |f'(z)|^2 \, .$$ In the last step we used the chain rule $\Delta (u\circ f)=(\Delta u \circ f) \cdot |f'|^2$ for the Laplace Operator $\Delta$ \[def:laplace\] and holomorphic functions $f$. We see that $T_{\lambda}(z):=\Delta (\log\lambda)(z)/\lambda(z)^2$ is conformally invariant.
\[def:curvature\] Let $\lambda (z) \, |dz|$ be a regular conformal pseudo–metric on $G$. Then $$\kappa_{\lambda}(z):=-\frac{\Delta (\log \lambda)(z)}{\lambda(z)^2}$$ is defined for all points $z \in G$ where $\lambda(z) > 0$. The quantity $\kappa_{\lambda}$ is called the (Gauss) curvature of $\lambda(z) \, |dz|$.
Our preliminary considerations can now be summarized as follows.
For every analytic map $w=f(z)$ and every regular conformal pseudo–metric $\lambda(w)\,|dw|$ the relation $$\label{eq:curv}
\kappa_{f^*\lambda}(z)=\kappa_{\lambda}(f(z))$$ is satisfied provided $\lambda(f(z)) \, |f'(z)|>0$.
For the [*euclidean metric*]{} $\lambda(z) \, |dz|:=|dz|$ one easily finds $\kappa_{\lambda}(z) = 0$. Note that every regular conformal metric $\lambda(z) \, |dz|$ with [*non–positive*]{} curvature gives rise to a [*subharmonic*]{} function $u(z):=\log \lambda(z)$ of class $C^2$ and vice versa.
Constant curvature {#subsec:constcurv}
------------------
In view of (\[eq:curv\]) conformal metrics $\lambda(z) \, |dz|$ with [*constant*]{} curvature are of particular relevance, since for such metrics curvature is an [*absolute*]{} conformal invariant, i.e., $\kappa_{f^*\lambda} \equiv
\kappa_{\lambda}$. In order to find all constantly curved conformal metrics, we set $u(z):=\log \lambda(z)$ and are therefore led to the problem of computing all solutions $u(z)$ to the equation $$\Delta u=-k \, e^{2 u} \, ,$$ where $k \in {{\mathbb{R}}}$ is a real \[ref:R\] constant. This equation is called [*Liouville’s equation*]{}.
If $k=0$, then the solutions of Liouville’s equation are precisely all harmonic functions. Thus, a regular conformal metric $\lambda(z) \, |dz|$ has zero curvature if and only if $\lambda=e^u$ for some harmonic function $u$.
We next consider the case of constant negative curvature $k<0$. We may normalize and choose $k=-1$. Thus we need to consider $$\Delta u= e^{2 u} \, .$$ We shall later find all solutions to this equation. For now it suffices to determine all [*radially symmetric*]{} solutions $u(z)=u(|z|)$, which can be computed rather easily. As a byproduct we obtain in a constructive way a number of important examples of conformal metrics. These metrics will play an ubiquitous r$\hat{\mbox{o}}$le in the sequel.
\[prop:1\] Let $\lambda(z) \, |dz|$ be a radially symmetric regular conformal metric defined on some open annulus centered at $z=0$ with constant curvature $-1$. Then one of the following holds.
- $\lambda$ is defined on a disk ${{\mathbb{D}}}_R:=\{z \in {{\mathbb{C}}}\, : \, |z|<R\}$ and \[def:DR\]
-- ----------------------------------------------------------------------------------------
$\displaystyle \lambda(z)=\lambda_{{{\mathbb{D}}}_R}(z):=\frac{2\, R}{R^2-|z|^2} \, .$
-- ----------------------------------------------------------------------------------------
- $\lambda$ is defined on a punctured disk ${{\mathbb{D}}}_{R}':=\{ z \in {{\mathbb{C}}}\,
: \, 0<|z|<R\}$ \[def:DR’\]and either
---- ----------------------------------------------------------------------------------------------
$\displaystyle \lambda(z) =\lambda_{{{\mathbb{D}}}'_R}(z):=\frac{1}{ |z| \log (R/|z|)}$
or
$\displaystyle \lambda(z) =\frac{2\, \alpha R^{\alpha} |z|^{\alpha-1}}{R^{2 \alpha}- |z|^{2
\alpha}} \, \qquad \text{ for some }\alpha \in (0, +\infty) \backslash \{ 1 \} \, .$
---- ----------------------------------------------------------------------------------------------
- $\lambda$ is defined on an annulus $A_{r,R}:=\{ z \in {{\mathbb{C}}}\, : \, r<|z|<R\}$ and \[def:ArR\]
-- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\displaystyle \lambda(z)=\lambda_{A_{r,R}}(z):=\frac{\pi}{\log (R/r)} \frac{1}{|z| \, \sin \left[ \pi \displaystyle \frac{\log (R/|z|)}{\log (R/r)} \right]} \, .$
-- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------
- $\lambda$ is defined on $\Delta_R:=\{ z \in {{\mathbb{C}}}\, : \, R<|z|<+\infty\}$ for some $R>0$ and either \[def:DeltaR\]
---- ----------------------------------------------------------------------------------------------
$ \displaystyle \lambda(z) =\lambda_{\Delta_R}(z):=\frac{1}{ |z| \log (|z|/R)}$
or
$\displaystyle \lambda(z) =\frac{2\, \alpha R^{\alpha} |z|^{\alpha-1}}{|z|^{2 \alpha}- R^{2
\alpha}} \, \qquad \text{ for some }\alpha \in (0, +\infty) \, .$
---- ----------------------------------------------------------------------------------------------
[**Proof.**]{} We give the main steps involved in the proof and leave the details to the reader. Since $\lambda$ is radially symmetric we get for $u(z):=\log \lambda(z)$ $$\Delta u= \frac{1}{r} \left( r u'(r) \right)' \, ,$$ where $r=|z|$ and $'$ indicates differentiation with respect to $r$. Thus Liouville’s equation $\Delta u=e^{2 u}$ transforms into the ODE $$(r u'(r))'= r \, e^{2 u(r)} \, .$$ We substitute $r=e^{x}$ and obtain for $v(x):=u(e^x)+x$ the ODE $$v''(x)= \, e^{2 v(x)} \, .$$ This ODE has $2 \, v'(x)$ as an integrating factor, so $$\left( v'(x)^2 \right)'= \left( e^{2 v(x)} \right)' \, .$$ Integration yields $$v'(x)=\pm \sqrt{ e^{2 \, v(x)}+c}$$ with some constant of integration $c \in {{\mathbb{R}}}$. The resulting two ODEs (one for each sign) are separable and can be solved by elementary integration. This leads to explicit formulas for $v(x)$ and thus also for $\lambda(z)=e^{u(|z|)}=e^{v(\log |z|)}/|z|$.
We call $$\label{eq:hypdef}
\lambda_{{{\mathbb{D}}}} (z)\, |dz|:=\frac{2}{1-|z|^2}\, |dz|$$ the hyperbolic metric for the unit disk ${{\mathbb{D}}}:=\{z \in {{\mathbb{C}}}\, : \, |z|<1\}$ and $\lambda_{{{\mathbb{D}}}}$ \[def:D\] the hyperbolic density of ${{\mathbb{D}}}$. The curvature of $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ is $-1$.
Some authors call $$\frac{ |dz|}{1-|z|^2}$$ the hyperbolic metric of ${{\mathbb{D}}}$. This metric has constant curvature $-4$.
Exercises for Section 2 {#exercises-for-section-2 .unnumbered}
=======================
1. \[exe:1\] Let $T$ be a conformal self–map of ${{\mathbb{D}}}$. Show that $T^*\lambda_{{{\mathbb{D}}}}\equiv \lambda_{{{\mathbb{D}}}}$.
2. \[exe:2\] Denote \[def:D’\] by ${{\mathbb{D}}}':={{\mathbb{D}}}\backslash \{ 0\}$ the punctured unit disk and let $$\lambda_{{{\mathbb{D}}}'}(z) \, |dz|:=\frac{|dz|}{ |z| \log(1/|z|)}
\, .$$ Find an analytic function $\pi : {{\mathbb{D}}}\to {{\mathbb{D}}}'$ such that $\pi^*\lambda_{{{\mathbb{D}}}'}=\lambda_{{{\mathbb{D}}}}$ and $\pi(0)=e^{-1}$.\
(Consider the “ODE” $$\lambda_{{{\mathbb{D}}}'}(\pi(x))\, |\pi'(x)|= \lambda_{{{\mathbb{D}}}}(x)$$ for $x \in (-1,1)$ assuming that $\pi(x)$ is real and positive and $\pi'(x)$ is real and negative.)
3. \[exe:2.3\] Find all radially symmetric regular conformal metrics defined on some open annulus centered at $z=0$ with constant curvature $+1$.
4. \[exe:pickeq\] Let $f : {{\mathbb{D}}}\to {{\mathbb{C}}}$ be an analytic map with $f(0)=0$ and $f^*\lambda_{{{\mathbb{D}}}} \equiv \lambda_{{{\mathbb{D}}}}$ in a neighborhood of $z=0$. Show that $f(z)=\eta z$ for some $|\eta|=1$.
5. \[exe:geodesic\_curvature\] (Geodesic curvature)
Let $\lambda(z) \, |dz|$ be a regular conformal metric on $D$ and $\gamma : [a,b] \to D$ a $C^2$–path. Then $$\kappa_{\lambda}(t,\gamma):=\frac{\Im \left[ \frac{d}{dt} \big( \log \gamma'(t) \big)+2 \frac{\partial \log \lambda}{\partial z}(\gamma(t)) \cdot \gamma'(t) \right]}{\lambda(\gamma(t)) \, |\gamma'(t)|} \, , \qquad t \in [a,b] \, ,$$ is called the geodesic curvature of $\gamma$ at $t$ for $\lambda(z) \, |dz|$. Show that if $f$ is an analytic map, then $\kappa_{f^*\lambda}(t,\gamma)=\kappa_{\lambda}(t,f \circ \gamma)$.
The Fundamental Theorem {#sec:ahlfors}
=======================
Ahlfors’ Lemma
--------------
The hyperbolic metric $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ has an important extremal property.
\[lem:ahlfors\] Let $\lambda(z) \, |dz|$ be a regular conformal pseudo–metric on ${{\mathbb{D}}}$ with curvature bounded above by $-1$. Then $\lambda(z) \le\lambda_{{{\mathbb{D}}}}(z)$ for every $z \in \mathbb{D}$.
Following A. Beardon and D. Minda we call Lemma \[lem:ahlfors\] the [ *Fundamental Theorem*]{}.
[**Proof.**]{} Fix $0<R<1$ and consider $\lambda_{{{\mathbb{D}}}_R}(z)= 2R/(R^2-|z|^2)$ on the disk ${{\mathbb{D}}}_R$. Then the function $u : {{\mathbb{D}}}_R \to {{\mathbb{R}}}\cup \{-\infty\}$ defined by $$u(z):=\log \left( \frac{\lambda(z)}{\lambda_{{{\mathbb{D}}}_R}(z)} \right) \, , \qquad z \in {{\mathbb{D}}}_R \, ,$$
tends to $-\infty$ as $|z|\to R$ and therefore attains its maximal value at some point $z_0 \in {{\mathbb{D}}}_R$ where $\lambda(z_0)>0$. Since $u$ is of class $C^2$ in a neighborhood of $z_0$, we get $$0 \ge \Delta u(z_0)=\Delta \log \lambda(z_0)-\Delta \log
\lambda_{{{\mathbb{D}}}_R}(z_0) \ge \lambda(z_0)^2-\, \lambda_{{{\mathbb{D}}}_R}(z_0)^2 \, .$$ Thus $u(z) \le u(z_0)\le 0$ for all $z\in {{\mathbb{D}}}_R$, and so $\lambda(z) \le \lambda_{{{\mathbb{D}}}_R}(z)$ for $z \in {{\mathbb{D}}}_R$. Now, letting $R \nearrow 1$, gives $\lambda(z) \le \lambda_{{{\mathbb{D}}}}(z)$ for all $z \in {{\mathbb{D}}}$.
\[lem:equality\] Let $\lambda(z) \, |dz|$ be a regular conformal pseudo–metric on ${{\mathbb{D}}}$ with curvature $\le -1$ and $\mu(z) \, |dz|$ a regular conformal metric on ${{\mathbb{D}}}$ with curvature $=-1$ such that $\lambda(z) \le \mu(z)$ for all $z \in {{\mathbb{D}}}$. Then either $\lambda<\mu$ or $\lambda \equiv \mu$.
In particular, if equality holds in Lemma \[lem:ahlfors\] for one point $z \in {{\mathbb{D}}}$, then $\lambda \equiv \lambda_{{{\mathbb{D}}}}$.
[**Proof.**]{} Assume $\lambda(z_0)=\mu(z_0)>0$ for some point $z_0 \in {{\mathbb{D}}}$. We may take $z_0=0$, since otherwise we could consider $\tilde{\lambda}:=T^*\lambda$ and $\tilde{\mu}:=T^*\mu$ for a conformal self–map $T$ of ${{\mathbb{D}}}$ with $T(0)=z_0$. Let $u(z):=\log (\mu(z)/\lambda(z))\ge 0$ and $$v(r):=\frac{1}{2 \pi} \int \limits_{0}^{2 \pi} u(r e^{i t}) \, dt \,
, \qquad V(r):=\int \limits_{0}^r v(\rho) \, d\rho \, .$$ Then $v(0)=0 \le v(r)=V'(r)$. To show $V \equiv 0$ observe $ \Delta u\le \mu(z)^2 \left(1- e^{-2 u} \right) \le C u$ in $|z| <R<1$ where $C:=2 \sup \{\mu(z)^2 \, : \, |z| < R\}<+\infty$ for small $R>0$. Hence an application of Green’s theorem, \[def:lm\] $$r v'(r)=\frac{r}{2 \pi} \frac{d}{dr} \int \limits_{0}^{2 \pi} u(r e^{it}) \, dt=
\frac{1}{2 \pi} \iint \limits_{|\zeta|<r}
\Delta u(\zeta) \, dm_{\zeta}=
\frac{1}{2 \pi} \int \limits_{0}^r \int \limits_{0}^{2 \pi} \Delta u(\rho e^{it}) \, dt
\, \rho \, d\rho\, ,$$ leads for all $0 \le r < R$ to the estimate $$r V''(r)-C r V(r) \le
r v'(r)-C \int \limits_{0}^r \rho \, v(\rho) \, d\rho=\frac{1}{2\pi}
\int \limits_{0}^{r} \int \limits_{0}^{2 \pi} \left( \Delta u(\rho e^{i t})-C u(\rho
e^{i t}) \right) \, dt \, \rho
\, d\rho \le 0 \, .$$ Thus $V''(r) \le C\, V(r)$, so $(V'(r)^2)' \le C (V(r)^2)'$. This implies $V'(r) \le \sqrt{C} \cdot V(r)$, i.e., $(V(r) e^{-\sqrt{C}\, r})' \le 0$ and therefore $0 \le
V(r)e^{-\sqrt{C} \, r} \le V(0) =0$. Hence $V \equiv 0$ and $v \equiv 0$ on $[0,R)$. Thus $u \equiv 0$ first on $|z|< R$ and therefore in all of ${{\mathbb{D}}}$, so $\lambda
\equiv \mu$.
Applications of Ahlfors’ Lemma
------------------------------
\[cor:pick\] Let $f : {{\mathbb{D}}}\to {{\mathbb{D}}}$ be an analytic function. Then $$\frac{|f'(z)|}{1-|f(z)|^2} \le \frac{1}{1-|z|^2} \, , \quad z \in \mathbb{D}\, .$$
[**Proof.**]{} Note that for a non–constant analytic function $f$, $(f^*\lambda_{{{\mathbb{D}}}})(z) \, |dz|$ is a regular conformal pseudo–metric on ${{\mathbb{D}}}$ with constant curvature $-1$. Now apply the Fundamental Theorem.
\[thm:nometriconC\] The complex plane ${{\mathbb{C}}}$ admits no regular conformal pseudo–metric with curvature $\le -1$.
[**Proof.**]{} Assume that there is a regular conformal pseudo–metric $\lambda(z) \, |dz|$ with curvature $\le -1$ on ${{\mathbb{C}}}$. For fixed $R>0$ consider $f_R(z):=R z$ for $z \in {{\mathbb{D}}}$. Then $(f_R^*\lambda)(z)
\, |dz|$ is a regular conformal pseudo–metric on ${{\mathbb{D}}}$ with curvature $\le
-1$. The Fundamental Theorem gives $\lambda(Rz) \, R =(f^*_R\lambda)(z)
\le \lambda_{{{\mathbb{D}}}}(z)$ for every $z \in
{{\mathbb{D}}}$, that is, $\lambda(w) \le \lambda_{{{\mathbb{D}}}}(w/R)/R$ for each $|w|<R$. Letting $R \to +\infty$ we thus obtain $\lambda \equiv 0$, which is not possible.
\[thm:liouvilleabstract\] Assume $G$ admits a regular conformal metric with curvature bounded above by $-1$. Then every entire function $f : {{\mathbb{C}}}\to G$ is constant.
[**Proof.**]{} Let $\lambda(w) \, |dw|$ be a regular conformal metric on $G$ with curvature bounded above by $-1$. If there would be a non–constant analytic function $f : {{\mathbb{C}}}\to G$, then $(f^*\lambda)(z) \, |dz|$ would be a regular conformal pseudo–metric on ${{\mathbb{C}}}$ with curvature $\le -1$, contradicting Corollary \[thm:nometriconC\].
Thus the exponential map shows that there is no regular conformal metric on the punctured plane ${{\mathbb{C}}}':={{\mathbb{C}}}\backslash \{ 0 \}$ \[def:C’\] with curvature bounded above by $-1$. However, there is such a metric on the twice–punctured plane.
\[thm:hyp\] The twice–punctured plane ${{\mathbb{C}}}'':=\mathbb{C}\backslash\{0,1\}$ carries a regular conformal metric with curvature $\le -1$. \[def:C”\] In particular, every domain $G \subset {{\mathbb{C}}}$ with at least two boundary points carries a regular conformal metric with curvature $\le -1$.
[**Proof.**]{} We first show that for ${\varepsilon}>0$ sufficiently small $$\tau(z):= {\varepsilon}\, \frac{\sqrt{1+ \left| z \right| ^{\frac{1}{3}}}}{ \left| z \right| ^{\frac{5}{6}}} \frac{
\sqrt{1+ \left| z-1 \right| ^{\frac{1}{3}}}}{\left| z-1 \right|^{\frac{5}{6}}}$$ defines a regular conformal metric $\tau(z) \, |dz|$ with curvature $\le -1$ on ${{\mathbb{C}}}''$.
A quick computation using polar coordinates gives for $r:= \left| z \right|$ $$\begin{aligned}
\Delta \, \log \frac{\sqrt{1+ \left| z \right|^{\frac{1}{3}}}}{\left| z \right| ^{\frac{5}{6}}} &=& \Delta \left( \frac{1}{2}\log
\left( 1+r^{\frac{1}{3}} \right) - \frac{5}{6} \log r \right)=
\frac{1}{2} \Delta \left[ \log \left( 1+r^{\frac{1}{3}} \right) \right] = \\
&=& \frac{1}{2} \left( \frac{\partial^2}{ \partial r^2}
\log (1+r^{\frac{1}{3}}) + \frac{1}{r}
\frac{\partial}{\partial r} \log(1+r^{\frac{1}{3}}) \right)=
\frac{1}{18} \frac{1}{(1+|z|^{\frac{1}{3}})^2 |z|^{\frac{5}{3}}}\, \end{aligned}$$ and analogously $$\begin{aligned}
\Delta \, \log \frac{\sqrt{1+ \left| z-1 \right|^{\frac{1}{3}}}}{\left| z-1 \right| ^{\frac{5}{6}}} &=&
\frac{1}{18} \frac{1}{(1+|z-1|^{\frac{1}{3}})^2 |z-1|^{\frac{5}{3}}}\, .\end{aligned}$$ Thus $$\kappa_{\tau}(z)= - \frac{1}{18 \, {\varepsilon}^2} \left[
\frac{|z-1|^{5/3}}{(1+|z|^{\frac{1}{3}})^3
(1+|z-1|^{\frac{1}{3}})}+ \frac{|z|^{5/3}}{(1+|z|^{\frac{1}{3}})
(1+ |z-1|^{\frac{1}{3}})^3} \right]
\le -1$$ for each $z \in {{\mathbb{C}}}''$, if ${\varepsilon}>0$ is small enough since $$\lim \limits_{z \to 0} \kappa_{\tau}(z)=-\frac{1}{36 \cdot{\varepsilon}^2}, \qquad
\lim \limits_{z \to 1} \kappa_{\tau}(z)=-\frac{1}{36 \cdot{\varepsilon}^2}, \qquad
\lim \limits_{|z| \to +\infty} \kappa_{\tau}(z)=-\infty.$$ If $a,b \in {{\mathbb{C}}}$ with $a\not=b$, then the pullback of $\tau(w) \, |dw|$ under $w=T(z)=(z-a)/(b-a)$ yields a regular conformal metric with curvature $\le -1$ on ${{\mathbb{C}}}\backslash \{ a,b\}$. Thus every domain $G$ with at least two boundary points carries a regular conformal metric with curvature $\le -1$ on $G$.
Every entire function $f : {{\mathbb{C}}}\to {{\mathbb{C}}}$ which omits two distinct complex numbers is constant.
[**Proof.**]{} Let $a$ and $b$ be two distinct complex numbers and assume that $f(z)\not=a$ and $f(z) \not= b$ for every $z \in {{\mathbb{C}}}$. By Theorem \[thm:hyp\] ${{\mathbb{C}}}\setminus \{ a , b \}$ carries a regular conformal metric with curvature $\le -1$, so $f$ must be constant in view of Corollary \[thm:liouvilleabstract\].
\[huber\] Let $G \subseteq {{\mathbb{C}}}$ be a domain which carries a regular conformal metric with curvature $\le -1$ and let $f : {{\mathbb{D}}}' \to
G$ be a holomorphic function. If there is a sequence $(z_n ) \subset
{{\mathbb{D}}}'$ such that $\lim_{n \to \infty} z_n=0$ and such that $\lim_{n \to \infty} f(z_n) $ exists and belongs to $G$, then $z=0$ is a removable singularity of $f$.
[**Proof.**]{} By hypothesis $G$ carries a conformal metric $\lambda(w) \, |dw|$ on $G$ with curvature $\le -1$. We may assume w.l.o.g. that $r_n:=|z_n|$ is monotonically decreasing to $0$ and that $f(z_n) \to 0 \in G$. We consider the closed curves $\gamma_n:=f(\partial {{\mathbb{D}}}_{r_n})$ in $G$. \[def:p\] Since $(f^*\lambda)(z) \, |dz|$ is a regular conformal pseudo–metric on ${{\mathbb{D}}}'$ with curvature $\le -1$, we get (see Exercise \[sec:ahlfors\].\[exe:punc\]) $$(f^*\lambda) (z) \le \lambda_{{{\mathbb{D}}}'}(z)=\frac{1}{|z| \log(1/|z|)} \, , \qquad z \in {{\mathbb{D}}}' \, ,$$ i.e., $$L_{\lambda}(\gamma_n) =L_{f^*\lambda}(\partial {{\mathbb{D}}}_{r_n})
\le L_{\lambda_{{{\mathbb{D}}}'}}(\partial {{\mathbb{D}}}_{r_n}) =
\frac{2 \pi}{\log(1/r_n)} \to 0 \qquad \text{ as } n \to \infty \, .$$ Now let $K$ be a closed disk of radius ${\varepsilon}>0$ centered at $w=0$ which is compactly contained in $G$. We may assume $f(z_n) \in K$ for any $n \in {{\mathbb{N}}}$. \[def:N\] Since $\lambda(w) \, |dw|$ is a conformal [*metric*]{}, its density is bounded away from zero on $K$, so $\lambda(w) \ge
c>0$ for all $w \in K$. Hence the euclidean length of $\gamma_n \cap K$ is bounded above by $L_{\lambda}(\gamma_n)/c$. In particular, $\gamma_n=f(\partial {{\mathbb{D}}}_{r_n}) \subset K$, i.e., $|f(z)| \le {\varepsilon}$ on $|z|=r_n$ for all but finitely many indices $n$. The maximum principle implies that $|f(z)| \le {\varepsilon}$ in a punctured neighborhood of $z=0$, so the singularity $z=0$ is removable.
If an analytic function $f : {{\mathbb{D}}}' \to {{\mathbb{C}}}$ has an essential singularity at $z=0$, then there exists at most one complex number $a$ such that the equation $f(z)=a$ has no solution.
[**Proof.**]{} Assume $a,b \not \in f({{\mathbb{D}}}')$ for $a\not=b$. If $c \in {{\mathbb{C}}}\backslash \{a,b\}$, then the theorem of Casorati–Weierstra[ß]{} tells us that there exists a sequence $( z_n)$ such that $z_n \to 0$ and $f(z_n) \to c$. This, however, contradicts Huber’s Theorem (Corollary \[huber\]), because ${{\mathbb{C}}}\setminus \{ a, b \}$ carries a regular conformal metric with curvature $\le -1$ by Theorem \[thm:hyp\].
Exercises for Section 3 {#exercises-for-section-3 .unnumbered}
=======================
1. \[exe:punc\] The Fundamental Theorem clearly holds for every disk ${{\mathbb{D}}}_R$, that is, $\lambda(z) \le \lambda_{{{\mathbb{D}}}_R}(z)$ for every regular conformal pseudo–metric $\lambda(z) \, |dz|$ with curvature $\le -1$ on ${{\mathbb{D}}}_R$.
Show that $\lambda(z) \le \lambda_{A_{r,R}}(z)$ for every regular conformal pseudo–metric $\lambda(z) \, |dz|$ with curvature $\le -1$ on the annulus $A_{r,R}$, see Proposition \[prop:1\]. Deduce that $\lambda(z) \le \lambda_{{{\mathbb{D}}}'}(z)$ for every regular conformal pseudo–metric $\lambda(z) \, |dz|$ with curvature $\le -1$ on ${{\mathbb{D}}}'$.
2. \[exe:3.2\] Let $\lambda(z) \, |dz|$ be a regular conformal metric with curvature $\le -1$ on a disk $K_R(z_0):=\{z \in {{\mathbb{C}}}\, : \, |z-z_0|<R\}$ \[def:dik\] with radius $R>$0 and center $z_0 \in {{\mathbb{C}}}$. Define $u(z):=\log \lambda(z)$ and $$v(r):=\frac{1}{2 \pi} \int \limits_{0}^{2 \pi} u\left(z_0+r e^{i t} \right) \, dt \, .$$ Show that $$v(r)-v(0) \ge \frac{e^{2 v(0)}}{4}\, r^2 \, \qquad \text{ for } 0 \le r < R\,.$$ (Hint: Compute $(r v'(r))'/r$ and recall Jensen’s inequality.)
3. \[exe:3.3\] Modify the proof of Lemma \[lem:equality\] to prove the following special case of the maximum principle of E. Hopf [@Hopf] (see also Minda [@Min87]):
[*Let $c : G \to {{\mathbb{R}}}$ be a non–negative bounded function and $u : G \to {{\mathbb{R}}}$ of class $C^2$ such that $\Delta u \ge c(z) u$ in $G$. If $u$ attains a non–negative maximum at some interior point of $G$, then $u$ is constant.*]{}
4. \[exe:3.4\] Use Lemma \[lem:equality\] and Exercise \[sec:curvature\].\[exe:pickeq\] to describe all cases of equality in Pick’s Theorem (Corollary \[cor:pick\]) for some $z \in {{\mathbb{D}}}$.
5. \[exe:robinson\] Let $z_1, \ldots , z_{n-1}$ be distinct points in ${{\mathbb{C}}}$ and $z_n=\infty$. Further, let $\alpha_1, \ldots \, ,\alpha_{n}$ be real numbers $< 1$ with $s:=\alpha_1+\cdots +\alpha_n>2$.
- Define for ${\varepsilon}>0$ and $\delta>0$ $$\tau(z):={\varepsilon}\prod \limits_{j=1}^{n-1}
\frac{\left[ 1+|z-z_j|^{\delta} \right]^{\frac{s-2}{(n-1) \delta}}}{|z-z_j|^{\alpha_j}} \, .$$ Show that if ${\varepsilon}$ and $\delta$ are sufficiently small, then $\tau(z) \, |dz|$ is a regular conformal metric with curvature $\le -1$ on ${{\mathbb{C}}}\backslash \{z_1, \ldots, z_{n-1}\}$ such that $|z-z_j|^{\alpha_j} \tau(z)$ is continuous and positive at $z=z_j$ for each $j=1, \ldots , n-1$ and $|z|^{2-\alpha_n} \tau(z)$ is continuous and positive at $z=\infty$.
- Let $f$ be a meromorphic function on ${{\mathbb{C}}}$. We say that $w$ is an exceptional value of order $m$ if the equation $f(z)=w$ has no root of multiplicity less than $m$. Prove the following result of Nevanlinna: [*Let $f$ be a meromorphic function on ${{\mathbb{C}}}$ with exceptional values $w_j$ of order $m_j$. If $$\sum \limits_{j} \left( 1-\frac{1}{m_j} \right)>2 \, ,$$ then $f$ is constant.*]{}
Notes {#notes .unnumbered}
=====
Lemma \[lem:ahlfors\] was established by Ahlfors [@Ahl38] in 1938. He used it to derive quantitative bounds in the theorems of Bloch and Schottky. Lemma \[lem:equality\] is due to M. Heins [@Hei62]. The proof given here is a simplified version of Heins’ method. Different proofs were also found by D. Minda [@Min87] and H. Royden [@Roy86]. An elegant direct treatment of the case of equality in Ahlfors’ lemma was given by H. Chen [@Ch01]. However, his method does not appear to be strong enough to prove the more general Lemma \[lem:equality\]. The proof of Theorem \[thm:hyp\] is from the paper [@MinScho3] of D. Minda and G. Schober, which in turn is based on the work of R. Robinson [@Rob39]. The statement of Exercise \[sec:ahlfors\].\[exe:robinson\] which generalizes Theorem \[thm:hyp\] is also due to Robinson. Huber’s Theorem (Corollary \[huber\]) was originally proved by using the Uniformization Theorem, see [@Hub]. The proof here is essentially that of M. Kwack [@Kwa].
The hyperbolic metric {#sec:hyperbolic}
=====================
SK–Metrics
----------
In many applications the class of [*regular*]{} conformal metrics is too restrictive to be useful. In analogy to the situation for subharmonic functions, we consider for a real–valued function $u$ the integral means $$v(r):=\frac{1}{2\pi} \int \limits_{0}^{2 \pi} u(z_0+r e^{it}) \, dt \, .$$ Assuming momentarily that $u$ is of class $C^2$ an application of Green’s theorem gives $$r v'(r)= \frac{1}{2 \pi} \iint \limits_{|\zeta|<r}
\Delta u(z_0+\zeta) \, dm_{\zeta} =
\frac{1}{2 \pi} \int \limits_{0}^r \int \limits_{0}^{2 \pi} \Delta u(z_0+\rho e^{it}) \, dt
\, \rho \, d\rho \, .$$ We differentiate this identity w.r.t. $r$ and get $$\Delta u(z_0)=\lim \limits_{r \to 0} \frac{1}{2 \pi} \int \limits_{0}^{2\pi} \Delta u(z_0+r
e^{it}) \, dt=
\lim \limits_{r \to 0} \frac{1}{r} \left( r v'(r)
\right)'\,,$$ so $(r v'(r))'=r \Delta u(z_0)+o(r)$. We integrate from $0$ to $r$ and obtain $v'(r)=(r/2) \Delta u(z_0)+o(r)$. Another integration from $0$ to $r$ yields $v(r)=u(z_0)+(r^2/4) \Delta u(z_0)+o(r^2)$ and therefore $$\Delta u(z_0)=\lim \limits_{r \to 0} \frac{4}{r^2} \left( \frac{1}{2 \pi} \int_0^{2 \pi } u(z_0 +r e^{it}) \, dt -u(z_0)
\right) \, .$$ If $u$ is not $C^2$, then this limit need not exist. We thus define:
\[def:genlap\] Let $u : G \to {{\mathbb{R}}}$ be a continuous function. Then the generalized lower Laplace Operator $\Delta u$ of $u$ at a point $z \in G$ is defined by $$\Delta u(z)= \liminf_{ r \to 0} \frac{4}{r^2} \left(
\frac{1}{2 \pi} \int_0^{2 \pi } u(z +r e^{it}) \, dt - u(z)
\right) \, .$$
Our preliminary considerations show that if $u$ is $C^2$ in a neighborhood of a point $z \in G$, then the generalized lower Laplace Operator coincides with the standard Laplace Operator at $z$. This justifies the use of the same symbol $\Delta$ for both operators.
\[def:gencurvature\] Let $\lambda(z) \, |dz|$ be a conformal pseudo–metric on $G$. Then the (Gauss) curvature of $\lambda(z) \, |dz|$ at a point $z \in G$ where $\lambda(z)>0$ is defined by $$\kappa_{\lambda}(z)= - \frac{\Delta (\log
\lambda)(z)}{\lambda(z)^2}\, .$$
For regular conformal metrics, Definition \[def:gencurvature\] is consistent with Definition \[def:curvature\].
A conformal pseudo–metric $\lambda(z) \, |dz|$ on $G$ is called SK–metric[^2] on $G$, if its curvature is bounded above by $-1$.
We now have all the technology to generalize the Fundamental Theorem to SK–metrics.
\[lem:fund\] Let $\lambda(z) \, |dz|$ be an SK–metric on ${{\mathbb{D}}}$. Then $\lambda(z) \le\lambda_{{{\mathbb{D}}}}(z)$ for every $z\in \mathbb{D}$. In particular, $\lambda_{{{\mathbb{D}}}}(z)\, |dz|$ is the (unique) maximal SK–metric on ${{\mathbb{D}}}$.
Theorem \[lem:fund\] can be proved in exactly the same way as Lemma \[lem:ahlfors\] by noting that for a continuous function $u$ the generalized lower Laplace Operator is always non–positive at a local maximum. One could also consider for $0<R<1$ the auxiliary function $$v(z):=\max \left\{ 0,\log \left( \frac{\lambda(z)}{\lambda_{{{\mathbb{D}}}_R}(z)} \right) \right\}\, .$$ The hypotheses guarantee that $v$ is [*subharmonic*]{} on ${{\mathbb{D}}}_R$ and $v=0$ on $|z|=R$. By the maximum principle for subharmonic functions we get $v \le 0$ in ${{\mathbb{D}}}_R$, so $\lambda(z) \le \lambda_{{{\mathbb{D}}}_R}(z)$ there. Now, let $R \to 1$ in order to arrive at the desired result.
The Perron method for SK–metrics
--------------------------------
We first discuss a number of simple, but very useful techniques for producing new SK–metrics from old ones.
\[lem:glue1\] Let $\lambda(z) \, |dz|$ and $\mu(z) \, |dz|$ be SK–metrics on $G$. Then $\sigma(z):= \max \{\lambda(z), \mu(z)\}$ induces an SK–metric $\sigma(z) \, |dz|$ on $G$.
[**Proof.**]{} Clearly, $\sigma$ is continuous in $G$. If $\sigma(z_0)=\lambda(z_0)$ for a point $z_0 \in G$, then $$\begin{aligned}
\Delta\log \sigma(z_0)&=& \liminf_{ r \to 0} \frac{4}{ r^2} \left(
\frac{1}{2 \pi} \int_0^{2 \pi } \log \sigma(z_0 +r e^{it}) \, d\!\; t - \log
\sigma(z_0)
\right)\\[2mm]
& \ge & \liminf_{ r \to 0} \frac{4}{ r^2} \left(
\frac{1}{2 \pi} \int_0^{2 \pi } \log \lambda(z_0 +r e^{it}) \, d\!\; t - \log
\lambda(z_0)
\right) \\[2mm] & \ge & \, \lambda(z_0)^2 = \, \sigma(z_0)^2 \, .\end{aligned}$$ Thus $\kappa_{\sigma}(z_0) \le -1$. Similarly, $\kappa_{\sigma}(z_0) \le -1$, if $\sigma(z_0)=\mu(z_0)$, so in either case $\kappa_{\sigma}(z_0) \le -1$.
From Lemma \[lem:glue1\] it follows that SK–metrics need not be smooth.
\[lem:glueing\] Let $\lambda(z)\, |dz|$ be an SK–metric on $G$ and let $\mu(z)\, |dz|$ be an SK–metric on an open subset $U$ of $G$ such that the “gluing condition” $$\limsup \limits_{ U \ni z \to \xi} \mu(z) \le \lambda(\xi)$$ holds for all $\xi \in \partial U \cap G$. Then $\sigma(z)\, |dz|$ defined by $$\sigma(z):=\begin{cases} \, \max \{\lambda(z), \mu(z)\} & \hspace{3mm} \, \text{for } z \in U \, , \\[2mm]
\, \lambda(z) & \hspace{3mm} \, \text{for } z \in G \backslash U
\end{cases}$$ is an SK–metric on $G$.
[**Proof.**]{} We first note that the gluing condition guarantees that $\sigma$ is continuous on $G$, so $\sigma(z) \, |dz|$ is a conformal pseudo–metric on $G$.
We need to compute the curvature $\kappa_{\sigma}(z_0)$ of $\sigma(z) \, |dz|$ at each point $z_0 \in G$ with $\sigma(z_0)>0$. If $z_0 \in U$, then $\kappa_{\sigma}(z_0) \le -1 $ by Lemma \[lem:glue1\]. If $z_0$ is an interior point of $G \backslash U$, then $\sigma(z)=\lambda(z)$ in a neighborhood of $z_0$, so $\kappa_{\sigma}(z_0)=\kappa_{\lambda}(z_0) \le -1$. Finally, if $z_0 \in \partial U \cap G$, then $\sigma(z_0)=\lambda(z_0)$ and thus $$\begin{aligned}
\Delta\log \sigma(z_0)&= & \liminf_{ r \to 0} \frac{4}{ r^2} \left(
\frac{1}{2 \pi} \int_0^{2 \pi } \log \sigma(z_0 +r e^{it}) \, d\!\; t - \log
\sigma(z_0)
\right)\\[2mm]
& \ge & \liminf_{ r \to 0} \frac{4}{ r^2} \left(
\frac{1}{2 \pi} \int_0^{2 \pi } \log \lambda(z_0 +r e^{it}) \, d\!\; t - \log
\lambda(z_0)
\right) \ge \lambda(z_0)^2 = \sigma(z_0)^2\end{aligned}$$ Consequently, $\kappa_{\sigma}(z_0) \le -1$.
By the Fundamental Theorem the hyperbolic metric $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ on the unit disk ${{\mathbb{D}}}$ is the maximal SK–metric on ${{\mathbb{D}}}$. We shall next show that every domain $G$ which carries at least one SK–metric carries a maximal SK–metric $\lambda_G(z) \, |dz|$ as well. Furthermore, this maximal SK–metric is a regular conformal metric with constant curvature $-1$.
The construction hinges on a modification of Perron’s method for subharmonic functions. An important ingredient of Perron’s method for subharmonic functions is played by the Poisson integral formula, which allows the construction of harmonic functions with specified boundary values on disks. We begin with the corresponding statement for regular conformal metrics with constant curvature $-1$.
\[thm:existence\] Let $K \subset {{\mathbb{C}}}$ be an open disk and $\varphi :\partial K \to (0, + \infty)$ a continuous function. Then there exists a unique regular conformal metric $\lambda(z)\, |dz|$ with constant curvature $-1$ on $K$ such that $\lambda$ is continuous on the closure $\overline{K}$ of $K$ \[def:closure\] and $\lambda(\xi)=\varphi(\xi)$ for all $\xi \in \partial K$.
For the somewhat involved proof of Theorem \[thm:existence\] we refer the reader to the Appendix.
We next consider increasing sequences of conformal metrics with constant curvature $-1$ and prove the following theorem of Harnack–type.
\[lem:monotone\] Let $\lambda_j(z)\, |dz|$, $j \in \mathbb{N}$, be a monotonically increasing sequence of regular conformal metrics with constant curvature $-1$ on $G$. Then $\lambda(z)\, |dz|$ for $\lambda(z):=\lim_{j \to \infty}
\lambda_j(z)$ is a regular conformal metric of constant curvature $-1$ on $G$.
[**Proof.**]{} Fix $z_0 \in G$ and choose an open disk $K:=K_R(z_0)$ which is compactly contained in $G$. By considering $T^*\lambda_j$ for $T(z)=R \,z+z_0$ we may assume $K={{\mathbb{D}}}$ and ${{\mathbb{D}}}_{R'} \subseteq G$ for some $R'>1$. The Fundamental Theorem (see Exercise \[sec:ahlfors\].\[exe:punc\]) shows $\lambda_j(z) \le \lambda_{{{\mathbb{D}}}_{R'}}(z)\le M<+\infty$ in ${{\mathbb{D}}}$, so $\lambda(z):=\lim_{j \to
\infty} \lambda_j(z)$ is well–defined and bounded in ${{\mathbb{D}}}$. We now consider to each $\lambda_j$ the function $u_j:= \log
\lambda_j$. Then by Remark \[rem:representation\], $$u_j(z)=h_j(z)- \frac{1}{2\pi} \iint \limits_{{{\mathbb{D}}}} g(z, \zeta) \, e^{2 u_j(\zeta)}\,
dm_{\zeta}\, , \quad z \in {{\mathbb{D}}}\, ,$$ where $h_j:{{\mathbb{D}}}\to {{\mathbb{R}}}$ is harmonic in ${{\mathbb{D}}}$ and continuous on $\overline{{{\mathbb{D}}}}$ with $h_j(\xi)= u_j(\xi)$ for $\xi \in \partial {{\mathbb{D}}}$ and $g$ denotes Green’s function of ${{\mathbb{D}}}$. The fact that $(u_j)$ is monotonically increasing and bounded above implies that $(h_j)$ forms a monotonically increasing sequence of harmonic functions bounded from above. Thus $(h_j)$ converges locally uniformly in ${{\mathbb{D}}}$ to a harmonic function $h$. Letting $j \to \infty$ we obtain by Lebesgue’s Theorem on monotone convergence $$u(z)=h(z)- \frac{1}{2\pi} \iint \limits_{{{\mathbb{D}}}} g(z, \zeta) \, e^{2 u(\zeta)}\,
dm_{\zeta}\, , \quad z \in {{\mathbb{D}}}\,$$ for $u(z)=\log \lambda(z)$. It follows from Remark \[rem:representation\] that $u$ is a $C^2$ solution to $\Delta u=e^{2u}$ in ${{\mathbb{D}}}$. Thus $\lambda(z) \, |dz|$ is a regular conformal metric of constant curvature $-1$ on $G$.
We finally prove an analog of the Poisson Modification of subharmonic functions for SK–metrics.
\[lem:mod\] Let $\lambda(z)\, |dz|$ be an SK–metric on $G$ and $K$ an open disk which is compactly contained in $G$. Then there exists a unique SK–metric $M_K\lambda(z)\, |dz|$ on $G$ with the following properties:
- $M_K\lambda(z) =\lambda(z)$ for every $z \in G\backslash K$
- $M_K\lambda(z)\, |dz| $ is a regular conformal metric on $K$ with constant curvature $-1$.
We call $M_K\lambda$ the modification of $\lambda$ on $K$.\
[**Proof.**]{} By Theorem \[thm:existence\] there exists a [*unique*]{} regular conformal metric $\nu(z)
\, |dz|$ with constant curvature $-1$ in $K$ such that $\nu$ is continuous on $\overline K$ and $\nu(\xi)=\lambda(\xi)$ for all $\xi \in \partial K$. Exercise \[sec:hyperbolic\].\[exe:ex\] shows $\lambda(z) \le \nu(z)$ for all $z \in K$. Thus, by the Gluing Lemma, $$M_K\lambda(z):= \begin{cases} \nu(z) \quad & \text{for } z \in K \\
\lambda(z) \quad& \text{for } z \in G \backslash K
\end{cases}$$ induces a (unique) SK–metric $M_K\lambda(z) \, |dz|$ with the desired properties.
\[rem:mod\] Note that $M_K[\lambda] \ge \lambda$ and $M_K[\lambda] \ge M_K[\mu]$ if $\lambda \ge \mu$.
A family $\Phi$ of (densities of) SK–metrics on $G$ is called a Perron family, if the following conditions are satisfied:
- If $\lambda \in \Phi$ and $\mu \in \Phi$, then $\sigma \in \Phi$, where $\sigma(z) := \max \{\lambda(z), \mu(z)\}$.
- If $\lambda \in \Phi$, then $M_K\lambda \in
\Phi$ for any open disk $K$ compactly contained in $G$.
\[thm:perron0\] Let $\Phi$ be a Perron family of SK–metrics on $G$. If $\Phi \not= \emptyset$, then $$\lambda_{\Phi}(z):=\sup_{\lambda \in \Phi} \lambda(z)$$ induces a regular conformal metric $\lambda_{\Phi}(z)\, |dz|$ of constant curvature $-1$ on $G$.
[**Proof.**]{} Fix $z_0 \in G$ and choose an open disk $K_R(z_0)$ in $G$. Then $(T^* \lambda)(0)\le \lambda_{{{\mathbb{D}}}}(0)=2 $ for $T(z)=R z+z_0$ by the Fundamental Theorem, so $\lambda(z_0) \le 2/R$ for every $\lambda \in
\Phi$ and $\lambda_{\Phi}$ is well–defined.
It suffices to show that $\lambda_{\Phi}(z) \, |dz|$ is regular and has constant curvature $-1$ in every open disk which is compactly contained in $G$. Now fix such an open disk $K$ and pick $z_0 \in K$. Let $(\lambda_n) \subset \Phi$ such that $\lambda_n(z_0) \to \lambda_{\Phi}(z_0)$ and let $\tilde{\lambda}_n \in \Phi$ denote the modification of the SK–metric $\max\{ \lambda_1, \ldots, \lambda_n\} \in \Phi$ on $K$. Then $\tilde{\lambda}_n(z_0) \to \lambda_{\Phi}(z_0)$ and $\tilde{\lambda}_n(z) \, |dz|$ has curvature $-1$ in $K$. Since the sequence $(\tilde{\lambda}_n)$ is monotonically increasing, it converges to a regular conformal metric $\tilde{\lambda}(z) \, |dz|$ with curvature $-1$ on $K$ by Lemma \[lem:monotone\].
We claim that $\tilde{\lambda}=\lambda_{\Phi}$ in $K$. By construction, $\tilde{\lambda} \le \lambda_{\Phi}$ in $K$ and $\tilde{\lambda}(z_0)=\lambda_{\Phi}(z_0)$. Assume, for a contradiction, that $\tilde{\lambda}(z_1) <\lambda_{\Phi}(z_1)$ for some $z_1 \in K$. Then $\tilde{\lambda}(z_1)<\lambda(z_1)$ for some $\lambda \in \Phi$. Let $\mu_n
\in \Phi$ denote the modification of $\max\{\lambda,\tilde{\lambda}_n\}\in \Phi$ in $K$. Then $\mu_n(z_0) \to \lambda_{\Phi}(z_0)$, $\mu_n \ge \tilde{\lambda}_n$ in $K$ and $(\mu_n)$ is monotonically increasing. By Lemma \[lem:monotone\] the sequence $(\mu_n)$ converges to a regular conformal metric $\mu(z) \, |dz|$ with curvature $-1$ on $K$ with $\tilde{\lambda}(z_0)=\mu(z_0)$ and $\tilde{\lambda} \le \mu$ in $K$. Lemma \[lem:equality\] shows that $\tilde{\lambda}=\mu$ in $K$, which contradicts $\tilde{\lambda}(z_1)<\lambda(z_1)
\le \mu_n(z_1) \to \mu(z_1)$.
The hyperbolic metric: Definition and basic properties
------------------------------------------------------
The family of all SK–metrics on a domain $G$ is clearly a Perron family. Thus we obtain as a special case of Theorem \[thm:perron0\] the following result.
\[thm:perron\] Let $\Phi_G$ be the Perron family of all SK–metrics on $G$. If $\Phi_G \not= \emptyset$, then $$\lambda_G(z):=\sup_{\lambda \in \Phi_G} \lambda(z)$$ induces a regular conformal metric $\lambda_{G}(z)\, |dz|$ of constant curvature $-1$ on $G$. In particular, $\lambda_G(z)\, |dz|$ is the (unique) maximal SK–metric on $G$.
We call $ \lambda_G(z)\, |dz|$ the hyperbolic metric on $G$. \[def:hypi\] By Theorem \[thm:hyp\] every domain with at least two boundary points carries a hyperbolic metric. Note the obvious, but important monotonicity property $\lambda_{D} \le \lambda_{G}$ if $G \subseteq D$.
The Fundamental Theorem shows that the hyperbolic metric of the unit disk is given by $$\lambda_{{{\mathbb{D}}}}(z) \, |dz| =\frac{2\, |dz|}{1-|z|^2} \, ;$$ the hyperbolic metric on the punctured unit disk ${{\mathbb{D}}}'$ is $$\lambda_{{{\mathbb{D}}}'}(z) \, |dz|=\frac{|dz|}{|z| \log (1/|z|)} \, ,$$ see Exercise \[sec:ahlfors\].\[exe:punc\] and Theorem \[thm:perron\].
In general, however, it is very difficult to find an explicit formula for the hyperbolic metric of a given domain. It is therefore important to obtain good [*estimates*]{} for the hyperbolic metric. We shall use the Gluing Lemma for this purpose and start with a lower bound for the hyperbolic metric of the twice–punctured plane ${{\mathbb{C}}}''$.
\[thm:min\] Let $$\log R:=\frac{1}{\min \limits_{|z|=1} \lambda_{{{\mathbb{C}}}''}(z)} \, .$$ Then $$\lambda_{{{\mathbb{C}}}''}(z) \ge \frac{1}{|z| \left( \log R+\big|\log|z|\big| \right) } \, , \qquad z \in {{\mathbb{C}}}'' \, .$$
We shall later find the exact value of $R$, see Exercise \[sec:hyperbolic\].\[exe:value\] and Corollary \[cor:val\].
[**Proof.**]{} We first consider the case $z \in {{\mathbb{D}}}'$. By definition of $R$, $$\lambda_{{{\mathbb{C}}}''}(z) \ge \frac{1}{\log R}=\frac{1}{|z| \log(R/|z|)}=\lambda_{{{\mathbb{D}}}_R'}(z) \quad \text{ for } z \in \partial {{\mathbb{D}}}\, .$$ Thus the Gluing Lemma guarantees that $$\sigma(z):=\begin{cases} \, \max \{\lambda_{{{\mathbb{C}}}''}(z), \lambda_{{{\mathbb{D}}}_R'}(z)\} & \hspace{3mm} \, \text{for } z \in {{\mathbb{D}}}' \, , \\[2mm]
\, \lambda_{{{\mathbb{C}}}''}(z) & \hspace{3mm} \,
\text{for } z \in {{\mathbb{C}}}''\backslash {{\mathbb{D}}}',
\end{cases}$$ induces an SK–metric on ${{\mathbb{C}}}''$, so $\sigma(z) \le \lambda_{{{\mathbb{C}}}''}(z)$, which implies $$\lambda_{{{\mathbb{C}}}''}(z) \ge \lambda_{{{\mathbb{D}}}_R'}(z)=\frac{1}{|z| \log(R/|z|)} \quad
\text{ for } z \in \overline{{{\mathbb{D}}}} \backslash \{ 0,1\}\, .$$ In a similar way (see Proposition \[prop:1\]) we get $$\lambda_{{{\mathbb{C}}}''}(z) \ge \lambda_{\Delta_{1/R}}(z)
=\frac{1}{|z| \log(R \, |z|)} \quad \text{ for } z \in {{\mathbb{C}}}'' \backslash {{\mathbb{D}}}' \, .$$ Combining both estimates completes the proof.
It is now easy to show that the behavior of the hyperbolic metric $\lambda_G(z) \, |dz|$ near an isolated boundary point mimics the behavior of the hyperbolic metric $\lambda_{{{\mathbb{D}}}'}(z) \, |dz|$ of the punctured unit disk near the origin.
\[cor:is\] Let $G$ be a domain with an isolated boundary point $z_0 \in {{\mathbb{C}}}$ and hyperbolic metric $\lambda_{G}(z) \, |dz|$. Then $$\lim \limits_{z \to z_0} \left( |z-z_0| \log \frac{1}{|z-z_0|} \right) \lambda_G(z)=1 \, .$$
[**Proof.**]{} We may assume $z_0=0$, ${{\mathbb{D}}}\backslash \{ 0\} \subset G \backslash \{0 \}$ and $1 \in \partial G$. Then $\lambda_{{{\mathbb{C}}}''}(z) \le \lambda_{G}(z) \le \lambda_{{{\mathbb{D}}}'}(z)$ for any $z \in {{\mathbb{D}}}'$, so Theorem \[thm:min\] leads to $$\frac{1}{|z| \log(R/|z|)} \le \lambda_{{{\mathbb{C}}}''}(z) \le
\lambda_{G}(z) \le \lambda_{{{\mathbb{D}}}'}(z)=\frac{1}{|z| \log (1/|z|)} \, , \qquad z \in {{\mathbb{D}}}' \,,$$ which proves the Corollary.
Completeness
------------
Given a conformal metric $\lambda(z)\, |dz|$ on $G$ we define an associated distance function by $$\d(z_0,z_1):= \inf_{\gamma} L_{\lambda}(\gamma) =\inf_{\gamma} \int_{\gamma} \lambda (z) \, |dz|\, ,$$ \[def:dist\] where the infimum is taken over all paths $\gamma$ in $G$ joining $z_0$ and $z_1$.
If $\lambda(z)\, |dz|$ is a conformal metric on $G$, then $(G, \d)$ is a metric space, see Exercise \[sec:hyperbolic\].\[exe:met\].
For instance, if we take the euclidean metric $\lambda(z) \, |dz|:=|dz|$ on $G={{\mathbb{C}}}$, then $\d$ is the euclidean distance, $\d(z_0,z_1)=|z_0-z_1|$. As another example, we briefly discuss the distance induced by the hyperbolic metric $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ of the unit disk.
\[ex:hypo\] We determine the distance $\d(z_0,z_1)$ associated to the hyperbolic metric $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ on the unit disk ${{\mathbb{D}}}$. We first deal with the special case $z_0=0$ and $z_1 \in {{\mathbb{D}}}\backslash \{ 0 \}$. Let $\gamma : [0,1] \to {{\mathbb{D}}}$ be a path connecting $0$ and $z_1$ with $\gamma(t)\not=0$ for all $t \in (0,1]$. Then $$\begin{aligned}
L_{\lambda_{{{\mathbb{D}}}}}(\gamma) &=&
\int \limits_{0}^{1} \frac{2\, |\gamma'(t)|}{1-|\gamma(t)|^2} \, dt
\ge
\int \limits_{0}^1 \frac{2\, \frac{d}{dt} |\gamma(t)|}{1-|\gamma(t)|^2} \, dt
\overset{s=|\gamma(t)|}{=}
\int \limits_{0}^{|z_1|} \frac{2\, ds}{1-s^2} = \log
\left( \frac{1+|z_1|}{1-|z_1|} \right) \, ,\end{aligned}$$ with equality if and only if $\gamma(t) \equiv z_1\, t$. Thus $$\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}(0,z_1)= \log \left( \frac{1+|z_1|}{1-|z_1|} \right)\, .$$
Now, let $z_0$ and $z_1$ be two distinct points in ${{\mathbb{D}}}$ and $T$ a conformal self–map of ${{\mathbb{D}}}$ with $T(z_0)=0$. By Exercise \[sec:curvature\].\[exe:1\] $ \lambda_{{{\mathbb{D}}}}(T(z)) \, |T'(z)|=\lambda_{{{\mathbb{D}}}}(z)$, so (\[eq:length\]) implies $L_{\lambda_{{{\mathbb{D}}}}}(\gamma)=L_{\lambda_{{{\mathbb{D}}}}}(T \circ \gamma)$ for each path $\gamma \subset {{\mathbb{D}}}$. This leads to $$\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}(z_0,z_1)=\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}(T(z_0),T(z_1))=\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}\left(0, \frac{z_1-z_0}{1-\overline{z_0} \, z_1} \right)
= \log \left( \frac{1+\left| \displaystyle \frac{z_1-z_0}{1-\overline{z_0} \,
z_1}\right|}{1-\displaystyle \left|
\frac{z_1-z_0}{1-\overline{z_0} \, z_1} \right|} \right)\, .$$
If $\lambda(z) \, |dz|$ is a conformal metric on $G$, we can equip $G$ with the ordinary euclidean metric and get the metric space $(G,|\cdot|)$ and we also can equip $G$ with the distance $\d$ coming from $\lambda(z) \, |dz|$ and get the metric space $(G,\d)$. Topologically, these two metric spaces are equivalent.
\[prop:topology\] Let $\lambda(z) \, |dz|$ be a conformal metric on $G$. Then the two metric spaces $(G,| \cdot |)$ and $(G,\d)$ have the same open and compact sets.
[**Proof.**]{} We will prove that the identity maps (i) ${\rm Id}:(G, | \cdot|) \to (G, \d)$ and (ii) ${\rm Id}:(G,
\d) \to (G, | \cdot|)$ are continuous.
\(i) Pick $z_0 \in G$ and choose $\varepsilon>0$. Since $\lambda$ is continuous on $G$, there exists a $\delta>0$ such that $K:=K_{\delta}(z_0)$ is compactly contained in $G$ and $$\delta \cdot \max \limits_{z \in K}\lambda(z) < \varepsilon\, .$$ Thus $\d(z, z_0) \le \int_{\gamma} \lambda(z) \, |dz| \le \delta \max_{z \in K} \lambda(z)< \varepsilon$ for all $z \in K$, where $\gamma$ is the straight line connecting $z_0$ and $z$.
\(ii) Fix $z_0 \in G$ and let $(z_n)$ be a sequence in $G$ with $\d(z_n,z_0) \to
0$ as $n \to \infty$. Further, let $\varepsilon >0$ such that $K:=\overline{K_{\varepsilon}(z_0)} \subset G$ and set $$m:= \min_{z \in K} \lambda(z)\, .$$ Then there is an integer $N$ such that $$\d(z_n, z_0) < m\, \varepsilon$$ for all $n\ge N$. We now assume that we can find some $j \ge N$ such that $| z_j -z_0| >
\varepsilon$. In particular, $z_j
\not \in K$. Let $\gamma:[0,1] \to G $ be a path joining $z_0$ and $z_j$. We set $\tau:=\inf\, \{ \, t \in [0,1]\, : \, \gamma(t) \not \in K\, \}$. Then $\tilde{\gamma}:=\gamma_{|_{[0,\tau]}}$ is a path in $K$ and we conclude $$L_{\lambda}(\gamma)=\int_{\gamma} \lambda(z)\, |dz| \ge m \int_{\tilde{\gamma}}
|dz| \ge m\, \varepsilon\,.$$ This shows $\d(z_j, z_0) \ge m \, \varepsilon$, contradicting our hypothesis. Thus $| z_n -z_0| \to 0$ whenever $\d(z_n,z_0) \to
0$.
However, even though the two metric spaces $(G,|\cdot|)$ and $(G,\d)$ are topologically equivalent, they are in general not [*metrically*]{} equivalent. For instance, in $({{\mathbb{D}}},|\cdot|)$ the boundary $\partial {{\mathbb{D}}}$ has distance $1$ from the center $z=0$, but in $({{\mathbb{D}}},\text{d}_{\lambda_{{{\mathbb{D}}}}})$ the boundary is infinitely away from $z=0$. This latter property turns out to be particularly useful, so we make the following definition.
A conformal metric $\lambda(z)\, |dz|$ on $G$ is called complete (for $G$), if $$\lim_{n \to \infty} \d (z_n,z_0)= +\infty$$ for every sequence $(z_n) \subset G$ which leaves every compact subset of $G$ and for some (and therefore for every) point $z_0 \in G$.
\[rem:complete\]
- If $\lambda(z) \, |dz|$ is a complete conformal metric for $G$, then $(G,\d)$ is a complete metric space in the usual sense, i.e., every Cauchy sequence in $(G,\d)$ converges in $(G,\d)$ and, by Proposition \[prop:topology\], then also in $({{\mathbb{C}}},| \cdot|)$ to some point [*in*]{} $G$.
- If $\lambda(z) \, |dz|$ is a complete conformal metric on $G$ and $\mu(z) \ge
\lambda(z)$ in $G$, then $\mu(z) \, |dz|$ is also complete (for $G$).
Obviously, the euclidean metric is complete for the complex plane ${{\mathbb{C}}}$ and the hyperbolic metric $\lambda_{{{\mathbb{D}}}}(z) \, |dz|$ is complete for the unit disk ${{\mathbb{D}}}$, see Example \[ex:hypo\].
\[ex:complete1\] The hyperbolic metric $\lambda_{{{\mathbb{D}}}'}(z) \, |dz|$ on the punctured disk ${{\mathbb{D}}}'$ is complete. By Exercise \[sec:curvature\].\[exe:2\] we have $$\lambda_{{{\mathbb{D}}}}(z)\, |dz|= \left( \pi^* \lambda_{{{\mathbb{D}}}'}\right)(z) \, |dz|\, ,$$ where $\pi : {{\mathbb{D}}}\to {{\mathbb{D}}}'$, $z \mapsto \text{ \rm exp}(- (1+z)/(1-z) )$, is locally one-to-one and onto. Now pick $z_0, z_1 \in {{\mathbb{D}}}'$ and let $\gamma : [a,b]
\to{{\mathbb{D}}}'$ be a path joining $z_0$ and $z_1$. Fix a branch $\pi ^{-1}$ of the inverse function of $\pi$ defined originally only in a neighborhood of $z_0=\gamma(a)$. Then $\pi^{-1}$ can clearly be analytically continued along $\gamma$ and we get the path $\tilde{\gamma}:= \pi^{-1} \circ \gamma \subset {{\mathbb{D}}}$. Thus we obtain $$\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}( \pi^{-1}(z_0), \pi^{-1}(z_1)) \le
L_{\lambda_{{{\mathbb{D}}}}}( \tilde{\gamma})= L_{\pi^*\lambda_{{{\mathbb{D}}}'}}(\tilde{\gamma}) \stackrel{\text{(\ref{eq:length})}}{=}
L_{\lambda_{{{\mathbb{D}}}'}}(\gamma)\, ,$$ so, in particular, $$\text{\rm d}_{\lambda_{{{\mathbb{D}}}}}( \pi^{-1}(z_0), \pi^{-1}(z_1)) \le \text{\rm
d}_{\lambda_{{{\mathbb{D}}}'}}(z_0, z_1)\, .$$ The desired result follows from the latter inequality since $|\pi^{-1}(z_1)| \to 1$ if $z_1 \to 0$ or $|z_1| \to 1$.
\[thm:hyp20\] The hyperbolic metric $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ is complete. In particular, the hyperbolic metric $\lambda_G(z)\, |dz|$ for any domain $G$ with at least two boundary points is complete.
[**Proof.**]{} By Proposition \[prop:topology\] it suffices to show that for fixed $z_0 \in {{\mathbb{C}}}''$ we have $ \text{d}_{{{\mathbb{C}}}''}(z_n,z_0) \to
+\infty$ for any sequence $(z_n) \subset {{\mathbb{C}}}''$ which converges either to $0$, $1$ or $\infty$.
If $z_n \to 0$, then by making use of the estimate $$\lambda_{{{\mathbb{C}}}''}(z) \ge \frac{1}{|z| \log(R/|z|)}=\lambda_{{{\mathbb{D}}}_{R}'}(z) \, ,\quad
z \in {{\mathbb{D}}}' \, ,$$ for some $R>0$ (see Theorem \[thm:min\]), we deduce that $\text{d}_{{{\mathbb{C}}}''}(z_n,z_0) \to +\infty$, since $\lambda_{{{\mathbb{D}}}'_R}(z) \, |dz|$ is complete for ${{\mathbb{D}}}'_R$, cf. Example \[ex:complete1\].
In order to deal with the other cases, we note that Exercise \[sec:hyperbolic\].\[exe:wert1/2\] gives us $$\lambda_{{{\mathbb{C}}}''}(z)= \lambda_{{{\mathbb{C}}}''}(T(z))\, |T'(z)| \, , \quad z \in {{\mathbb{C}}}''\, ,$$ for $T(z)=1/z$ and $T(z)=1-z$. Hence in view of (\[eq:length\]), $$\begin{aligned}
\text{d}_{{{\mathbb{C}}}''}(z_1,z_0)&=&\text{d}_{{{\mathbb{C}}}''}\left(\frac{1}{z_1},\frac{1}{z_0}\right)\, , \\[2mm]
\text{d}_{{{\mathbb{C}}}''}(z_1,z_0)&=&\text{d}_{{{\mathbb{C}}}''}(1-z_1,1-z_0)\, . \end{aligned}$$ This immediately shows $\text{d}_{{{\mathbb{C}}}''}(z_n,z_0) \to + \infty$ if $z_n \to
\infty$ and $\text{d}_{{{\mathbb{C}}}''}(z_n,z_0)\to + \infty$ if $z_n \to 1$. The desired result follows.
As an application, we now make use of the completeness of the hyperbolic metric to prove Montel’s extension of the theorems of Picard.
The set of all holomorphic functions $f : {{\mathbb{D}}}\to {{\mathbb{C}}}''$ is a normal family.
[**Proof.**]{} Let $\lambda:=\lambda_{{{\mathbb{C}}}''}$. For every holomorphic function $f : {{\mathbb{D}}}\to {{\mathbb{C}}}''$, the Fundamental Theorem shows $f^*\lambda \le \lambda_{{{\mathbb{D}}}}$, so $$\text{d}_{\lambda} (f(z_1),f(z_2))\le \text{d}_{\lambda_{{{\mathbb{D}}}}}(z_1,z_2) \, , \qquad z_1, z_2
\in {{\mathbb{D}}}\, .$$ Thus the family ${\cal F}:=\{f : ({{\mathbb{D}}},\text{d}_{\lambda_{{{\mathbb{D}}}}}) \to ({{\mathbb{C}}}'',\d) \text{ holomorphic}\}$ is equicontinuous at each point of ${{\mathbb{D}}}$. We distinguish two cases.
1. Case: For each $z \in {{\mathbb{D}}}$ the set $\{f(z) \, : \, f \in {\cal F}\}$ has compact closure in $({{\mathbb{C}}}'',\d)$. Then Ascoli’s theorem (see [@Edw Thm. 0.4.11]) shows that every sequence in ${\cal F}$ has a subsequence which converges locally uniformly in ${{\mathbb{D}}}$ to some function in ${\cal F}$.
2. Case: There exists a point $z_0 \in {{\mathbb{D}}}$ such that the closure of $\{f(z_0) \, : \, f \in {\cal F}\}$ is not compact in $({{\mathbb{C}}}'',\d)$. Thus there is a sequence $(f_j) \subset {\cal F}$ such that $(f_j(z_0))$ converges to either $0$ or $1$ or $\infty$. It suffices to consider the case $f_j(z_0) \to 0$. Since $\d(f_j(z),f_j(z_0)) \le \text{d}_{\lambda_{{{\mathbb{D}}}}}(z,z_0)$ and $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ is complete, it follows that $f_j(z) \to 0$ for each $z \in {{\mathbb{D}}}$. This convergence is actually locally uniform in ${{\mathbb{D}}}$, since otherwise one could find points $z_j \in {{\mathbb{D}}}$ such that $z_j \to z_1 \in {{\mathbb{D}}}$ and $|f_j(z_j)| \ge {\varepsilon}>0$. But then, once again, the completeness of $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ would imply $+\infty
\leftarrow\d(f_j(z_j),f_j(z_0)) \le \text{d}_{\lambda_{{{\mathbb{D}}}}}(z_j,z_0) \to
\text{d}_{\lambda_{{{\mathbb{D}}}}}(z_1,z_0)<+\infty$. Contradiction!
Exercises for Section 4 {#exercises-for-section-4 .unnumbered}
=======================
1. \[exe:4.1\] Show that there exists no SK–metric on the punctured plane ${{\mathbb{C}}}'$.
2. \[exe:ex\] Let $\lambda(z) \, |dz|$ be a conformal pseudo–metric on $G$. Show that the following are equivalent:
- $\lambda(z) \, |dz|$ is an SK–metric on $G$.
- If $D$ is compactly contained in $G$ and $\mu(z) \, |dz|$ is a regular conformal metric with constant curvature $-1$ on $D$ such that $$\limsup \limits_{z \to \xi} \frac{\lambda(z)}{\mu(z)} \le 1 \quad \text{ for every } \xi \in \partial D \, ,$$ then $\lambda \le \mu$ in $D$.
3. \[exe:4.3\] Show that the pullback of an SK–metric under a non–constant analytic map is again an SK–metric.
4. \[exe:4.4a\] Let $\lambda(z) \, |dz|$ be an SK–metric on ${{\mathbb{D}}}'$ such that $$\limsup_{z \to 0} \lambda(z) \, |z|^{\alpha} < + \infty$$ for some $\alpha <1$. Let $$\lambda_{\alpha}(z)= \frac{2\, (1-\alpha) |z|^{- \alpha}}{1- |z|^{2\, (1-
\alpha)}}\, .$$ Show that $$\lambda(z) \le \lambda_{\alpha}(z)$$ for all $z \in {{\mathbb{D}}}'$.
5. \[exe:4.4\] Let $$\lambda_{n}(z) =\frac{2\, \left(1+\frac{1}{n} \right) \, |z|^{\frac{1}{n}}}{1- |z|^{2
(1+\frac{1}{n})}} \, .$$ Show that $(\lambda_{n})$ is a monotonically increasing sequence of densities of SK–metrics in ${{\mathbb{D}}}$, whose limit $\lambda$ does [*not*]{} induce an SK–metric on ${{\mathbb{D}}}$.
6. \[exe:heins\] Use the Gluing Lemma to prove the following theorem of M. Heins ([@Hei62 Theorem 18.1]): [*Let $G$ be a domain with a hyperbolic metric $\lambda_{G}(z) \, |dz|$ and let $D \subseteq G$ denote a component of the complement of a compact subset $K \subset G$. Then $\lambda_{G}/\lambda_{D}$ has a positive lower bound on $D \backslash V$ for any neighborhood $V$ of $K$.*]{}
7. \[exe:wert1/2\] Let $T$ be one of the Möbiustransformations $$1/z \, , \qquad 1-z \, , \qquad \frac{z}{z-1} \, .$$ Show that $T^* \lambda_{{{\mathbb{C}}}''}=\lambda_{{{\mathbb{C}}}''}$ and deduce $\lambda_{{{\mathbb{C}}}''}(1/2)=4 \, \lambda_{{{\mathbb{C}}}''}(-1)$.
8. \[exe:schwarzpicard\]
(The Schwarz–Picard Problem on the sphere)
A point $z_0 \in \hat{{{\mathbb{C}}}}:={{\mathbb{C}}}\cup \{ \infty\}$ is called a conical singularity of order $\alpha \le 1$ of an SK–metric $\lambda(z) \, |dz|$ defined in a punctured neighborhood of $z_0$ if $$\begin{array}{rcll}
\log \lambda(z) &\!\!=\!\!& \begin{cases} -\alpha \log |z-z_0|+O(1) \hspace{3.2cm} & \text{ as } z \to z_0 \not=\infty \\ -(2-\alpha) \log|z|+O(1) & \text{ as } z \to z_0=\infty \,
\end{cases}\, & \text{ if } \alpha <1 \\[6mm]
\log \lambda(z) & \!\!=\!\!& \begin{cases} - \log |z-z_0|-\log \left( -\log |z-z_0| \right)+O(1) & \text{ as } z \to z_0 \not=\infty \\ - \log|z|+ \log \log |z|+O(1) & \text{ as } z \to z_0=\infty \,
\end{cases}\, & \text{ if } \alpha=1 \, .
\end{array}$$ Now, let $n \ge 3$ distinct points $z_1, \ldots \, ,z_{n-1},\infty \in \hat{{{\mathbb{C}}}}$ and real numbers $\alpha_1, \ldots , \alpha_n \le 1$ be given such that $$\label{eq:gaussbonnet}
\sum \limits_{j=1}^n \alpha_j >2 \, .$$ Choose $\delta>0$ such that $$\min \limits_{k\not=j} |z_j-z_k|>\delta \quad \text{ and } \quad |z_j|<1/\delta \, \quad \text{ for } \quad j=1, \ldots, n-1 \, .$$ Let $$\begin{aligned}
&\lambda_j(z)= \begin{cases}
\displaystyle 2\, (1- \alpha_j) \,
\frac{\delta^{1-\alpha_j}\, |z-z_j|^{-\alpha_j}}{\delta^{2(1-\alpha_j)} -|z-z_j|^{2(1-\alpha_j)}} \qquad \qquad &
\text{if } \, \alpha_j<1\\[5mm]
\displaystyle \frac{1}{|z-z_j|\, \log(\delta/|z-z_j|)} & \text{if } \, \alpha_j=1
\end{cases}
\intertext{for $j=1, \ldots, n-1$, and}
&\lambda_n(z)= \begin{cases}
\displaystyle 2\, (1- \alpha_n)\, \frac{\delta^{1-\alpha_n} |z|^{-
\alpha_n}} {\delta^{2(1-\alpha_n)} \, |z|^{2(1-\alpha_n)}-1} \qquad \qquad &
\text{if } \, \alpha_n<1\\[5mm]
\displaystyle \frac{1}{|z|\, \log(\delta \, |z|)} & \text{if } \, \alpha_n=1\, .
\end{cases}\end{aligned}$$
- Denote by $\Phi$ the family of all densities of SK–metrics $\lambda(z) \, |dz|$ on the $n$–punctured plane $G:={{\mathbb{C}}}\backslash \{z_1, \ldots ,z_{n-1}\}$ such that $$\limsup_{z \to z_j} \frac{\lambda(z)}{\lambda_j(z)} \le 1$$ for all $j=1, \ldots, n$. Show that $\Phi$ is a non–empty Perron family.
- Show that $\lambda_{\Phi}(z):=\sup_{\lambda \in \Phi} \lambda(z)$ induces the uniquely determined regular conformal metric of constant curvature $-1$ on $G$ with conical singularities of order $\alpha_j$ at $z_j$ for $j=1, \ldots, n$.
(Note: Condition (\[eq:gaussbonnet\]) is also necessary (Gauss–Bonnet).)
9. \[exe:app\] Let $\Omega_n:={{\mathbb{C}}}'' \backslash \{e^{2 \pi i k/n} \, : \, k=1, \ldots , n-1\}$ for each positive integer n and let $f_n(z)=z^n$. Show that $\lambda_{\Omega_n}(z) \, |dz| = (f_n^*\lambda_{{{\mathbb{C}}}''})(z) \, |dz|$.
10. \[exe:value\] The aim of this exercise is to show that $\min \limits_{|z|=1} \lambda_{{{\mathbb{C}}}''}(z)=\lambda_{{{\mathbb{C}}}''}(-1)$.
For $\eta \in \partial {{\mathbb{D}}}$ let $\lambda_{\eta}(z):=\lambda_{{{\mathbb{C}}}\backslash \{0, \eta\}}(z)$.
- Show that $\lambda_{\overline{\eta}}(z)=\lambda_{\eta}(\overline{z})$.
- Use the Gluing Lemma to show that $\lambda_{\overline{\eta}}(z) \le
\lambda_{\eta}(z)$ for all $\Im z>0$ if $\Im \eta>0$.
(Hint: Consider $\max \{ \lambda_{\overline{\eta}}(z),\lambda_{\eta}(z)\}$ on the upper half plane and $\lambda_{\eta}(z)$ in the lower half plane.)
- Now fix $\theta \in (-\pi,0)$ and let $\eta:=e^{-i \theta/2}$. Verify that $$\lambda_{{{\mathbb{C}}}''}(-e^{i \theta})\overset{(a)}{=}\lambda_{\eta}(-\overline{\eta})
\overset{(b)}{\ge} \lambda_{\overline{\eta}}(-\overline{\eta})\overset{(a)}{=}\lambda_{{{\mathbb{C}}}''}(-1) \,.$$
- Show in addition that $\theta \mapsto \lambda_{{{\mathbb{C}}}''}(r e^{i \theta})$ is strictly decreasing on $(0,\pi)$ and strictly increasing on $(-\pi,0)$ for each $r>0$.
11. \[exe:met\] Let $\lambda(z) \, |dz|$ be a conformal metric on $G$. Show that $(G,\d)$ is a metric space.
Notes {#notes-1 .unnumbered}
=====
Most of the material of this section and much more can be found in M. Heins [@Hei62], see also S. Smith [@Smi86]. We again wish to emphasize the striking similarities of regular conformal metrics of constant curvature $-1$ and SK–metrics with harmonic and subharmonic functions, see Ransford [@Ransford]. Theorem \[thm:min\] was proved by J. A. Hempel [@Hem79] (see also D. Minda [@Min87b]). Different proofs for Corollary \[cor:is\] were given by J. Nitsche [@Nit57], M. Heins [@Hei62 Section 18], A. Yamada [@Yam1], S. Yamashita [@Yam2] and D. Minda [@Min97]. The metric space $(G,\d)$ is an example of a path metric space in the sense of Gromov [@Gro99]. The converse of Remark \[rem:complete\] (a) is the Hopf–Rinow Theorem. It says that $(G,\d)$ is a complete metric space if and only if every closed and bounded set in $(G,\d)$ is compact. See Gromov [@Gro99 Chapter 1] for a quick proof of the Hopf–Rinow Theorem. For more information about normal families of analytic functions and conformal metrics beyond Montel’s Big Theorem we refer to Grauert and Reckziegel [@GR65]. The Schwarz–Picard Problem (Exercise \[sec:hyperbolic\].\[exe:schwarzpicard\]) was first solved by Poincaré [@Poi1898] in the case $\alpha_1=\ldots=\alpha_n=1$. The general case (even on a compact Riemann surface instead of the sphere) was treated for instance by Picard [@Pic1893; @Pic1905], Bieberbach [@Bie16], Lichtenstein [@Li], Heins [@Hei62], McOwen [@McO88; @McO93] and Troyanov [@Tro90]. See Lehto, Virtanen & Väisäla [@Leh2], Hempel [@Hem79], Weitsman [@Wei79] and Minda [@Min87b] for Exercise \[sec:hyperbolic\].\[exe:value\]. There one can find much more information about monotonicity properties of the hyperbolic metric.
Constant Curvature {#sec:constcurv}
==================
The pullback $(f^*\lambda_{{{\mathbb{D}}}})(z) \, |dz|$ of the hyperbolic metric $\lambda_{{{\mathbb{D}}}}(w) \, |dw|$ under a locally univalent analytic function $w=f(z)$ from $G$ to ${{\mathbb{D}}}$ is a regular conformal metric $\lambda(z) \, |dz|$ on $G$ with constant curvature $-1$. We shall prove the following local converse.
\[thm:liouville\] Let $\lambda(z)\, |dz|$ be a regular conformal metric of constant curvature $-1$ on a simply connected domain $G$. Then there exists a locally univalent analytic function $f: G \to \mathbb{D}$ such that $$\label{eq:liouville2}
\lambda(z) = \frac{2\, |f'(z)|}{1-|f(z)|^2} \, , \qquad z \in G \, .$$ If $g:G \to \mathbb{D}$ is another locally univalent analytic function, then $$\lambda(z) = \frac{2\, |g'(z)|}{1-|g(z)|^2} \, , \qquad z \in G \, ,$$ if and only if $g=T\circ f$, where $T$ is a conformal self–map of ${{\mathbb{D}}}$.
Thus on simply connected domains conformal metrics of constant curvature $-1$ and bounded locally univalent functions can be identified. More precisely, if $G$ is a simply connected domain with $z_0 \in G$, let $$\Lambda(G):=\{ \lambda \, : \, \lambda(z) \, |dz| \text{ regular conformal metric with } \kappa_{\lambda} \equiv -1 \text{ on } G \}$$ and $${\cal H}_0(G):=\{ f\, : \, f:G \to \mathbb{D} \text{ analytic and locally
univalent with } f(z_0)=0, f'(z_0)>0\} \, .$$ Then the map $$\Psi: {\cal H}_0(G) \to \Lambda(G)\, , \qquad f \mapsto \lambda=\frac{2\ |f'|}{1-|f|^2}\, ,$$ is one–to–one.
\[rem:21\] The proof of Theorem \[thm:liouville\] below will show that when $G$ is not simply connected, then the function $f$ can be analytically continued along any path $\gamma \subset G$ and (\[eq:liouville2\]) holds along the path $\gamma$. We call the (multivalued and locally univalent) function $f$ the developing map of the constantly curved metric $\lambda(z) \,
|dz|$. The developing map is uniquely determined by the metric up to postcomposition with a conformal self–map of ${{\mathbb{D}}}$.
The conformal metric $$\lambda(z) \, |dz|=\frac{|dz|}{\sqrt{|z|} \, (1-|z|)}$$ on the punctured unit disk ${{\mathbb{D}}}'$ has constant curvature $-1$. Its developing map (modulo normalization) is the multivalued analytic function $f(z)=\sqrt{z}$.
Proof of Theorem \[thm:liouville\]
----------------------------------
The proof of Theorem \[thm:liouville\] will be split into several lemmas which are of independent interest. We start off with the following preliminary observation, see Remark \[rem:representation\].
\[rem:34\] If $\lambda(z)\, |dz|$ is a regular conformal metric of constant curvature $-1$, then the function $\lambda$ is of class $C^{\infty}$.
In order to find for a conformal metric $\lambda(z) \, |dz|$ with constant curvature $-1$ a holomorphic function $f$ such that the representation formula (\[eq:liouville2\]) holds, we first associate to $\lambda(z) \, |dz|$ a [*holomorphic*]{} auxiliary function.
\[lem:liou0\] Let $\lambda(z)\, |dz| $ be a regular conformal metric of constant curvature $-1$ on $G$ and $u(z):=\log \lambda(z)$. Then $$\frac{\partial u^2}{\partial z^2}(z) - \left( \frac{\partial u}{\partial z}(z) \right)^2$$ is holomorphic in $G$.
[**Proof.**]{} Since $u$ is a solution to $\Delta u=e^{2 u}$, we can write $$\frac{\partial^2 u}{\partial z \partial \overline{z}}(z)=\frac{1}{4} \, e^{2 u(z)}
\, .$$ In view of Remark \[rem:34\], we are allowed to differentiate this identity with respect to $z$. Hence $$\frac{\partial^3 u}{\partial z^2 \partial \overline{z}}=\frac{1}{2} \,
e^{2 u} \, \frac{\partial u}{\partial z}=2 \,
\frac{\partial^2 u}{\partial z \partial \overline{z}} \, \frac{\partial
u}{\partial z}=\frac{\partial}{\partial \overline{z}} \left[ \left( \frac{\partial
u}{\partial z} \right)^2 \right] \, ,$$ and therefore $$\frac{\partial}{\partial \overline{z}} \left[ \frac{\partial^2u}{\partial z^2}(z)-\left( \frac{\partial u}{\partial z}(z)
\right)^2 \right] \equiv 0 \, .$$
Let $\lambda(z)\, |dz| $ be a regular conformal metric and $u(z):=\log
\lambda(z)$. Then the function $$\label{eq:schwarz}
S_{\lambda}(z):= 2 \left[ \frac{\partial u^2}{\partial z^2}(z) - \left( \frac{\partial u}{\partial z}(z) \right)^2 \right]$$ is called the Schwarzian derivative of $\lambda(z)\, |dz| $.
The appearance of the constant $2$ in this definition is motivated by
\[l1\] If $\lambda(z) \, |dz|$ has the form $$\lambda(z)=\frac{2 \, |f'(z)|}{1-|f(z)|^2} \, ,$$ then the Schwarzian derivative of the metric $\lambda(z) \, |dz|$ is equal to the Schwarzian derivative of the analytic function $f$, that is $$\label{eq:liouvivp0}
\displaystyle S_f(z):=\left( \frac{f''(z)}{f'(z)} \right)'-\frac{1}{2}\left( \frac{f''(z)}{f'(z)} \right)^2=
S_{\lambda}(z) \, .$$
Thus, to find for a conformal metric $\lambda(z) \, |dz|$ with constant curvature $-1$ a holomorphic function $f$ such that (\[eq:liouville2\]) holds, first one computes the Schwarzian derivative $S_{\lambda}$ and then solves the Schwarzian differential equation (\[eq:liouvivp0\]) for $f$. In order to guarantee that this program works one needs to know that conformal metrics with constant curvature $-1$ are (modulo a normalization) uniquely determined by their Schwarzian derivatives.
\[lem:liou1\] Let $\lambda(z) \, |dz|$ and $\mu(z) \, |dz|$ be regular conformal metrics of constant curvature $-1$ on $D$ such that $$S_{\lambda}(z)=S_{\mu}(z)$$ for every $z \in D$ and $$\lambda(z_0)=\mu(z_0) \quad \text{and} \quad \frac{\partial \lambda}{\partial z} (z_0) = \frac{\partial \mu}{\partial z} (z_0)$$ for some point $z_0 \in D$. Then $\lambda(z) = \mu(z)$ for every $z \in D$.
[**Proof.**]{} (a) Let $u(z)=\log \lambda(z)$ and $v(z)=\log \mu(z)$. Then $$\label{eq:nitsche1}
\frac{\partial u^2}{\partial z^2}(z) - \left( \frac{\partial u}{\partial z}(z) \right)^2=\frac{\partial v^2}{\partial z^2}(z) - \left( \frac{\partial v}{\partial z}(z) \right)^2$$ for every $z \in D$ and $$\label{eq:nitsche2}
u(z_0)=v(z_0) \quad \text{and} \quad \frac{\partial u}{\partial z} (z_0) =
\frac{\partial v}{\partial z} (z_0) \, .$$
We first show that $$\label{eq:nitsche3}
\frac{\partial^{k+j} u}{\partial z^k \partial \bar{z}^j}(z_0)=
\frac{\partial^{j+k} v}{\partial z^k \partial \bar{z}^j} v(z_0)$$ for $k,j=0,1, \ldots$. By Lemma \[lem:liou0\] both sides of equation (\[eq:nitsche1\]) are holomorphic functions in $D$ and it follows inductively from (\[eq:nitsche1\]) and (\[eq:nitsche2\]) that (\[eq:nitsche3\]) holds for $j=0$ and each $k=0,1, \ldots$. Since $u$ is real–valued (\[eq:nitsche3\]) is valid for $k=0$ and every $j=0,1, \ldots$. But then the partial differential equation $\Delta u=e^{ 2u}$ implies that (\[eq:nitsche3\]) holds for all $k,j=0,1, \ldots$.
\(b) We note that $ v_1(z):=e^{-u(z)}$ and $v_2(z):=e^{-v(z)}$ are (formal) solutions of the linear equation $$v_{zz}+\frac{S_{\lambda}(z)}{2} \, v=0 \, .$$ Thus we consider the auxiliary function (Wronskian) $$W(z):=\frac{\partial v_1}{\partial z}(z) \, v_2(z) -v_1(z) \, \frac{\partial
v_2}{\partial z}(z) \, .$$ Then a straightforward calculation, using $S_{\lambda}=S_{\mu}$, shows $$\frac{\partial W}{\partial z}(z) =0 \, ,$$ so that $W$ is antiholomorphic in $D$. But (a) implies that $$\frac{\partial^k W}{\partial \bar{z}^k}(z_0)=0$$ for each $k=0,1, \ldots$, so $W \equiv 0$ in $D$. Consequently, $$\frac{\partial}{\partial z} \left( \frac{v_1(z)}{v_2(z)} \right)
=\frac{W(z)}{v_2(z)^2} \equiv 0 \, ,$$ i.e. $v_1/v_2$ is antiholomorphic in $D$. Since $v_1/v_2$ is real–valued in $D$ it has to be constant $v_1(z_0)/v_2(z_0)=1$ by (\[eq:nitsche2\]). Therefore $u \equiv v$.
[**Proof of Theorem \[thm:liouville\].**]{} Let $u(z):=\log \lambda(z)$. Lemma \[lem:liou0\] implies that $S_{\lambda}(z)$ is a holomorphic function in $G$. Fix $z_0 \in G$. Then, as it is well–known (see [@Leh p. 53] or [@Lai93]), the initial value problem $$\label{eq:liouvivp}
\begin{array}{rl}
& \displaystyle S_f(z) =
S_{\lambda}(z) \, \\[4mm]
& f(z_0)=0 \, , \quad \displaystyle f'(z_0)=\frac{\lambda(z_0)}{2} \, , \displaystyle \quad f''(z_0)=\frac{\partial
\lambda}{\partial z}(z_0) \, ,
\end{array}$$ has a unique meromorphic solution $f : G
\to {{\mathbb{C}}}$. Let $D$ be the connected component of the set $\{z \in G \, : \, f(z) \in {{\mathbb{D}}}\ \text{ and } f'(z) \not=0\}$ which contains $z_0$ and let $$\mu(z):= \frac{2\, |f'(z)|}{1-\displaystyle |f(z)|^2}\, , \qquad z \in D
\, .$$ Example \[l1\] and the choice of the initial conditions for $f$ guarantee that the hypotheses of Lemma \[lem:liou1\] are satisfied in $D$, so $\lambda(z)=\mu(z)$ for each $z \in D$. It is not difficult to show that this implies that $D$ is closed (and open) in $G$, i.e., $D=G$ and (\[eq:liouville2\]) holds for each $z \in G$.
Now assume that $$\lambda(z)=\frac{2\, |g'(z)|}{1- |g(z)|^2} \, , \qquad z \in G \,
,$$ for some locally univalent holomorphic function $g : G \to {{\mathbb{D}}}$. There is a conformal self–map $T$ of ${{\mathbb{D}}}$ with $T(g(z_0))=0$ and $(T \circ g)'(z_0)>0$. Then, by Exercise \[sec:curvature\].\[exe:1\], $$\frac{2\, |(T \circ g)'(z)|}{1-\, |(T \circ g)(z)|^2}
=\frac{2\, |T'(g(z))|}{1-\, |T (g(z))|^2} \, |g'(z)|=
\frac{2\, |g'(z)|}{1- |g(z)|^2}=\lambda(z)
\, ,
\qquad z \in G \, .$$ Thus $T \circ g$ is a solution of the initial value problem (\[eq:liouvivp\]) and therefore has to be identical to $f$ by uniqueness of this solution, so $f= T \circ g$.
Applications to the hyperbolic metric
-------------------------------------
Theorem \[thm:liouville\] is of fundamental importance and has many applications. For instance, it can be used to analyze the Schwarzian of the hyperbolic metric near isolated boundary points.
\[thm:sch\] Let $G$ be a domain with an isolated boundary point $z_0$ and hyperbolic metric $\lambda_G(z) \, |dz|$. Then $S_{\lambda_G}$ has a pole of order $2$ at $z=z_0$ and $$S_{\lambda_G}(z)=\frac{1}{2 \, (z-z_0)^2}+\frac{C}{z-z_0}+ \ldots \,$$ for some $C \in {{\mathbb{C}}}$.
[**Proof.**]{} We may assume $z_0=0$ and ${{\mathbb{D}}}' \subseteq G$. Let $f_n(z)=z^n$ and $\lambda_n(z) \, |dz|:=\left(f^*_n\lambda_G\right)(z) \, |dz|$. Then by Corollary \[cor:is\], $$\lambda_n(z)=\lambda_G(z^n)\, n\, |z|^{n-1}= \frac{1}{|z| \, \log(1/|z|)} \,
\mu(z^n)\, ,$$ where $\mu$ is a continuous function at $z=0$ with $\mu(0)=1$. This shows that $\lambda_n \to \lambda_{{{\mathbb{D}}}'}$ locally uniformly in ${{\mathbb{D}}}'$ as $n
\to \infty$. Exercise \[sec:constcurv\].\[exe:schwarztransform\] and Liouville’s Theorem imply $$S_{\lambda_G}(z^n) \, n^2 z^{2n-2}+\frac{1-n^2}{2 \, z^2}= S_{\lambda_n}(z)
\to S_{\lambda_{{{\mathbb{D}}}'}}(z)=\frac{1}{2 \, z^2}$$ locally uniformly in ${{\mathbb{D}}}'$. Now a comparison of the Laurent coefficients on both sides immediately completes the proof.
We now apply Liouville’s Theorem to [*complete*]{} conformal metrics of curvature $-1$.
\[thm:liouvillecomplete\] Let $\lambda(z)\, |dz|$ be a regular conformal metric of constant curvature $-1$ on $G$. Then the following are equivalent.
- $\lambda(z) \, |dz|$ is complete.
- Every branch of the inverse of the developing map $f : G \to {{\mathbb{D}}}$ can be analytically continued along any path $\gamma \subset {{\mathbb{D}}}$.
- $\lambda(z) \, |dz|$ is the hyperbolic metric of $G$.
[**Proof.**]{}
\(a) $\Longrightarrow$ (b): Let $\lambda(z) \, |dz|$ be a complete conformal metric of constant curvature $-1$ in $G$. By Liouville’s Theorem we have $$\lambda(z) \, |dz|=\frac{2\, |f'(z)|}{1-|f(z)|^2} \, |dz|$$ for some locally univalent (multivalued) function $f : G \to {{\mathbb{D}}}$. Without loss of generality we may assume $0 \in G$ and $f(0)=0$. We claim that the branch $f^{-1}$ of the inverse function of $f$ with $f^{-1}(0)=0$ can be continued analytically along every path $\gamma : [a,b] \to {{\mathbb{D}}}$ with $\gamma (a)=0$.
If this were false, then there exists a number $\tau \in (a,b)$ such that $f^{-1}$ can be continued analytically along $\gamma : [a,\tau) \to {{\mathbb{D}}}$, but not further. Let $\tau_n \in [a,\tau)$ with $\tau_n \to \tau$ and let $\gamma_n:=\gamma|_{[a,\tau_n]}$. Note that $$\int \limits_{\gamma_n} \lambda_{{{\mathbb{D}}}}(w) \, |dw| \le
\int \limits_{\gamma} \lambda_{{{\mathbb{D}}}}(w) \, |dw| \le C$$ for some constant $C>0$. Let $w_n=\gamma(\tau_n)$ and $z_n=f^{-1}(w_n)$. Then we get $$\text{d}_{\lambda}(z_n,0) \le \int \limits_{f^{-1} \circ
\gamma_n} \lambda(z) \, |dz|=\int \limits_{\gamma_n} \lambda_{{{\mathbb{D}}}}(w)
\, |dw| \le C\, .$$ As $\lambda(z) \, |dz|$ is complete for $G$, the sequence $(z_n)$ stays in some compact subset of $G$, so some subsequence converges to a point $z_{\infty} \in G$. We may thus assume $z_n \to z_{\infty} \in G$. Hence $\gamma (\tau) \leftarrow w_n=f(z_n) \rightarrow f(z_{\infty})$, i.e. $f(z_{\infty})=\gamma(\tau)$. Since $f$ is locally univalent, it maps a neighborhood of $z_{\infty}$ univalently onto a disk $K_r(\gamma(\tau))$, that is $f^{-1}$ can be continued analytically to this disk. This, however, contradicts our hypothesis.
\(b) $\Longrightarrow$ (c): Note that $g:=f^{-1}: {{\mathbb{D}}}\to G$ is holomorphic and single–valued by the Monodromy Theorem. Hence the Fundamental Theorem gives us $$(g^{\, *}\lambda_G) (z) \le \lambda_{{{\mathbb{D}}}}(z)\,, \quad z \in {{\mathbb{D}}}\,.$$ But, by construction, $$\lambda_{{{\mathbb{D}}}} (z)= (g^{ *}\lambda)(z)\,, \quad z \in {{\mathbb{D}}}\,.$$ Thus $\lambda_G \le \lambda$ in $G$ and therefore $\lambda\equiv \lambda_G$ in $G$.
\(c) $\Longrightarrow$ (a): This is Theorem \[thm:hyp20\].
Let $G$ be a domain with at least two boundary points. Then there exists a locally univalent, surjective and analytic function $\pi: {{\mathbb{D}}}\to G$ such that every branch of $\pi^{-1}$ can be analytically continued along any path in $G$ ($\pi$ is called a universal covering of G). If $\tau$ is another universal covering of G, then $ \tau= \pi \circ T$ for some conformal self–map $T$ of ${{\mathbb{D}}}$.
[**Proof.**]{} $G$ carries an SK–metric, see Theorem \[thm:hyp\], and thus possesses a hyperbolic metric $\lambda_G(w) \, |dw|$ by Theorem \[thm:perron\], which is complete by Theorem \[thm:hyp20\]. Theorem \[thm:liouvillecomplete\] implies that the every branch $\pi$ of the inverse of the developing map $f : G \to {{\mathbb{D}}}$ can be analytically continued along any path in ${{\mathbb{D}}}$. By the Monodromy Theorem $\pi$ is an analytic function from ${{\mathbb{D}}}$ into $G$. It follows from the construction that $\pi$ is locally univalent. Since $f$ can also be analytically continued along any path in $G$ (see Remark \[rem:21\]) it follows that $\pi$ is onto. If $\tau$ is another such function, then it is easy to see that $\tau=\pi \circ T$ for some conformal self–map $T$ of ${{\mathbb{D}}}$.
Let $\lambda_G(z)\, |dz| $ be the hyperbolic metric on $G$ and $\pi : {{\mathbb{D}}}\to
G$ a universal covering. Then $( \pi^* \lambda_G)(z)\, |dz|= \lambda_{{{\mathbb{D}}}}(z) \, |dz|$ by construction.
Let $G \subsetneq {{\mathbb{C}}}$ be a simply connected domain. Then there exists a conformal map $f$ from $G$ onto ${{\mathbb{D}}}$, which is uniquely determined up to postcomposition with a conformal self–map of ${{\mathbb{D}}}$.
[**Proof.**]{} If $G$ is simply connected, then the developing map $f$ is a single–valued analytic function and therefore a conformal map from $G$ onto ${{\mathbb{D}}}$.
The twice–punctured plane
-------------------------
In case of the twice–punctured plane it is possible to derive explicit formulas for the hyperbolic metric and its developing map.
\[thm:agard\] Let $$K(z) :=\frac{2}{\pi} \int \limits_0^1 \frac{dt}{\sqrt{(1-t^2) \, (1-z t^2)}} \, .$$ Then $$\lambda_{{{\mathbb{C}}}''}(z)\, |dz|=\frac{|dz|}{\pi \, |z| \, |1-z| \, \Re \left[ K(z) K(1-\overline{z})\right]}$$ and the developing map of $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ is given by $$z \mapsto \frac{K(1-z)-K(z)}{K(1-z)+K(z)}\, .$$
The proof of Theorem \[thm:agard\] relies on Liouville’s Theorem and the following lemma.
\[lem:aga1\] Let $\lambda(z) \, |dz|=\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ be the hyperbolic metric of ${{\mathbb{C}}}''$. Then $$S_{\lambda}(z)=\frac{1}{2} \left[ \frac{1}{z^2}+\frac{1}{(z-1)^2}+\frac{1}{z \, (1-z)} \right] \, .$$
[**Proof.**]{} Theorem \[thm:sch\] shows that $S_{\lambda}$ is analytic in ${{\mathbb{C}}}''$ with poles of order $2$ at $z=0$ and $z=1$, so $$S_{\lambda}(z)=\frac{1}{2 \, z^2}+\frac{c_1}{z}+\frac{1}{2 \, (z-1)^2}
+\frac{c_2}{z-1}+r(z)=\frac{1}{2 \, z^2}+\frac{1}{2 \, (z-1)^2}
+\frac{(c_1+c_2) z-c_1}{z (z-1)}+r(z) \, ,$$ where $c_1, c_2 \in {{\mathbb{C}}}$ and $r$ is analytic on ${{\mathbb{C}}}$. We analyze $S_{\lambda}(z)$ at $z=\infty$. Let $f(z)=1/z$ and observe that $f^*\lambda=\lambda$, so $S_{\lambda}(z)=S_{f^*\lambda}(z)=S_f+S_{\lambda}(f(z)) \, f'(z)^2=S_{\lambda}(1/z)/z^4$ in view of Exercise \[sec:constcurv\].\[exe:schwarztransform\]. Hence $$\lim \limits_{z \to \infty} z^2 S_{\lambda}(z)=\lim \limits_{z \to \infty} \frac{1}{z^2} S_{\lambda}(1/z)=\lim \limits_{z \to 0} z^2 S_{\lambda}(z)=\frac{1}{2} \, .$$ This forces $r \equiv 0$ and $c_1=-c_2=1/2$.
[**Proof of Theorem \[thm:agard\].**]{} We consider the simply connected domain $G={{\mathbb{C}}}\backslash ((-\infty,0] \cup [1,+\infty))$. Liouville’s Theorem gives us an analytic function $f : G \to {{\mathbb{D}}}$ such that $$\frac{2 \, |f'(z)|}{1-|f(z)|^2}=\lambda_{{{\mathbb{C}}}''}(z) \, , \qquad z \in G \, .$$ By Example \[l1\] and Lemma \[lem:aga1\], we know that $f$ is a solution to $$\label{eq:s34}
S_f(z)=\frac{1}{2} \left[ \frac{1}{z^2}+\frac{1}{(z-1)^2}+\frac{1}{z \, (1-z)} \right]\,.$$ We consider the hypergeometric differential equation $$\label{eq:hypgeometric}
z (1-z) \, u''+\left( 1-2 \,z\right) \, u'-\frac{1}{4} \, u=0 \, ,$$ since every solution of (\[eq:s34\]) can be written as a fractional linear transformation of two linearly independent solutions $u_1$ and $u_2$ of (\[eq:hypgeometric\]), see [@Neh52 p. 203 ff.]. It is well–known and easy to prove (see Nehari [@Neh52 p.206 ff.]) that $u_1(z)=K(z)$, $u_2(z)=K(1-z)$ are two linearly independent solutions of (\[eq:hypgeometric\]) in $G$. Hence $$f(z)=\frac{a \, u_2(z)+b \, u_1(z)}{c \, u_2(z)+d \, u_1(z)}$$ for $a,b,c,d \in {{\mathbb{C}}}$ with $ad-bc\not=0$ and thus $$\lambda_{{{\mathbb{C}}}''}(z)=2 |ad-bc| \, \frac{|u_2'(z) u_1(z)-u_2(z) u_1'(z)|}{|c \,u_2(z)+
d \, u_1(z)|^2-|a \, u_2(z)+b \, u_1(z)|^2} \, , \qquad z \in G \, .$$ Now, consider the Wronskian $w:=u_2'u_1-u_2 u_1'$ of $u_1$ and $u_2$, which is a solution to $$w'=-\frac{1-2 z}{z \, (1-z)} \, w \, .$$ As $u_1$ is holomorphic in a neighborhood of $z=0$ with $u_1(0)=1$ and $$u_2(z)=-\frac{1}{\pi} \log z + h_1(z) \, \log z+ h_2(z)\, ,$$ where $h_1$ and $h_2$ are analytic in a neighborhood of $z=0$ with $h_1(0)=0$, cf. [@AB 15.5.16 and 15.5.17]), we obtain
$$u_2'u_1-u_2 u_1'=- \frac{1}{\pi}\, \frac{1}{z \, (1-z)} \, .$$ Consequently, if we set $\beta:= |ad-bc|$ and $\gamma:=\overline{c} d-b\overline{a}$, we get $$\label{eq:aga2}
\lambda_{{{\mathbb{C}}}''}(z)= \frac{2\, \beta}{ \pi } \frac{1}{ |z| \, |1-z|} \frac{1}{\left( |c|^2-|a|^2
\right) |u_2(z)|^2+\left(
|d|^2-|b|^2 \right) |u_1(z)|^2+2 \Re \big( \gamma \, u_1(z) \overline{u_2(z)} \big) } \, ,$$ for every $z \in G$. Now, if we let $z \to 0$, then by the asymptotics of $\lambda_{{{\mathbb{C}}}''}(z)$ at $z=0$ described in Corollary \[cor:is\] we see that $|a|=|c|$. A similar analysis for $z \to 1$ gives $|b|=|d|$. Hence $$\label{eq:89}
\lambda_{{{\mathbb{C}}}''}(z)=\frac{\beta}{\pi }\, \frac{1}{|z| \, |1-z| \, \Re \left(
\gamma \, u_1(z) \, \overline{u_2(z)} \right) } \, .$$
Applying Corollary \[cor:is\] once more shows $\beta=
\Re(\gamma)$. In addition, the continuity of $\lambda_{{{\mathbb{C}}}''}(z)$ in ${{\mathbb{C}}}''$ implies that $\gamma$ is real. We thus arrive at $$\lambda_{{{\mathbb{C}}}''}(z)\, |dz|=\frac{|dz|}{\pi \, |z| \, |1-z| \, \Re \left[ K(z) K(1-\overline{z})\right]} \, .$$
Now, we are left to show that some developing map $F : {{\mathbb{C}}}'' \to {{\mathbb{D}}}$ of $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ has the asserted form. Here we will make essential use of the fact that with $f$ also $T \circ f$ is a developing map for $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$, when $T$ is a conformal self–map of ${{\mathbb{D}}}$. We have shown above that $$f(z)= \frac{a\, u_2(z) +b \, u_1(z) }{c \, u_2(z) + d u_1(z)}$$ with $|a|=|c|$, $|b|=|d|$ and $\overline{c} d-b\overline{a}=|ad -bc| >0$ is a developing map for $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$. Therefore, $$\tilde{F}(z)=\frac{u_2(z) + B\, u_1(z)}{ u_2(z) + D\, u_1(z)}$$ is also a developing map for $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ where $B=b/a$ and $D=d/c$. Furthermore, $|B|=|D|$ and $D-B>0$. This implies $\Re(B)<0<\Re(D)$. Thus, $z_0:=(1+B)/(1+D) \in {{\mathbb{D}}}$ and if $$T(z)= \frac{1-\overline{z_0}}{1- z_0 }\, \frac{z -z_0}{1-\overline{z_0}
z}\, ,$$ then $$F(z) =(T \circ \tilde{F})(z) = \frac{u_2(z) -u_1(z)}{u_2(z)+u_1(z)}= \frac{K(1-z)-K(z)}{K(1-z)+K(z)} \,$$ is the desired developing map of $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$.
\[cor:val\] The density of the hyperbolic metric $\lambda_{{{\mathbb{C}}}''}(z) \, |dz|$ at $z=-1$ has the value $$\lambda_{{{\mathbb{C}}}''}(-1)=\frac{\Gamma \left( \frac{3}{4} \right)^4}{\pi^2} \approx \,0.22847329 .$$
[**Proof.**]{} Since $K(1/2)=\sqrt{\pi}/\Gamma(3/4)^2$ (see [@AB]), Exercise \[sec:hyperbolic\].\[exe:wert1/2\] and Theorem \[thm:agard\] give $$\lambda_{{{\mathbb{C}}}''}(-1)=\frac{\lambda_{{{\mathbb{C}}}''}(1/2)}{4} =\frac{1}{\pi \, K(1/2)^2} =
\frac{\Gamma \left( \frac{3}{4} \right)^4}{\pi^2}
\, .$$\
{width="12cm"}
Exercises for Section 5 {#exercises-for-section-5 .unnumbered}
=======================
1. \[exe:5.1\] Let $\lambda(z) \, |dz|$ be a regular conformal metric with constant curvature $+1$ on a simply connected domain $G$. Show that $$\lambda(z)=\frac{2 \, |f'(z)|}{1+|f(z)|^2}$$ for some locally univalent meromorphic function $f$ on $G$.
2. \[exe:schwarztransform\] Let $\lambda(w) \, |dw|$ be a regular conformal pseudo–metric on $G$ and $f : D \to G$ an analytic map. Show that $$S_{f^*\lambda}(z)=S_f(z)+S_{\lambda}(f(z)) \, f'(z)^2\,.$$
Notes {#notes-2 .unnumbered}
=====
Theorem \[thm:liouville\] was apparently first stated by Liouville [@Lio1853], but his proof is certainly not complete by today’s standards. The proof given here is an adaption of the method of Nitsche [@Nit57], whose presentation contains some gaps as well. Liouville’s theorem is also proved in Bieberbach [@Bie16] and a very elegant geometric proof has been given by D. Minda (unpublished). Theorem \[thm:sch\] is Theorem 1 (iii) in Minda [@Min97], but can also be deduced from Nitsche’s result in [@Nit57]. The methods of Minda and Nitsche are different from the short proof given above.
Appendix
========
In this appendix we construct conformal metrics with constant curvature $-1$ on disks with prescribed (continuous) boundary values (Theorem \[thm:existence\]). It clearly suffices to consider the case of the unit disk ${{\mathbb{D}}}$. In this situation, Theorem \[thm:existence\] is equivalent to the following result.
\[solgen\] Let $\psi : \partial {{\mathbb{D}}}\to
{{\mathbb{R}}}$ be a continuous function. Then there exists a uniquely determined function $u \in C(\overline{{{\mathbb{D}}}} ) \cap C^2({{\mathbb{D}}})$ such that \[ref:diff\] \[def:C\] $$\label{eq:pdesing}
\begin{array}{rccc}
\Delta u &=& e^{2 u} & \, \, \text{in }\, \, {{\mathbb{D}}}, \\[2mm]
u &=& \psi & \, \, \text{on } \, \, \partial {{\mathbb{D}}}.
\end{array}$$
We shall use the following basic facts about the Poisson equation $\Delta u=q$ in ${{\mathbb{D}}}$.
\[thm:Gilbarg\] Let $q : {{\mathbb{D}}}\to {{\mathbb{R}}}$ be a bounded and continuous function. Then $$w(z):= -\frac{1}{2 \pi} \iint \limits_{{{\mathbb{D}}}} g(z, \zeta) \, q(\zeta) \, d
m_{\zeta}$$ where $g(z, \zeta)=\log \left(\left|1 - \overline{\zeta} z\right| / \left|z- \zeta
\right|\right) $ is Green’s function for ${{\mathbb{D}}}$, belongs to $C(\overline{{{\mathbb{D}}}}) \cap C^1({{\mathbb{D}}})$. \[ref:Diff\] If, in addition, $q$ is continuously differentiable in ${{\mathbb{D}}}$ then $w \in
C^{2}({{\mathbb{D}}})$ and solves the boundary value problem $$\begin{array}{rccc}
\Delta w &=& q &\, \, \text{in } \, \, {{\mathbb{D}}}, \\[2mm]
w &=& 0 & \, \, \text{on } \, \, \partial {{\mathbb{D}}}.
\end{array}$$ Moreover, if $q \in C^k({{\mathbb{D}}})$ for some $k \ge 1$, then $w \in C^{k+1}({{\mathbb{D}}})$.
\[rem:representation\] Theorem \[thm:Gilbarg\] shows that every $C^2$ solution to (\[eq:pdesing\]) has the form $$\label{eq:form}
u(z)=h(z)-\frac{1}{2 \pi}\iint \limits_{{{\mathbb{D}}}} g(z,\zeta) \,
e^{2 u(\zeta)} \, dm_{\zeta}\,, \qquad z \in {{\mathbb{D}}}\, ,$$ where $h$ is continuous on $\overline{{{\mathbb{D}}}}$, harmonic in ${{\mathbb{D}}}$ and coincides with $\psi$ on $\partial {{\mathbb{D}}}$. If, conversely, a bounded and continuous function $u$ on ${{\mathbb{D}}}$ has the form (\[eq:form\]) with a harmonic function $h$ in ${{\mathbb{D}}}$ which is continuous on $\overline{{{\mathbb{D}}}}$ and has boundary values $\psi$, then $u \in
C^2({{\mathbb{D}}})$ and solves (\[eq:pdesing\]). In particular, every $C^2$ solution to $\Delta u=e^{2u}$ is of class $C^{\infty}$.
[**Proof of Theorem \[thm:Gilbarg\].**]{} For the differentiability properties of $w$ we refer to [@GT77 p. 50 ff]. So, it remains to show that $$\lim \limits_{z \to \xi} \iint \limits_{{{\mathbb{D}}}} g(z,\zeta) \, q(\zeta) \,
dm_{\zeta}=0\, .$$
Since $\zeta \mapsto g(z, \zeta)$ has no integrable majorant independent of $z$, we cannot make profit of the fact that $z \mapsto g(z, \zeta)$ vanishes continuously on the boundary of ${{\mathbb{D}}}$. We need a more refined argument. For this we pick $\tau \in \partial {{\mathbb{D}}}$ and choose ${\varepsilon}>0$. Then we can find $0<
\varrho < 1/4$ such that for any $z\in {{\mathbb{D}}}\cap K_{\varrho/2}(\tau)$ $$\begin{aligned}
\displaystyle \iint \limits_{{{\mathbb{D}}}\cap K_{\varrho} (\tau)} \log \left| \frac{1-
\overline{\zeta}z}{z- \zeta}\right| \, dm_{\zeta} & \le & (\log 2)\, \pi \,
\varrho^2 + \iint \limits_{ K_{\varrho} (\tau)} \log \frac{1}{|z-
\zeta|}\, dm_{\zeta} \\[1mm]
\displaystyle & \le & (\log 2)\, \pi \,\varrho^2 + \iint \limits_{ K_{2
\varrho} (z)} \log \frac{1}{|z- \zeta|} \, dm_{\zeta} \le (\log 2)\, \pi
\,\varrho^2 +4\, \pi \, \varrho \le {\varepsilon}\, .\end{aligned}$$
If $z \in {{\mathbb{D}}}\cap K_{\varrho/2}(\tau)$ and $\zeta \in {{\mathbb{D}}}\backslash
K_{\varrho}( \tau)$ then $$\displaystyle{\left| \frac{1- \overline{\zeta}z}{z- \zeta} \right| \le
\frac{|1- \overline{\zeta} \tau| + | z- \tau|}{|\tau- \zeta|- |z -\tau|}
= \frac{1 + \frac{| z- \tau|}{|\tau- \zeta|} }{1- \frac{|z
-\tau|}{|\tau- \zeta|} } \le \frac{1 + \frac{| z- \tau|}{\varrho}
}{1- \frac{|z
-\tau|}{\varrho} } }\, .$$ This implies that $$\iint \limits_{{{\mathbb{D}}}\backslash K_{\varrho} (\tau)} \log \left| \frac{1-
\overline{\zeta}z}{z- \zeta}\right| \, dm_{\zeta} \le \pi \, \log
\left( \frac{\varrho + |z - \tau|}{\varrho -|z - \tau|}
\right) \le {\varepsilon}$$ if $z \in {{\mathbb{D}}}\cap K_{\varrho/2}(\tau)$ is sufficiently close to $\tau$. The result follows.
The next lemma provides an important equicontinuity property of Green’s function.
\[lem:equicontinuous\] Let $0< \varrho<1 $. Then for every $z_1,z_2 \in
\overline{{{\mathbb{D}}}}_{1-\varrho}$ with $|z_1-z_2|< \varrho/4$, $$\label{uniform}
\iint \limits_{{{\mathbb{D}}}}\, |g(z_1, \zeta)-g(z_2, \zeta)| \, dm_{\zeta} \le
\, \pi \, ( 5/\varrho+4 \, \varrho) \, |z_1-z_2|\, .$$
[**Proof.**]{} Let $z_1,z_2 \in \overline{{{\mathbb{D}}}}_{1- \varrho}$ such that $|z_1-z_2| < \varrho/4
$. We first observe that $$|g(z_2, \zeta)-g(z_1,\zeta)|
\le \displaystyle \left| \log \left| \frac{z_1- \zeta}{z_2 - \zeta}\right| \right| + \left| \log \left| \frac{1-\overline{\zeta}z_2}{1- \overline{\zeta}z_1} \right| \right| \displaystyle \le \left| \log\left| \frac{z_1- \zeta}{z_2 - \zeta}\right| \right| + \log
\left(1+\frac{|z_1-z_2|}{\varrho} \right) \, .
$$ Thus we have $$\iint \limits_{{{\mathbb{D}}}} |g(z_1, \zeta)-g(z_2,\zeta)| \, dm_{\zeta} \le \pi \,
\frac{|z_1-z_2|}{\varrho} + \iint \limits_{{{\mathbb{D}}}} \left| \log\left|
\frac{z_1- \zeta}{z_2 - \zeta}\right| \right| \, dm_{\zeta}\, .$$
Now let $M=(z_1+z_2)/2$ be the midpoint of the line segment joining $z_1$ and $z_2$ and let $K=K_{ \varrho /2}(M)$. Then we obtain $$\begin{array}{l}
\displaystyle{\iint \limits_{{{\mathbb{D}}}} \left| \log\left|
\frac{z_1- \zeta}{z_2 - \zeta}\right| \right| \, dm_{\zeta}} = \displaystyle{ \iint
\limits_{K } \left| \log\left|
\frac{z_1- \zeta}{z_2 - \zeta}\right| \right| \, dm_{\zeta} + \iint
\limits_{{{\mathbb{D}}}\backslash K } \left| \log\left|
\frac{z_1- \zeta}{z_2 - \zeta}\right| \right| \, dm_{\zeta} }\\[8mm]
\quad \le \displaystyle{ \iint
\limits_{K } \left[ \log \left( 1 + \frac{|z_2-z_1|}{|z_1 - \zeta|} \right) +
\log \left( 1 + \frac{|z_2-z_1|}{|z_2 - \zeta|} \right) \right] dm_{\zeta} + \iint
\limits_{{{\mathbb{D}}}\backslash K } \log \left( 1+ \frac{4\, |z_1 -z_2|}{\varrho}
\right) dm_{\zeta}}\\[8mm]
\quad \le \displaystyle{ \iint \limits_{K_{\varrho }(z_1)} \log \left( 1 +
\frac{|z_2-z_1|}{|z_1 - \zeta|} \right)\, dm_{\zeta}+ \iint\limits_{K_{\varrho}(z_2)}
\log \left( 1 + \frac{|z_2-z_1|}{|z_2 - \zeta|} \right) \, dm_{\zeta}+
\frac{4}{\varrho} \, \pi |z_1-z_2| }\\[8mm]
\quad \displaystyle{\le 4\, \pi \, \varrho\, |z_1-z_2| \, + \, \frac{4}{\varrho} \,
\pi |z_1-z_2| }\, . \hfill \blacksquare
\end{array}$$
We are now in a position to give the
[**Proof of Theorem \[solgen\].**]{}
[**Uniqueness:**]{} Assume $u$ and $v$ are two solutions of the boundary value problem (\[eq:pdesing\]). Then $s(z):= \max\{0, u(z)-v(z)\}$ is a non–negative subharmonic function in ${{\mathbb{D}}}$ and $s(z)=0$ on $|z|=1$. By the maximum principle for subharmonic functions it follows that $u(z) \le v(z)$ for every $z \in {{\mathbb{D}}}$. Switching the r$\hat{\mbox{o}}$les of $u$ and $v$ gives $u(z)=v(z)$ for $z \in
{{\mathbb{D}}}$.
[**Existence:**]{} Suppose for a moment $u \in C(\overline{{{\mathbb{D}}}})
\cap C^2({{\mathbb{D}}})$ is a solution to (\[eq:pdesing\]). Then Theorem \[thm:Gilbarg\] implies that $$u(z)=h(z)-\frac{1}{2 \pi}\iint \limits_{{{\mathbb{D}}}} g(z,\zeta) \,
e^{2 u(\zeta)} \, dm_{\zeta}\,, \qquad z \in {{\mathbb{D}}}\, ,$$ where $h$ is continuous on $\overline{{{\mathbb{D}}}}$, harmonic in ${{\mathbb{D}}}$ and coincides with $\psi$ on $\partial {{\mathbb{D}}}$.
This suggests to introduce the operator $$T[u](z):= h(z)-\frac{1}{2 \pi}\iint \limits_{{{\mathbb{D}}}} g(z,\zeta)\,
e^{2 u(\zeta)} \, dm_{\zeta},$$ and to apply Schauder’s fixed point theorem.
To set the stage for Schauder’s theorem, let $X$ be the Fréchet space of all real–valued continuous functions in ${{\mathbb{D}}}$ equipped with the (metriziable) compact–open topology, and let $$M:=\{ u\in X \, : \, m \le u(z) \le h(z) \, \, \text{for all}\, z \in {{\mathbb{D}}}\}\,, \text{ where } \, m:=\inf_{z \in {{\mathbb{D}}}} T[h](z)\, .$$ Note that $m>-\infty$. In order to be able to apply Schauder’s fixed point theorem we need to check the following properties:
The set $M$ is closed and convex (in $X$). The operator $T : M \to X$ is continuous, maps $M$ into $M$, and $T(M)$ is precompact.
Clearly, $M$ is closed and convex. Next, we will prove $T[M]$ is precompact, by showing $T[M]$ is a locally equicontinuous family and the set $\{Tu(z) \, : \, u \in M , \, z \in {{\mathbb{D}}}\}$ is bounded in the reals. For that pick $0< \varrho <1$, let $B_{1-\varrho}:= \{ z \in {{\mathbb{C}}}: |z| \le 1- \varrho \}$ and fix ${\varepsilon}>0$. Since $h$ is continuous on $\overline{{{\mathbb{D}}}}$ there exists a constant $\delta'>0$ such that $|h(z_1)-h(z_2)|<{\varepsilon}/2$ for all $z_1, z_2 \in
{{\mathbb{D}}}$ with $|z_1-z_2|<\delta'$. We now define $$\delta:=\min \left\{ \delta', \frac{ {\varepsilon}}{ C \left( 5/\varrho +4
\varrho \right)}, \frac{\varrho}{4} \right\} \, ,$$ where $C:= \sup_{\zeta \in {{\mathbb{D}}}} e^{2 h(\zeta)}$. Then by Lemma \[lem:equicontinuous\] we have for all $z_1,z_2 \in B_{1-\varrho}$ with $|z_1-z_2|<\delta$ and for every $u \in M$: $$\begin{aligned}
|T[u](z_2)-T[u](z_1)| & \le &
|h(z_2)-h(z_1)|+\frac{1}{2 \pi} \iint \limits_{{{\mathbb{D}}}} \left|
g(z_2,\zeta)-g(z_1,\zeta)\right| \, e^{2 u(\zeta)} \,
dm_{\zeta}\\
& \le & |h(z_2)-h(z_1)|+ \frac{1}{2 \pi} \, C \iint \limits_{{{\mathbb{D}}}} \left|
g(z_2,\zeta)-g(z_1,\zeta)\right| \,
dm_{\zeta}\\
& \le & \frac{{\varepsilon}}{2}+ \frac{1}{2 \pi} \, C\, \pi (5/\varrho + 4\, \varrho)
\cdot \delta \le {\varepsilon}.\end{aligned}$$ Thus $T[M]$ is a locally equicontinuous set of functions on ${{\mathbb{D}}}$. Moreover, for all $u \in M$ and all $z \in \overline{{{\mathbb{D}}}}$ $$\label{oli}
T[h](z) \le T[u](z) \le h(z) \, ,
$$ which implies $$\min \limits_{\zeta \in \overline{{{\mathbb{D}}}}} T[h](\zeta) \le T[u](z) \le
\max \limits_{\zeta \in \overline{{{\mathbb{D}}}}} |h(\zeta)| \quad \text{for every} \, u
\in M\, \, \text{and for all} \, z \in \overline{{{\mathbb{D}}}}\,
.$$ This shows $\{Tu(z) \, : \, u \in M, \, z \in {{\mathbb{D}}}\}$ is bounded, so $T[M]$ is a precompact subset of $X$. Note, estimate (\[oli\]) also gives $T[M] \subseteq M$.
It remains to prove that $T : M \to M$ is continuous. Let $(u_k)$ be a sequence of functions in $M$ which converges locally uniformly in ${{\mathbb{D}}}$ to $u \in M$. We have to show that the sequence $(T[u_k])$ converges locally uniformly in ${{\mathbb{D}}}$ to $T[u]$.
To do this, choose $0<\varrho<1$ and fix ${\varepsilon}>0$. We shall deduce that $|T[u_k](z)-T[u](z)|<{\varepsilon}$ for all $z \in B_{1-\varrho}$ and all $k \ge \tilde{k}$ for some $\tilde{k}$ independent of $z$.
For notational simplicity we set
$$C_1:= \frac{1}{ \pi} \sup \limits_{\zeta \in {{\mathbb{D}}}} e^{2
h(\zeta)}\, , \qquad C_2 := \sup_{z \in {{\mathbb{D}}}}\iint \limits_{{{\mathbb{D}}}} g(z, \zeta) \, dm_{\zeta}\, , \qquad
C_3 := 2 \sup \limits_{\zeta \in {{\mathbb{D}}}} |h(\zeta)|,$$
Now we choose $0<r <\varrho/2$ such that $$\label{eq:ex1}
C_1\, C_3 \, \log \left(\frac{2}{r} \right) \, \pi\, (2r -r^2) < \frac{{\varepsilon}}{2}\, .$$ Further, we can find an index $\tilde{k} \in {{\mathbb{N}}}$ such that $$\label{eq:ex2}
\sup_{z \in B} \left|u_k(z)-u(z) \right| <\frac{{\varepsilon}}{2 C_1 C_2}, \qquad
k \ge \tilde{k}\, ,$$ where $B:= B_{1-r}$. We now obtain by equations (\[eq:ex1\]) and (\[eq:ex2\]) for $z \in B_{1-\varrho}$
$$\begin{array}{l}
\displaystyle\left| T[u_k](z)-T[u](z) \right| = \left| \frac{1}{2\pi} \iint \limits_{{{\mathbb{D}}}}g(z,
\zeta) \, (e^{2u_k(\zeta)} -e^{2u(\zeta)}) dm_{\zeta}\right|\\[6mm]
\qquad \displaystyle\le \frac{1}{ \pi} \, \sup \limits_{\zeta \in \overline{{{\mathbb{D}}}}} e^{2
h(\zeta)}\cdot \iint \limits_{{{\mathbb{D}}}} g(z,\zeta) \left|
u(\zeta)-u_k(\zeta) \right| \, dm_{\zeta} \\[2mm]
\qquad \displaystyle \le C_1 \sup_{z \in B} \left|u_k(z)-u(z) \right| \iint \limits_{{{\mathbb{D}}}} g(z,
\zeta) \, dm_{\zeta} \, + \, C_1 \, 2 \, \sup_{\zeta \in {{\mathbb{D}}}} |h(\zeta)| \iint
\limits_{{{\mathbb{D}}}\backslash B} g(z, \zeta) \, dm_{\zeta} \\
\qquad \displaystyle \le \frac{{\varepsilon}}{2} + C_1 \, C_3 \iint \limits_{{{\mathbb{D}}}\backslash B}
\log\left(\frac{2}{r} \right) \,
dm_{\zeta}< {\varepsilon}\, .
\end{array}$$
Thus $T:M \to M$ is continuous.
Now Schauder’s fixed point theorem gives us a fixed point $u \in M$ of $T$, i.e., $$\label{eq:fixedpoint}
u(z)=T[u](z)=h(z)-\frac{1}{2 \pi} \iint \limits_{{{\mathbb{D}}}} g(z,\zeta) \, e^{2
u(\zeta)} \, dm_{\zeta}.$$ We claim $u$ belongs to $C(\overline{{{\mathbb{D}}}})\cap C^2({{\mathbb{D}}})$ and is a solution of (\[eq:pdesing\]).
Indeed, since $\zeta \mapsto e^{2 u(\zeta)}$ is bounded and continuous in ${{\mathbb{D}}}$, the function $u$ belongs to $
C(\overline{{{\mathbb{D}}}})\cap C^1({{\mathbb{D}}})$ by Theorem \[thm:Gilbarg\]. This implies that the function $\zeta \mapsto e^{2 u(\zeta)}$ belongs to $C^1({{\mathbb{D}}})$ and applying Theorem \[thm:Gilbarg\] again proves that $u \in C(\overline{{{\mathbb{D}}}}) \cap
C^2({{\mathbb{D}}})$ solves (\[eq:pdesing\]).
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S. Yamashita, Sur allures de la densité de Poincaré et ses dérivées au voisinage d’un point frontieré, [*Kodai Math. J*]{}. (1993), [**16**]{}, 235–243.
[^1]: Throughout, all paths are assumed to be continuous and piecewise continuously differentiable.
[^2]: “SK is intended to convey curvature subordinate to $-1$”, see [@Hei62].
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---
author:
- |
Pak Hin Li[^1], Nicholas Ryder[^2],\
Robert S. Strichartz[^3], Baris Evren Ugurcan[^4]
title: Extensions and their Minimizations on the Sierpinski Gasket
---
Introduction
============
In classical analysis one often wants to study finite variants of the Whitney extension theorem, in which data at a finite set of points in Euclidean space (or domains in Euclidean space or Riemannian manifolds) is prescribed, such as values and certain derivatives of a function. The problem is to find a function with the prescribed data that minimizes (or comes close to minimizing) some prescribed norm. There are many such problems depending on the nature of the data and the chosen norm. We give just a few references to recent work in this area ([@F], [@F2], [@FI]).\
The purpose of this paper is to initiate the study of analogous problems on fractals. For the most part we restrict attention to the “poster child” of all fractals, the Sierpinski gasket (SG). Some of our results are “generic” and extend easily to Kigami’s class of post-critically finite (PCF) fractals [@Ki][@Str]. But we also include a number of results that deal specifically with the geometry of $SG$. We are interested in minimizing two Sobolev types of norms. The first, treated in section 2, is ${\mathcal{E}}(u) + \lambda \int |u|^2 d \mu$ for some fixed $\lambda \geq 0$, where ${\mathcal{E}}$ denotes the standard self-similar energy on $SG$. For $\lambda = 0$ this amounts to minimizing energy, while for $\lambda>0$ we are minimizing the analog of the $H^1$ Sobolev norm in Euclidean space. For this minimization we prescribe values of the function on a fixed finite set $E$. We prove existence and uniqueness of a minimizer, and show how to express the solution in terms of the resolvent of the Laplacian, which was studied in detail in [@resolvent]. In the case of $\lambda =0$ the minimizers are analogs of continuous piecewise linear functions on the line, and in fact are continuous piecewise harmonic functions on the complement of $E$. The energy of the minimizer is expressible as $${\mathcal{E}}(u) = \sum_{\{x,y\}\subset E}c_{x,y}(u(x)- u(y))^2$$ for certain “conductance” coefficients $c_{x,y}$ depending on the set $E$. In the case when the points in $E$ are all junction points in $SG$, we give estimates for the size of the coefficients, and characterize the pairs of points with $c_{x,y} = 0$ in terms of path connectedness properties of $E$. We study in detail three specific families of sets $E$ for which we compute the coefficients explicitly. The most challenging of those we call the “bottom row”. For fixed $m$, we take $E$ to be the equally spaced $2^m + 1$ points along the bottom line in $SG$. The results we obtain show that this is a good discrete approximation of the continuous problem of computing the energy of functions that are harmonic in the complement of the bottom line that was studied in [@OS].\
The second type of Sobolev norm, considered in section 3, is $$T_{\zeta}(u) = \int |\Delta_\zeta u|^2 d \zeta$$ where $\zeta$ is any finite positive continuous measure on $SG$ and $\Delta_\zeta$ is the associated Laplacian ([@Ki],[@Str]). We are most interested in the cases $\zeta = \mu,$ the standard self-similar measure, and $\zeta = \nu$, the Kusuoka measure. This norm is the analog of the Sobolev $H^2$ norm on the line. Here we consider two types of data: prescribing function values on a finite set $E$, or prescribing those values and also the values of normal derivatives at a set $F$ of junction points $(F=E$ is most interesting). For the first problem we prove existence and uniqueness of the minimizer under the additional geometric hypothesis that the set $E$ is not contained in a straight line in the harmonic coordinates realization of $SG$. A simple counterexample shows that some such hypothesis is needed for uniqueness. We obtain a similar existence and uniqueness theorem for the second type of data. For the analogous problem on the line, the minimizers are $C^1$ piecewise cubic splines, so it is not surprising that on $SG$ the minimizers are piecewise biharmonic functions (solutions of $\Delta_\zeta^2u = 0$). In the case where $\zeta = \mu$ and $E$ consists of all junction points of a fixed level, we are able to give an explicit local formula for $T_\mu(u)$ in terms of the data. If we consider the minimization of $T_\mu(u)$ with just the values prescribed on all junction points of a fixed level, the problem is no longer local. In this case we are able to give an explicit linear formula relating the discrete Laplacian and the continuous Laplacian for minimizers. In principle all that is needed to obtain an explicit solution is to solve these equations, but we do not have any insight into the form of the solution. In a similar vein, we are able to express the minimizer for any data set $E$ in terms of integrals of the Green’s function, but again this involves solving a certain system of linear equations.\
\
We mention a couple of interesting open problems relating to this work:
1\) Minimize the functional $\|\Delta u\|_\infty$ with prescribed values and normal derivatives. Of course the minimizer will not be unique. The analogous problem on the line is local and was solved in [@G] in terms of piecewise quadratic functions with one additional cut point in each interval between points in $E$. On $SG$ the analog of quadratic functions is solutions of $\Delta u = constant$. It is not at all clear what “piecewise” should mean in this context.
2\) Study analogous problems on non-PCF fractals, such as the Sierpinski carpet or products of $SG$.
Minimizing Generalizations of Energy
====================================
In this section we start with a straightforward example which sets up many methods used later. We are concerned with the space ${\text{dom}}{\mathcal{E}}$.
We consider the functional ${\mathcal{E}}_\lambda: u \mapsto \mathcal{E}(u) + \lambda \int u^2 d \zeta$ defined on ${\text{dom}}{\mathcal{E}}$ for $\lambda \geq 0$. Notice that given $u \in {\text{dom}}\mathcal{E}$ we have that $u$ is continuous and thus is bounded in this space so ${\mathcal{E}}_\lambda(u) < \infty$.
We apply the following well known theorem:
\[hilbertconvexminimum\] If $X$ is a non-empty closed convex subset of a Hilbert space $H$, there exists a unique point $y \in X$ that minimizes the norm of all points in $X$.
We first establish the following fact which we use throughout the paper.
\[lemma:supenergy\] Given $u \in {\text{dom}}{\mathcal{E}}$, if $u(q) = 0$ for some $q \in SG$, then $$\label{equation:supenergy}
\|u\|^2_\infty \leq C \mathcal{E}(u).$$
We first note the following fact (check [@Str] for reference) $$|u(x) - u(y)| \leq \mathcal{E}(u)^{1/2}R(x,y)^{1/2}$$ where $R(x,y)$ denotes the resistance metric. Since $SG$ is compact with respect to effective resistance metric, this implies $$\|u\|^2_\infty \leq C \mathcal{E}(u).$$
Now we set $X = \{ u \in {\text{dom}}{\mathcal{E}}~|~u(x_i) = a_i\}$ for all $i$ where $\{x_1, \dots, x_n\}$ is any set in $SG$ . We consider the Hilbert space ${\text{dom}}{\mathcal{E}}/ \text{const.}$ and the projection $\tilde{X}$ of X to this space.
For $\lambda = 0$, $\tilde{X}$ is closed and convex in $ {\text{dom}}{\mathcal{E}}/ \text{const.}$.
The convexity follows immediately. We consider a sequence $u_n \to u$ with $u_n \in \tilde{X}$. We pick representatives $\widetilde{u_n}$ of $u_n$ such that $\widetilde{u_n} \in X$ and $\tilde{u}$ of $u$ such that $\tilde{u}(x_1) = a_1$. We can apply Lemma \[lemma:supenergy\] to $\tilde{u} - \widetilde{u_n}$. Since ${\mathcal{E}}(\tilde{u} - \widetilde{u_n}) \to 0$ we get $\|\tilde{u}-\widetilde{u_n}\|_\infty \to 0$. Thus $\tilde{u} \in X$ so $\widetilde{X}$ is closed.
For $\lambda > 0$, $X$ is closed and convex in the Hilbert space $({\text{dom}}{\mathcal{E}}, {\mathcal{E}}_\lambda)$.
The convexity follows immediately.\
We consider a sequence $u_n \to u$ with $u_n \in X$. We set $v_n = u_n - u_n(x_1) = u_n - a_1$ and $v = u - u(x_1)$. Since ${\mathcal{E}}_\lambda(u_n - u) \to 0$ we get ${\mathcal{E}}(u_n - u) \to 0$. Applying Lemma \[lemma:supenergy\] to $v_n - v$ we get $\|v_n - v\|_\infty \to 0$.\
Since ${\mathcal{E}}_\lambda (u_n-u) \to 0$ we also have $\int (u_n - u)^2 d \zeta \to 0.$ $$\begin{aligned}
\int(u_n - u)^2 d \zeta &= \int ((v_n - v) + (a_1 - u(x_1)))^2 d \zeta \\
&= \int (v_n - v)^2 d \zeta + 2(a_1 - u(x_1))\int(v_n - v) d \zeta + \int (a_1 - u(x_1))^2 d \zeta .\\\end{aligned}$$ Since $\|v_n - v\|_\infty \to 0$, the above implies $a_1 = u(x_1)$. Thus we can use Lemma \[lemma:supenergy\] to $u_n - u$ to see $\|u_n - u\|_\infty \to 0$, so $X$ is closed.
Thus we get both existence and uniqueness of the minimum.
Construction
------------
To find an explicit construction, we use the functions $G_{\lambda} $ constructed in [@resolvent]. Thus we restrict our attention to the case where we have Dirichlet boundary conditions and we only consider the standard measure. We use the notation ${\text{dom}}{\mathcal{E}}_0 = \{u \in {\text{dom}}{\mathcal{E}}~|~ u|_{V_0} \equiv 0\}$.\
First we establish a necessary condition that parallels the Euler Lagrange equations in smooth analysis.
\[lemma:eulerlagrange\] Suppose that we have $u$ which minimizes ${\mathcal{E}}_\lambda$. Then we get$$\label{equation:eulerlagrange}{\mathcal{E}}(u,v) = -\lambda \int u v d\mu$$ for all $v \in {\text{dom}}{\mathcal{E}}_0$ with $v|_E \equiv 0.$
Suppose $u$ minimizes ${\mathcal{E}}_\lambda$ with respect to the constraints. Let $v \in {\text{dom}}{{\mathcal{E}}}_0$ with $v|_E \equiv 0.$ Given $t \in {\mathds{R}}$ we have $u+tv \in \text{dom}{\mathcal{E}}_0$ with $u+tv|_E \equiv u|_E$. We compute $${\mathcal{E}}_\lambda(u+tv) = \mathcal{E}(u+tv,u+tv) + \lambda \int (u+tv)^2 d \mu$$ $$= \mathcal{E}(u) + 2 t {\mathcal{E}}(u,v) + t^2 {\mathcal{E}}(v) + \lambda\left(\int u^2 d\mu + 2 t \int u v d \mu + t^2 \int v^2 d \mu\right).$$ Since $u$ is a minimizer, if we view the function $f(t)={\mathcal{E}}_\lambda(u+tv)$, we must have $t = 0$ at a minimum. Thus we apply single variable calculus to show the stated result.
Now we use the functions $G_{\lambda}$ constructed in $\cite{resolvent}.$ A quick calculation shows that $G_{\lambda}$ satisfies (\[equation:eulerlagrange\]).
\[lemma:construction\] (Construction) Given a set $E = \{ x_1, x_2, \dots, x_n \} \subset SG$ and $a_i$ then we can guarantee unique $c_i$ such that $f(x) = \sum_i c_i G_\lambda(x,x_i)$ satisfies $f(x_i) = a_i.$ Furthermore this function is the unique minimizer of ${\mathcal{E}}_\lambda.$
We have a necessary condition for the minimizer. We show that there exists a unique function satisfying this neccessary conditions. Then the existence of a minimizer guarantees this function is in fact the minimizer. First we show uniqueness. Suppose $u_1, u_2$ both satisfy (\[equation:eulerlagrange\]), then we get $${\mathcal{E}}(u_1 - u_2) = -\lambda \int (u_1 - u_2)^2 d\mu.$$ Thus we have ${\mathcal{E}}(u_1 - u_2) = 0$ which means they must differ by a harmonic function, but the Dirichlet boundary conditions guarantee that this harmonic function is identically zero so $u_1 = u_2$.\
To show existence, we set up the following linear system: Let $G$ be defined by $[G]_{i,j} = G_\lambda(x_j,x_i).$ Now setting a vector of $c_i$ and $a_i$ we obtain the system $$G c = a.$$ This is a linear map from equal dimension vector spaces. We look at the kernel of this map. If we have $G y = 0$ then we know $\tilde{f}(x) = \sum y_i G_\lambda(x,x_i)$ with $\tilde{f}(x_i) = 0$. Note that this function satisfies (\[equation:eulerlagrange\]) for all appropriate $v$ so we can apply uniqueness to show $\tilde{f} \equiv 0$ since the constant zero function also satisfies (\[equation:eulerlagrange\]). By Fundamental Theorem of Linear Algebra, injectivity implies surjectivity so we can guarantee the desired $c_i$.\
Thus we have found the unique function satisfying (\[equation:eulerlagrange\]) which therefore is the minimizer.
Now we compute the value of this minimizer: $${\mathcal{E}}_\lambda(u) = \sum_{i,j} c_i c_j {\mathcal{E}}_\lambda(G_\lambda(x,x_i),G_\lambda(x,x_j))$$ $$= \sum_{i,j} c_i c_j G_\lambda(x_i,x_j) = \sum_i c_i a_i$$ $$= a \cdot c = G^{-1}a \cdot a.$$
In fact, we can construct the minimizer when we do not consider the Dirichlet boundary conditions. We sketch the construction here:\
For $\lambda \ge 0$, $\lambda$ is not a Dirichlet eigenvalue so the space of eigenfunctions is always three dimensional and it is completely determined by the boundary values on $V_0$. We denote the eigenfunctions by $u_i$ with $u_i(q_i)=1, u_i(q_{i-1})=0, u_i(q_{i+1})=0$ and $\Delta u_i=\lambda u_i$.\
In the case $E\subset \text{SG}\setminus V_0$, let $u=\sum_it_iu_i(x)+\sum_j c_j G_{\lambda}(x_j,x)$. For all $v\in \text{dom} {\mathcal{E}}$ with $v|_E=0$, we have ${\mathcal{E}}_{\lambda}(u,v)=\sum_{k}v(q_k)\left(\sum_i t_i \partial_n u_i(q_k)+\sum_j c_j \partial_n G_{\lambda}(q_k,x_j)\right)$. Now suppose that there exist $t_i,c_j$ such that $u=\sum_it_iu_i(x)+\sum_j c_j G_{\lambda}(x_j,x)$, $u(x_i)=0$ and $\partial_nu(q_k)=0$. Then we have ${\mathcal{E}}_{\lambda}(u,v)=0$ for all $v$ with $v|_E=0$. This would imply that $u=0$.\
Given $a_i$, we can always construct a function $u$ such that $u(x)=\sum_it_iu_i(x)+\sum_j c_j G_{\lambda}(x_j,x)$, $u(x_i)=a_i$ and also $\sum_i t_i \partial_n u_i(q_k)+\sum_j c_j \partial_n G_{\lambda}(q_k,x_j)=0$ for $k=0,1,2$. Then from the uniqueness property we can guarantee that this $u$ is the unique minimizer.
Arbitrary Sets of Junction Points
---------------------------------
Now we restrict our attention to arbitrary sets $E \subset V_m$ and the standard energy i.e. $\lambda=0$. We are guaranteed that given the minimizer $u$ we have $${\mathcal{E}}(u) = \sum_{\{x,y\}\subset E}{c_{x,y}(u(x)-u(y))^2}.$$ We seek some basic properties for the coefficients $c_{x,y}$, which may be interpreted as conductances regarding $E$ as an electrical resistance network.
### Notation
Throughout this section we consider different minimizing sets $\Gamma' \subset \Gamma \subset V_m$.\
Given a minimizing set $\Gamma$, we let $c^\Gamma_{x,y}$ denote the coefficient for $(u(x)-u(y))^2$ in the minimizing form. If there is no ambiguity about which set we are looking at, we will use the shorthand $c^\Gamma_{x,y} = c_{x,y}$.
We define an electrical path in $\Gamma' \subset V_m$ to be a sequence of points $x_1 \to ... \to x_n$ where $x_i \in \Gamma'$ and $c^{\Gamma'}_{x_i,x_{i+1}} > 0$
### Estimates on Restricted Coefficients
\[lemma:subgraph\] Given any $\Gamma' \subset \Gamma \subset V_m$ and points $x, y \in \Gamma' \subset \Gamma$, then $c^{\Gamma'}_{x,y} \geq c^{\Gamma}_{x,y}$.
It suffices to show this for the case where $\Gamma'$ and $\Gamma$ differ by a point. Let $\Gamma = \{x_1, ..., x_{n+1}\}$ and $\Gamma' = \{x_1, ..., x_n\}$ and let $u(x_i) = y_i$ so we have $$E(u) = \sum_{1 \leq i < j \leq n}{c_{i,j}(y_i - y_j)^2} + \sum_{i = 1}^{n}{c_{n+1,i}(y_i - y_{n+1})^2}.$$ We differentiate with respect to $y_{n+1}$ to find the conductances for the subgraph. $$\partial_{y_{n+1}} E(u) = \sum_{i=1}^{n}{2 c_{i,n+1} (y_{n+1} - y_i)} = 0~~\text{so}$$ $$y_{n+1} \sum_{i=1}^{n}{c_{i,n+1}}= \sum_{i=1}^{n}{c_{i,n+1} y_i}~~\text{so}$$ $$y_{n+1} = \frac{\sum_{i=1}^{n}{c_{i,n+1} y_i}}{ \sum_{i=1}^{n}{c_{i,n+1}}}.$$ Thus the energy for the subgraph is
$$E(u) = \sum_{1 \leq i < j \leq n}{c_{i,j}(y_i - y_j)^2} + \sum_{i = 1}^{n}{c_{n+1,i}(y_i - \frac{\sum_{j=1}^{n}{c_{j,n+1} y_j}}{ \sum_{j=1}^{n}{c_{j,n+1}}})^2}.$$ To simplify notation let the points be $x_1$ and $x_2$. We want to show the conductance increases. We have $$c^{\Gamma'}_{x_1, x_2} = \frac{-1}{2}\partial_{y_1y_2}E(u)$$ $$= c_{1,2} + c_{1,n+1}\left(1-\frac{c_{1,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\right)\left(\frac{c_{2,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\right) + c_{2,n+1}\left(1-\frac{c_{2,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\right)\left(\frac{c_{1,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\right)$$ $$- \sum_{j=3}^{n}\left(c_{j,n+1}\frac{c_{1,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\frac{c_{2,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}\right)$$ $$= c_{1,2} + \frac{c_{1,n+1}c_{2,n+1}}{(\sum_{i=1}^{n}{c_{i,n+1}})^2}\left(\sum_{i=1}^{n}{c_{i,n+1}} - c_{1,n+1} + \sum_{i=1}^{n}{c_{i,n+1}} - c_{2, n+1} - \sum_{i=3}^{n}{c_{i,n+1}}\right)$$ $$= c_{1,2} + \frac{c_{1,n+1}c_{2,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}.$$ Thus we obtain $$\label{equation:conductance}
c^{\Gamma'}_{x_1, x_2} = c_{1,2} + \frac{c_{1,n+1}c_{2,n+1}}{\sum_{i=1}^{n}{c_{i,n+1}}}.$$
\[corollary: neighbors\] Given any subset $A\subseteq V_m$, if $ x,y \in A$ such that $x \underset{m}{\sim} y$, then $$(\frac{5}{3})^m \leq c^A_{x,y} \leq R(x,y)^{-1} \leq 4(\frac{5}{3})^m$$where $R(x,y)$ denotes the effective resistance between $x$ and $y.$
We have that $\Gamma_{x,y} \subset A \subset V_m$ where $\Gamma_{x,y} = \{x, y\}$ is the restriction to just $x$ and $y$. On $V_m$, by Lemma \[lemma:subgraph\], we have $$(\frac{5}{3})^m = c^{V_m}_{x,y} \leq c^A_{x,y}.$$
We also have $$c^A_{x,y}\leq c^{\Gamma_{x,y}}_{x,y} = R(x,y)^{-1}$$ where $R(x,y)$ is the effective resistance metric so we have $R(x,y)^{-1} \leq 4(\frac{5}{3})^m$ from (1.6.6) in [@Str] since $x \underset{m}{\sim} y$.
### Zero Coefficients
Throughout this section we let $\Gamma = V_m$ be our ambient space. We set out to prove the following theorem:
\[zerocoefficients\]
Given $x, y \in \Gamma' \subset \Gamma,$ $c^{\Gamma'}_{x,y} = 0$ if and only if every path in $\Gamma$ from $x$ to $y$ intersects $\Gamma'$.
If $x, y \in \Gamma' \subset \Gamma$ and every path from $x$ to $y$ intersects $\Gamma'$, then $c^{\Gamma'}_{x,y} = 0$.
Let $\Gamma' = \{x,y, x_1, x_2, x_3, \dots, x_n \}$. We define the propogation set of y as $$P(y): = \{g \in \Gamma~|~ \text{there is a path P from } y \text{ to } g \text{ with } P \cap \Gamma' = \{y\} ~\}~.$$ Now we partition the edges as follows: $$F = \{\{a,b\} ~|~ a \sim b \text{ with one of } a \text{ or } b \text{ in } P(y) \}$$ $$G = \{\{a,b\} ~|~ a \sim b \text{ with } a,b \notin P(y) \}~.$$ From the definition of $P(y)$ we immediately get the fact that given $e_F \in F, e_G \in G$ if $x \in e_F \cap e_G$ then $x \in \Gamma'$. Thus we have for each $z \in \Gamma \setminus \Gamma'$ the edges containing $z$ are contained either all in $F$ or all in $G$. Thus we get a partition of $\Gamma \setminus\Gamma'$ into $$\Gamma'_F = \{x \in \Gamma \setminus\Gamma' | x \in z \text{ for some } z \in F \}$$ $$\Gamma'_G = \{x \in \Gamma \setminus\Gamma' | x \in z \text{ for some } z \in G \}.$$ Then we can express the energy as $${\mathcal{E}}(u) =\left( \frac{5}{3} \right)^m \left( \sum_{F} (u(a) - u(b))^2 + \sum_{G} (u(m) - u(n))^2 \right)~.$$
Therefore, since we have partition of $\Gamma \setminus \Gamma'$ we can minimize each sum individually to get functions ${\mathcal{E}}_F$ and ${\mathcal{E}}_G$ on $\Gamma'$ with $${\mathcal{E}}(u) = {\mathcal{E}}_F + {\mathcal{E}}_G$$ where ${\mathcal{E}}_F$ depends on $u(y)$, not on $u(x)$ and ${\mathcal{E}}_G$ depends on $u(x)$ not on $u(y)$. Hence, $\partial_{u(x)u(y)} {\mathcal{E}}(u) = 0$. So, $c_{x,y}^{\Gamma'} = 0$.
To prove the converse of this, we will use a reduction argument to deal with the situation where $\Gamma'$ differs from $\Gamma$ by one point. We establish the appropriate machinery to make this reduction:
\[lemma:gamma\] Given $\Gamma = \{x_1, x_2, \dots, x_n, x_{n+1} \}$ and $\Gamma' = \{ x_1, x_2, \dots, x_n\}$, suppose that $c_{1, n+1}^{\Gamma}> 0$ and $c_{n+1, 2}^{\Gamma}>0$. Then, $c_{1, 2}^{\Gamma'} >0$.
This follows from (\[equation:conductance\]), since $$c_{1, 2}^{\Gamma'} = c_{1, 2}^{\Gamma} + \frac{c_{1,n+1}^{\Gamma} c_{2,n+1}^{\Gamma}}{\sum_{j=1}^{n} c_{j,n+1}^{\Gamma}} >0~.$$
\[lemma: electrical path\] Consider an electrical path $x_{i_0} \rightarrow x_{i_1} \rightarrow x_{i_2} \rightarrow x_{i_3} \rightarrow \dots \rightarrow x_{i_n}$ and $p \in \Gamma \setminus \{x_{i_0}, x_{i_n}\}$. Denote $\Gamma'$ the restricted graph for $\Gamma \setminus \{p\}$. If $p$ is not on the path, then we have an electrical path of the same length and with the same points. If $p=x_{i_j}$ is on the path, then we have shorter electrical path in $\Gamma'$, $x_{i_0} \rightarrow x_{i_1} \rightarrow x_{i_2} \rightarrow \dots \rightarrow x_{i_{j-1}} \rightarrow x_{i_{j+1}} \rightarrow \dots \rightarrow x_{i_n}$.
If $p$ is not in the path then we know from Lemma \[lemma:subgraph\] that $c^{\Gamma'}_{i_j,i_{j+1}} \geq c^{\Gamma}_{i_j,i_{j+1}} > 0$ so the same points form an electrical path.\
If $p = x_{i_j}$ is on the path then it follows from the above observation that we only need to show $c^{\Gamma'}_{i_{j-1},i_{j+1}} > 0$. Since $c^{\Gamma}_{i_{j-1},i_{j}}, c^{\Gamma}_{i_j,i_{j+1}} > 0$ by Lemma \[lemma:gamma\] we have $c^{\Gamma'}_{i_{j-1},i_{j+1}} > 0$. We therefore have a shorter path $x_{i_0} \rightarrow x_{i_1} \rightarrow x_{i_2} \rightarrow \dots \rightarrow x_{i_{j-1}} \rightarrow x_{i_{j+1}} \rightarrow \dots \rightarrow x_{i_n}$.
With this tool, we can easily make the reduction and use our previous calculation to show the converse:
Let $\Gamma' = \{x_1, x_2, \dots, x_n \}$ and $\Gamma = \{x_1, x_2, \dots, x_n, x_{n+1}, x_{n+2}, \dots, x_{n+k} \}$. Let $x,y$ be two distinct points in $\Gamma'$. Suppose there is a path from $x$ to $y$ such that the path does not intersect any other points in $\Gamma'$. Then $c_{x,y}^{\Gamma'}>0$.
We prove here a stronger result, namely that this holds for all electrical paths satisfying our criteria. By Corollary \[corollary: neighbors\], we know every path is an electrical path, so it suffices to show this.\
For notation we set $x=x_{i_0}$ and $y=x_{i_t}$. Let $x_{i_0} \rightarrow x_{i_1} \rightarrow x_{i_2} \rightarrow \dots \rightarrow x_{i_t}$ be a path connecting $x_{i_0}$ and $x_{i_t}$ where all $x_{i_j}$ are in $\Gamma \setminus \Gamma'$,where $1\le j \le t-1$. Define $\Gamma^{(j)}=\{x_1, x_2, \dots, x_n, x_{n+1},\dots,x_{n+j} \}$. Then from Lemma \[lemma: electrical path\], we can inductively get an electrical path from $x$ to $y$ which has length less than or equal to $j+1$ for the restricted graph $\Gamma^{(j)}.$\
We have $c_{x,y}^{\Gamma'} \ge c_{x,y}^{\Gamma^{(1)}}$. As a result, we only need to concentrate on the case $\Gamma^{(1)}.$ This case follows directly from Lemma \[lemma:gamma\].
### Lower bounds
Let $x_0\rightarrow x_1 \rightarrow x_2 \rightarrow \dots \rightarrow x_n \rightarrow x_{n+1} $ be a path in $V_m$ such that $x_i \neq x_j$ for all $i \neq j$ and $x_i \underset{m}{\sim} x_j$ if and only if $|i-j|=1$. Define $V_m^0 = V_m$ and $V_m^i = V_m \setminus \{x_1, \dots ,x_i\}~.$ We use the notation $c^i_{z_1,z_2} = c^{V_m^i}_{z_1,z_2}~$, and let $$N_i= \# \{y\in V_m^{i-1}~|~c_{y,x_i}^{i-1}>0\}~,$$
$$M_i=\max \{c_{y,x_i}^{{i-1}}~|~ y\in V_m^{i-1}, c_{y,x_i}^{i-1} >0\},$$
$$m_i=\min \{c_{y,x_i}^{{i-1}}~|~ y\in V_m^{i-1}, c_{y,x_i}^{{i-1}} >0\}~.$$ Then we obtain the relationships $$1 \le N_i \le i+3,$$ $$\left(\frac{5}{3}\right)^m\frac{m_i}{N_i M_i}\le m_{i+1},$$ $$4\left(\frac{5}{3}\right)^m\left(1+\frac{M_i}{N_i m_i}\right) \ge M_{i+1}.$$
Let $y$ be a point such that $c_{y,x_i}^{{i-1}} > 0$.\
We claim that either $c_{y,x_{i+1}}^{{i-1}} = 0$ or $y \underset{m}{\sim} x_{i+1}$. To see this suppose $c_{y,x_{i+1}}^{{i-1}} > 0$ and they are not neighbors. Then we know we have a path from $y$ to $x_{i+1}$ where the interior of the path is contained in $\{x_1, \dots, x_{i-1}\}$ by Theorem \[zerocoefficients\]. Now we know the second point in the path and $y$ are neighbors, implying that we have $x_j \underset{m}{\sim} x_{i+1}$ for some $j\leq i-1$, contradicting our choice of path. Thus either they are neighbors or $c_{y,x_{i+1}}^{{i-1}} = 0$.\
From (\[equation:conductance\]) we have:
$$\begin{aligned}
c_{y,x_{i+1}}^{{i}}=c_{y,x_{i+1}}^{{i-1}}+ \frac{c_{y,x_{i}}^{{i-1}}c_{x_{i+1},x_{i}}^{{i-1}}}{{\sum_{z\in V_m^i}c_{x_{i},z}^{{i-1}}}}~.\end{aligned}$$
It follows that $$\begin{aligned}
4\left( \frac{5}{3} \right)^m + \frac{4\left( \frac{5}{3} \right)^m M_i}{N_i m_i} \geq c_{y,x_{i+1}}^{i} \geq \frac{\left( \frac{5}{3} \right)^m m_i}{N_i M_i}.\end{aligned}$$
So we have $$\left(\frac{5}{3}\right)^m\frac{m_i}{N_i M_i}\le m_{i+1}$$ and $$4 \left(\frac{5}{3}\right)^m\left(1+\frac{M_i}{N_i m_i}\right) \ge M_{i+1}~.$$
To show $1 \le N_i \le i+3$, note that we obviously have $1 \le N_i$ by Corollary \[corollary: neighbors\]. By Lemma \[lemma:gamma\] and Corollary \[corollary: neighbors\] we obtain $N_{i+1} \le N_i +1$. By induction, it follows that $N_i \leq i+3$. Thus we get the stated bounds.
There is a sequence $a_N$ such that for arbitrary $m\in \mathbb{N}$ and arbitrary subset $A\subset V_m$, given any two points $x,y \in A$ with $d(x,y) = N$ we have $$a_N(\frac{5}{3})^m \le c_{x,y}^A$$ where $d(x,y)$ denotes the length of shortest path in $V_m$ from $x$ to $y$ without intersecting any other points in $A$.
Let us define two sequences $\{a_n\},\{b_n\}$ with $a_1=1$ and $b_1=1$. The two sequences satisfy the recursive relations $$a_{i+1}=\frac{a_i}{(i+3)b_i}$$ and $$b_{i+1}=4\left(1+\frac{b_i}{a_i}\right)~.$$ For two given points $x,y\in A \subset V_m$, let $x\rightarrow x_1 \rightarrow x_2 \rightarrow \dots \rightarrow x_{N-1} \rightarrow y$ be a shortest path connecting them without intersecting any other points in $A$. Then this path satisfies all the conditions of the previous lemma. It is easy see that $a_i\left(\frac{5}{3}\right)^m \le m_i$ and $M_i \le \left(\frac{5}{3}\right)^m b_i$. This follows from the previous lemma and the recursive definition of $a_n$ and $b_n$.\
So we have $a_N (\frac{5}{3})^m \le c_{x,y}^{N-1}\le c_{x,y}^A$ as desired.
Specific Sets
-------------
Now we turn to specific sets of interest in $V_m$.
### 2-Set
We define the following set as follows: $$\beta_0 = \{q_0, q_1, q_2, p_2 = F_2 F_1 q_0,p_1 = F_1 F_0 q_2,p_0= F_0 F_2 q_1 \},$$ $$\beta_m = \bigcup_{i}{F_i \beta_{m-1}}.$$
![$\beta_0$ and $\beta_1$[]{data-label="figure:2set"}](2_Set_Level_2_Graphic.pdf "fig:") ![$\beta_0$ and $\beta_1$[]{data-label="figure:2set"}](2_Set_Level_3_Graphic.pdf "fig:")
A direct computation yields the following.
For $\beta_0$, the coefficients for $i \neq j$ are $$c_{q_i,p_i}=410/159~,$$ $$c_{q_i,q_{j}}=5/53~,$$ $$c_{q_i, p_{j}} = 20/53,$$ $$c_{p_i,p_{j}}=80/53~.$$
We consider level $m>0$ and we let $\tilde{c}$ be the coefficients established for $\beta_0$.\
If $x$ and $y$ are in different $m$ cells, we have $$c_{x,y} = 0 ~.$$ If $x=F_w(\tilde{x})$ and $y=F_w(\tilde{y})$ for some $w$ such that $|w|=m$, then $$c_{x,y}=(\frac{5}{3})^{m} \tilde{c}_{\tilde{x},\tilde{y}}~.$$
When $x$ and $y$ are in different $m$-cells it is easy to see by Theorem \[zerocoefficients\] that $c_{x,y} = 0$. For $m>0,$ $V_m \subset \beta_m$, we can use the electrical network model of conductances. We glue the network $F_w(SG)$ with the same graph and conductance and multiply the conductances in $\beta_0$ by $(5/3)^m$. Therefore, we obtain the stated conductances for higher levels.
### New Level
Here we consider $V_n \setminus V_{n-1}$. Let $\Gamma_n$ represent the graph representation of $V_n$ and let $d(\cdot, \cdot)$ denote the graph distance. We denote by $c_{x,y}^n$ the conductance between $x$ and $y$ as elements in $\Gamma_n$.\
![The New Level Set on Level 2 with an arbitrary ordering used to describe the coefficients[]{data-label="figure:newlevel"}](New_Level_Energy_Minimization_Labeling_Level_2.pdf)
For each $x_i=\frac{q_{i-1}+q_{i+1}}{2}=F_{i-1}(q_{i+1}) \in V_1 \setminus V_0$ let $D^i_n$ denote the set of its four neighbors lying in $V_n$.
In this situation we cannot immediately extend from the first level. In fact we need the first 2 levels before we can use induction. We obtain the coefficients for the first 2 levels through direct computation and then extend.
For the first level, when we extend the values to $V_1 \setminus V_0$, all conductances are equal to $5/2$.
For the second level, as seen in Figure \[figure:newlevel\], from a simple calculation we have the following 3 types:
1. $c^2_{1,4} = c^2_{2,8} = c^2_{6,9} = 25/6$.
2. $c^2_{4,7} = c^2_{1,7} = c^2_{3,9} = c^2_{3,6} = c^2_{5,8} = c^2_{5,2} = 125/36$.
3. $c^2_{3,4}= c^2_{3,7}=c^2_{3,5}=c^2_{3,8}=c^2_{5,7}=c^2_{5,1}=c^2_{5,9}=c^2_{7,6}=c^2_{7,2}=c^2_{4,6}=c^2_{1,2}=c^2_{8,9}= 25/36$.
We assume now that $n \geq 3$ and we obtain the following lemmas:
\[lemma:czero\] If $x$ and $y$ are two points such that $d(x,y) \geq 3$ then $c_{x,y}^n = 0$.
This follows immediately from Theorem \[zerocoefficients\].
![The neighbors of a junction point. Here we see $\{p_j\} = D^i_n$[]{data-label="figure:junctionpoint"}](New_Point_Schematic2.pdf)
(Four-neighbors) \[lemma:fourneigh\] Let $a, b \in D_n^i$. If $d(a,b) = 1$ then $c_{a,b}^{n} = (\frac{5}{3})^n \frac{5}{4}$. In the case $d(a,b) = 2$, we have $c_{a,b}^{n} = (\frac{5}{3})^n \frac{1}{4}$.
Follows from a direct calculation after observing figure \[figure:junctionpoint\].
For $n \geq 3$, let $x,y \in F_i(V_{n-1}\setminus V_{n-2}) \setminus (D_n^{i+1} \cup D_n^{i-1})$. Then, $c_{x,y}^{n} = \frac{5}{3}c_{x,y}^{n-1}$.
Follows from the local nature of the calculation of energy due to Lemma \[lemma:czero\].
Bottom Row
----------
Here we consider the set $E$ which is the bottom row of $V_n$. We use the natural ordering on $E$ from left to right to write $x_0, \dots, x_{2^n}$ and denote the values attained at $x_i$ by $t_i$. Unlike the last two examples, this set has a very complicated structure. To develop a method to determine the coefficients we add the top point of $V_n$ to $E$ and denote the value at this point $q_0$ by $a$. This allows us to establish a recurrence between coefficients. We then relate these coefficients to our original problem without the top point.\
After determining the coefficients for the quadratic form, we will see that the energy of the minimizer has a unique structure expressible in terms of Haar functions that provides a discrete analog of the continuous result established in [@OS].
### Bottom Row with Top Point
Throughout this section we denote the coefficients for level $n$ as $c^n_{x,y}.$ Also we use the shorthand $c^n_{x_i,x_j} = c^n_{i,j}$. We get the following relation between the energy of the minimizers on different levels:
$$\label{equation: bottomlinerecurse}
{\mathcal{E}}_{n+1}(u) = \frac{5}{3}\left((a-b)^2 + (b-c)^2 + (c-a)^2 + {\mathcal{E}}_n(u \circ F_1) + {\mathcal{E}}_n(u \circ F_2)\right),$$
where $b=u(\frac{1}{2}(q_0+q_1))$ and $c=u(\frac{1}{2}(q_0+q_2))$. Now we want to minimize the energy with respect to $b$ and $c$ to get the quadratic form in terms of $a$ and $x_i$.
\[lemma:bottomlinetoppointrecurse\]
We set $a_n = c^n_{q_0, x_0} = c^n_{q_0,x_{2^n }}$. Then $$c^n_{q_0, x_j} = 2 a_n \text{ for } 0 < j < 2^n$$ and $$a_n = \frac{7\cdot 5^n}{3^n + 6\cdot 10^n}~.$$
The energy is minimized with
$$\begin{aligned}
b & = \frac{(3+2^{n+1}a_n)a + (2+2^{n+1}a_n)(a_n)( t_0+ 2\sum_{i=1}^{2^n-1}{t_i} + t_{2^n})}{(1+ 2^{n+1} a_n)(3+ 2^{n+1} a_n)}\\
&~~~~ +\frac{a_n( t_{2^n}+ 2\sum_{i=2^n+1}^{2^{n+1}-1}t_i + t_{2^{n+1}})}{(1+ 2^{n+1} a_n)(3+ 2^{n+1} a_n)} \\
c & =\frac{(3+2^{n+1}a_n)a + a_n( t_0+ 2\sum_{i=1}^{2^n-1}{t_i} + t_{2^n})}{(1+ 2^{n+1} a_n)(3+ 2^{n+1} a_n)} \\
&~~~~ +\frac{ (2+2^{n+1}a_n)(a_n) ( t_{2^n}+ 2\sum_{i=2^n+1}^{2^{n+1}-1}t_i + t_{2^{n+1}})}{(1+ 2^{n+1} a_n)(3+ 2^{n+1} a_n)}~.
\end{aligned}$$
We use induction and obtain the base case by direct computation. We omit this computation.\
To use induction, we compute the derivative with respect to $b$ to obtain $$\frac{\partial {\mathcal{E}}_{n+1}}{\partial b} = \frac{5}{3}(2(b-a) + 2(b-c)) + \frac{5}{3} \frac{\partial {\mathcal{E}}_n}{ \partial b}.$$ From the inductive hypothesis we have $$\frac{\partial {\mathcal{E}}_n}{ \partial b} = 2 a_n ((b-t_0) + (b-t_{2^n})) + 4 a_n(\sum_{i=1}^{2^n-1} (b-t_i)).$$
Then, we obtain a system of linear equations from $\frac{\partial {\mathcal{E}}_{n+1}}{\partial b} = 0,
\frac{\partial {\mathcal{E}}_{n+1}}{\partial c} = 0:
$
$$\left(\begin{matrix}
2+2^{n+1}a_n&-1 \\
-1& 2+2^{n+1}a_n
\end{matrix}\right)
\left(\begin{matrix}
b\\
c
\end{matrix}\right)
=\left(\begin{matrix}
a + a_n( t_0+ 2\sum_{i=1}^{2^n-1}{t_i} + t_{2^n }) \\
a + a_n( t_{2^n}+ 2\sum_{i=2^n+1}^{2^{n+1}-1}t_i + t_{2^{n+1}})
\end{matrix}\right).$$ We solve the above linear system to obtain the desired $b$ and $c$.\
Upon examination we see $-2 c_{q_0, x_0} = \partial_{at_0} {\mathcal{E}}_{n+1} = \frac{1}{2} \partial_{at_j} {\mathcal{E}}_{n+1} = -c_{q_0, x_j}$ for $0<j < 2^n$.\
To calculate the explicit formula for $a_n$, we get $a_0=1$ and the following recurrence: $$2 a_{n+1} = \partial_{at_0} {\mathcal{E}}_{n+1} = \frac{-10 a_n}{3(2^{n+1}a_n+1)}.$$ Solving this recurrence with our appropriate inital starting condition, we get $$a_n = \frac{7\cdot 5^n}{3^n + 6\cdot 10^n}.$$
By examining (\[equation: bottomlinerecurse\]) we derive several facts about the coefficients, using $$c^{n+1}_{y,z} = \frac{-1}{2}\partial_{u(y),u(z)} {\mathcal{E}}_{n+1}.$$
We set $b_n = c^n_{0, 2^n}$. Then $$\begin{aligned}
c^n_{0, j} = 2 b_n && \text{for}~~~~2^{n-1} < j < 2^n, \\
c^n_{j, 2^n} = 2 b_n && \text{for}~~~~0 < j < 2^{n-1} \\
c^n_{i, j} = 4 b_n && \text{for}~~~~0 < i < 2^{n-1}, 2^{n-1} < j < 2^n. \end{aligned}$$ Also, we have $$b_{n} = \frac{49\cdot 25^{
n }}{(5 \cdot 3^{n} + 16 \cdot 10^{n}) (3^n + 6 \cdot 10^{n})}.$$
\[lemma:proved\] We set $l_n = c^n_{0, 1}$. Then we get the recurrence $$l_1 = \frac{20}{9},$$ $$l_{n+1} = \frac{5}{3} l_{n} + \frac{98 \cdot 25^{n+1} (10 \cdot 3^{n+1} + 39 \cdot 10^{n+1})}{(3^{n+1} + 6 \cdot 10^{n+1}) (5 \cdot 3^{n+1} +
16 \cdot 10^{n+1}) (5 \cdot 3^{n+1} + 9 \cdot 10^{n+1})}$$ for $n\ge 1$.
We set $m_n = c^n_{{2^{n-1} -1}, {2^{n-1}}} = c^n_{{2^{n-1}}, {2^{n-1}}+1}$. Then we get the following formula in terms of $l_n$: $$m_1 = l_1 = \frac{20}{9},$$ $$m_{n+1} = \frac{5}{3} l_{n} + \frac{294\cdot 25^{n+1}}{(3^{n+1} + 6 \cdot 10^{n+1}) (5 \cdot 3^{n+1} + 9 \cdot 10^{n+1})}$$ for $n \ge 1$.
Given $0 < i < j < 2^{n}$ we have $$c^{n+1}_{i,j} = \frac{5}{3}c^{n}_{i,j} +
\frac{196\cdot 25^{n+1} (10 \cdot 3^{n+1} + 39\cdot 10^{n+1})}{(3^{n+1} + 6 \cdot 10^{n+1}) (5\cdot 3^{n+1} + 16 \cdot 10^{n+1}) (5 \cdot 3^{n+1} + 9\cdot 10^{n+1})}$$ for $n \ge 1$.
It is easy to see that if we fix two indices $i, j$ then the asymptotic behavior of $c^n_{i,j}$ is $(\frac{5}{3})^n$. If we look at the relative position of points, instead, with $0 \leq i < 2^{n-1}$ and $ 2^{n-1} < j \leq 2^n$ we see the asymptotic behavior of $c^n_{i,j}$ is $(\frac{1}{4})^n$.
### Bottom Row without Top Point
Here we let $\tilde{c}^n_{i,j}$ be the coefficient of $(u(x_i) - u(x_j))^2$ in the minimizing form. We obtain analogous formulas for this case by noting that we minimize the energy by setting $$a = \frac{t_0 + 2(t_1 + \dots + t_{2^n - 1}) + t_{2^n}}{2^{n+1}}.$$
We have $$\tilde{c}^n_{0,2^n} = c^n_{0,2^n} + \frac{a_n}{2^{n+1}},$$ $$\left\{ \begin{array}{lr}
\tilde{c}^n_{0,j} = c^n_{0,j} + \frac{a_n}{2^{n}} & 0 < j < 2^n,\\
\tilde{c}^n_{i,j} = c^n_{i,j} + \frac{a_n}{2^{n-1}} & 0 < i < j < 2^n.
\end{array} \right.$$
\[corollary:farpointcoefficient\] We set $\tilde{b}_n = \tilde{c}_{0,2^n}^{n}$. Then $$\begin{aligned}
\tilde{c}_{0, j}^n = 2 \tilde{b}_n && 2^{n-1} < j < 2^n, \\
\tilde{c}_{j, {2^n}}^n = 2 \tilde{b}_n && 0 < j < 2^{n-1}, \\
\tilde{c}_{i, j}^n = 4 \tilde{b}_n && 0 < i < 2^{n-1}, 2^{n-1} < j < 2^n .\end{aligned}$$ Also we have $$\tilde{b}_n = \frac{a_n}{2^{n+1}} + b_n.$$ More specifically, $$\tilde{b}_n = \frac{35 \cdot 5^n}{10 \cdot 6^{n} + 32 \cdot 20^{n}}.$$
Since $a_n$ grows at a rate of $(\frac{1}{2})^n$, the previous lemma shows that we add only a term of order $(\frac{1}{4 })^n$ to the coefficients. Since all the asymptotics from the case including the top point were at least this, we get identical asymptotics.
### Haar Structure
Following [@OS], we prove a discrete equivalent of those results showing that the energy of the minimizer can be expressed in a specific form using Haar functions. We first introduce the notation used in that paper.
We define the Haar functions as follows $$\Psi_{n,k}(t) = \left\{
\begin{array}{lr}
2^{n/2} & t \in [\frac{k}{2^n},\frac{k+1/2}{2^{n}})\\
-2^{n/2} & t \in (\frac{k+1/2}{2^n},\frac{k+1}{2^{n}}]\\
0 & ~~\text{otherwise}~.
\end{array}
\right.$$
Given a function $u$ with values $u(x_i) = t_i$ on the bottom row of $V_m$ we define a function $f_m(u)$ on the bottom line which is a piecewise constant interpolation: $$\label{equ:fm}
f_m(u) = t_0 \chi_{[0,\frac{1}{2^{m+1}}]} + \sum_{i=1}^{2^m - 1}{t_i \chi_{[\frac{2i - 1}{2^{m+1}},\frac{2i + 1}{2^{m+1}}]}} + t_{2^m}\chi_{[\frac{2^{m+1} - 1}{2^{m+1}},1]}.$$ Using the $L^2$ inner product we define $$\label{equ:inner}
D^m_{n,k}(u) = <\Psi_{n,k},f_m(u)>.$$
When there is no ambiguity about which function we are using we use $D^m_{n,k}= D^m_{n,k}(u)$. These give us weighted averages of dyadic groups of values on the bottom row. We give some examples below: $$D^m_{0,0} = (2^{-\frac{0}{2}-1})\left(\frac{(t_0 + 2 t_1 + \dots + 2 t_{2^{m-1} - 1} + t_{2^{m-1}}) - (t_{2^{m-1}} + 2 t_{2^{m-1} + 1} + \dots + 2 t_{2^{m} -1} + t_{2^m})}{2^m}\right),$$
$$D^m_{m-1,0} = (2^{-\frac{m-1}{2}-1})\left(\frac{(t_0 + t_1) - (t_1 + t_2)}{2}\right),~~{D}^m_{m-1,1} = (2^{-\frac{m-1}{2}-1})\left(\frac{(t_2 + t_3) - (t_3 + t_4)}{2}\right),$$ $$D^m_{m,0} =(2^{-\frac{m}{2}-1}) (t_0 - t_1)~~\text{and}~~D^m_{m,1} = (2^{-\frac{m}{2}-1})(t_1 - t_2).$$ We prove the following about the energy of the minimizer:
\[theorem:haarstructure\]
$$\begin{aligned}
{\mathcal{E}}(u) & = \sum_{i=0}^{m}\gamma^m_i\sum_{j=0}^{2^{m-i} - 1}{(D^m_{m-i,j})^2},\end{aligned}$$
where $$\gamma_{0}^1 = 4,$$ $$~~~~~~~~\gamma^m_n = \frac{10}{3}\gamma^{m-1}_n~\text{ for }0 \leq n < m ~\text{and}~ m\geq 2,$$ $$\gamma^m_m = \frac{70\cdot 10^{m}}{5 \cdot 3^m + 16\cdot 10^m}~~\text{for}~ m \geq 1.$$
We see that $\gamma_i^m \approx \left(\frac{10}{3}\right)^{m-i}$ and $\lim_{m\to\infty} \gamma_{m}^m=\frac{35}{8}.$ This is a discrete analog of Theorem 2.4 of [@OS].
We start off showing some other properties of the minimizer. We begin by creating a sort of bump function which allows us to vary our functions harmonically away from the bottom row and the top point while keeping our initial conditions on bottom row. When we refer to the bottom row throughout this section, we mean the subset of $V_m$ on the bottom row for the appropriate $V_m$ in the context.\
\[lemma:integralharmonicextension\] We define $\Phi_m$ to be the unique function for which $\Phi_m = 0$ on the bottom row of $V_m$, $\Phi_m(q_0) = 1$ and $\Phi_m$ is harmonic away from the bottom row of $V_m$ and the top point. Set $$~~~~~~~~b_m = \partial_n \Phi_m(q_0)~~\text{and}$$ $$c_m = \Phi_m(\frac{q_0 + q_1}{2}) = \Phi_m(\frac{q_0 + q_2}{2}).$$
Then for $m \ge 1$ we have $$~~~~~~~~b_m = \frac{14\cdot 10^m}{3^m + 6\cdot 10^m}~~\text{and}$$ $$c_m = \frac{5\cdot 3^m + 9 \cdot 10^m}{5\cdot 3^m + 30 \cdot 10^m}.$$
Because $\Phi_m \circ F_0$ is harmonic we have $$b_m = \frac{5}{3}(2 - 2 c_m).$$ Furthermore we can combine the matching normal derivative conditions at $\frac{q_0 + q_1}{2}$ and the fact that $\Phi_m \circ F_1 = c_m \Phi_{m-1}$ to get $$(\frac{5}{3})^2(2 c_m - 2 c_m c_{m-1}) + (\frac{5}{3})(2c_m - c_m - 1) = 0.$$ Thus we get the recurrence $$c_m =\frac{-3}{-13 + 10 c_{m-1}}$$ and since $c_1=\frac{1}{3}$ so we can solve to get the stated closed form for both $c_m$ and $b_m$.
Now we find the exact weighted average of the values on the bottom row and discover some other interesting properties of the minimizer.
\[Lemma:harmonicbottom\] Let $u$ be a function harmonic away from the bottom row of $V_m$. Then $\partial _nu(q_0)=0, u(q_0) = \int_{SG}{u d\mu}$ and $$\label{equation:bottomlineintegral}
\int_{\text{SG}}u d\mu=\frac{1}{2^{m+1}}\left(u(x_0)+2u(x_1) + \dots + 2u(x_{2^m - 1}) + u(x_{2^m})\right)=u(q_0).$$
We prove this by induction. For the base case we look at $V_0$ and we see that $$u(q_0) = \frac{1}{2}(u(q_1) + u(q_2))$$ which implies that $\partial_nu(q_0) = 0$ and (\[equation:bottomlineintegral\]) is satisfied trivially.\
Now suppose we have the result for $V_{m-1}$. We use the notation $u(x_i) = t_i$ for the values on the bottom row. Let $u_1$ be the function on $V_{m-1}$ with prescribed values $u_1(x_i) = t_i$ for $0 \leq i \leq 2^{m-1}$ with $\partial_n u_1(q_0) = 0$ which is harmonic away from the bottom row of $V_{m-1}$. Similarly let $u_2$ be the function with $u_2(x_i) = t_{2^{m-1} + i}$ for $0 \leq i \leq 2^{m-1}$ which also satisfies $\partial_n u_2(q_0) = 0$ and is harmonic away from the bottom row of $V_{m-1}$. We set $$a = \int_{SG} u_1 d \mu,$$ $$b = \int_{SG} u_2 d \mu$$ and define a function $u$ piecewise on $V_1$ by: $$u \circ F_1 = u_1 + x \Phi_{m-1},$$ $$u \circ F_2 = u_2 + y \Phi_{m-1},$$ $$\label{equation:uf0}
u \circ F_0 = \left(\frac{a+b+x+y}{2}\right) h_0 + (a+ x) h_1 + (b+y) h_2.$$ In order for this to be harmonic away from the bottom row, we must have the proper gluing conditions. Thus we set out to pick $x,y$ to meet these conditions. From the inductive hypothesis we see we get continuity at $\frac{q_0 + q_1}{2}$ and $\frac{q_0 + q_2}{2}$ by $$~~~~~~~~u \circ F_1(q_0) = u_1(q_0) + x = \int_{SG}{u_1 d \mu} + x = v \circ F_0(q_1)~~\text{and}$$ $$u \circ F_2(q_0) = u_2(q_0) + y = \int_{SG}{u_2 d \mu} + y = v \circ F_0(q_2).$$ All that remains to show is matching normal derivatives: $$~~~~~~~~\partial_n (u \circ F_1)(q_0) + \partial_n (u \circ F_0) (q_1) = 0 ~~\text{and}$$ $$\partial_n (u \circ F_2)(q_0) + \partial_n (u \circ F_0) (q_2) = 0 .$$ This gives us a system of linear equations. Solving it we get $$\label{equation:matrix}
\left(\begin{matrix} b_{m-1} + \frac{3}{2} & -\frac{3}{2} \\
-\frac{3}{2} & b_{m-1} + \frac{3}{2} \end{matrix} \right) \left( \begin{matrix} x \\ y \end{matrix} \right) = \frac{3}{2}(a-b) \left(\begin{matrix} 1 \\ -1 \end{matrix} \right).$$ From Lemma \[lemma:integralharmonicextension\] we know $b_{m-1} > 0$ , so we can solve this system. Specifically, we have $x+y = 0$ and $x= C_m(a-b)$, $y=-C_m(a-b)$ for some constant $C_m$. Thus we have found a unique function which is harmonic away from the bottom row with specified values. Note from our set up that $$\partial_n u(q_0) = \frac{5}{3}\left(2(u \circ F_0)(q_0) - (u \circ F_0)(q_1) - (u \circ F_0)(q_2)\right) = 0.$$ Now we see $$\int_{SG}{u d\mu} = \frac{1}{3}\sum_i{\int_{SG}{u \circ F_i d \mu}}$$ $$=\frac{1}{3}(a + b + \frac{1}{3}(a+x + b+ y+ \frac{a+b+x+y}{2}))$$ $$= \frac{a+b}{2} = u(q_0).$$ Thus we have all the stated properties.
One can show in fact $F_0^n(\frac{q_1 + q_2}{2}) = \int u d \mu$ for all $n >0$. Since we see from above $x = -y = C_m(a-b)$ for some constant $C_m$, we also get ${\mathcal{E}}(u \circ F_0) = \tilde{C}_m (a-b)^2$ for some $\tilde{C}$.
Using the same set up as the previous lemma, we obtain an inductive argument to prove Theorem \[theorem:haarstructure\].
For $m=1$, namely for $V_1$, we obtain $${\mathcal{E}}(u) = C_1 \left( (t_0 - t_1)^2 + (t_1 - t_2)^2\right) + C_2(t_0 - t_2)^2.$$ A direct calculation shows that the energy of the function that harmonically extends the values $t_0, t_1, t_2$ on the bottom row of $V_1$ has $C_1 = \frac{5}{2}$ and $C_2 = \frac{1}{4}$, where
$$\begin{aligned}
(t_1 - t_2)^2 &= 8 (D_{1,1}^1)^2 \\
(t_0 - t_1)^2 &= 8 (D_{1,0}^1)^2 \\
(t_0 - t_1)^2 &= 16 (D_{0,0}^1)^2.\end{aligned}$$
Therefore in this case, we obtain $\gamma_0^1 = 4, \gamma_{1}^1 = 20$. Now, we suppose it holds in the case $V_{m-1}$ and prove it for $V_{m}$: From the self similar identity and the proof of Lemma \[Lemma:harmonicbottom\], we get $$\begin{aligned}
\label{equ:Eu}
{\mathcal{E}}(u) &= \frac{5}{3}\left({\mathcal{E}}(u \circ F_0) + {\mathcal{E}}(u \circ F_1) + {\mathcal{E}}(u \circ F_2)\right) \nonumber \\
&=\frac{5}{3}\left({\mathcal{E}}(u \circ F_0) + {\mathcal{E}}(u_1 + x \Phi_{m-1}) + {\mathcal{E}}(u_2 + y \Phi_{m-1}) \right) \nonumber \\
&=\frac{5}{3} \left( {\mathcal{E}}(u_1) + {\mathcal{E}}(u_2) + {\mathcal{E}}(u \circ F_0) + 2 x {\mathcal{E}}(u_1,\Phi_{m-1}) \right. \nonumber\\
&\left. + x^2 {\mathcal{E}}(\Phi_{m-1}) + 2 y {\mathcal{E}}(u_2,\Phi_{m-1}) + y^2 {\mathcal{E}}(\Phi_{m-1}) \right).\end{aligned}$$
From the induction hypothesis, we have ${\mathcal{E}}(u_l) = \sum_{i=0}^{m-1}{\gamma}^{m-1}_i\sum_{j=0}^{2^{m-i} - 1}{ {D}^{m-1}_{m-1-i,j}(u_l)^2}$ for $l=1,2$. Also by equations (\[equ:fm\]) and (\[equ:inner\]) we have $$\begin{aligned}
D^{m-1}_{m-1-i,j}(u_1)&=2^{\frac{1}{2}}D^m_{m-i,j}(u) \label{equ1:D}\\
D^{m-1}_{m-1-i,j}(u_2)&=2^{\frac{1}{2}}D^m_{m-i,j+2^{m-1-i}}(u) \label{equ2:D}.\end{aligned}$$
We want to obtain $\gamma_{0,n}^m$ for $0 \leq n < m$. If we plug the equations (\[equ1:D\]) and (\[equ2:D\]) in ${\mathcal{E}}(u_1)+ {\mathcal{E}}(u_2)$ we obtain
$$\begin{aligned}
\label{equ:E1plusE2}
{\mathcal{E}}(u_1)+ {\mathcal{E}}(u_2) &= \sum_{i=0}^{m-1}{\gamma}^{m-1}_i\sum_{j=0}^{2^{m-1-i} - 1}{{D}^{m-1}_{m-1-i,j}(u_1)^2}+\sum_{i=0}^{m-1}{\gamma}^{m-1}_i\sum_{j=0}^{2^{m-1-i} - 1}{{D}^{m-1}_{m-1-i,j}(u_2)^2}\\ &= 2\sum_{i=0}^{m-1}{\gamma}^{m-1}_i\sum_{j=0}^{2^{m-i} - 1}{{D}^m_{m-i,j}(u)^2}.\end{aligned}$$
Now, what we want to show is the remaining part in equation (\[equ:Eu\]) is a multiple of $D^m_{0,0}(u)^2.$ We set $a = u_1(q_0)$ and $b = u_2(q_0)$. By the right side of equation (\[equation:bottomlineintegral\]) we have $$\begin{aligned}
(2^{-2})(a-b)^2 &= (2^{-2})4^{-m}\big((t_0 + 2 t_1 + \dots + 2 t_{2^{m-1} - 1} + t_{2^{m-1}}) - (t_{2^{m-1}} + 2 t_{2^{m-1} + 1} + \dots + 2 t_{2^{m} -1} + t_{2^m})\big)^2\\
&=D^m_{0,0}(u)^2.\end{aligned}$$ Thus we wish to show first that $$\frac{5}{3}\left({\mathcal{E}}(u \circ F_0) + 2 x {\mathcal{E}}(u_1,\Phi_{m-1}) + 2 y {\mathcal{E}}(u_2,\Phi_{m-1}) + (x^2 + y^2) {\mathcal{E}}(\Phi_m)\right) = k(a-b)^2$$ for some constant $k$ which only depends on $m$.\
From equations (\[equation:uf0\]) and (\[equation:matrix\]) , we have that ${\mathcal{E}}(u \circ F_0) = \tilde{C}_m (a-b)^2$ and also $x^2 + y^2 = 2 C_m^2(a-b)^2$ where $C_m,\tilde{C_m}$ only depend on $m$. Thus all that remains is to show $$\begin{aligned}
2 x {\mathcal{E}}(u_1,\Phi_{m-1}) + 2 y {\mathcal{E}}(u_2,\Phi_{m-1}) &= 2 C_m (a-b) \left( {\mathcal{E}}(u_1,\Phi_{m-1}) - {\mathcal{E}}(u_2,\Phi_{m-1}) \right) \\\end{aligned}$$ is a multiple of $(a-b)^2$. By the polarization identity, we get $$\begin{aligned}
{\mathcal{E}}(u_1,\Phi_{m-1}) &= \frac{1}{4}\left({\mathcal{E}}(u_1 + \Phi_{m-1}) - {\mathcal{E}}(u_1 - \Phi_{m-1})\right). \\\end{aligned}$$ For $l=1,2$, $u_l \pm \Phi_{m-1}$ are harmonic away from the bottom row and the top point. They agree on the bottom row so we have a lot of cancellation. We let $a_i$ denote the coefficients from Lemma \[lemma:bottomlinetoppointrecurse\]. Then, $$\begin{aligned}
4 {\mathcal{E}}(u_1,\Phi_{m-1}) &= \left({\mathcal{E}}(u_1 + \Phi_{m-1}) - {\mathcal{E}}(u_1 - \Phi_{m-1})\right) \\
&=a_{m-1}(a+1 - t_0)^2 + 2a_{m-1} \left(\sum_{i=1}^{2^{m-1} -1}(a+1 - t_i)^2\right) +a_{m-1}(a+1 - t_{2^{m-1}})^2 \\
&-\left(a_{m-1}(a-1 - t_0)^2 + 2a_{m-1}\left( \sum_{i=1}^{2^{m-1} -1}(a-1 - t_i)^2\right) +a_{m-1}(a-1 - t_{2^{m-1}})^2\right)\\
&=4a_{m-1}(a - t_0) + 8a_{m-1} \left(\sum_{i=1}^{2^{m-1} -1}(a - t_i)\right) + 4a_{m-1}(a - t_{2^{m-1}}) \\
&= 2^n a_{m-1} a - 4a_{m-1}\left(t_0 + 2 \left(\sum_{i=1}^{2^{m-1} -1}t_i\right) + t_{2^{m-1}}\right)\\
&= (2^m a_{m-1} - 4 a_{m-1})a.\end{aligned}$$ Similarly, we get $$4 {\mathcal{E}}(u_2,\Phi_{m-1}) = (2^m a_{m-1} - 4 a_{m-1})b.$$ Thus we have $$\begin{aligned}
2 x {\mathcal{E}}(u_1,\Phi_{m-1}) + 2 y {\mathcal{E}}(u_2,\Phi_{m-1}) &= 2 C_m (a-b) \left( {\mathcal{E}}(u_1,\Phi_m) - {\mathcal{E}}(u_2,\Phi_m) \right) \\
&=\tilde{k} (a-b)^2.\end{aligned}$$ So, we can write $${\mathcal{E}}(u) = \frac{5}{3} \left( 2\sum_{i=0}^{m-1}{\gamma}^{m-1}_i\sum_{j=0}^{2^{m-i} - 1}{{D}^m_{m-i,j}(u)^2} + 4 \tilde{k} D_{0,0}^m(u)^2 \right)$$ Since $D_{0,0}^m(u)^2$ is the only term that includes $t_0$ and $t_{2^m}$, we have $-2 c_{x_0, x_{2^m}} = \partial_{t_0, t_{2^m}} {\mathcal{E}}= \frac{-10}{3} 4^{-m} \tilde{k}$. We note that $4^{-m-1}\gamma^m_m = \frac{-1}{2}\partial_{t_0,t_{2^m}}{\mathcal{E}}$ and from Corollary \[corollary:farpointcoefficient\] we get the stated closed form.
Laplacian
=========
Given a positive, finite measure $\zeta$ on $SG$ we now define a functional $T_\zeta(u) = \int (\Delta_\zeta u)^2 d \zeta$ on the space ${\text{dom}}_{L^2}\Delta_\zeta.$ We first consider the constraints $u(x_i) = a_i$ and $\partial_n u(y_j) = b_j$ for all $x_i \in E$ and $y_i \in F$, with $E,F$ finite subsets of $V_m$. We need the following assumption on $E$:
There exist 3 points $x_1, x_2, x_3 \in E$ such that $$H=\begin{pmatrix}
h_0(x_1)&h_1(x_1) &h_2(x_1) \\
h_0(x_2)&h_1(x_2) &h_2(x_2)\\
h_0(x_3)&h_1(x_3) &h_2(x_3)
\end{pmatrix}$$ is invertible, where $h_j$ denotes the harmonic function with boundary values $h_j(q_i) = \delta_{ij}$.
We see from the theory of harmonic coordinates on the Sierpinski Gasket [@Ki03] that this is equivalent to guaranteeing $E$ is not contained in a line in the harmonic Sierpinski Gasket. In the situation where $E$ is contained on a line we can see that we should not expect uniqueness. For example, there is a one dimensional space of skew symmetric harmonic functions which vanish on the line through the middle of $SG$. See [@LS] for a complete discussion of this example.
\[theorem:laplacian1\] (Existence and Uniqueness) Set $$Y = \{u \in {\text{dom}}_{L^2}\Delta_\zeta ~|~ u(x_i) = a_i\}.$$ Then we have a unique minimizer in $Y$ minimizing $T_\zeta$.
Convexity holds trivially. We consider the space $$\tilde{A} = \text{dom}_{L^2} \Delta_{\zeta} / H(K)$$ where $H(K)$ is the space of harmonic functions with the norm $\|\cdot\| = \sqrt{\int_{SG}{|\Delta_\zeta u|^2 d \zeta}}$. Let us define $\tilde{Y} = Y / H(K)$. We want to show that $\tilde{Y}$ is closed.\
We first consider the situation in which we have Dirichlet boundary conditions and then observe we can easily extend to the general situation. We denote the space $\text{dom}_{L^2}\Delta_0=\{u\in \text{dom}_{L^2}\Delta_\zeta:u|_{V_0}=0 \}$. Note that the space $\tilde{A} \cong {\text{dom}}_{L^2}\Delta_0$.\
In the case that $Y \subset dom_{L^2} \Delta_0$, we want to show that $Y$ is closed in $dom_{L^2}\Delta_0$ with respect to the norm $|| u || :=\sqrt{ \int (\Delta_\zeta u)^2 d \zeta} $. By $(\ref{equation:supenergy})$, we know that $||u||_{\infty}\le C{\mathcal{E}}(u)^{\frac{1}{2}}$ for some constant $C$. By [@SU]\*[Lemma 4.6]{}, we have $$\mathcal{E}(u) \leq C_0 || \Delta_\zeta u ||_2^2.$$ Combining the two inequalities yields $$|| u ||_{\infty}^2 \leq C_1 || \Delta_\zeta u ||_2^2$$ where $C,C_0,C_1$ are positive constants. Given $u_n \rightarrow u$ in $dom_{L^2}\Delta_0$ with $u_n$ in $Y$, the above inequality implies that $ u(x_i) = a_i$ for any $i$ and $u$ vanishes on the boundary. Therefore, $Y$ is closed.\
To extend the result, we view our space as $\tilde{A}$. Now suppose $\widetilde{u_n}\rightarrow \tilde{u}$, where $\widetilde{u_n}$ are in $\tilde{Y}$ and $u_n\in Y$ for all $n$. We note we can pick representatives with Dirichlet boundary conditions $w_n$ for each $u_n$. Since $H$ is invertible, we know that there exists a harmonic function $v$ such that $u(x_i)=a_i$ for $i=1,2,3$ where $u=w+v$ and $w$ has Dirichlet boundary conditions. Since $||w_n-w||_{\infty}\rightarrow 0$ and $u(x_i)=a_i$ we have $\lim_{n\rightarrow \infty} v_n(x_i)=v(x_i)$ for $i=1,2,3$. We note that using the invertibilty of $H$ we obtain $\lim_{n\rightarrow \infty} v_n(q_i)=v(q_i)$ for $i=0,1,2$. By the maximum principle for harmonic function, we know that $||v_n-v||_{\infty}\rightarrow 0$ as $n\rightarrow \infty$. As a result, we have $||u_n-u||_{\infty}\rightarrow 0$ as $n \to \infty$ so $u\in Y$ and $\tilde{u} \in \tilde{Y}$. Therefore, we again get that $\tilde{Y}$ is closed.
When we add in the normal derivatives constraint, we require a new condition on the measure, $$\label{equation:measure}
\|\Delta_\zeta u\|^2_2 \geq C_w \|\Delta_\zeta(u\circ F_w)\|^2_2$$ for every word $w$ and some constant $C_w$ depending only on $w$. We now show that both the standard self-similar measure, $\mu$, and the Kusuoka measure, satisfy (\[equation:measure\]).
\[lemma:measureproperty\] Both the self similar measure and the Kusuoka measure, $\nu$, satisfy (\[equation:measure\]). Furthermore we can establish estimates on $C_w$.
In the case of the self similar measure, it follows immediately from the self similar identity on the Laplacian which can be found in [@Str]. In this case, $C_w = r_w \mu_w^{-1} = \left(\frac{25}{3}\right)^{|w|}$.\
We have the following “self-similar” identity for the Kusuoka measure ([@RCS], Theorem 2.3), $$\begin{aligned}
\label{equ2}
||\Delta_{\nu} u||_{2}^2 &= \int |\Delta_{\nu} u|^2 d \nu \\ &= \sum_{w} \int Q_w |(\Delta_{\nu} u) \circ F_w|^2 d \nu = \sum_{w} \int \frac{1}{Q_w} |\Delta_{\nu}(u \circ F_w)|^2 d \nu \geq C_w ||\Delta_{\nu}(u \circ F_w)||_{2}^2,\end{aligned}$$ where $C_w$ is a constant depending on $w$ that we explore as follows.
We have $Q_j = \frac{1}{15}+ \frac{12}{25}\frac{d\nu_j}{d\nu}$ and $\nu_j$ and $Q_w$ are as defined in [@RCS]. By [@RCS]\*[Theorem 3.5]{}, we have $ 0 \leq \frac{d\nu_j}{d\nu} \leq \frac{2}{3}$. Therefore, we obtain $\frac{1}{15} \leq Q_j \leq \frac{29}{75}$ in which case by [@RCS]\*[Corollary 2.4]{}, for a general word $w$, we have $$\left( \frac{1}{15} \right)^{|w|} \leq Q_w \leq \left( \frac{29}{75} \right)^{|w|}$$ so
$$\left( \frac{75}{29} \right)^{|w|} \leq C_w \leq 15^{|w|}.$$
Now we add in the normal derivative constraints and show we are guaranteed a minimizer.
\[lemma:energyconvergence\] Let everything be as in Theorem \[theorem:laplacian1\], so $Y=\{u\in dom_{L^2}\Delta_\zeta :u(x_i)=a_i\}$. Define $\tilde{Y}=Y/H(K)$. Suppose $\tilde{u}_n \rightarrow \tilde{u} \in \tilde{Y}$ and $u_n,u\in Y$. If we set $z_n = u_n - u$ then we have ${\mathcal{E}}(z_n)\rightarrow 0$ as $n \rightarrow \infty $.
As in Theorem \[theorem:laplacian1\], we write $u_n=w_n+v_n$ and $u=w+v$ where $w_n, w$ are Dirichlet functions and $v, v_n$ are harmonic. Since $w_n$ and $w$ have Dirichlet boundary conditions, we can use [@SU]\*[Lemma 4.6]{} to conclude that $${\mathcal{E}}(w_m-w) \leq C || \Delta_\zeta (w_m-w) ||_{2}^2.$$ We can use this to get ${\mathcal{E}}(w_n - w)\rightarrow 0$ when $n\rightarrow \infty$. Also since $\lim_{n\rightarrow \infty} v_n(q_i)=v(q_i)$ for $i=0,1,2$ we have ${\mathcal{E}}(v_n-v)\rightarrow 0$ as $n\rightarrow \infty$. Hence, the result follows immediately from the triangle inequality.
\[theorem:laplacian2\] (Existence and Uniqueness) Set $$X = \{u \in {\text{dom}}_{L^2}\Delta_\zeta ~|~ u(x_i) = a_i, \partial_n u(y_j) = b_j\}.$$ Then for any measure $\zeta$ satisfying (\[equation:measure\]) there is a unique minimizer in $X$ for $T_\zeta$.
To extend the result, we are only concerned with showing that $\tilde{X}$ is closed as a subset of $\tilde{Y}$ (convexity is again obvious). Let $u_n \rightarrow u$ as $n \rightarrow \infty$ in $||\cdot||$ with $u_n \in Y$. We put $z_m = u_m - u$. We pick $y_j = F_{w}q_i$ for some $w$ and $q_i$. By using the Gauss-Green formula, we have $${\mathcal{E}}(z_m \circ F_{w}, h_i) = - \int \Delta_\zeta (z_m \circ F_{w}) h_i d \zeta + \partial_n (z_m \circ F_{w}) (q_i).$$ So we obtain $$|\partial_n (z_m \circ F_{w}) (q_i)| \leq |{\mathcal{E}}(z_m \circ F_{w}, h_i)| + |\int \Delta_\zeta (z_m \circ F_{w}) h_i d \zeta| \leq \tilde{C}(\mathcal{E}^{1/2}(z_m) + ||\Delta_\zeta z_m ||_2)$$ for some constant $\tilde{C} >0$ from the self-similar identity for energy and (\[equation:measure\]). We apply Lemma \[lemma:energyconvergence\] to see that as $m \to \infty$ we have $(\mathcal{E}^{1/2}(z_m) + ||\Delta_\zeta z_m ||_2) \to 0$ so $|\partial_n (z_m \circ F_{w}) (q_i)| \to 0$. Thus taking the limit as $m \rightarrow \infty$ yields $\partial_n u(y_j) = b_j$.
Note that we may view this as a subset of the value only constraints, and thus the results are fully generalized. We do not know what happens in the situation when $E$ does not satisfy our assumption and we impose boundary conditions.
Construction
------------
In this section, we assume $E = F$ for simplicity and assume the measure $\zeta$ satisfies the appropriate conditions to guarantee the existence and uniquness of a minimizer. We continue as before by finding Euler Lagrange equations for a necessary condition for the minimizer.\
\[lemma:eulerlagrangelaplacian\] Given $u \in {\text{dom}}_{L^2}\Delta_\zeta$ which minimizes $T_\zeta$, we must have $$\label{eulerlagrangelaplacian}\int \Delta_\zeta u \Delta_\zeta v d \zeta = 0$$ for all $v \in {\text{dom}}_{L^2}\Delta_\zeta$ with $v|_{E} \equiv 0$ and $\partial_n v|_{E} \equiv 0.$
Suppose $u$ is the minimum. Given any $v \in {\text{dom}}_{L^2}\Delta_\zeta$ with $v|_{E} \equiv 0$ and $\partial_n v|_E \equiv 0$ we have that $u + tv$ also satisfies the constraints for any $t \in {\mathds{R}}$. We consider $f(t) = T_{\zeta}(u+tv)$ and notice that since $u$ is a minimum by hypothesis we have $f'(0) = 0$. Since $$T_{\zeta}(u + tv) = \int (\Delta_\zeta u)^2 d \zeta + 2 t \int \Delta_\zeta u \Delta_\zeta v d \zeta + t^2 \int (\Delta_\zeta v)^2 d \zeta,$$ we have $$f'(0) = \int \Delta_\zeta u \Delta_\zeta v d \zeta = 0.$$
We can apply this along with the Dirichlet boundary conditions to immediately get uniqueness:
\[lemma:laplacianuniqueness\] (Uniqueness) If $u_1$ and $u_2$ both satisfy (\[eulerlagrangelaplacian\]) then $u_1 = u_2$
Specifically, since $u_1(x_i) = u_2(x_i) = a_i$ we get $(u_1 - u_2)|_{E} \equiv 0$ and likewise $\partial_n(u_1-u_2)|_E \equiv 0$. Therefore, we can set $v = u_1 - u_2$ and use linearity of the integral and Laplacian to obtain $$\int (\Delta_\zeta(u_1 - u_2))^2 d \zeta = 0.$$ This gives $\Delta_\zeta(u_1 - u_2) = 0$ almost everywhere, which implies ${\mathcal{E}}(u_1 - u_2) = 0.$ So the functions differ by a harmonic function $h$. By our assumption that $H$ is invertible we immediately get that $h=0$ so $u_1 = u_2$.
\[lemma: laplacianconstruction\] If $u$ is a piecewise biharmonic function on $m$-cells and $v \in {\text{dom}}_{L^2}{\Delta_\zeta}$ then we have $$\label{equation:biharmonic}\int \Delta_\zeta u \Delta_\zeta v d \zeta = \sum_{|w|=m}{r_w^{-1}\sum_{V_0}{\left( (\Delta_\zeta u \circ F_w )\partial_n(v \circ F_w) - (v \circ F_w)\partial_n (\Delta_\zeta u \circ F_w)\right)}}.$$
Let $v \in {\text{dom}}_{L^2}{\Delta_\zeta}$. Then $$\int \Delta_\zeta v \Delta_\zeta u d \zeta= \sum_{|w|=m}{\mu_w \int (\Delta_\zeta v \circ F_w)(\Delta_\zeta u \circ F_w)}.$$ Since $\Delta_\zeta u \circ F_w$ is harmonic, Gauss-Green implies this equals $$\sum_{|w|=m}{\mu_w (\mu_w r_w)^{-1}\left(-{\mathcal{E}}(v \circ F_w, \Delta_\zeta u \circ F_w) +\sum_{V_0}{ (\Delta_\zeta u \circ F_w) \partial_n(v \circ F_w)}\right)}.$$ We can apply Gauss-Green again since $v \circ F_w \in {\text{dom}}{\mathcal{E}}$ to obtain $$\sum_{|w|=m}{r_w^{-1}\left(\int (v \circ F_w) \Delta_\zeta(\Delta_\zeta u \circ F_w) d \zeta -\sum_{V_0}{(v \circ F_w)\partial_n (\Delta_\zeta u \circ F_w)} +\sum_{V_0}{(\Delta_\zeta u \circ F_w )\partial_n(v \circ F_w)}\right)}$$ and since $\Delta_\zeta u \circ F_w$ is harmonic we finally obtain the equation (\[equation:biharmonic\]).
\[corollary: biharmonicminimizer\] If $E=V_m$ then the minimizer must be piecewise biharmonic.
Consider the piecewise biharmonic function $f$ that satisfies the constraints of Lemma \[lemma:eulerlagrangelaplacian\]. This implies $f$ satisfies equation (\[eulerlagrangelaplacian\]). Then, Lemma \[lemma:laplacianuniqueness\] implies that it is the minimizer.
In the case where $E=V_m$ and $\zeta = \mu$, we can give a concrete way to calculate the energy. When $u$ is biharmonic, we have $\Delta u = \sum_{i=0}^{2} c_i h_i$ for some constants $c_i$. We know that a biharmonic function is completely determined by its values and the normal derivatives at $V_0$. We put $a_i = u(q_i),$ $b_i = \partial_n u(q_i),$ and $d_i = 2a_i - a_{i+1} - a_{i-1} - b_i$. We can interpret these $d_i$ as a measurement of how much the function differs from a harmonic function since all the $d_i$ are 0 if and only if the function is harmonic. By using equation (\[matrixeq\]) we can solve $c_i$ in terms of $a_i$ and $b_i$. Reorganizing the equation in terms of $d_i$ yields $$\begin{aligned}
\Theta(a_i,b_i) := \int |\Delta u|^2 d \mu &=\int \left( \sum_{i=0}^{2} c_i h_i\right)^2 d \mu \\
&= 90 \sum_{i} a_i^2 +
30 \sum_{i\neq j}a_i b_j +
11 \sum_{i}b_i^2 - 90 \sum_{i < j} a_i a_j -
60 \sum_{i}a_i b_i - 8 \sum_{i < j}b_i b_j\\
&=11 \sum_{i}d_i^2- 8 \sum_{i < j}d_i d_j.
\end{aligned}$$ Using the self similar identity we get for $V_m$: $$\int|\Delta u|^2 d\mu = \left(\frac{25}{3}\right)^{m}\sum_{|w|=m} \Theta(F_w q_i,(\frac{5}{3})^m \partial_n F_w q_i).$$
\
\
We use Lemma \[lemma: laplacianconstruction\], to show that there is a unique function which satisfies the following properties, and that function is the minimizer. Namely the function is piecewise biharmonic on $V_m$ and its Laplacian is harmonic away from $E$. The exact conditions are the following,
1. $u(x_i) = a_i,$
2. $\partial_n u(x_i) = b_i,$
3. $u \text{ is piecewise biharmonic on } V_m,$
4. $\Delta_\zeta u \text{ is continuous at } y \text{ for } y \in V_m \setminus E,$
5. $\Delta_\zeta u \text{ has matching normal derivatives at } y \text{ for } y \in V_m \setminus E.$
From Lemma \[lemma: laplacianconstruction\] we get that given a function $u$ which satisfies (1-5) must satisfy (\[eulerlagrangelaplacian\]) and from Lemma \[lemma:laplacianuniqueness\] we have that $u$ must be the only function satisfying (1-5). Thus we only need to show the existence of a function $u$ which satisfies these properties. To show this we use the guaranteed existence of a minimizer and conclude that the minimizer must satify these properties.
\[lemma: minimizerpiecewisebiharmonic\] Let $E \subset V_m$. The minimizer $u$ must be piecewise biharmonic on $V_m$.
Let $\tilde{u}$ be the minimizer subject to the constraints $\tilde{u}(z) = u(z)$ and $\partial_n \tilde{u}(z) = \partial_n u(z)$ for all $z\in V_m$. Since these constraints include the original ones, we must have $T_{\zeta}(u)\leq T_{\zeta}(\tilde{u})$. Furthermore since $u$ satisfies the constraints for the new problem, we have $T_{\zeta}(\tilde{u}) \leq T_{\zeta}(u)$. By uniqueness of the minimizer we immediately get $u = \tilde{u}$.\
By Corollary \[corollary: biharmonicminimizer\], we have $u = \tilde{u}$ is piecewise biharmonic.
$\Delta u$ is continuous and has matching normal derivaties at all $y \in V_m \setminus E$, where $u$ is the minimizer.
Since $u$ is piecewise biharmonic, we consider the piecewise biharmonic functions $v_z,w_z \in {\text{dom}}_{L^2} \Delta$ for $z \in V_m \setminus E$ which satisfy $$v_z|_{V_m} \equiv 0,$$ $$\partial_n v_z|_{V_m \setminus \{z\}} \equiv 0,$$ $$\partial_n v_z(z) = 1,$$ and $$w_z|_{V_m \setminus \{z\}} \equiv 0,$$ $$w_z(z) = 1,$$ $$\partial_n w_z|_{V_m} \equiv 0.$$ In equation (\[equation:biharmonic\]) we put $v=v_z$ and by Lemma \[lemma:eulerlagrangelaplacian\] we get the continuity at $z$. For the second part, this time plugging in $v=w_z$ in equation (\[equation:biharmonic\]) we obtain the matching normal derivatives at $z$. Hence, the result.
Hence, we have fully characterized the unique minimizer as the function which is piecewise biharmonic on $V_m$ and has a Laplacian which is harmonic away from $E$.
Constraining Values Only
------------------------
In this section, we only fix the values on $E.$ When $E \subset V_*$ we can view this as minimizing over the normal derivatives using our previous model. We have already seen existence and uniqueness. Also, the previous section guarantees that the minimizer is piecewise biharmonic. Here we find some properties of the minimizer. Throughout this section we assume $\zeta = \mu$. We write $\Delta = \Delta_{\mu}$ and $T = T_{\mu}$.
First we get the equivalent Euler Lagrange equation as before; in this case we allow more sample functions $v$ since we have relaxed the constraints. The proof is the same as for Lemma \[lemma:eulerlagrangelaplacian\].
If $u$ is a minimizer of $T$ with respect to the constraints $u(x_i) = a_i$ then $u$ satisfies $$\label{equation:eulerlagrange2}\int \Delta u \Delta v d \mu = 0$$ for all $v \in {\text{dom}}_{L^2}\Delta$ with $v|_E \equiv 0$.
Using this we get two new properties of the minimizer $u$.
The minimizer satisfies $u \in {\text{dom}}\Delta$ and $\Delta u|_{V_0} = 0.$
Define the piecewise biharmonic function $v_z$ with $$v_z|_{V_m} \equiv 0,$$ $$\partial_n v_z|_{V_m \setminus \{z\}} \equiv 0,$$ $$\partial_n v_z(F_w q_i) = 1.$$
For $z \in V_m \setminus V_0$ we have $z = F_w q_i = F_{w'} q_j$. Then applying Lemma \[lemma: laplacianconstruction\] and (\[equation:eulerlagrange2\]) we get $\Delta (u \circ F_w)(q_i) = \Delta (u \circ F_{w'})(q_j).$ This means that $\Delta u$ is continuous so $u \in {\text{dom}}\Delta$.\
For $z=q_i \in V_0$ we apply Lemma \[lemma: laplacianconstruction\] and (\[equation:eulerlagrange2\]) again to get $\Delta u (q_i) = 0$.
### Calculating the Minimizing Form {#section:calculations}
Using the continuity and the Dirichlet boundary conditions of $\Delta u$, we can relate $\Delta_m$ and $\Delta,$ giving us a system of equations to solve in order to calculate $T(u)$. Throughout this section, we continue to assume that $\zeta=\mu$.\
We calculate here some basic facts to develop a system of equations to find $T(u)$. Recall that $h_i$ is the standard basis for harmonic functions on SG . From [@SU] we have that
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~M_{i,j} = \int h_i h_j d \mu~~\text{is given explicitly as}$$ $$M_{i,i} = \frac{7}{45},$$ $$~~~~~~~~~~~~~M_{i,j} = \frac{4}{45} ~~\text{for}~i\neq j.$$
Now we relate the boundary data of a biharmonic function to its Laplacian. Suppose the biharmonic function satisfies $u(q_i) = a_i$ , $\partial_n u(q_i) = b_i$ and $\Delta u(q_i)=c_i. $ Given such a $u$ we have $$\label{matrixeq}
\left(
\begin{matrix}
2 & -1 & -1 \\
-1 & 2 & -1 \\
-1 & -1 & 2 \\
\end{matrix}\right)\left(
\begin{matrix}
a_0 \\
a_1 \\
a_2 \\
\end{matrix}\right)
+
\left( \begin{matrix}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1 \\
\end{matrix}\right)
\left(
\begin{matrix}
b_0 \\
b_1 \\
b_2 \\
\end{matrix} \right)
= -\frac{1}{45}\left(
\begin{matrix}
7 & 4 & 4 \\
4 & 7 & 4 \\
4 & 4 & 7 \\
\end{matrix}\right)\left(
\begin{matrix}
c_0 \\
c_1 \\
c_2 \\
\end{matrix}\right),$$
from which it follows that
$$\label{equ:partialuqi}
\partial_n u(q_i) = 2u(q_i) - u(q_{i+1}) - u(q_{i-1}) + \frac{1}{45}(7\Delta u(q_i) + 4 \Delta u(q_{i+1}) + 4 \Delta u(q_{i-1}))$$
and $$\Delta u(q_i) = -15(2u(q_i) - u(q_{i+1}) - u(q_{i-1}))+11\partial_nu(q_i) - 4\partial_nu(q_{i+1}) -4\partial_nu(q_{i-1}).$$
Given any $z=F_w(q_i)=F_{w'}(q_{j}) \in V_m \setminus V_0$, we have $$\Delta_m u(z) = \frac{1}{45\cdot5^m}(14 \Delta u(z) + 4 \sum_{y \underset{m}{\sim} z} \Delta u (y)).$$
Replacing $u$ by $u\circ F_w$ in equation (\[equ:partialuqi\]), we have
$$-\partial_n (u \circ F_w)(q_i) = - 2 (u\circ F_w)(q_i) + (u\circ F_w)(q_{i-1}) + (u\circ F_w)(q_{i+1})$$ $$- \frac{1}{45} (7 \Delta (u\circ F_w)(q_i)+ 4 \Delta (u\circ F_w)(q_{i-1}) + 4 \Delta (u\circ F_w) (q_{i+1}))$$ and
$$-\partial_n (u \circ F_{w'})(q_j) = - 2 (u\circ F_{w'})(q_j) + (u\circ F_{w'})(q_{j-1}) + (u\circ F_{w'})(q_{j+1})$$ $$- \frac{1}{45} (7 \Delta (u\circ F_{w'})(q_j)+ 4 \Delta (u\circ F_{w'})(q_{j-1}) + 4 \Delta (u\circ F_{w'}) (q_{j+1})).$$
Then, summing them up and using $\Delta(u \circ F_w)=r_{w} \mu_{w}\Delta u \circ F_w$, $r_w=\left(\frac{3}{5} \right)^m~ \text{and}~ \mu_w=\left(\frac{1}{3}\right)^m$ yields the result.
Then we have $|V_m\setminus V_0|$ number of equations and we know that $\Delta u|_{V_0}=0$ so we can solve the above system of equations and express $\Delta u |_{V_m\setminus V_0}$ in terms of $u|_{V_m}$. By Gauss-Green, the fact that $\Delta u$ is piecewise harmonic, and the fact $\Delta u$ vanishes on the boundary we have $$\begin{aligned}
\int_{SG} |\Delta u(x)|^2 d \mu(x)=-{\mathcal{E}}(\Delta u,u)&=-\sum_{|w|=m}r_w^{-1}{\mathcal{E}}(\Delta u \circ F_w,u \circ F_w)\\
&=-r^{-m}\sum_{|w|=m}{{\mathcal{E}}_0(\Delta u \circ F_w, u \circ F_w)}\\
&=-r^{-m}\sum_{V_m \setminus V_0}{\Delta u(x) \left(\sum_{y \underset{m}{\sim} x}{u(y) - u(x)}\right)}\\
&=r^{-m}\sum_{V_m \setminus V_0}\Delta u \Delta_m u.\end{aligned}$$
### Generalization to all of $SG$
In this section, we present a construction for the minimizer when $E = \{x_1, \dots, x_n\} \subset SG$ is not restricted to the junction points. As before we require Dirichlet boundary conditions. This construction parallels the construction for the energy minimizer. Note that Theorem \[theorem:laplacian1\] generalizes immediately to all $E \subset SG$ so we get uniqueness of the minimizer.
\[generallaplacianeuler\] Suppose $u(x)=\sum_{i=1}^{n} c_i \int G(x,y)G(x_i,y) d\mu (y)$ for some constants $c_i$. Then for all $v \in dom_{L^2}\Delta$ such that $v|_{V_0 \cup E}=0$,we have $$\int \Delta u \Delta v d \mu=0.$$
For such $v$, we have $$\int \Delta u \Delta v d \mu=\int (-\sum c_i G(x_i,x))(\Delta v(x)) d\mu(x)=\sum_{i} c_i v(x_i)=0.$$
Thus if we can find a function of the above form satisfying the constraints, then we get a function which satisfies the neccessary condition so it is the unique minimizer.
$$[G]_{i,j}=\int G(x_i,y)G(x_j,y) d \mu (y)$$ is invertible. Thus we can find $c_i$ such that $u=\sum_{i=1}^{n} c_i \int G(x,y)G(x_i,y) d \mu (y)$ satisfies the constraints.
Suppose we have $c_i$ with $u=\sum_{i=1}^{n} c_i \int G(x,y)G(x_i,y) d\mu (y)$ such that $u_{E \cup V_0}=0$. Then we apply Lemma \[generallaplacianeuler\] with $v=u$ to get $\int |\Delta u|^2 d \mu=0.$ This implies $u$ is a harmonic function but it vanishes at boundary so it is zero. This proves the injectivity of $G$, which gives invertibility (and surjectivity). We can solve for $c_i$ such that $u(x_i) = a_i$.
[9]{}
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**Pak Hin Li**\
<span style="font-variant:small-caps;">Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong.</span>\
<span style="font-variant:small-caps;">E-mail:</span> **pakhinbenli@gmail.com**
**Nicholas Ryder**\
<span style="font-variant:small-caps;">Department of Mathematics, Rice University, 6100 Main St, Houston, TX 77005 USA.</span>\
<span style="font-variant:small-caps;">E-mail:</span> **nick.ryder@rice.edu**
**Robert S. Strichartz**\
<span style="font-variant:small-caps;">Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.</span>\
<span style="font-variant:small-caps;">E-mail:</span> **str@math.cornell.edu**
**Baris Evren Ugurcan**\
<span style="font-variant:small-caps;">Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.</span>\
<span style="font-variant:small-caps;">E-mail:</span> **beu4@cornell.edu**
[^1]: Research supported by the Mathematics Department of the Chinese University of Hong Kong and the Undergraduate Summer Research Fellowship from the office of Academic Links of the Chinese University of Hong Kong.
[^2]: Research supported by the National Science Foundation through the Research Experiences for Undergraduates(REU) Program at Cornell.
[^3]: Research supported by the US NSF grant DMS-1162045.
[^4]: Research supported by the US NSF grant DMS-1156350.
|
---
abstract: '[[ We examine the prospects for probing heavy top quark-antiquark ($\ttbar$) resonances at the upgraded LHC in pp collisions at $\roots$ = 14 TeV. Heavy $\ttbar$ resonances ($\zprime$ bosons) are predicted by several theories that go beyond the standard model. We consider scenarios in which each top quark decays leptonically, either to an electron or a muon, and the data sets correspond to integrated luminosities of $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ and $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$. We present the expected 5$\sigma$ discovery potential for a $\zprime$ resonance as well as the expected upper limits at 95% C.L. on the $\zprime$ production cross section and mass in the absence of a discovery.]{}]{}'
author:
- '[ Ia Iashvili,$^{1}$ Supriya Jain,$^{1}$ Avto Kharchilava,$^{1}$ Harrison B. Prosper$^{2}$ ]{}'
date: 'September 25, 2013'
title: |
Discovery potential for heavy $\ttbar$ resonances in dilepton+jets final states in pp collisions at $\roots = 14$ TeV\
A Snowmass 2013 Whitepaper
---
Introduction {#sec:intro}
============
An important goal of the LHC research programs is to deepen our understanding of electroweak symmetry breaking. Electroweak symmetry breaking in the standard model (SM) is closely associated with the existence of a neutral Higgs boson. Therefore, the discovery of a new boson [@Aad:2012tfa; @Chatrchyan:2012ufa] with properties consistent with those of the SM Higgs boson is clearly a monumental development. However, the top quark, by far the heaviest known fundamental particle, has a mass close to the electroweak scale, which suggests that it too may play a role in electroweak symmetry breaking. This alone provides ample motivation for the continued intense scrutiny of the top quark in all of its manifestations.
A generic prediction of many models that go beyond the standard model (BSM) is the existence of at least one heavy neutral boson, referred to generically as a $\zprime$, that preferentially decays to a $\ttbar$ pair and that appears as a resonant structure superimposed on the SM $\ttbar$ continuum production. These models include coloron models [@laneref31; @theory_hill_parke; @laneref32; @jainharris], models based on extended gauge theories with massive color-singlet Z-like bosons [@theory_rosner; @theory_lynch; @theory_carena], and models in which a pseudoscalar Higgs boson may couple strongly to top quarks [@theory_dicus]. Furthermore, various extensions of the Randall-Sundrum model [@theory_randall_sundrum] with extra dimensions predict Kaluza-Klein excitations of gluons [@theory_agashe], or gravitons [@theory_davoudiasl], both of which can have enhanced couplings to $\ttbar$ pairs. The recent observation of forward-backward asymmetry in $\ttbar$ production at the Tevatron [@cdf_ttbar_asymmetry1; @d0_ttbar_asymmetry1; @cdf_ttbar_asymmetry2; @d0_ttbar_asymmetry2] has inspired new models [@theory_bai; @theory_frampton; @theory_gresham; @theory_antunano; @theory_alvarez] that explain the observation by positing new physics at the TeV scale. The latter can manifest itself as a broad enhancement over the SM $\ttbar$ production at high invariant mass. The top quark, and $\ttbar$ production in particular, is a powerful probe of potential new physics.
Direct searches for heavy $\ttbar$ resonances have been performed at the Tevatron and the LHC. No such resonances have been found. The Tevatron experiments probed the mass range up to $\sim$900 GeV [@cdfZprime; @d0Zprime], while the LHC experiments have set sub-pb limits on the $\ttbar$ resonance production cross section in the mass range of 1–3 TeV depending on the $\zprime$ width, and have excluded the existence of a narrow width $\zprime$ ($\Gamma_{\zprime} = 0.012M_{\zprime}$) below $\Mzprime$ = 2.1 TeV at 95% C.L.[@cms_ttbar_resonance1; @atlas_ttbar_resonance1; @atlas_ttbar_resonance2; @cms_ttbar_resonance2; @cms_ttbar_resonance3; @cms_ttbar_resonance4].
The null results indicate that $\ttbar$ resonances, if they exist, must have masses in the TeV range or higher. In this paper, we examine how high a mass can be expected to be probed using $\zprime$ $\to$ $\ttbar$ $\to$ W$^+$b W$^-$ $\mathrm{\bar{b}}$ production in $\pp$ collisions at the upgraded LHC operating at $\roots$ = 14 TeV. We consider final states in which both W bosons decay to leptons (electron or muon), that is, final states comprising two high $\pt$ leptons of opposite charge (e$^+$e$^-$, $\mu^+\mu^-$, or e$^\pm\mu^\mp$), at least two jets from the hadronization of $\mathrm{b/\bar{b}}$ quarks, and missing transverse momentum due to escaping neutrinos. Top quarks from the decay of a heavy $\zprime$ are expected to be highly boosted leading to decay products that may not be spatially well separated. Consequently, we expect events would contain a non-isolated lepton from W$\to \ell \nu$ decay that is partially or fully overlapped with the b-quark jet from t$\to$ Wb decay. The dominant (irreducible) background is the $\ttbar$ continuum production. Other SM processes contributing to the background are the production of single top quarks, $\rm{Z}/\gamma^*/\rm{W}$+jets, and dibosons (WW, WZ, and ZZ). We consider two potential data sets, one corresponding to $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ and the other to $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$, as anticipated by the end of Run 2 of the upgraded LHC and by the end of the High Luminosity LHC (HL-LHC) runs, respectively.
Signal and Background Samples {#sec:modeling}
=============================
This study considers four different $\zprime$ mass hypotheses, $\Mzprime$ = 2, 3, 4 and 5 TeV, and assumes a resonance width of $\Gamma_{\zprime} = 0.012M_{\zprime}$. For each mass hypothesis, signal event samples are generated using the [pythia]{} program [@Pythia8]. The expected signal yields are computed using the leading-order (LO) cross sections for a leptophobic $\zprime$ [@jainharris] scaled by a K-factor of 1.3 [@ZprimeKfactor] to approximate the cross section at next-to-leading-order (NLO). The SM background samples are generated using the [Madgraph]{} event generator [@SMbkg] and higher-order and non-perturbative effects are approximated using [pythia]{} through its parton showering and hadronization models. The LO cross sections for the background processes are obtained from the event generator and corrected for NLO effects [@snowmass_mc]. The detector response to the simulated events is computed using the “Combined Snowmass LHC detector" [@SnowmassSimulation], which is implemented in the [Delphes-3]{} fast simulation program [@delphes]. The [Delphes-3]{} program can be used to model (to an accuracy of about 10 – 20%) the projected performance of future ATLAS [@atlas] and CMS [@cms] detectors at the upgraded LHC. The program also supports the simulation of additional pp interactions per bunch crossing (that is, in-time pile-up). We use samples that correspond to two different luminosity and pile-up (PU) scenarios at $\roots$ = 14 TeV: $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$, with an average number of pile-up events of $<\rm{PU}>$ = 50 events per bunch crossing (LHC Run 2), and $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$, with $<\rm{PU}>$ = 140 events per bunch crossing (HL-LHC).
Event Selection and Yields {#sec:selection}
==========================
We select $\zprime \rightarrow \ttbar \rightarrow 2\ell+2\nu+\bbbar$ candidate events by requiring two oppositely charged leptons, each with $\pt > 20$ GeV and pseudorapidity $|\eta| < 2.4$, and at least two jets within $|\eta| < 2.4$ and with $\pt > 30$ GeV. In addition, events are required to have $\met > 30$ GeV and at least one b-tagged jet, where the b-tagging efficiency is assumed to be $\sim 65\%$ [@snowmass_mc]. In order to reduce the background from low-mass dilepton resonances, events are rejected if the dilepton mass $M_{\ell\ell} < 12$ GeV. The remaining events are split into three disjoint categories depending on the lepton flavors, the $ee$, $\mu\mu$, and $e\mu$ channels. In the $ee$ and $\mu\mu$ channels, the contribution from Z+jets production is suppressed by vetoing events with $76 < M_{\ell\ell} < 106$ GeV. We refer to the sample at this stage as the “pre-selected" sample.
Starting with the pre-selected sample, selection cuts are optimized using the Random Grid Search (RGS) method [@rgs] and the signal significance measure $S/\sqrt{B}$, where S is the expected number of $\zprime$ signal events with $\Mzprime = 2$ TeV, and $B$ is the total expected background. Since the signal-to-background separation power increases with the hypothesized $\zprime$ mass, the set of cuts optimized for $\Mzprime = 2$ TeV also yields good discrimination between signal and background for higher $\Mzprime$ values. The selection optimization is performed separately for the $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ and $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$ scenarios.
The kinematic variables used in the RGS procedure are the transverse momenta of the two leading leptons and the two leading jets, and the missing transverse momentum. In addition, we use two highly discriminating variables. The first is the separation between the lepton and the closest jet in the space $\Delta R=\sqrt{\Delta\eta^2 + \Delta\phi^2}$, where $\Delta\eta$ and $\Delta\phi$ are the pseudorapidity and the azimuthal angle differences, respectively, between the lepton and jet. The boosted top quarks from the decay of a heavy $\zprime$ produce a lepton and b-quark that are close together in space. We therefore expect $\Delta R$ to be smaller on average for the signal than for the background processes, which, unlike the signal, do not contain highly boosted particles. Figure \[fig:dRJetLep1\_b1\] shows an example of the distribution of the $\Delta R$ between the leading lepton and the closest jet in the $ee$ channel in the pre-selected sample.
The other highly discriminating variable is a mass variable $M$. The mass variable $M$ is computed from the four-momenta of the two leading leptons, the two leading jets and a four-momentum formed from the $p_x$ and $p_y$ components of the missing transverse momentum with the $p_z$ set to zero. The distributions of the mass variable for the backgrounds and for the signal with $\zprime$ masses of 2 TeV and 3 TeV, after *all* selections, are shown in Figs. \[fig:invmass300\] and \[fig:invmass3000\] for the $\int\mathcal{L}\rm{dt} = 300$ $\rm{fb^{-1}}$ and 3000 $\rm{fb^{-1}}$ scenarios, respectively. A heavy $\zprime$ produces higher values of $M$ than the background processes. Table \[tab:selections\] summarizes the final selection cuts obtained from the RGS for the two luminosity scenarios. The expected event yields are given in Table \[tab:yields\].
![Distribution of $\Delta R$ between the leading lepton and closest jet in the $ee$ channel in the pre-selected cuts sample. Shown are contributions from the SM background processes and the $\zprime$ signal assuming $\Mzprime$ 2 TeV and $\int\mathcal{L}\rm{dt} = 300$ $\rm{fb^{-1}}$ luminosity. \[fig:dRJetLep1\_b1\]](fig1){width="50.00000%"}
------------------------------------------- ---------------------------------------------------- -----------------------------------------------------
LHC luminosity scenario $\int\mathcal{L}\rm{dt} = 300$ $\rm{fb^{-1}}$ $\int\mathcal{L}\rm{dt} = 3000$ $\rm{fb^{-1}}$
\[3mm\] Leading lepton $p_T>$ 100 GeV 100 GeV
Second leading lepton $p_T>$ 30 GeV 20 GeV
Leading jet $p_T>$ 175 GeV 550 GeV
Second leading jet $p_T>$ 150 GeV 100 GeV
$\met >$ 95 GeV 35 GeV
$\Delta R(\rm {lepton, closest~jet} ) < $ 0.6 1.2
$M >$ 1500 GeV –
------------------------------------------- ---------------------------------------------------- -----------------------------------------------------
: Summary of the final selection cuts obtained from the RGS for the two LHC luminosity scenarios at $\roots=14$ TeV.
\[tab:selections\]
[|l|r|r|]{}LHC luminosity scenario & $\int\mathcal{L}\rm{dt} = 300$ $\rm{fb^{-1}}$ & $\int\mathcal{L}\rm{dt} = 3000$ $\rm{fb^{-1}}$\
\
$\zprime$ $\Mzprime$ = 2 TeV & 1395 & 22534\
$\zprime$ $\Mzprime$ = 3 TeV & 446 & 5955\
$\zprime$ $\Mzprime$ = 4 TeV & 85.7 & 1118\
$\zprime$ $\Mzprime$ = 5 TeV & 14.5 & 184\
\
& 17599 & 427058\
single top & 2044 & 50545\
$\rm{W}/\rm{Z}/\gamma^*$+jets & 2545 & 81740\
Diboson & 163 & 6384\
Total background & 22351 & 565727\
\[tab:yields\]
![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 300 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass300\]](fig2 "fig:"){width="33.00000%"} ![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 300 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass300\]](fig3 "fig:"){width="33.00000%"} ![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 300 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass300\]](fig4 "fig:"){width="33.00000%"}
![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 3000 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass3000\]](fig5 "fig:"){width="33.00000%"} ![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 3000 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass3000\]](fig6 "fig:"){width="33.00000%"} ![Distributions of the mass variable $M$ for the $ee$, $\mu\mu$, and $e\mu$ channels for 3000 $\rm{fb^{-1}}$ after selection cuts are applied. \[fig:invmass3000\]](fig7 "fig:"){width="33.00000%"}
Expected Discovery Reach and Limits {#sec:results}
====================================
In order to quantify the expected 5$\sigma$ discovery or 95% C.L. exclusion limit for a $\zprime$ resonance, we use the Bayesian method [@bayesian] implemented in the statistical software package [theta]{} [@theta]. A multi-Poisson likelihood, constructed from the binned mass distributions of all three channels ($ee$, $\mu\mu$, and $e\mu$), is combined with a flat prior for the signal cross section. The following systematic uncertainties are accounted for in the signal and background models, assuming full correlation across channels: 10% in the cross section normalization for each background process, 10% in the b-tagging efficiency, and 2% in the jet-energy scale.
Figure \[fig:limits\] (left) shows the $\zprime$ production cross section times the branching fraction to $\ttbar$ ($\sigma_{\zprime}\mathcal{B}$), as a function of $\Mzprime$, that would yield a signal with a statistical significance of 5$\sigma$ at $\roots = 14$ TeV, that is, a discovery, with integrated luminosities $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ and $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$. The cross section times branching fraction, $\sigma_{\zprime}\mathcal{B}$, ranges from 6 – 300 (2 – 60 ) fb with $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ ($\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$) for the mass range 2–5 TeV. Comparing these with the theoretical prediction for the production cross section of a leptophobic $\zprime$ yields the expected $\zprime$ discovery mass reach of 2.8 TeV with $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ and 4.1 TeV with $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$.
Figure \[fig:limits\] (right) shows expected 95% C.L. limits on $\sigma_{\zprime}\mathcal{B}$ as a function of $\Mzprime$ for the two luminosity scenarios. The expected limits range from 2 – 100 (1–20) fb with $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ ($\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$) for the mass range 2–5 TeV. Comparing these with the predicted production cross section for a leptophobic $\zprime$ shows that we can expect to exclude the existence of a $\zprime$ with mass $< 4.4$ (4.7) TeV at 95% C.L. with $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ ($\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$) should we fail to make a discovery.
![Required $\sigma_{\zprime}\mathcal{B}$ for a 5$\sigma$ observation (left) and upper limits at 95% C.L. on $\sigma_{\zprime}\mathcal{B}$ (right) as a function of $\Mzprime$ for narrow-width, leptophobic $\zprime$ resonances. Also shown is the theoretical prediction for the $\zprime$. \[fig:limits\]](fig8 "fig:"){width="48.00000%"} ![Required $\sigma_{\zprime}\mathcal{B}$ for a 5$\sigma$ observation (left) and upper limits at 95% C.L. on $\sigma_{\zprime}\mathcal{B}$ (right) as a function of $\Mzprime$ for narrow-width, leptophobic $\zprime$ resonances. Also shown is the theoretical prediction for the $\zprime$. \[fig:limits\]](fig9 "fig:"){width="48.00000%"}
Summary {#sec:summary}
=======
We have assessed the potential for finding evidence of a leptophobic $\zprime$ boson in $\zprime \rightarrow \ttbar \rightarrow 2\ell+2\nu+\bbbar$ decays in $\pp$ collisions at $\roots$ = 14 TeV. Two sets of hypothetical data, simulated using [pythia]{}, [Madgraph]{} and [Delphes]{}, have been analyzed assuming an integrated luminosity of $\int\mathcal{L}\rm{dt} = 300~\rm{fb^{-1}}$ with an average number of events per bunch crossing (pile-up) of $\rm{<PU>}=$ 50, and $\int\mathcal{L}\rm{dt} = 3000~\rm{fb^{-1}}$ with $\rm{<PU>}=$ 140. For the lower (higher) integrated luminosity, our study indicates that it is possible to discover a $\zprime$ up to a mass 2.8 (4.1) TeV with a statistical significance of 5$\sigma$. Should we fail to make a discovery, the existence of a $\zprime$ with mass $< 4.4$ (4.7) TeV can be excluded at 95% C.L. using data associated with the lower (higher) integrated luminosity scenario.
Acknowledgments {#sec:acknowledge .unnumbered}
===============
The authors would like to thank James Dolen, John Stupak, Sergei Chekanov, and Johannes Erdmann for their help in setting up tools as well as providing the samples for this study.
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|
---
address: |
DPNC, University of Geneva,\
24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland\
e-mail: Arno.Straessner@cern.ch
author:
- ARNO STRAESSNER
title: MEASUREMENT OF THE W BOSON MASS AT LEP
---
Introduction
============
At LEP, W bosons are produced in the reaction ${\mathrm{e^+e^-}}{\rightarrow}{\mathrm{W^+W^-}}$ with the subsequent decay of the W’s into quark pairs, ${\mathrm{q} \overline{\mathrm{q^\prime}}}$, or a lepton and a neutrino, ${\ell \nu}$. About 40000 W pair events are registered by the four experiments ALEPH, DELPHI, L3 and OPAL, corresponding to a total luminosity of $2.8$ $\mathrm{fb}^{-1}$.
One of the main goals of the LEP programme is to determine the mass of the W boson, ${M_{\mathrm{W}}}$, from the reconstructed invariant mass spectra. Involved techniques are used to obtain an optimal statistical precision. However, in the fully hadronic channel, systematic uncertainties are important, like final state interactions (FSI) between the hadronically decaying W bosons. These uncertainties reduce the sensitivity of this channel. The recent activities concentrate on increasing the weight of the hadronic events to obtain a globally more precise result on the LEP W mass.
Extraction of the W Mass
========================
In the semi-leptonic and fully hadronic WW decay channels the complete four-fermion final states, shown in Figure \[fig:spectrum\] [[-0.25em a]{}]{}, can be reconstructed. The most sensitive observable for the W mass measurement is the invariant mass of the decaying W bosons. To improve the resolution on this quantity, a kinematic fitting procedure is applied. The reconstructed four-momenta of the final state fermions are varied within their resolution and kinematic constraints, like energy-momentum conservation and equal invariant masses in each event, are imposed. In fully hadronic events the resolution is diluted due to the different pairing combinations of the final state jets. In general, the most probable pairings according to the kinematics are chosen. Figure \[fig:spectrum\] [[-0.25em b]{}]{} shows an example of the invariant mass spectrum measured in the ${\mathrm{q q q q}}$ channel.
In ${\ell \nu \ell \nu}$ final states the event kinematic can not be reconstructed completely and the leptonic energy spectrum and a pseudo-mass are chosen as optimal W mass estimators.
To obtain W mass and width, three basic extraction methods are used: in the Monte Carlo reweighting procedure, the underlying ${M_{\mathrm{W}}}$ and ${\Gamma_{\mathrm{W}}}$ values of the Monte Carlo prediction are varied and compared to the measured spectra; in the convolution method the theoretically predicted spectra are folded with resolution functions and fitted to data; in the Breit-Wigner method the resonance curve is fitted to data and possible bias is calibrated with Monte Carlo samples. All methods exploit information from several event observables, including the invariant masses obtained in kinematic fits with different constraints, as well as their uncertainties.
With the current techniques an equal statistical precision of $32\MeV$ and $35\MeV$ is reached in the ${{\mathrm{q q} \ell \nu}}$ and ${\mathrm{q q q q}}$ channel, respectively.
Systematic Uncertainties
========================
A common source of systematic uncertainty in the ${{\mathrm{q q} \ell \nu}}$ and ${\mathrm{q q q q}}$ channels is the theoretical description of photon radiation that is used in the Monte Carlo simulation. In total, the uncertainty on initial state radiation (ISR), final state radiation (FSR) and ${\cal{O}(\alpha)}$ electroweak corrections amounts to $8\MeV$ in all channels. The current numbers are based on reweighting of Monte Carlo events according to possible differences in the theoretical prediction of the photon spectrum. However, recent comparisons between two different Monte Carlo generators, YFSWW [@yfsww] and RacoonWW [@racoon], show larger differences. The uncertainty is therefore expected to be underestimated and may increase to $10-15\MeV$ for the final LEP W mass measurement.
Another common systematic error source is the description of the hadronisation of quarks. The uncertainty is mainly derived from the difference between the hadronisation models Pythia, Herwig and Ariadne [@frag-models]. Especially the simulated rate of heavy baryons influences the jet masses and therefore the derived invariant masses. A reweighting to the measured baryon rates is performed by Opal. Delphi also compares mixed Lorentz boosted Z decays, that are arranged to reproduce W-pair decay kinematics, to the hadronisation models. Since the agreement between the different models and data have improved, the current systematic uncertainty of $18-19\MeV$ may be reduced in future.
A similar reduction of systematic uncertainty, which is not yet included in the LEP combined W mass value, is originating from the LEP beam energy uncertainty. The final LEP energy calibration has improved [@elep] with respect to the current calibration applied. It will result in a much smaller uncertainty of only $10\MeV$, with only a small change of the average LEP beam energies. The LEP experiments also performed a cross-check of the LEP energy measurement by reconstructing the Z boson mass in radiative fermion-pair events, as shown in Figure \[fig:zret\]. Both Opal and L3 obtain results in good agreement with the LEP energy calibration [@zret].
![a) Invariant Z mass spectrum reconstructed in ${\mathrm{q} \overline{\mathrm{q}}}\gamma$ events. b) Difference between the LEP beam energy and the beam energies determined by Opal in an analysis of radiative fermion-pair events. \[fig:zret\]](l3_qq_mz.ps "fig:"){width="0.49\linewidth"} ![a) Invariant Z mass spectrum reconstructed in ${\mathrm{q} \overline{\mathrm{q}}}\gamma$ events. b) Difference between the LEP beam energy and the beam energies determined by Opal in an analysis of radiative fermion-pair events. \[fig:zret\]](opal_mz_plot.ps "fig:"){width="0.49\linewidth"}
Z decays at high centre-of-mass energies as well as in Z peak calibration data are analysed to test the detector simulation. Energy and angular measurement of leptons and jets in data are compared to Monte Carlo simulation. Possible differences are corrected and the remaining uncertainty on the difference is taken as systematic uncertainty, which amounts to $14\MeV$ on the combined W mass result.
--------------------------------------- ---------------------------------- --------------------------- ------------------------
Source
${{\mathrm{q q} \ell \nu}}$ ${\mathrm{q q q q}}$ ${\mathrm{ffff}}$
ISR/FSR 8 8 8
Hadronisation 19 18 18
LEP Beam Energy 17 17 17
Detector Systematics 14 10 14
Colour Reconnection – 90 9
BE Correlations – 35 3
Other 4 5 4
Total Systematic 31 101 31
Statistical 32 35 29
Total 44 107 43
Statistical in absence of Systematics 32 28 21
--------------------------------------- ---------------------------------- --------------------------- ------------------------
: Systematic uncertainties in the W mass measurement.\[tab:syst\]
Final State Interactions
========================
A special complication in reconstructing the invariant masses appears in the fully hadronic channel. Hadronic FSI may introduce cross-talk between the two decaying W bosons of one event. Bose-Einstein Correlations (BEC) lead to an increased production of identical bosons, like pions and kaons, close in phase space. Colour Reconnection (CR) changes the colour flow between the four quarks in the final state. By reconnecting colour strings between the previously colour-neutral di-quark systems from each W decay, momentum is transferred between the W’s and the hadronisation process is altered.
As listed in Table \[tab:syst\], BEC and CR may change the reconstructed W mass by $35\MeV$ and $90\MeV$, respectively. The increase of CR effects with centre-of-mass energy is taken into account.
At LEP, FSI effects are measured also in other observables, which are used to constrain the various BEC and CR models. The BEC measurements in W pairs mainly concentrate on charged pion production. Assuming a spherical and gaussian shaped source emitting the pions, the two-particle correlation function $C$ can be written as: $$\begin{aligned}
C(Q_{\pi\pi}) = 1 - \lambda \exp(-R^2 Q^2_{\pi\pi}) \; ,\end{aligned}$$ where $Q_{\pi\pi}$ is the square of the four-momentum difference of the two pions, $-(p_{\pi,1}-p_{\pi,2})^2$. The parameter $\lambda$ is the correlation strength and $R$ is the inverse radius of the source. When analysing semi-hadronic W events, it is found that the BEC between pions coming from the same W boson agree very well with the correlations observed in Z decays, if $\mathrm{Z}{\rightarrow}\mathrm{bb}$ is suppressed.
Important for the mass measurement are, however, the correlations between pions from different W bosons. If there are no such correlations the two-particle density in fully hadronic events can be split into three terms: $$\begin{aligned}
\rho^{\mathrm{WW}} (1,2) = \rho^{\mathrm{W+}}(1,2) + \rho^{\mathrm{W-}}(1,2) +
2 \rho^{\mathrm{W^+}}(1)\rho^{\mathrm{W^-}}(2) \; .\end{aligned}$$ The first two terms on the right-hand side of the equation are the density functions for pions coming from the same W. They can be determined in semi-hadronic events. The last term describes the case when one pion comes from one W boson and the second from the other W boson. This part can be constructed from a sample of mixed semi-hadronic event: $\rho^{\mathrm{W^+}}(1)\rho^{\mathrm{W^-}}(2) =
\rho^{\mathrm{W^+W^-}}_{\mathrm{mix}}$. If the equation holds, i.e. in absence of BE correlations between two W’s, the following ratio and difference of densities $$\begin{aligned}
D &=& \frac{\rho^{\mathrm{WW}}}{2 \rho^{\mathrm{W}} +
2 \rho^{\mathrm{W^+W^-}}_{\mathrm{mix}}} \\
\Delta\rho &=& \rho^{\mathrm{WW}}- 2 \rho^{\mathrm{W}} -2
\rho^{\mathrm{W^+W^-}}_{\mathrm{mix}}\end{aligned}$$ are equal to 1 and 0, respectively. Figure \[fig:bec\] [[-0.25em a]{}]{} shows the quantity $D$ as a function of $Q$ as measured by Delphi. The Delphi data are consistent with moderate BEC between pions from different W’s. However, the combination of all LEP experiments prefers the absence of those correlations [@ewwg-note], as shown in Figure \[fig:bec\] [[-0.25em b]{}]{}. Monte Carlo studies show that the corresponding W mass shift is proportional to the BEC strength, so that the systematic uncertainty on ${M_{\mathrm{W}}}$ due to BEC can in future be reduced to $10-15\MeV$ using the direct BEC measurement.
![a) The Delphi measurement of the $D$ parameter shows a preference for BEC, visible as an enhancement of $D$ for low $Q$. b) BEC results from the LEP experiments and their combination. The individual results are normalised such that the maximal BEC effect yields a value of 1 and the absence of BEC a value of 0. \[fig:bec\]](be_delphi_dfit.ps "fig:"){width="0.45\linewidth"} ![a) The Delphi measurement of the $D$ parameter shows a preference for BEC, visible as an enhancement of $D$ for low $Q$. b) BEC results from the LEP experiments and their combination. The individual results are normalised such that the maximal BEC effect yields a value of 1 and the absence of BEC a value of 0. \[fig:bec\]](belep_summer03_4LEP.eps "fig:"){width="0.35\linewidth"}
A similar approach is made in the reduction of the CR uncertainty. Different models [@cr] predict CR, for example the Sj[ö]{}strand-Khose (SK) models 1 and 2, the Herwig model, or the Ariadne models 1-3. CR mainly manifests in a distortion of the W mass distribution. Effects on particle multiplicities are too small to be detected in data [@aleph-ch]. The only other observable sensitive to CR is the particle flow in the regions between the quark jets. Figure \[fig:cr\] [[-0.25em a]{}]{} shows the angular distribution of particles in ${\mathrm{q q q q}}$ events, after rescaling the angle of the particle to the next jet in such a way that each of the four jets is positioned at an integer angular value. A ratio $R$ is then calculated between the particle flow of the regions that connect two jets of one W (A+B) and the region that connects two jets of different W’s (C+D): $$\begin{aligned}
R=\frac{d\phi/dN(A+B)}{d\phi/dN(C+D)}\end{aligned}$$ To quantify the difference in the two regions A+B and C+D, the $R$ distribution is integrated. A variable $r$ is then constructed as the ratio of integrals $R_N$ obtained in data and in a Monte Carlo without CR: $$\begin{aligned}
r(\mathrm{data})=R_N(\mathrm{data})/R_N(\mathrm{MC\ witout\ CR})\end{aligned}$$ Figure \[fig:cr\] [[-0.25em b]{}]{} shows the result of the LEP experiments, using all data. In case all LEP experiments are combined, the SK1 model with 100% reconnection would yield a value of $r(\mathrm{SK1-100\%})=0.891$, while a value of $r(\mathrm{data})=0.969\pm0.015$ is obtained in the LEP measurement. Data therefore excludes strong CR effects as predicted by SK1 with 5.2$\sigma$. The preferred values of the SK1 model parameter $k_i$ are in the range $[0.39,2.13]$ [@cr-lep].
![a) Particle flow measured by L3 in the regions inside the W’s (A.B) and between W’s (C,D). b) W mass bias due to CR for different jet reconstruction methods. Restricting the jet cone size and the energy threshold reduces the CR effect. \[fig:cr\]](l3_cr_abcd.eps "fig:"){width="0.49\linewidth"} ![a) Particle flow measured by L3 in the regions inside the W’s (A.B) and between W’s (C,D). b) W mass bias due to CR for different jet reconstruction methods. Restricting the jet cone size and the energy threshold reduces the CR effect. \[fig:cr\]](delphi_cr_all_final.eps "fig:"){width="0.49\linewidth"}
The upper boundary of $k_i$ is used to estimate the systematic uncertainty on ${M_{\mathrm{W}}}$ due to CR. Averaged over all centre-of-mass energies one yields a possible mass shift of $90\MeV$. This shift is larger than those observed in other models, like Ariadne 2 or Herwig.
A further constraint on the CR effects can be put by the mass measurement itself. An indication that CR can not be strong comes from the comparison of the W mass measured in ${\mathrm{q q q q}}$ and ${{\mathrm{q q} \ell \nu}}$ events: $$\begin{aligned}
\Delta{M_{\mathrm{W}}}&=&{M_{\mathrm{W}}}({\mathrm{q q q q}})-{M_{\mathrm{W}}}({{\mathrm{q q} \ell \nu}})=+22\pm43\MeV \;,\end{aligned}$$ where systematic errors due to possible FSI are removed. The observed value is compatible with zero.
In addition, the W mass bias due to CR can also be reduced by modifying the jet algorithms that are used to reconstruct the W decay products. When reducing the cone size of the jets or by restricting the energy range of the objects clustered to a jet, the CR mass bias can be reduced, as shown in Figure \[fig:cr\] [[-0.25em b]{}]{}. Recent investigations have shown that this is the case for all CR models used. If one now varies the cone size or the energy cut-off, the mass shift observed in data and for the CR models can be compared. This result can also be combined with the particle flow measurement since the correlations between the measurements are small [@delphi-cr].
The four LEP experiments foresee to perform such an alternative mass analysis with reduced sensitivity to CR effects. This will enhance the weight of the ${\mathrm{q q q q}}$ channel in the LEP combination, which is currently only 0.10.
Results
=======
![a) Measurements of ${M_{\mathrm{W}}}$ by the four LEP experiments, using a common systematic uncertainty for FSI. b) LEP result compared and combined with the measurements in pp-collisions. The direct measurement of ${M_{\mathrm{W}}}$ is in good agreement with the indirect result obtained in fits to the remaining electroweak data. \[fig:res\]](mw_lep.eps "fig:"){width="0.35\linewidth"} ![a) Measurements of ${M_{\mathrm{W}}}$ by the four LEP experiments, using a common systematic uncertainty for FSI. b) LEP result compared and combined with the measurements in pp-collisions. The direct measurement of ${M_{\mathrm{W}}}$ is in good agreement with the indirect result obtained in fits to the remaining electroweak data. \[fig:res\]](w04_plot_mw.eps "fig:"){width="0.40\linewidth"}
Figure \[fig:res\] [[-0.25em a]{}]{} shows the individual results obtained by the LEP experiments. With the current systematic errors and all correlations properly included, the LEP W mass measurement split into ${{\mathrm{q q} \ell \nu}}$ and ${\mathrm{q q q q}}$ channel yields [@lep-mw]: $$\begin{aligned}
{M_{\mathrm{W}}}({{\mathrm{q q} \ell \nu}})&=&80.411\pm0.032\mathrm{(stat.)}\pm0.030\mathrm{(syst.)}\GeV \\
{M_{\mathrm{W}}}({\mathrm{q q q q}})&=&80.420\pm0.035\mathrm{(stat.)}\pm0.101\mathrm{(syst.)}\GeV \; ,\end{aligned}$$ with a correlation coefficient of 0.18. The combined mass value for all channels is $$\begin{aligned}
{M_{\mathrm{W}}}({\mathrm{ffff}})&=&80.412\pm0.029\mathrm{(stat.)}\pm0.031\mathrm{(syst.)}\GeV \; ,\end{aligned}$$ with a good $\chi^2/\mathrm{d.o.f}$ of 28.2/33. Including the result derived from the cross-section measurement at the W-pair production threshold does not change numerically the above result: $$\begin{aligned}
{M_{\mathrm{W}}}^{\mathrm{LEP}}&=&80.412\pm0.042\GeV \; .\end{aligned}$$
The method of direct reconstruction is also well suited to measure the width of the W boson, ${\Gamma_{\mathrm{W}}}$. A combined fit to the LEP data yields [@lep-mw]: $$\begin{aligned}
{\Gamma_{\mathrm{W}}}^{\mathrm{LEP}}&=&2.150\pm 0.068\mathrm{(stat.)}\pm0.060\mathrm{(syst.)}\GeV \; ,\end{aligned}$$ and agrees well with the Standard Model (SM) prediction of ${\Gamma_{\mathrm{W}}}=2.099$ using the LEP W mass cited above.
As shown in Figure \[fig:res\] [[-0.25em b]{}]{}, the LEP W mass agrees well with the measurement at $\mathrm{p\bar{p}}$ colliders [@ewwg-note]. The combination of these direct ${M_{\mathrm{W}}}$ measurements is also in good agreement with the indirect determination from the other electroweak data [@ewwg-note-new]. The result obtained in $\nu$N scattering by NUTEV, which is derived from the measurement of the electroweak mixing angle, $\sin^2\theta_w$, deviates from the LEP result by 2.9 $\sigma$. However, there is no systematic effect found that may explain this difference.
The W mass is an important parameter in the Standard Model. The precise measurement of ${M_{\mathrm{W}}}$ probes the SM at the level of its radiative corrections. A comparison of the direct and indirect W mass determinations [@ewwg-note-new] is shown in Figure \[fig:contours\] [[-0.25em a]{}]{} in the plane of W mass and top quark mass, ${M_{\mathrm{t}}}$, that is measured at the Tevatron. Good agreement between the two sets of measurements is observed. Also shown is the SM prediction for various values of the mass of the Higgs boson, ${M_{\mathrm{H}}}$. The measurements prefer small values of ${M_{\mathrm{H}}}$.
In a supersymmetric extension of the theory, the Minimal Supersymmetric Standard Model (MSSM), the preferred ${M_{\mathrm{W}}}$-${M_{\mathrm{t}}}$ area differs slightly from the SM prediction. Additional radiative correction terms, involving supersymmetric particles, shift the ${M_{\mathrm{W}}}$-${M_{\mathrm{t}}}$ band to larger values of ${M_{\mathrm{W}}}$. The MSSM prediction [@mssm-weiglein] is shown in Figure \[fig:contours\] [[-0.25em b]{}]{}. The experimental precision of the electroweak measurements is not accurate enough to decide between the two models. However, an increased precision is expected from measurements at the Tevatron, the future LHC and linear colliders. The electroweak data gives also motivation to continue the search for the Higgs boson in a low mass range, as the last missing piece of the SM, and for supersymmetric particles at the Tevatron and at LHC experiments.
![ a) Contour curves for the direct and indirect measurement of ${M_{\mathrm{W}}}$ and ${M_{\mathrm{t}}}$ compared to the SM prediction for different values of ${M_{\mathrm{H}}}$. b) The direct ${M_{\mathrm{W}}}$ and ${M_{\mathrm{t}}}$ measurements compared to the MSSM prediction. The exclusion limits from searches for new particles at LEP are taken into account in the calculation. The MSSM and SM areas overlap in the region where the mass of the SM Higgs boson is in the MSSM range, [*i.e.*]{} for ${M_{\mathrm{H}}}\stackrel{<}{\sim}130\GeV$. \[fig:contours\] ](w04_mt_mw_contours.eps "fig:"){width="0.49\linewidth"} ![ a) Contour curves for the direct and indirect measurement of ${M_{\mathrm{W}}}$ and ${M_{\mathrm{t}}}$ compared to the SM prediction for different values of ${M_{\mathrm{H}}}$. b) The direct ${M_{\mathrm{W}}}$ and ${M_{\mathrm{t}}}$ measurements compared to the MSSM prediction. The exclusion limits from searches for new particles at LEP are taken into account in the calculation. The MSSM and SM areas overlap in the region where the mass of the SM Higgs boson is in the MSSM range, [*i.e.*]{} for ${M_{\mathrm{H}}}\stackrel{<}{\sim}130\GeV$. \[fig:contours\] ](MWMT04.cl.eps "fig:"){width="0.49\linewidth"}
Conclusion
==========
The mass of the W boson is measured at LEP to ${M_{\mathrm{W}}}^{\mathrm{LEP}}=80.412\pm0.042\GeV$ and the width to ${\Gamma_{\mathrm{W}}}^{\mathrm{LEP}}=2.150\pm0.091\GeV$. It is an important contribution to the tests of the Standard Model and its supersymmetric extensions. Exploiting the recent results on systematic uncertainties, the ${M_{\mathrm{W}}}$ measurement is expected to reach a final accuracy of about $35\MeV$, completing the many precision tests of the SM performed at LEP.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank the experiments ALEPH, DELPHI, L3 and OPAL for making their most recent and preliminary results available. I also like to thank the LEP Electroweak and WW Working Groups as well as Georg Weiglein and collaborators for preparing their results in form of nice graphs and plots.
References {#references .unnumbered}
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S. Jadach, W. Placzek, M. Skrzypek, B. F. L. Ward and Z. Was, Comp. Phys. Comm. [**140**]{} (2001) 432.
A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, Comp. Phys. Comm. 153 (2003) 462.
T. Sj[ö]{}strand [*et al.*]{}, Comp. Phys. Comm. [**135**]{} (2001) 238; L. L[ö]{}nnblad, Comp. Phys. Comm. [**71**]{} (1992) 15; G. Corcella [*et al.*]{}, hep-ph/0011363; G.Marchesini [*et al.*]{}, Comp. Phys. Comm. [**67**]{} (1992) 465.
The LEP Energy Working Group, Memo “Final LEP 2 centre-of-mass energy values”, see [ http://lepecal.web.cern.ch/LEPECAL/]{}.
The ALEPH Collaboration, Internal Note 2003-002; The OPAL Collaboration, Internal Note PN 520; The L3 Collaboration, Phys. Lett. [**B 585**]{} (2004) 42.
The LEP Collaborations and the LEP Electroweak Working Group, Internal Note LEPEWWG/2003-02.
T. Sj[ö]{}strand and V.A. Khoze, Z. Phys. [**C 62**]{} (1994) 281; L. L[ö]{}nnblad, Z. Phys. [**C 70**]{} (1996) 107; G. Corcella [it et al.]{}, JHEP [**01**]{} (2001) 010.
The ALEPH Collaboration, Internal Note 2002-015.
The LEP Collaborations and the LEP WW Working Group, Internal Note LEPEWW/FSI/2002-02.
The DELPHI Collaboration, Internal Note 2003-021-CONF-641.
The LEP Collaborations and the LEP WW Working Group, Internal Note LEPEWWG/MASS/2003-01.
The LEP Collaborations and the LEP Electroweak Working Group, Internal Note LEPEWWG, to be published, see [http://lepewwg.web.cern.ch/LEPEWWG/plots/winter2004/]{}.
S. Heinemyer and G. Weiglein, “The MSSM in the Light of Precision Data”, hep-ph/0307177.
|
---
abstract: 'We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group $G$ on a Poisson manifold $M$, we find an explicit description of the lifted hamiltonian action on the symplectic groupoid $\Sigma(M)$. We give applications of these results to the integration of Poisson quotients $M/G$, Lu-Weinstein quotients $\mu^{-1}(e)/G$ and Poisson homogeneous spaces $G/H$.'
address:
- |
Departamento de Matemática\
Instituto Superior Técnico\
1049-001 Lisboa\
Portugal
- |
Instituto de Ciencias Matemáticas\
CSIC-UAM-UC3M-UCM\
C/ Serrano 123, 28006 Madrid\
Spain
author:
- Rui Loja Fernandes
- David Iglesias Ponte
title: 'Integrability of Poisson-Lie group actions'
---
[^1]
Introduction {#sec:introduction .unnumbered}
============
Let $(M,\pi)$ be a Poisson manifold, $G$ a Lie group and assume that $G$ acts on $M$ by Poisson diffeomorphisms. Such an action usually does not admit a momentum map (e.g., if the action is transverse to the symplectic leaves). However, there is a symplectization functor which turns this action into a hamiltonian action. More precisely, associated with an integrable Poisson manifold $(M,\pi)$ there is a canonical symplectic groupoid $\Sigma(M){\rightrightarrows}M$ and the action lifts to a hamiltonian action on the symplectic groupoid $(\Sigma(M),\Omega)$ with momentum map $J:\Sigma(M)\to{\mathfrak{g}}^*$ (see [@Fer; @FerOrRa; @Xu0] and references therein).
One should think of $J$ as a canonical momentum map which is attached to the Poisson action, and which always exists. This map satisfies $$J(x\cdot y)=J(x)+J(y),$$ and, in fact, it is the Lie groupoid morphism that integrates the Lie algebroid morphism $j:T^*M\to{\mathfrak{g}}^*$ canonically associated with the action (here we view ${\mathfrak{g}}^*$ as an abelian Lie algebra). The momentum map $J$ is relevant, e.g., to understand the reduced space $M/G$. Namely, $\Sigma(M)//G:=J^{-1}(0)/G$ is a symplectic groupoid integrating the Poisson quotient $M/G$ (which, however, does not need to coincide with $\Sigma(M/G)$; see [@FerOrRa]).
Our aim is to understand how this theory extends to actions of Poisson-Lie groups. Suppose $G\times M\to M$ is a Poisson action of a Poisson-Lie group $G$, with associated Lie bialgebra $({\mathfrak{g}},{\mathfrak{g}}^*)$. The map $j:T^*M\to {\mathfrak{g}}^*$ is still a Lie algebroid morphism and it integrates to a Poisson groupoid morphism $J:\Sigma(M)\to G^*$, where $G^*$ is the 1-connected, dual Poisson-Lie group of $G$. We would like to lift the $G$-action on $M$ to a hamiltonian action on $\Sigma(M)$ with momentum map $J:\Sigma(M)\to G^*$. As was observed by Lu [@Lu1], any Poisson map $J$ into $G^*$ determines a *local* hamiltonian $G$-action with momentum map $J$, so there exists a *local* hamiltonian $G$-action on $\Sigma(M)$.
Recall that a Poisson-Lie group $G$ is said to be [*complete*]{} if the infinitesimal dressing action of ${\mathfrak{g}}^*$ on $G$ can be integrated to a global action $G\times G^*\to G$. Our main result is the following:
\[thm:main\] Let $G$ be a complete Poisson-Lie group, $M$ an integrable Poisson manifold and $G\times M\to M$ a Poisson action. There exists a lifted Poisson action of $G$ on the symplectic groupoid $\Sigma(M)$, which is hamiltonian with a canonical momentum map $J:\Sigma(M)\to G^*$.
Our proof of Theorem \[thm:main\] takes advantage of the description of $\Sigma(M)$ in terms of cotangent paths (see [@CrFe; @CrFe2]) to explicitly construct the lifted $G$-action. In this respect, it is important to observe that in the situation described by Theorem \[thm:main\], elements of $G$ *do not act* as groupoid automorphims. In fact, we have the following result:
\[prop:main:aux:intro\] Let $G$ be a connected, complete Poisson-Lie group, and ${\mathcal{G}}{\rightrightarrows}M$ a source-connected symplectic groupoid. For a hamiltonian action $G\times{\mathcal{G}}\to {\mathcal{G}}$ with momentum map $J:{\mathcal{G}}\to G^*$ such that $J(M)={e}$, the following are equivalent:
(i) $J:{\mathcal{G}}\to G^*$ is a groupoid morphism: $$J(x\cdot y)=J(x)\cdot J(y),\quad x,y\in{\mathcal{G}}^{(2)}.$$
(ii) The twisted multiplicativity property holds: $$\label{eq:twisted}
g(x\cdot y)=(gx)\cdot (g^{J(x)}y),\quad x,y\in{\mathcal{G}}^{(2)}, g\in G,$$ where we denote by $g^u$ the right dressing action of $u\in G^*$ on an element $g\in G$.
When the Poisson structure on $G$ vanishes we have $G^*={\mathfrak{g}}^*$. In this case, $G^*$ acts trivially on $G$, so $G$ is always complete. Also, the twisting disappears and the action is by groupoid automorphisms, so we recover the results mentioned above. The twisted multiplicativity property was also observed by Lu in [@Lu2].
It is easy to check that, under the assumptions of Proposition \[prop:main:aux:intro\], the twisted multiplicativity property implies that there is an induced $G$-action on the units $M$ and that this action is Poisson. Hence, our results establish a 1:1 correspondence: $$\framebox[4.7 cm]{\parbox{4.5 cm}{Poisson actions on integrable Poisson manifolds}}\, \longleftrightarrow \,
\framebox[7.2 cm]{\parbox{7 cm}{Twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids}}$$
We illustrate Theorem \[thm:main\] with a few applications to the problem of integrating various classes of Poisson manifolds. The first simple application is:
\[thm:Poisson:quotients\] Let $G$ be a complete Poisson-Lie group, $(M,\pi)$ an integrable Poisson manifold and $G\times M\to M$ a proper and free Poisson action. The reduced space $$\Sigma(M)//G:=J^{-1}(e)/G$$ is a symplectic groupoid integrating the Poisson manifold $M/G$.
In general, $\Sigma(M)//G\not=\Sigma(M/G)$. We will see that the failure in this equality can be controlled in much the same way as in the case of actions by Poisson diffeomorphisms [@FerOrRa] (see Theorem \[thm:int:reduct:alt\] below).
The problem of integrating the Poisson quotient $M/G$ has already been discussed by several authors. The first results in this direction are due to Xu [@Xu1]. Later, Lu in [@Lu3], Stefanini in [@Ste] and Bonechi *et al.* [@BCST] derive results on integration based on more complicated procedures, using various notions of action algebroids/groupoids and doubles. Our approach gives a clear explanation for the fact that the completeness of $G$ implies the existence of a global action on $\Sigma(M)$, rather than just a local one, a problem circumvented by these authors.
Our second application of Theorem \[thm:main\] is to the integration of the Poisson manifold obtained by reduction of a hamiltonian action $G\times M\to M$ with equivariant momentum map $\mu:M\to G^*$. If the action is proper and free, Lu [@Lu1] has shown that the Marsden-Weinstein type quotient $\mu^{-1}(e)/G$ is a Poisson submanifold of $M/G$. In general, a Poisson submanifold of an integrable Poisson manifold does not need to be integrable (see [@CrFe2]), and when it is integrable, its symplectic groupoid need not be a subgroupoid of the symplectic groupoid of the ambient Poisson manifold. We will give a simple condition that guarantees the integrability of $\mu^{-1}(e)/G$ and yields a symplectic groupoid integrating $\mu^{-1}(e)/G$ which is a symplectic subgroupoid of $\Sigma(M)//G$. In the case where $G$ is a 1-connected, simple, compact Lie group, a classical result of Alekseev [@Alek] states that one can gauge transform the Poisson structure so that the Poisson action becomes an action by Poisson diffeomorphisms. Our methods allow us to describe what happens to this operation at the level of the lifted action on the symplectic groupoid.
Our last application of Theorem \[thm:main\] is to the integrability of Poisson homogeneous spaces. If $G$ is any Poisson-Lie group, the action of $G$ on itself by left translations is Poisson. Hence, the theorem shows that it lifts to a hamiltonian $G$-action on $\Sigma(G)$ with a momentum map $J:\Sigma(G)\to G^*$. If $H\subset G$ is a closed, connected, coisotropic subgroup, the quotient $G/H$ is a Poisson homogeneous space ([@Lu2]). The coisotropy condition is equivalent to the annihilator ${\mathfrak{h}}^\perp\subset{\mathfrak{g}}^*$ of the Lie algebra of $H$ being a Lie subalgebra. We will say that the pair $(G,H)$ is *relatively complete* if ${\mathfrak{h}}^\perp\subset{\mathfrak{g}}^*$ integrates to a closed subgroup $H^\perp\subset G^*$ and the right dressing action ${\mathfrak{g}}^*\to{\ensuremath{\mathfrak{X}}}(G)$ restricted to ${\mathfrak{h}}^\perp$ integrates to an action of $H^\perp$ on $H$.
Our methods then lead to a simple proof of the following theorem, which improves results of [@BCST] (see, also, [@Lu3] for a more general approach to the integration of Poisson homogeneous spaces):
Let $G$ be a Poisson-Lie group and let $H\subset G$ be a closed, coisotropic subgroup, such that $(G,H)$ is relatively complete. Then $J^{-1}(H^\perp )/H{\rightrightarrows}G/H$ is a symplectic groupoid integrating the Poisson homogeneous space $G/H$.
The results presented in this paper are part of a wider picture: since Poisson-Lie groups are the group-like objects in the category of Poisson groupoids, one should expect them to appear as the *group of symmetries* of such objects. A systematic study of symmetries of Poisson groupoids and their infinitesimal counterparts, Lie bialgebroids, will be the subject of a separate publication [@FP].
The rest of this paper is organized as follows. In Section \[sec:background\] we review a few notions and facts we will need to state and prove our results. Section \[sec:actions\] contains a proof of Theorem \[thm:main\] and other results concerning Poisson actions. Section \[sec:applications\] contains the three applications of Theorem \[thm:main\] mentioned above.
**Acknowledgments.** The authors would like to thank the following institutions for their hospitality and support: Erwin Schrödinger International Institute for Mathematical Physics (R.L.F. and D.I.P.), Consejo Superior de Investigaciones Científicas (R.L.F.) and Instituto Superior Técnico (D.I.P.). They would also like to thank Yvette Kosmann-Schwarzbach and the anonymous referees for comments and corrections on a preliminary version of this paper.
Basic Notions {#sec:background}
=============
In this section we give a short review of all the basic notions we will need: Poisson and symplectic groupoids, Lie bialgebra(oid)s and Poisson actions.
Poisson and symplectic groupoids {#subsec:groupoids}
--------------------------------
Let ${\mathcal{G}}$ be a Lie groupoid over $M$. We denote by ${\mathbf{s}}$ and ${\mathbf{t}}$ the source and target maps, by ${\mathbf{m}}:{\mathcal{G}}^{(2)}\to{\mathcal{G}}$ the multiplication (defined on the space ${\mathcal{G}}^{(2)}$ of pairs of composable arrows), by $\textbf{i}:{\mathcal{G}}\to{\mathcal{G}}$ the inverse map, and by ${\varepsilon}:M\to {\mathcal{G}}$ the identity section. Our convention for the groupoid multiplication is such that, given two arrows $x,y\in{\mathcal{G}}$, the product $x\cdot y:={\mathbf{m}}(x,y)$ is defined provided ${\mathbf{s}}(x)={\mathbf{t}}(y)$. Also, if $m\in M$ we write $1_m:={\varepsilon}(m)$ for the unit arrow over $m$, and if $x\in{\mathcal{G}}$ we write $x^{-1}:=\textbf{i}(x)$ for the inverse arrow. We denote the groupoid by ${\mathcal{G}}{\rightrightarrows}M$.
We will be interested in Lie groupoids ${\mathcal{G}}{\rightrightarrows}M$ carrying Poisson structures on the space of arrows and on the space of units. A *Poisson groupoid* is a pair $({\mathcal{G}},\Pi)$, where ${\mathcal{G}}$ is a Lie groupoid and $\Pi\in{\ensuremath{\mathfrak{X}}}^2({\mathcal{G}})$ is a multiplicative Poisson structure. This means that the graph of the groupoid multiplication $$\operatorname{graph}({\mathbf{m}}):=\{(x,y,x\cdot y) \,|\, {\mathbf{s}}(x) = {\mathbf{t}}(y)\}$$ is a coisotropic submanifold of ${\mathcal{G}}\times {\mathcal{G}}\times\bar{{\mathcal{G}}}$ ([@We]). When $\Pi$ is non-degenerate, so $\Omega =\Pi^{-1}$ is a symplectic form, the multiplicativity condition amounts to: $$\label{eq:mult:symp}
{\mathbf{m}}^*\Omega=\pi_1^*\Omega+\pi_2^*\Omega,$$ where $\pi_i:{\mathcal{G}}^{(2)}\to{\mathcal{G}}$ are the projections on each factor. In this case, we say that the pair $({\mathcal{G}},\Omega)$ is a *symplectic groupoid*. A *morphism of Poisson groupoids* is a Lie groupoid morphism $\Phi:({\mathcal{G}}_1,\Pi_1)\to ({\mathcal{G}}_2,\Pi_2)$ which is also a Poisson map.
For this paper, the two most important examples are the following:
Lie groups are precisely the Lie groupoids for which the space of units reduces to a single object. For a Lie group $G$, a Poisson structure $\pi_G$ is multiplicative iff the multiplication $m:G\times G\to G$ is a Poisson map (where we consider the Poisson structure $\pi_G\oplus\pi_G$ on $G\times G$). In this case, one calls $(G,\pi_G)$ a *Poisson-Lie group*.
Let $(M,\pi)$ be a Poisson manifold. Its *Weinstein groupoid* $\Sigma(M){\rightrightarrows}M$ is defined as (see [@CrFe2]): $$\Sigma(M):=\frac{\{\text{cotangent paths}\}}{\{\text{cotangent homotopies}\}},$$ where multiplication is concatenation of cotangent paths. If $p:T^*M\to M$ denotes the projection, the source and target maps are given by: $${\mathbf{s}}([a])=p(a(0)),\quad {\mathbf{t}}([a])=p(a(1)).$$
A Poisson manifold $(M,\pi)$ is called integrable if its Weinstein groupoid $\Sigma(M)$ is smooth (in which case, one has $\dim \Sigma(M)=2\dim M$). The obstructions to integrability were determined in [@CrFe; @CrFe2]. When $(M,\pi)$ is integrable, $\Sigma(M)$ carries a natural multiplicative symplectic form $\Omega$. Moreover, the source (respectively, target map) is a Poisson (resp. anti-Poisson map).
Alan Weinstein in [@We] observed that the properties of the source/target maps in the last example are by no means exceptional: given a Poisson groupoid $({\mathcal{G}},\Pi)$ with base $M$ there exists a unique Poisson structure $\pi$ on $M$, such that ${\mathbf{s}}:{\mathcal{G}}\to M$ is a Poisson map and ${\mathbf{t}}:{\mathcal{G}}\to M$ is anti-Poisson.
Lie bialgebras and Lie bialgebroids {#subsec:bialgebroids}
-----------------------------------
Now let us go to the infinitesimal level. We will denote by $A$ a Lie algebroid with bundle projection $p:A\to
M$, anchor $\#:A\to TM$, and Lie bracket $[~,~]_A$ on its space of sections. The $A$-differential forms are $\Omega^\bullet(A):=\Gamma(\wedge^\bullet A^*)$ and they form a complex with the $A$-differential ${\mathrm d}_A$ (see, e.g., [@Mc2]). Our conventions are such that if ${\mathcal{G}}{\rightrightarrows}M$ is a Lie groupoid, then its Lie algebroid $A=A({\mathcal{G}})$ has $A_x:=\operatorname{Ker}{\mathrm d}_{1_x}{\mathbf{s}}$ and $\#_x:={\mathrm d}_{1_x}{\mathbf{t}}$. Moreover, $\Gamma(A)={\ensuremath{\mathfrak{X}}}(A)$ is identified with the space ${\ensuremath{\mathfrak{X}}}_r({\mathcal{G}})$ of right invariant vector fields on ${\mathcal{G}}$ and we will denote by ${\overrightarrow}{X}\in {\ensuremath{\mathfrak{X}}}_r({\mathcal{G}})$ the right invariant vector field corresponding to $X\in \Gamma(A)$. Similarly, one obtains identifications of ${\ensuremath{\mathfrak{X}}}^\bullet(A)$ and $\Omega^\bullet(A)$ with the spaces ${\ensuremath{\mathfrak{X}}}^\bullet_r({\mathcal{G}})$ and $\Omega^\bullet_r({\mathcal{G}})$ of right invariant multivector fields and differential forms on ${\mathcal{G}}$ (note that a right invariant differential form is a ${\mathbf{s}}$-foliated differential form on ${\mathcal{G}}$). Under these identifications, the bracket $[~,~]_A$ and the differential ${\mathrm d}_A$ are identified with the Schouten bracket on right invariant multivector fields and the de Rham differential on right invariant differential forms.
We recall the following basic proposition, due to Weinstein [@We]:
\[prop:induced-bialgebroid\] If $({\mathcal{G}},\Pi)$ is a Poisson groupoid then it induces a Lie algebroid structure on $A({\mathcal{G}})^*$, the dual of the Lie algebroid, whose $A({\mathcal{G}})^*$-differential is characterized by $$\label{eq:dual-differential}
{\overrightarrow}{{\mathrm d}_{A({\mathcal{G}})^*}X} = -[{\overrightarrow}{X},\Pi] ,\qquad (X\in {\ensuremath{\mathfrak{X}}}(A)).$$
This leads to the infinitesimal counterpart of a Poisson groupoid, i.e., the notion of a *Lie bialgebroid*. This is pair of Lie algebroid structures $(A,A^*)$, on a vector bundle $A\to M$ and on its dual bundle $A^*\to M$, such that for any $X,\ Y\in {\ensuremath{\mathfrak{X}}}(A)$, $${\mathrm d}_{A^*}[X, Y]_A = {\boldsymbol{\pounds}}_{X}{\mathrm d}_{A^*}Y-{\boldsymbol{\pounds}}_{Y}{\mathrm d}_{A^*}X.$$ Just like for a Poisson groupoid, if $(A, A^*)$ is a Lie bialgebroid over $M$, there exists a Poisson structure $\pi _M$ on $M$ which is characterized by $$\pi _M (df ,dg )= \# ({\mathrm d}_{A^*}f) (g)=\langle {\mathrm d}_{A^*}f, {\mathrm d}_A g\rangle , \qquad (f,g \in C^\infty (M)).$$ A *morphism* of Lie bialgebroids $\phi:(A_1,A_1^*)\to (A_2,A_2^*)$ is a Lie algebroid morphism $\phi:A_1\to A_2$ which is also a Poisson map (note that $A_i$ has a fiberwise linear Poisson structure, being the dual of the Lie algebroid $A_i^*$).
If $({\mathcal{G}},\Pi)$ is a Poisson groupoid, it follows from Proposition \[prop:induced-bialgebroid\] that $(A({\mathcal{G}}),A({\mathcal{G}})^*)$ is a Lie bialgebroid. Conversely, it is proved in [@McXu2] that any Lie bialgebroid structure $(A,A^*)$, where $A$ can be integrated to a Lie groupoid, can actually be integrated to a Poisson groupoid $({\mathcal{G}}(A),\Pi )$. Here ${\mathcal{G}}(A)$ is just the source 1-connected Lie groupoid integrating $A$. In this situation, the Poisson structures on $M$ induced by $({\mathcal{G}}(A),\Pi)$ and $(A,A^*)$ coincide. Similarly, Poisson groupoid morphisms $\Phi:{\mathcal{G}}_1\to {\mathcal{G}}_2$ are in 1:1 correspondence with Lie bialgebroid morphisms $\phi:(A_1,A_1^*)\to (A_2,A_2^*)$, provided ${\mathcal{G}}_1$ is source 1-connected.
Note that the notion of Lie bialgebroid is symmetric: if $(A,A^*)$ is a Lie bialgebroid so is $(A^*,A)$. On the other hand, at the level of groupoids things are more subtle: for example, in a Lie bialgebroid $(A,A^*)$ it is possible that $A$ is integrable while $A^*$ is not.
It is well known that if $(M,\pi)$ is a Poisson manifold, then $A=T^*M$ is Lie algebroid with anchor $\pi^\sharp:T^*M\to TM$ and Lie bracket: $$[{\alpha},{\beta}]={\boldsymbol{\pounds}}_{\pi^\sharp({\alpha})}{\beta}-{\boldsymbol{\pounds}}_{\pi^\sharp({\beta})}{\alpha}-{\mathrm d}(\pi({\alpha},{\beta})).$$ When one consider $A^*=TM$ with its canonical Lie algebroid structure, the pair $(T^*M,TM)$ becomes a Lie bialgebroid. While $A^*=TM$ is always integrable, $A=T^*M$ does not have to be integrable. Its integrability is equivalent to the integrability of $(M,\pi)$. When $(M,\pi)$ is integrable, $(\Sigma(M),\Omega)$ is the source 1-connected symplectic groupoid integrating the Lie bialgebroid $(T^*M,TM)$.
If $(G,\pi_G)$ is a Poisson-Lie group, the corresponding Lie bialgebroid is just a Lie bialgebra $({\mathfrak{g}},{\mathfrak{g}}^*)$. According to our conventions, ${\mathfrak{g}}$ is the space of right invariant vector fields on $G$. We can also identify ${\mathfrak{g}}^*$ with the space of right invariant 1-forms on $G$. The bracket on 1-forms induced by $\pi_G$ (see the previous example) preserves the right invariant forms, and it induces the Lie bracket $[~,~]_{{\mathfrak{g}}^*}$ on ${\mathfrak{g}}^*$.
The 1-connected Lie group integrating ${\mathfrak{g}}^*$, denoted $G^*$, is called the dual Poisson-Lie group: its Lie bialgebra is $({\mathfrak{g}}^*,{\mathfrak{g}})$.
Poisson actions {#subsec:poisson:actions}
---------------
Let $(G,\pi_G)$ be a Poisson-Lie group and let $(M,\pi)$ be a Poisson manifold. Recall that a smooth action $\Psi:G\times M\to M$ is called a *Poisson action* if $\Psi$ is a Poisson map. Here the product $G\times M$ is equipped with the direct sum Poisson structure $\pi_G\oplus\pi$.
For a smooth action $\Psi:G\times M\to M$ of a Lie group on a manifold $M$, we will denote by $\psi:{\mathfrak{g}}\to{\ensuremath{\mathfrak{X}}}(M)$ the corresponding infinitesimal Lie algebra action defined by: $$\psi(\xi)_a=\left.\frac{{\mathrm d}}{{\mathrm d}t}\exp(t\xi)a\right|_{t=0}\quad (\xi\in{\mathfrak{g}}).$$ According to our conventions, ${\mathfrak{g}}$ is identified with the space of right invariant vector fields on $G$ and it follows that $\psi:{\mathfrak{g}}\to {\ensuremath{\mathfrak{X}}}(M)$ is a Lie algebra homomorphism. The following characterization of Poisson actions is due to Lu [@Lu; @Lu1]:
\[prop:Poisson:actions:1\] Let $(G,\pi_G)$ be a connected Poisson-Lie group and let $(M,\pi)$ be a Poisson manifold. For a smooth action $\Psi:G\times M\to M$ the following two properties are equivalent:
(i) The action $\Psi$ is Poisson;
(ii) Setting $\delta:={\mathrm d}_e\pi:{\mathfrak{g}}\to{\mathfrak{g}}\wedge{\mathfrak{g}}$, the infinitesimal action satisfies: $${\boldsymbol{\pounds}}_{\psi(\xi)}\pi=(\psi\wedge\psi)\delta(\xi),\quad (\xi\in{\mathfrak{g}}).$$
The map $\delta$ is just (the dual of) the Lie bracket on ${\mathfrak{g}}^*$. Hence, the proposition leads to a definition of an infinitesimal action of a Lie bialgebra $({\mathfrak{g}},{\mathfrak{g}}^*)$ on a Poisson manifold $(M,\pi)$.
Let $(G,\pi_G)$ be a Poisson-Lie group with Lie bialgebra $({\mathfrak{g}},{\mathfrak{g}}^*)$. According to our conventions, we can identify ${\mathfrak{g}}^*$ with the space of right invariant 1-forms on $G$. The map $\lambda:{\mathfrak{g}}^*\to{\ensuremath{\mathfrak{X}}}(G)$ which to a right invariant 1-form $\eta\in{\mathfrak{g}}^*$ associates the vector field $\pi_G^\sharp(\eta)$ is a Lie algebra morphism and so defines an (left) infinitesimal action of ${\mathfrak{g}}^*$ on $G$. Using Proposition \[prop:Poisson:actions:1\], one checks that this is an infinitesimal action of $({\mathfrak{g}}^*,{\mathfrak{g}})$ on the Poisson manifold $(G,\pi_G)$, called the *left dressing action*. Similarly, the identification of ${\mathfrak{g}}^*$ with the left invariant 1-forms on $G$, leads to Lie algebra anti-morphism $\rho:{\mathfrak{g}}^*\to{\ensuremath{\mathfrak{X}}}(G)$ and hence to a *right dressing action*. Switching the roles of $G$ and $G^*$ we also obtain left/right dressing actions of ${\mathfrak{g}}$ on $G^*$. If one of the infinitesimal dressing actions is complete so is the other. We say that $(G,\pi_G)$ is a *complete Poisson-Lie group* if the right dressing action $\rho:{\mathfrak{g}}^*\to{\ensuremath{\mathfrak{X}}}(G)$ integrates to a (Poisson) right action of $(G^*,\pi_{G^*})$ on $(G,\pi_G)$.
There is another useful characterization of Poisson actions, due to Xu [@Xu0]:
\[prop:Poisson:actions:2\] Let $(G,\pi_G)$ be a connected Poisson-Lie group and let $(M,\pi)$ be a Poisson manifold. For a smooth action $\Psi:G\times M\to M$ define $j:T^*M\to{\mathfrak{g}}^*$ by $$\langle j({\alpha}),\xi\rangle=\langle {\alpha},\psi(\xi)\rangle,\quad (\xi\in{\mathfrak{g}}).$$ Then the following two properties are equivalent:
(i) The action $\Psi$ is Poisson;
(ii) The map $j:T^*M\to{\mathfrak{g}}^*$ is a Lie bialgebroid morphism.
Hamiltonian actions {#subsec:hamiltonian}
-------------------
Let $\Psi:G\times M\to M$ be a Poisson action. A smooth map $\mu:M\to G^*$ is called a *momentum map* for the action if: $$\label{eq:momentum}
\psi(\xi)=\pi^{\sharp}(\mu^*\xi^R)\quad (\xi\in{\mathfrak{g}}).$$ Here, $\xi^R\in\Omega^1(G^*)$ is the right invariant 1-form on $G^*$ with value $\xi\in{\mathfrak{g}}$ at the identity $e\in G^*$. Observe that when $\pi_G=0$, so that $G^*={\mathfrak{g}}^*$, the momentum map condition reduces to the usual condition. These generalized momentum maps were first studied by Lu in [@Lu; @Lu1].
From the definition, it is evident that the left/right dressing actions of $G$ on $G^*$ have momentum map $G^*\to G^*$ the identity map. Similarly, the left/right dressing actions of $G^*$ on $G$ have momentum map $G\to G$ the identity. These actions satisfy versions of the twisted multiplicativity property . For example, the (left) dressing action $G\times G^*\to G^*$ satisfies: $$\label{eq:twist:dress}
g(u_1\cdot u_2)=(g u_1)\cdot(g^{u_1} u_2),\qquad (g\in G,\ u_1,u_2\in G^*).$$
A proof of the following basic fact can be found in [@Lu1]:
Let $(G,\pi_G)$ be a connected and complete Poisson-Lie group. A momentum map $\mu:M\to G^*$ for a Poisson action $G\times
M\to M$ is $G$-equivariant (relative to the left dressing action of $G$ on $G^*$) if and only if it is a Poisson map.
We will say that a Poisson action $G\times M\to M$ is a *hamiltonian action* if it admits an equivariant momentum map $\mu:M\to G^*$. Lu has also shown that the usual Marsden-Weinstein symplectic reduction extends to these hamiltonian actions.
In order to explain this fact, let $G\times M\to M$ be a hamiltonian action on a Poisson manifold, with momentum map $\mu:M\to G^*$. If $u\in G^*$, denote by $G_u$ the isotropy group of $u$ for the the left dressing action of $G$ on $G^*$. Then we have the following result (see Lu [@Lu1]):
Let $G\times M\to M$ be a proper and free hamiltonian action, with momentum map $\mu:M\to G^*$. For each $u\in G^*$, the level set $\mu^{-1}(u)$ carries a natural Dirac structure $L_u$, the space $\mu^{-1}(u)/G_u$ carries a natural Poisson structure and we have a commutative diagram: $$\xymatrix{
&M\ar[dr]\\
\mu^{-1}(u)\ar[ur]\ar[dr]& &M/G\\
&\mu^{-1}(u)/G_u\ar[ur]}$$ where the inclusions are backward Dirac maps and the projections are forward Dirac maps (see [@BuRa] for the definition of these classes of maps).
If one starts with a hamiltonian action on a *symplectic* manifold $(S,\omega)$ the reduced spaces $\mu^{-1}(u)/G_u$ are also symplectic. In fact, their connected components are the symplectic leaves of the quotient Poisson manifold $S/G$.
If we start with a Poisson-Lie group $(G,\pi_G)$ and a Poisson manifold $(M,\pi)$, any Poisson map $\mu:M\to
G^*$ determines an infinitesimal Poisson action $\psi:{\mathfrak{g}}\to{\ensuremath{\mathfrak{X}}}(M)$ by setting: $$\psi(\xi):=\pi^{\sharp}(\mu^*\xi^R)\quad (\xi\in{\mathfrak{g}}).$$ Integration gives a *local* Poisson action with equivariant momentum map $\mu$.
Integration of Poisson actions {#sec:actions}
==============================
In this section, we will prove Theorem \[thm:main\] and other results concerning the integration of Poisson actions.
Poisson actions on symplectic groupoids {#subsec:twisted:action}
---------------------------------------
Before considering the problem of lifting a Poisson action on $M$ to a Poisson action on the symplectic groupoid $\Sigma(M)$, we discuss actions on symplectic groupoids and how the twisted multiplicativity property arises.
\[prop:main:aux\] Let $(G,\pi_G)$ be a connected, complete Poisson-Lie group, and let ${\mathcal{G}}{\rightrightarrows}M$ be a source-connected symplectic groupoid. If $G\times{\mathcal{G}}\to {\mathcal{G}}$ is a hamiltonian action with momentum map $J:{\mathcal{G}}\to G^*$ such that $J(M)={e}$, the following are equivalent:
(i) $J:{\mathcal{G}}\to G^*$ is a groupoid morphism: $$J(x\cdot y)=J(x)\cdot J(y),\quad x,y\in{\mathcal{G}}^{(2)}.$$
(ii) The twisted multiplicativity property holds: $$\label{eq:twisted:2}
g(x\cdot y)=(gx)\cdot (g^{J(x)}y),\quad x,y\in{\mathcal{G}}^{(2)}, g\in G,$$ where we denote by $g^u$ the right dressing action of $u\in G^*$ on an element $g\in G$.
Denote by $\psi:{\mathfrak{g}}\to{\ensuremath{\mathfrak{X}}}({\mathcal{G}})$ the infinitesimal ${\mathfrak{g}}$-action. For the proof, we remark that the multiplicativity property of the symplectic form $\Omega$, when evaluated at $(x,y)$ on the pair $(\psi(\xi)_x,\psi(\operatorname{Ad}^*J(x)\cdot \xi)_y),(v,0)\in T_{(x,y)}{\mathcal{G}}^{(2)}$ yields: $$\label{eq:mult:aux}
\Omega_{x\cdot y}({\mathrm d}_{(x,y)}{\mathbf{m}}(\psi(\xi)_x,\psi(\operatorname{Ad}^* J(x)\cdot \xi)_y),{\mathrm d}_x R_{y}v)=\Omega_x(\psi(\xi)_x,v).$$ where ${\mathbf{m}}:{\mathcal{G}}^{(2)}\to{\mathcal{G}}$ is the groupoid multiplication and $R_y$ denotes right translation by the element $y\in{\mathcal{G}}$ (here $v$ is any vector tangent to the source fiber at $x$).
Now, since the Lie group $G$ is connected, the twisted multiplicativity property is equivalent to its infinitesimal version, which reads: $$\label{eq:twisted:inft}
\psi(\xi)_{x\cdot y}={\mathrm d}_{(x,y)}{\mathbf{m}}(\psi(\xi)_x,\psi(\operatorname{Ad}^* J(x)\cdot \xi)_y),\quad x,y\in{\mathcal{G}}^{(2)}, \xi\in{\mathfrak{g}}.$$ So if this condition holds, we conclude from that $$\Omega_{x\cdot y}(\psi(\xi)_{x\cdot y},{\mathrm d}_x R_{y}v)=\Omega_x(\psi(\xi)_x,v),$$ for any vector $v$ tangent to the source fiber at $x$. In other words, $i_{\psi(\xi)}\Omega=J^*\xi^R$ is a right invariant 1-form, for all $\xi\in{\mathfrak{g}}$. But if $J^*\xi^R$ is a right invariant 1-form, for all $\xi\in{\mathfrak{g}}$ and $J(M)=e$, then $J:{\mathcal{G}}\to G^*$ must be a groupoid homomorphism.
Conversely, assume that $J:{\mathcal{G}}\to G^*$ is a groupoid homomorphism. Then: $$\label{eq:momentum:Poisson}
\psi(\xi)=\pi^{\sharp} (J^*\xi^R),$$ where $\pi=\Omega^{-1}$ is a multiplicative Poisson structure. Let $\xi \in {\mathfrak{g}}$ and $(x,y)\in{\mathcal{G}}^{(2)}$. Since $J$ is a groupoid morphism, we have $$\label{eq:cotangent:morphism}
J^*(\theta _1\star_{G^*} \theta _2)=(J^*\theta _1)\star_{{\mathcal{G}}}
(J^*\theta _2), \qquad (\theta _1\star \theta _2)\in
(T^*(G^*))^{(2)},$$ where $\star _{{\mathcal{G}}}$ (respectively, $\star _{G^*}$) denotes the groupoid multiplication in $T^*{\mathcal{G}}$ (respectively, $T^*(G^*)$). Now, using , and the fact that $(\xi ^R)_{uv}=(\xi^R)_u\star _{G^*}(\operatorname{Ad}^*J(x)\cdot\xi^R)_v$ for $\xi\in {\mathfrak{g}}$ and $u,v\in G^*$, we deduce $$\begin{aligned}
\psi (\xi )_{x\cdot y}&=\pi ^\sharp ( J^* \xi ^R)_{x\cdot y}
=\pi ^\sharp ( J^*(( \xi ^R)_{J(x)} \star_{G^*} ((\operatorname{Ad}^*J(x)\cdot \xi )^R)_{J(y)}))\\
&=\pi ^\sharp ( (J^* \xi ^R)_x \star_{{\mathcal{G}}} (J^* (\operatorname{Ad}^*J(x)\cdot \xi )^R)_y )\\
&={\mathrm d}_{(x,y)}{\mathbf{m}}(\pi^\sharp (J^* \xi ^R)_x,\pi ^\sharp (J^*(\operatorname{Ad}^*J(x)\cdot \xi)^R)_y)\\
&={\mathrm d}_{(x,y)}{\mathbf{m}}(\psi (\xi )_x,\psi (\operatorname{Ad}^*J(x)\cdot \xi )_y) .\end{aligned}$$ Here, we have also used that $\pi ^\sharp :T^*{\mathcal{G}}\to T{\mathcal{G}}$ is a groupoid morphism, i.e., $${\mathrm d}{\mathbf{m}}(\pi^\sharp (\eta _1),\pi ^\sharp (\eta _2))=\pi ^\sharp
(\eta _1\star_{{\mathcal{G}}} \eta _2),\qquad (\eta _1,\eta _2)\in
(T^*{\mathcal{G}})^{(2)}.$$ Therefore, the infinitesimal condition is satisfied and, as a consequence, the twisted multiplicativity condition holds.
Our next remark is even more general.
\[prop:general\] Let $(G,\pi_G)$ be a complete Poisson-Lie group, $G\times {\mathcal{G}}\to{\mathcal{G}}$ a smooth action on a Lie groupoid, and $J:{\mathcal{G}}\to G^*$ a groupoid morphism. If the action satisfies the twisted multiplicativity property , then there is an induced action on the Lie algebroid $A$ of ${\mathcal{G}}$. Moreover, if ${\mathcal{G}}$ is source 1-connected the $G$-action on ${\mathcal{G}}$ is completely determined by $J$ and the induced $G$-action on $A$.
Note that in this proposition there is no assumption about a symplectic or Poisson structure on ${\mathcal{G}}$. Also, the induced action on $A$, in general, *is not* by Lie algebroid automorphisms.
First, we remark that the twisted multiplicativity property and the fact that $J$ is a homomorphism imply that, for any $x\in{\mathcal{G}}$, we have: $$\left\{
\begin{array}{l}
g\, x=g\,(1_{{\mathbf{t}}(x)}\cdot x)=(g\,1_{{\mathbf{t}}(x)})\cdot(g\,x)\\
g\, x=g\,(x\cdot 1_{{\mathbf{s}}(x)})=(g\,x)\cdot(g^{J(x)}\,1_{{\mathbf{s}}(x)})
\end{array}\right.
\quad\Rightarrow\quad
\left\{
\begin{array}{l}
g\,1_{{\mathbf{t}}(x)}=1_{{\mathbf{t}}(g\,x)}\\
g^{J(x)}\,1_{{\mathbf{s}}(x)}=\,1_{{\mathbf{s}}(g\,x)}
\end{array}\right.$$ Therefore, we have an induced $G$-action on $M$ such that: $$g\,1_m=1_{g\,m}.$$ Moreover, we also find that: $${\mathbf{t}}(g\,x)=g\,{\mathbf{t}}(x),\quad {\mathbf{s}}(g\,x)=g^{J(x)}{\mathbf{s}}(x).$$ We will also need the identity: $$(g\, x)^{-1}=g^{J(x)}\,x^{-1},$$ whose proof is straightforward from and the fact that $J$ is a homomorphism.
The previous identities show that the $G$-action sends ${\mathbf{t}}$-fibers to ${\mathbf{t}}$-fibers, but does not preserve source fibers. However, we can consider a new $G$-action on ${\mathcal{G}}$ defined by: $$g\odot x:=(g\,x^{-1})^{-1},$$ which does preserve ${\mathbf{s}}$-fibers, and induces the same action on the identity section. Hence, we have an induced $G$-action on the Lie algebroid $A$ of ${\mathcal{G}}$, by vector bundle automorphisms (but, in general, not Lie algebroid automorphisms), defined by: $$g\,a:=\left.\frac{{\mathrm d}}{{\mathrm d}t}g\odot \gamma(t)\right|_{t=0},\quad (g\in G, a\in A_m)$$ where $\gamma(t)$ is any curve lying in the source fiber ${\mathbf{s}}^{-1}(m)$ with $\gamma(0)=1_m$ and $\dot{\gamma}(0)=a$.
If ${\mathcal{G}}$ has source 1-connected fibers, then we can identify an element $x\in{\mathcal{G}}$ with the homotopy class $[x(t)]$, where $x(t)$ is any ${\mathbf{s}}$-path, i.e., a path lying in the source fiber through $x$ and such that $x(0)=1_{{\mathbf{s}}(x)}$ and $x(1)=x$ (see [@CrFe]). Then we can identify ${\mathcal{G}}$ with the Weinstein groupoid ${\mathcal{G}}(A)$ consisting of $A$-paths modulo $A$-homotopy. This identification can be done at the level of paths by setting: $$x(t) \longmapsto a(t):=\left.\frac{{\mathrm d}}{{\mathrm d}s}x(s)\cdot x(t)^{-1}\right|_{s=t}.$$ Using this identification, we transport the $G$-action on ${\mathcal{G}}$ to an action on ${\mathcal{G}}(A)$: if $x$ is represented by the ${\mathbf{s}}$-path $x(t)$ then $g\,x$ is represented by the ${\mathbf{s}}$-path: $$\bar{x}(t):=g^{J(x)J(x(t))^{-1}}\,x(t).$$ In fact, we find ${\mathbf{s}}(\bar{x}(t))=g^{J(x)}\,{\mathbf{s}}(x(t))=g^{J(x)}{\mathbf{s}}(x)={\mathbf{s}}(g\,x)$ and $\bar{x}(1)=g\,x$. Then we compute the $A$-path associated to $\bar{x}(t)$, $$\begin{aligned}
\bar{a}(t):=&\left.\frac{{\mathrm d}}{{\mathrm d}s}\bar{x}(s)\cdot \bar{x}(t)^{-1}\right|_{s=t}\\
=&\left.\frac{{\mathrm d}}{{\mathrm d}s} \left(g^{J(x)J(x(s))^{-1}}\,x(s)\right)\cdot \left(g^{J(x)J(x(t))^{-1}}\,x(t)\right)^{-1}\right|_{s=t}\\
=&\left.\frac{{\mathrm d}}{{\mathrm d}s} \left(g^{J(x)J(x(t))^{-1}}\,\left(x(s)\cdot x(t)^{-1}\right)^{-1}\right)^{-1}\right|_{s=t}\\
=&\left.\frac{{\mathrm d}}{{\mathrm d}s} \left(g^{J(x)J(x(t))^{-1}}\,
\left(x(s)\cdot
x(t)^{-1}\right)\right)\right|_{s=t}=g^{J(x)J(x(t))^{-1}}a(t).\end{aligned}$$ This last expression shows that the action of $G$ on ${\mathcal{G}}$ is completely determined by $J$ and the action of $G$ on $A$, as claimed.
Lifting of local Poisson actions {#subsec:lifted:action:local}
--------------------------------
Let us now consider the problem of lifting a Poisson action on $M$ to a Poisson action on the symplectic groupoid $\Sigma(M)$.
Given a Poisson action $\Psi:G\times M\to M$, it follows from Proposition \[prop:Poisson:actions:2\] that the induced map $j:T^*M\to{\mathfrak{g}}^*$ is a Lie bialgebroid morphism from $(T^*M,TM)$ to $({\mathfrak{g}}^*,{\mathfrak{g}})$. Integrating this morphism (see [@Xu0 Theorem 5.5]), we conclude that:
\[cor:integ:J\] Let $\Psi:G\times M\to M$ be a Poisson action of a Poisson-Lie group $(G,\pi_G)$ on an integrable Poisson manifold $(M,\pi)$. The Lie bialgebroid morphism $j:T^*M\to{\mathfrak{g}}^*$ integrates to a morphism of Poisson groupoids $J:\Sigma(M)\to
G^*$.
At the level of cotangent paths, the map $J$ is simply given by the formula: $$J([a])=[j\circ a]$$ (see [@CrFe], where it is explained how to integrate morphisms of Lie algebroids to morphisms of Lie groupoids in terms of cotangent paths).
Since $J:\Sigma(M)\to G^*$ is a Poisson map and a groupoid morphism, we conclude from Proposition \[prop:main:aux\] that:
\[prop:local:act\] Let $\Psi:G\times M\to M$ be a Poisson action of a Poisson-Lie group $(G,\pi_G)$ on a Poisson manifold $(M,\pi)$. There exists a local hamiltonian action of $G$ on $\Sigma(M)$ with momentum map $J:\Sigma(M)\to G^*$ which satisfies the infinitesimal twisted multiplicativity property .
Later, we will give an explicit expression for this local action (see Remark \[rem:local:action\]). The following example shows that, in general, the lifted action will not be a *global action*.
Let $G$ be any Poisson-Lie group which is not complete. The action of $G$ on itself by left translations $G\times G\to G$ is a Poisson action. The lifted (local) action on $\Sigma(G)$ is not a global action. In fact, observe that the identity $e\in G$ is a fixed point for the Poisson structure where the isotropy Lie algebra is ${\mathfrak{g}}^*$. Hence, the corresponding isotropy group is: $$\Sigma(G)_e={\mathbf{s}}^{-1}(e)={\mathbf{t}}^{-1}(e)\simeq G^*.$$ The restriction of $J:\Sigma(G)\to G^*$ to this isotropy group is an isomorphism, so if the lifted action were a global action, the dressing action would have to be complete.
Lifting to global Poisson actions {#subsec:lifted:action}
---------------------------------
Our main result states that if $(G,\pi_{G})$ is a complete Poisson-Lie group, then the lifted action is a global action. In the sequel, we will assume that $G$ is complete and will denote by $g^u$ the right dressing action of an element $u\in G^*$ on an element $g\in G$.
\[thm:lift:actions\] Let $(G,\pi_G)$ be a complete Poisson-Lie group, $(M,\pi)$ an integrable Poisson manifold and $G\times M\to
M$ a Poisson action. Let $J:\Sigma(M)\to G^*$ be the integration of $j:T^*M\to{\mathfrak{g}}^*$. Then there exists a lifted hamiltonian action of $(G,\pi_G)$ on the symplectic groupoid $\Sigma(M)$ with momentum map $J$, such that:
1. $J$ is equivariant: $$J(g\,x)=g\,J(x),\qquad (g\in G, x\in\Sigma(M)).$$
2. The action is twisted multiplicative: $$g\,(x\cdot y)=(g\, x)\cdot (g^{J(x)}\, y),\qquad (g\in G,(x,y)\in\Sigma(M)^{(2)}).$$
Let $a:I\to T^*M$ be a cotangent path and define a new path $\bar{a}:I\to T^*M$ by: $$\label{cotg:path}
\overline{a}(t):=g^{J(x)J(x(t))^{-1}}\, a(t).$$ In this formula, we use the lifted cotangent action of $G$ on $T^*M$ and $x(t)$ denotes the element in $\Sigma(M)$ which, for a fixed $t\in I$, is defined by the cotangent path $s\mapsto t\,a(st)$. The motivation for this definition can be found in the proof of Proposition \[prop:general\].
One now checks that:
(a) For any cotangent path $a(t)$, the path $\bar{a}(t)$ defined by is also a cotangent path.
(b) If $a_{{\varepsilon}}$ is a cotangent homotopy, then the corresponding family $\bar{a}_{\varepsilon}$ defined by is also a cotangent homotopy.
This means that formula leads to a map $G\times \Sigma(M)\to\Sigma(M)$ by setting at the level of cotangent homotopy classes: $$\label{eq:action} g\, [a(t)]=[g^{J(x)J(x(t))^{-1}}\, a(t)].$$
Using this formula, we will show that $J:\Sigma(M)\to G^*$ is $G$-equivariant, i.e., that: $$\label{eq:equivariance} J(g\, x)=g\, J(x), \quad (x\in\Sigma(M),\
g\in G),$$ where on the left-hand side $g$ acts by and on the right-hand side $g$ acts by the (left) dressing action on $G^*$. In order to prove one starts by remarking that, since $G^*$ is simply-connected, one can identify $G^*$ with paths $\xi:I\to {\mathfrak{g}}^*$ up to ${\mathfrak{g}}^*$-homotopy. This is a special instance of the general construction mentioned in the proof of Proposition \[prop:general\]: given an element $u\in G^*$ we first identify it with (the homotopy class of) a path $u(t)\in G^*$ starting at the identity $u(0)=e$ and ending at $u(1)=u$. Then we associate to it (the ${\mathfrak{g}}^*$-homotopy class of) a path in the Lie algebra $\xi:I\to {\mathfrak{g}}^*$ by setting: $$\xi(t)=\left.\frac{{\mathrm d}}{{\mathrm d}s}u(s)u(t)^{-1}\right|_{s=t}.$$ By the formula proved at the end of Proposition \[prop:general\], under this identification, the dressing action $G\times G^*\to G^*$ is given at the level of ${\mathfrak{g}}^*$-paths by: $$g\, [\xi(t)]=[\operatorname{Ad}^*g^{u(1)u(t)^{-1}}\, \xi(t)].$$ Using this relation, we see that if $x=[a(t)]\in\Sigma(M)$ and $g\in G$, then: $$\begin{aligned}
J(g\, x)&=[j(g^{J(x)J(x(t))^{-1}}\, a(t))]\\
&=[\operatorname{Ad}^*g^{J(x)J(x(t))^{-1}}\, j(a(t))]\\
&=g\, [j(a(t))]=g\, J(x),\end{aligned}$$ so the equivariance follows.
We now check that defines a $G$-action, i.e., that:
(a) If $e\in G$ is the identity element, then $e\, x=x$, for all $x\in\Sigma(M)$;
(b) If $g,h\in G$, then $g\,(h\, x)=(gh)\, x$, for all $x\in\Sigma(M)$;
In order to prove that (a) holds, one observes that the identity element $e\in G$ is fixed by the right dressing action of $G^*$, so if $x=[a(t)]\in\Sigma(M)$ we find: $$e\, [a(t)]=[e^{J(x)J(x(t))^{-1}}\, a(t)]=[a(t)],$$ and (a) follows.
To prove (b), we first observe that if $x=[a(t)]\in\Sigma(M)$ so that $h\, x=[h^{J(x)J(x(t))^{-1}}\, a(t)]$, then: $$J(h\, x)=h\, J(x),\quad J((h\, x)(t))=h^{J(x)J(x(t))^{-1}}J(x(t)).$$ It then follows from the twisted multiplicativity of the left dressing action of $G$ on $G^*$ that: $$\begin{aligned}
g\, (h\, x)&=g\,(h\, [a(t)])\\
&=[g^{J(h\, x)J((h\, x)(t))^{-1}}h^{J(x)J(x(t))^{-1}}\, a(t)]\\
&=[g^{(h\, J(x))(h^{J(x)}\, J(x(t))^{-1})}h^{J(x)J(x(t))^{-1}}\, a(t)]\\
&=[g^{h\, (J(x) J(x(t))^{-1})}h^{J(x)J(x(t))^{-1}}\, a(t)].\end{aligned}$$ Using the twisted multiplicativity of the right dressing action of $G^*$ on $G$ we obtain: $$\begin{aligned}
g\,(h\, x)&=g\,(h\, [a(t)])\\
&=[g^{h\, (J(x) J(x(t))^{-1})}h^{J(x)J(x(t))^{-1}}\, a(t)]\\
&=[(gh)^{J(x)J(x(t))^{-1}}\, a(t)]\\
&=(gh)\,[a(t)]=(gh)\, x,\end{aligned}$$ so (b) holds.
Finally, we need to check that the $G$-action is hamiltonian with momentum map $J$. By Proposition \[prop:local:act\] we have a local hamiltonian $G$-action on $\Sigma(M)$ with momentum map $J$. This local action satisfies the twisted multiplicativity property and the induced action on the Lie algebroid $A(\Sigma(M))=T^*M$ is the cotangent lifted action. On the other hand, the $G$-action also satisfies the twisted multiplicativity property and induces the same action on $T^*M$. By the uniqueness property proved in Proposition \[prop:general\], the two actions must coincide, so we conclude that the lifted action is hamiltonian with momentum map $J$.
\[rem:local:action\] Assume that $G$ is not a complete Poisson-Lie group. The identity $e\in G$ is a fixed point for the infinitesimal right dressing action of $G^*$ on $G$. It follows that, for any element $u\in G^*$, the (local) dressing action $g^u$ is defined for $g$ in a sufficiently small neighborhood of $e\in G$ (which depends on $u$). Similarly, if $u:I\to G^*$ is any path starting at $u(0)=e\in G^*$, a compactness argument shows that $g^{u(1) u(t)^{-1}}$ is defined provided $g$ is sufficiently close to $e\in G^*$. It follows that, when $G$ is not complete, formula still defines a *local action* of $G$ on $\Sigma(M)$. The proof of the Theorem shows that this action will be twisted multiplicative and hamiltonian, with momentum map $J:\Sigma(M)\to G^*$. In other words, it is the local action given by Proposition \[prop:local:act\].
Applications {#sec:applications}
============
In this section we will consider three applications of Theorem \[thm:lift:actions\]. The first application is to the integration of a Poisson quotient $M/G$. The second application concerns hamiltonian actions and the integration of Lu-Weinstein quotients. The third application is to the integration of Poisson homogeneous spaces.
Integrability of Poisson quotients {#subsec:quotients}
----------------------------------
We now turn to the integrability of Poisson quotients. We start with the following remark:
\[lift:actions:proper\] Let $G\times M\to M$ be a Poisson action of a complete Poisson-Lie group on an integrable Poisson manifold. The lifted action $G\times\Sigma(M)\to
\Sigma(M)$ is proper (respectively, free) if and only if the original action $G\times M\to M$ is proper (respectively, free).
For the proof, we use the following simple fact: Let a Lie group $G$ act smoothly on manifolds $P$ and $Q$ and let $\phi:P\to Q$ be a $G$-equivariant map. If the action on $Q$ is free (respectively, proper), then the action on $P$ is free (respectively, proper).
Now, we just need to observe that the identity section ${\varepsilon}:M\to\Sigma(M)$ and the target map ${\mathbf{t}}:\Sigma(M)\to M$ are $G$-equivariant maps.
From now on we will assume that the action $G\times M\to M$ is proper and free, so that the lifted action is also proper and free. For a Poisson action $G\times M\to M$ the space of $G$-invariant functions is a Poisson subalgebra $C^\infty(M)^G\subset C^\infty(M)$. It follows that if the action is proper and free, this space can be identified with $C^\infty(M/G)$, so that $M/G$ has a reduced Poisson structure $\pi_\text{red}$ such that the quotient map $M\to M/G$ is a Poisson map.
Using the lifted action one can construct a symplectic groupoid integrating $M/G$:
\[thm:int:M/G\] Let $(G,\pi_G)$ be a complete Poisson-Lie group, $(M,\pi)$ an integrable Poisson manifold and $G\times M\to M$ a Poisson action which is proper and free. Then the symplectic reduced space $(J^{-1}(e)/G,\Omega_\text{red})$ is a symplectic groupoid over $M/G$ which integrates the reduced Poisson manifold $(M/G,\pi_\text{red})$.
By Proposition \[lift:actions:proper\], the lifted action of $G$ on $\Sigma(M)$ is proper and free. It follows that its momentum map $J:\Sigma(M)\to G^*$ is a groupoid morphism which is a submersion. Therefore, its kernel $J^{-1}(e)\subset\Sigma(M)$ is a Lie subgroupoid. The equivariance of the momentum map implies that $J^{-1}(e)$ is $G$-invariant. Moreover, the twisted multiplicativity property guarantees that $G$ acts on $J^{-1}(e)$ by groupoid automorphisms. We conclude that the quotient $J^{-1}(e)/G$ is a Lie groupoid over $M/G$.
Now, by the Lu-Weinstein reduction theorem, there exists a reduced symplectic form $\Omega_\text{red}$ on $J^{-1}(e)/G$. It follows from the multiplicativity of $\Omega$ that $\Omega_\text{red}$ is a multiplicative 2-form, so that $(J^{-1}(e)/G,\Omega_\text{red})$ is a symplectic groupoid over $M/G$ and the source map $s:J^{-1}(e)/G\to M/G$ is a Poisson morphism. Hence, $(J^{-1}(e)/G,\Omega_\text{red})$ integrates $(M/G,\pi_\text{red})$.
Theorem \[thm:int:M/G\] raises a natural question: does $\Sigma(M/G)$ coincide with the symplectic reduction: $$\Sigma(M)//G:=J^{-1}(e)/G\, ?$$ Since we already know that this is a symplectic groupoid integrating the quotient $M/G$, the question amounts to deciding whether the source fibers are 1-connected or not. This problem can be handled by the method used in [@FerOrRa] for the case $\pi_G=0$.
First, we observe that the source fibers of $J^{-1}(e)$ need not be connected. Let $J^{-1}(e)^0$ be the unique source connected Lie subgroupoid of $\Sigma (M)$ integrating the Lie algebroid $j^{-1}(0)\subset T^*M$: $$J^{-1}(e)^0=\{[a]\in \Sigma (M): j(a(t))=0,\forall t\in[0,1]\}.$$
If $G$ is connected, $J^{-1}(e)^0$ is $G$-invariant. The action of $G$ on $J^{-1}(e)^0$ is by automorphisms, so we have a groupoid morphism $\Phi:J^{-1}(e)^0 \to J^{-1}(e)^0/G$ which induces the Lie algebroid morphism $\phi:j^{-1}(0)\to j^{-1}(0)/G\cong T^*(M/G)$. On the other hand, the Lie algebroid morphism $\phi$ integrates to a morphism of source 1-connected groupoids $\widehat{\Phi}: {\mathcal{G}}(j^{-1}(0)) \to \Sigma (M/G)$ which covers the homomorphism $\Phi$ (here, ${\mathcal{G}}(j^{-1}(0))$ denotes the source 1-connected groupoid integrating $j^{-1}(0)$). This yields a commutative diagram: $$\label{eq:diag:com:symp}
\newdir{ (}{{}*!/-5pt/@^{(}}
\xymatrix{
K_M\ar[d]^{\widehat{\Phi}}\ar@<-2pt>@{ (->}[r]& {\mathcal{G}}(j^{-1}(0))\ar[d]^{\widehat{\Phi}}\ar[r]^{\widehat{p}}& J^{-1}(e)^0\ar[d]^{\Phi}\\
K_{M/G}\ar@<-2pt>@{ (->}[r]& \Sigma (M/G)\ar[r]^p& J^{-1}(e)^0/G}$$ where $K_M$ and $K_{M/G}$ are group bundles over $M$ and $M/G$, respectively, with discrete fibers. Now the same argument as in [@FerOrRa] shows that the group bundle $K_M$ measures how symplectization and reduction fail to commute. More precisely, we recover [@Ste Proposition 5.3]:
\[thm:int:reduct:alt\] Let $(G,\pi_G)$ be a connected complete Poisson-Lie group, $(M,\pi)$ an integrable Poisson manifold and $G\times M\to M$ a Poisson action which is proper and free. Then symplectization and reduction commute if and only if the discrete groups $$K_{m}:=\frac{\{a:I\to j^{-1}(0) \mid a \text{\rm ~is a cotangent loop such that }a\sim 0_m\}}
{\{\text{\rm cotangent homotopies with values in } j^{-1}(0)\}}$$ are trivial, for all $m\in M$.
We refer to [@FerOrRa] for a detailed proof.
Integration of hamiltonian actions {#subsec:int:hamiltonian}
----------------------------------
Let us turn now to the study of hamiltonian actions $G\times M\to
M$. The following remark is due to Xu [@Xu0]:
If the action $G\times M\to M$ is a hamiltonian action with momentum map $\mu:M\to G^*$ , then the momentum map of the lifted action $J:\Sigma(M)\to G^*$ is exact: $$J(x)=\mu({\mathbf{t}}(x)\mu({\mathbf{s}}(x))^{-1}.$$
For our next proposition we need the definition of an *action groupoid* which we now briefly recall. Let $\Psi:G\times M\to M$ be an action of a Lie group $G$ on a manifold $M$ and let $\psi:{\mathfrak{g}}\to{\ensuremath{\mathfrak{X}}}(M)$ be the corresponding infinitesimal Lie algebra action. The action groupoid associated to $\Psi$ is the groupoid whose elements are pairs $(g,m)\in G\times M$ viewed as arrows from $m$ to $g\cdot m$. We will denote it by $G\ltimes
M$. The associated Lie algebroid, denoted by ${\mathfrak{g}}\ltimes M$, is the trivial vector bundle ${\mathfrak{g}}\times M\to M$ with anchor map $(\xi,m)\mapsto \psi (\xi )_m$ and Lie bracket characterized by $$[\tilde{\xi},\tilde{\eta}]_A=\widetilde{[\xi ,\eta]},\quad (\xi,\eta\in{\mathfrak{g}})$$ where $\tilde{\xi}\in \Gamma({\mathfrak{g}}\ltimes M)$ denotes the constant section associated with an element $\xi\in{\mathfrak{g}}$.
One checks easily that the momentum map $\mu:M\to G^*$ defines a Lie algebroid morphism $\psi:{\mathfrak{g}}\ltimes M\to T^*M$ by setting: $$\label{eq:momentum:ham} \psi(\xi,m) = (\mu^*\xi^R)_m.$$ Now we have:
Assume that the Lie algebroid morphism $\psi:{\mathfrak{g}}\ltimes M\to T^*M$ defined by integrates to a groupoid morphism $\Psi:G\ltimes M\to \Sigma(M)$. Then the lifted $G$-action on $\Sigma(M)$ is a twisted inner action, i.e., it is given by: $$\label{eq:action:inner}
g\,x=\Psi(g,{\mathbf{t}}(x))\cdot x\cdot \Psi(g^{J(x)},{\mathbf{s}}(x))^{-1}.$$
One checks that formula defines a $G$-action on $\Sigma(M)$ which is twisted multiplicative (relative to $J:\Sigma(M)\to G^*$). The corresponding $G$-action induced on the Lie algebroid $A=T^*M$ is just the cotangent lifted action of $G\times M\to M$. By Proposition \[prop:general\], it follows that must coincide with the lifted $G$-action on $\Sigma(M)$.
Let us assume now that the hamiltonian action $G\times M\to M$ is proper and free. Then the quotient $M//G:=\mu^{-1}(e)/G$ is a Poisson submanifold of $M/G$. Is this Poisson submanifold integrable? Which symplectic groupoid integrates it? The following result gives an answer to these questions:
Let $G\times M\to M$ be a proper and free hamiltonian action with momentum map $\mu:M\to G^*$ and assume that the Lie algebroid morphism $\psi:{\mathfrak{g}}\ltimes M\to T^*M$ given by integrates to a groupoid morphism $\Psi:G\ltimes M\to \Sigma(M)$. Then, there exists a hamiltonian action of $G\times G$ on $\Sigma(M)$ which is proper and free, and the symplectic quotient, $$\Sigma(M)//G\times G:=J^{-1}(e)|_{\mu^{-1}(e)}/G\times G \subset \Sigma(M)//G,$$ is a symplectic subgroupoid integrating the Poisson submanifold $M//G\subset M/G$.
We will only sketch a proof of this result. Further details will be available in [@FP]. First one defines an action of $G\times G$ on $\Sigma(M)$ by setting: $$(g_1,g_2)\,x=\Psi(g_1,{\mathbf{t}}(x))\cdot x\cdot \Psi(g_2^{J(x)},{\mathbf{s}}(x))^{-1}.$$ This action is hamiltonian, with momentum map: $$\bar{\mu}:\Sigma(M)\to G^*\times G^*,\quad x\mapsto (\mu({\mathbf{t}}(x)),\mu({\mathbf{s}}(x))^{-1}).$$ Observe that the restriction of this action to the diagonal of $G\times G$ yields the lifted $G$-action.
Next one checks that the $(G\times G)$-action on $\Sigma(M)$ is proper, free, and that $(e,e)\in G^*\times G^*$ is a regular value of the momentum map. Hence, we have the symplectic quotient: $$\Sigma(M)//G\times G=\bar{\mu}^{-1}((e,e))/G\times G=J^{-1}(e)|_{\mu^{-1}(e)}/G\times G.$$ Note that $J^{-1}(e)|_{\mu^{-1}(e)}\subset \Sigma(M)$ is a Lie subgroupoid. One can show that its product structure descends to the quotient $\Sigma(M)//G\times G$, so that this is a symplectic groupoid. Finally, to complete the proof, one verifies that its source (respectively, target) map is a Poisson (respectively, anti-Poisson) map.
The quotient $M//G$ is still defined when $e\in G^*$ is a regular value of $\mu:M\to G^*$ and the action on the level set $\mu^{-1}(e)/G$ is proper and free. In this case, one can check that the groupoid given in the proposition stills gives an integration of $M//G$. However, now $M/G$ need not be a smooth manifold and it may not make sense to speak of the groupoid $\Sigma(M)//G$.
For a compact Poisson-Lie group $G$, a result of Ginzburg and Weinstein [@GiWe] states that there is a Poisson diffeomorphism $e:{\mathfrak{g}}^*\to G^*$. Moreover, for hamiltonian actions of compact Poisson-Lie groups we have the following result of Alekseev [@Alek]:
Let $G$ be a 1-connected, simple, compact Poisson-Lie group and let $G\times M\to M$ be a Poisson action with momentum map $\mu:M\to G^*$. There is a Poisson structure on $M$, gauge equivalent to the original one, such that $G$ acts by Poisson diffeomorphisms with momentum map $e^{-1}\circ\mu:M\to{\mathfrak{g}}^*$.
Our description of the lifted $G$-action allows us to explain this result at the level of the symplectic groupoid. Let us denote by $\pi$ the original Poisson structure on $M$ and by $\tilde{\pi}$ the gauge equivalent Poisson structure. This means that $\pi$ and $\tilde{\pi}$ have the same symplectic leaves and that there is a closed 2-form $B\in\Omega^2(M)$ such that the symplectic structures on a leaf $S$ differ by the pullback of $B$ to $S$ (see [@BuRa]). It follows ([@BuRa Theorem 4.1]) that the symplectic groupoids of $(M,\pi)$ and $(M,\tilde{\pi})$ have the same groupoid structure, while the symplectic forms are related by: $$\tilde{\Omega}=\Omega+{\mathbf{t}}^*B-{\mathbf{s}}^*B.$$ Our next result describes the lifted $G$-action on the symplectic groupoid $(\Sigma(M),\tilde{\Omega})$:
For a 1-connected, simple, compact Poisson-Lie group $G$ acting on an integrable Poisson manifold $(M,\pi)$, the $G$-action on $(M,\tilde{\pi})$ lifts to a hamiltonian $G$-action on $(\Sigma(M),\tilde{\Omega})$ with momentum map $e^{-1}\circ\mu\circ{\mathbf{t}}-e^{-1}\circ\mu\circ{\mathbf{s}}:\Sigma(M)\to{\mathfrak{g}}^*$. Moreover, this action is inner and is explicitly given by: $$g\,x=\Psi(g,{\mathbf{t}}(x))\cdot x\cdot \Psi(g,{\mathbf{s}}(x))^{-1}.$$
The proof is more or less straightforward. We refer to [@FP] for more details.
Poisson homogeneous spaces {#subsec:Poisson:homogeneous:spaces}
--------------------------
Let $(G,\pi _G)$ be a Poisson-Lie group. We say that a Poisson manifold $(M,\pi )$ is a *Poisson homogeneous space* if there exists a Poisson action $G\times M\to M$ which is transitive (see [@D]). In this section, we will give a simple description of the symplectic groupoid $\Sigma (M)$, in the case where the Poisson structure vanishes at some point, by applying Theorem \[thm:lift:actions\].
Let $m_0\in M$ be the point where the Poisson structure vanishes. We identify $M$ with $G/H$, where $H=G_{m_0}$ is the isotropy group at $m_0$. Under this identification, the Poisson action of $G$ on $G/H$ is by left translations and the projection $q:G\to G/H\cong M$ becomes a Poisson map. Then $H$ is a closed subgroup of $G$ with Lie algebra ${\mathfrak{h}}$ such that its annihilator ${\mathfrak{h}}^\perp$ is a Lie subalgebra of ${\mathfrak{g}}^*$, that is, $H$ is a coisotropic subgroup of $G$. In the sequel we will assume that $H$ is connected.
Our final aim is to describe the symplectic groupoid integrating the Poisson manifold $G/H$ as some kind of quotient. This will be possible under the following completeness assumption (compare with [@BCST]):
Let $G$ be a Poisson-Lie group and let $H\subset G$ be a closed, connected, coisotropic subgroup with Lie algebra ${\mathfrak{h}}\subset{\mathfrak{g}}$. We say that the pair $(G,H)$ is *relatively complete* if the annihilator ${\mathfrak{h}}^\perp\subset{\mathfrak{g}}^*$ integrates to a closed subgroup $H^\perp\subset G^*$ and the right dressing action ${\mathfrak{g}}^*\to {\ensuremath{\mathfrak{X}}}(G)$ restricted to ${\mathfrak{h}}^\perp$ integrates to an action of $H^\perp$ on $H$.
As above, we let $\Sigma(G)$ be the symplectic groupoid of the Poisson manifold $G$ and we let $J:\Sigma(G)\to
G^*$ be the momentum map for the local action of $G$ on $\Sigma(G)$ obtained by lifting the action of $G$ on itself by left translations (recall that we are not assuming that $G$ is complete, so this is only a local action). We have:
\[thm:Poisson:hom\] Let $G$ be a Poisson-Lie group and let $H\subset G$ be a closed, connected, coisotropic subgroup such that the pair $(G,H)$ is relatively complete. Then $(J^{-1}(H^\perp )/H,\Omega _{\operatorname{red}})$ is a symplectic groupoid integrating the Poisson homogeneous space $G/H$.
The action by left translations of $G$ on itself is free, so the lifted action of $G$ on $\Sigma(G)$ is also free and $J:\Sigma(G)\to G$ is a submersion. By assumption, $H^\perp\subset G^*$ is a closed subgroup, so it follows that $J^{-1}(H^\perp)\subset\Sigma(G)$ is a Lie subgroupoid. Moreover, using that $J$ is a Poisson submersion and that $H^\perp$ is coisotropic, so is $J^{-1}(H^\perp)$.
By our relative completeness assumption, the action of $H$ on $G$ by left translations can be integrated to a *global action* of $H$ on $J^{-1}(H^\perp)\subset\Sigma(G)$. This follows from the explicit formula , which only uses the dressing action of $H^\perp$ on $H$.
Now the twisted multiplicativity property shows that the quotient $J^{-1}(H^\perp)/H$ inherits a groupoid structure over $G/H$. This quotient is also a symplectic manifold, and the source (respectively, target) map is a Poisson (respectively, anti-Poisson) map, so we conclude that it is a symplectic groupoid integrating $G/H$.
(see also [@Lu3 Remark 5.13]) Let us consider the case where $G$ is a complete Poisson-Lie group. Then the symplectic groupoid $\Sigma(G)$ is isomorphic to the transformation groupoid $G^*\times G$, associated with the left dressing action of $G^*$ on $G$, denoted by $(u,g)\mapsto {}^ug$. The lift to the symplectic groupoid $\Sigma(G)$ of the action of $G$ on itself by left translations, is given by: $$g\cdot (u,h)=(^gu,g^uh).$$ The momentum map $J:\Sigma(G)\to G^*$ is the projection on the factor $G^*$ and Theorem \[thm:Poisson:hom\] says that: $$J^{-1}(H^\perp )/H =(H^\perp \times G)/H,$$ is a symplectic groupoid integrating $G/H$.
We do not know whether Poisson homogeneous spaces of the form $G/H$ are always integrable. Observe that $G/H$ is integrable whenever the map $q:G\to G/H$ is a complete Poisson map. This follows from a criteria for integrability due to Crainic and Fernandes (see [@CrFe2 Theorem 8]) which states that a Poisson manifold is integrable if and only if it admits a complete symplectic realization. For example, when $H$ is compact, the quotient map is complete so $G/H$ is integrable, However, in general, even when $G/H$ is integrable, the pair $(G,H)$ may fail to be relatively complete, so to construct the symplectic groupoid of $G/H$ one requires more complicated procedures that the one given in Theorem \[thm:Poisson:hom\].
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, Momentum mappings and reduction of Poisson actions, in *Symplectic geometry, groupoids, and integrable systems* (Berkeley, CA, 1989), 209–-226, Math. Sci. Res. Inst. Publ., 20, Springer, New York (1991).
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[^1]: R.L.F. is partially supported by the Fundação para a Ciência e a Tecnologia (FCT/Portugal)and project PTDC/MAT/098936/2008. D.I.P. is partially supported by CSIC through a “JAE-Doc" research contract, by MICYT (Spain) Grants MTM2007-62478 and S-0505/ESP/0158 of the CAM. Both authors are partially supported by CSIC/FCT grant 2007PT0014.
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abstract: 'In this paper we propose Spatial PixelCNN, a conditional autoregressive model that generates images from small patches. By conditioning on a grid of pixel coordinates and global features extracted from a Variational Autoencoder (VAE), we are able to train on patches of images, and reproduce the full-sized image. We show that it not only allows for generating high quality samples at the same resolution as the underlying dataset, but is also capable of upscaling images to arbitrary resolutions (tested at resolutions up to $50\times$) on the MNIST dataset. Compared to a PixelCNN++ baseline, Spatial PixelCNN quantitatively and qualitatively achieves similar performance on the MNIST dataset.'
bibliography:
- 'references.bib'
---
Introduction
============
[1]{} ![ Random samples generated from PixelCNN++ and Spatial PixelCNN. Samples on the top row of each group are generated at $28 \times 28$ resolution, while the bottom are $56 \times 56$ resolution. All three models generate high quality samples at $28 \times 28$ resolution. Notably, Spatial PixelCNN qualitatively performs on par with PixelCNN++, despite training on $8 \times 8$ patches. Compared to training on $16 \times 16$ patches, Spatial PixelCNN trained on $8 \times 8$ patches produces more coherent $56 \times 56$ resolution samples. []{data-label="fig:random-samples"}](without_vae_28x28_samples.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Random samples generated from PixelCNN++ and Spatial PixelCNN. Samples on the top row of each group are generated at $28 \times 28$ resolution, while the bottom are $56 \times 56$ resolution. All three models generate high quality samples at $28 \times 28$ resolution. Notably, Spatial PixelCNN qualitatively performs on par with PixelCNN++, despite training on $8 \times 8$ patches. Compared to training on $16 \times 16$ patches, Spatial PixelCNN trained on $8 \times 8$ patches produces more coherent $56 \times 56$ resolution samples. []{data-label="fig:random-samples"}](with_vae_coord_8x8_samples.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Random samples generated from PixelCNN++ and Spatial PixelCNN. Samples on the top row of each group are generated at $28 \times 28$ resolution, while the bottom are $56 \times 56$ resolution. All three models generate high quality samples at $28 \times 28$ resolution. Notably, Spatial PixelCNN qualitatively performs on par with PixelCNN++, despite training on $8 \times 8$ patches. Compared to training on $16 \times 16$ patches, Spatial PixelCNN trained on $8 \times 8$ patches produces more coherent $56 \times 56$ resolution samples. []{data-label="fig:random-samples"}](with_vae_coord_16x16_large_samples.png "fig:"){width="1.0\linewidth"}
{width="1.0\linewidth"}
Generative image modeling has elicited much excitement in the past few years. Much of the enthusiasm is centered on a small set of popular techniques. These include variational inference, through the use of the reparameterization trick [@Kingma:2013aa; @pmlr-v32-rezende14], integral probability metrics [@goodfellow2014generative; @Arjovsky:2017aa; @nowozin2016f], and autoregressive, explicit density estimation [@oord2016pixel]. Despite the success of these techniques, it is still challenging to generate high-resolution images, especially for datasets with large variability [@nguyen2017plug; @odena2016conditional], though that is readily changing [@karras2017progressive].
In this paper, we propose Spatial PixelCNN, a conditional autoregressive PixelCNN [@oord2016pixel] capable of generating large images from small patches. We combine the strengths of three components: (1) an autoregressive model—specifically, PixelCNN++ [@salimans2017pixelcnn++]—to capture local image statistics from patches; (2) a latent variable model—in our case a variational autoencoder [@Kingma:2013aa]—to capture the global structures in images; and (3) spatial locations of each pixel in an image. Intuitively, an image is modeled autogressively pixel-by-pixel. Each pixel sampled is conditioned on (1) a subset of previously sampled pixels in its neighborhood; (2) a latent code that encodes global image statistics; and (3) 2-D spatial coordinates indicating its location in the image. This spatial conditioning enables us to reduce the coupling of each pixel from the resolution of the image. At generation time, our approach takes a target resolution as input, and outputs an image of arbitrary size (e.g. $224\times224$ images from $8\times8$ training patches, Fig. \[fig:concept\]).
While performing impressively, state-of-the-art super-resolution techniques [@ledig2016photo; @tyka2017semantic] (a) require a large dataset of high-resolution images—which may not be always available in practice; (b) have a fixed-sized output. We instead explore learning a generic upscaling function from only low-resolution images, in the absence of high-resolution images. Compared to existing image generative models that output fixed-sized images, our approach can be trained on small patches as opposed to full-sized images, thus requiring much less GPU memory—an important implication for scalability. Our method also enables the possibility of training on a dataset of images of mixed resolutions.
Our contributions are summarized as follows:
1. We show the effects of spatial conditioning, using pixel coordinates, on generating high-resolution images.
2. We show quantitative and qualitative evidence that Spatial PixelCNN models full-sized $28\times28$ MNIST [@lecun1998gradient] images relatively well compared to a baseline PixelCNN++ [@salimans2017pixelcnn++] model, despite being trained on $8\times8$ patches.
3. We show that remarkably, Spatial PixelCNN is capable of generating coherent images at arbitrary resolutions for the MNIST dataset.
Related Work
============
[0.2]{} ![ Examples showing the need for coordinates and a global latent code: (*a*) vanilla PixelCNN++ trained on $20 \times 20$ patches, displaying a juxtaposition of two distinct digits; (*b*) PixelCNN++ with spatial conditioning trained on $20 \times 20$ patches achieves $1.48$ bits per dimension and, models $28 \times 28$ MNIST; (*c*) when trained on $4 \times 4$ patches, spatial conditioning alone does not confer global coherence, despite achieving $1.42$ bits per dimension; (*d*) and while conditioning on a latent code helps with global coherence, without coordinates it does not provide scale. []{data-label="fig:image-defects"}](pixelcnn_20x20_sample.png "fig:"){width="1.0\linewidth"}
[0.2]{} ![ Examples showing the need for coordinates and a global latent code: (*a*) vanilla PixelCNN++ trained on $20 \times 20$ patches, displaying a juxtaposition of two distinct digits; (*b*) PixelCNN++ with spatial conditioning trained on $20 \times 20$ patches achieves $1.48$ bits per dimension and, models $28 \times 28$ MNIST; (*c*) when trained on $4 \times 4$ patches, spatial conditioning alone does not confer global coherence, despite achieving $1.42$ bits per dimension; (*d*) and while conditioning on a latent code helps with global coherence, without coordinates it does not provide scale. []{data-label="fig:image-defects"}](pixelcnn+coords_20x20_sample.png "fig:"){width="1.0\linewidth"}
[0.2]{} ![ Examples showing the need for coordinates and a global latent code: (*a*) vanilla PixelCNN++ trained on $20 \times 20$ patches, displaying a juxtaposition of two distinct digits; (*b*) PixelCNN++ with spatial conditioning trained on $20 \times 20$ patches achieves $1.48$ bits per dimension and, models $28 \times 28$ MNIST; (*c*) when trained on $4 \times 4$ patches, spatial conditioning alone does not confer global coherence, despite achieving $1.42$ bits per dimension; (*d*) and while conditioning on a latent code helps with global coherence, without coordinates it does not provide scale. []{data-label="fig:image-defects"}](pixelcnn+coords_4x4_sample.png "fig:"){width="1.0\linewidth"}
[0.2]{} ![ Examples showing the need for coordinates and a global latent code: (*a*) vanilla PixelCNN++ trained on $20 \times 20$ patches, displaying a juxtaposition of two distinct digits; (*b*) PixelCNN++ with spatial conditioning trained on $20 \times 20$ patches achieves $1.48$ bits per dimension and, models $28 \times 28$ MNIST; (*c*) when trained on $4 \times 4$ patches, spatial conditioning alone does not confer global coherence, despite achieving $1.42$ bits per dimension; (*d*) and while conditioning on a latent code helps with global coherence, without coordinates it does not provide scale. []{data-label="fig:image-defects"}](pixelcnn+vae+coords_16x16_reconstruction.png "fig:"){width="1.0\linewidth"}
Pixel Recurrent Neural Networks [@oord2016pixel] (PixelRNN), are a class of powerful generative model. PixelRNN is an explicit generative model which can be trained to directly maximize the likelihood of the training data. Here, the likelihood $p({\mathbf{x}})$ of each 2-D image ${\mathbf{x}}\in \mathbb{R}^{W\times H}$ is decomposed via chain rule into:
$$p({\mathbf{x}}) = \prod_{j}^{n}\prod_{i}^{n}{p(x_{j \ast n+i}|{\mathbf{x}}_{<j \ast n+i})}
\label{eq:autoregressive-pixelcnn}$$
which is the product of every pixel $x_{j \ast n+i}$, conditioned on all previous pixels ${\mathbf{x}}_{<j \ast n+i}$ in the row-major order—a left to right, top to bottom order of columns and rows $(i, j)$.
It is this grounding in RNNs, that helps explain the motivation to train on only a patch of an image at time. RNNs have been used extensively for sequence modeling [@Krause:2016aa; @theis2015generative; @kim2016character]. Often in the area of natural language processing (NLP), the corpus of text being trained is too large to feed the entire sequence of characters or words to the model at once. In order to circumvent that limitation, training involves a truncated form of backpropagation through time [@werbos1990backpropagation; @graves2013generating]. Our approach of training on patches can be seen as a similar trade-off. Rather than conditioning each pixel on all previous pixels in a given image, we only condition on a local window of pixels.
Pixel Convolutional Neural Networks (PixelCNN) [@oord2017neural], are a reformulation of PixelRNNs, that exploit masked convolutions to parallelize the the autoregressive computation. The approach of using masked convolutions for autoregressive density estimation has also been exploited for text [@kalchbrenner2016neural], audio [@oord2016wavenet], and videos [@pmlr-v70-kalchbrenner17a].
Thus, there is reason to believe sequential modeling of pixels in an image can be achieved through training on patches utilizing a PixelCNN. Though to allow recreating the original image, there are several additional factors which are crucial to producing an adequate result.
Variational Autoencoders
------------------------
Variational autoencoders [@Kingma:2013aa] are a form of latent variable model. They are trained to encode their input ${\mathbf{x}}$ into a latent code ${\mathbf{z}}$. The true posterior $p({\mathbf{z}}|{\mathbf{x}})$ is often intractable, so an approximate posterior $q({\mathbf{z}}|{\mathbf{x}})$ is computed by optimizing a variational lower bound on the log-likelihood of the data. Often this latent code is interpretable and encodes a global representation of ${\mathbf{x}}$.
[@gulrajani2016pixelvae; @chen2016variational] have evaluated integrating a PixelCNN decoder into a VAE framework. [@gulrajani2016pixelvae] note that it is critical to ensure the receptive field of the PixelCNN is smaller than the input image ${\mathbf{x}}$. Otherwise, a powerful PixelCNN decoder alone can model the entire image distribution, rendering the latent code ${\mathbf{z}}$ from the VAE unused. When evaluated on binarized MNIST [@larochelle2011neural], both report the successful combination yields state-of-the-art results.
Spatial PixelCNN indeed fits the criteria needed to ensure utilization of the latent code ${\mathbf{z}}$, as it only trains on a patch of an image at a time. In fact the inclusion of a latent variable model, in this case a VAE, is necessary to ensure the model has a global representation of the images.
Conditioning on Pixel Coordinates
---------------------------------
Compositional Pattern Producing Networks (CPPNs) [@stanley2007compositional] are feedforward networks that utilize a wide array of transfer functions and are often trained via evolutionary algorithms. CPPNs can encode 2-D images [@nguyen2015deep], 3-D objects [@clune2011evolving] and even weights of another target network [@stanley2009hypercube].
To encode a 2-D color image, a CPPN performs a transformation $f:\mathbb{R}^{2} \to \mathbb{R}^3$ parameterized by a network which takes as input a pair of coordinates $(i, j)$ and outputs 3 color values (e.g. RGB) [@secretan2008picbreeder]. A pair of coordinates $(i, j)$ is often computed by evenly sampling a pre-defined range e.g. $[-1, 1]$. The coordinates are assembled into a grid, each corresponding to a pixel in the training image. At image generation time, an image can be rendered at an arbitrarily large resolution by simply sampling more coordinates within the specified range, and querying the CPPN for the associated color value.
CPPNs have been shown to impose a strong spatial prior yielding images of highly regular patterns [@secretan2008picbreeder; @nguyen2015deep]. It is this spatial regularity from CPPNs that we exploit in this work. We find that conditioning Spatial PixelCNN on a grid of coordinates is crucial for ensuring a coherent image is generated, and also confers an ability to upscale images.
Other High-Resolution Image Generation Methods
----------------------------------------------
Since directly modeling high resolution images is challenging, an effective approach is to generate an image in hierarchical stages of increasing resolutions [@zhang2016stackgan; @Denton:2015aa; @karras2017progressive]. While producing impressive results, this approach outputs fixed-sized images, and requires storing the entire image in GPU memory at once—this is problematic when the training images exceed GPU memory capacity. Spatial PixelCNN instead allows for choosing the target image size at generation time, and only processes a small patch of the image at a time.
To cut down on GPU memory requirements, [@tyka2017semantic] devises upscaling image tiles, and stitching them together to produce the final result. While requiring less memory, the approach still retains the reliance on generating a fixed-sized output.
Our framework most closely resembles [@ha2017latent], which combines spatial coordinates, and the latent code from a VAE, trained as a GAN end-to-end on full-sized images. Our model differs in two important ways: (1) Spatial PixelCNN conditions each pixel on the latent code *and* a local neighborhood of preceding pixels rather than assuming conditional independence given the latent code; (2) Spatial PixelCNN is trained on patches rather than full-sized images.
Methods
=======
In this section we define the mathematical formulation of our model. We additionally describe how we generate images of a target size from the trained model.
Spatial PixelCNN
----------------
Our proposed model is based on a modified version of PixelCNN++ [@salimans2017pixelcnn++]. In order to reduce cumbersome mathematical notation, we restate Eq. \[eq:autoregressive-pixelcnn\] as:
$$p({\mathbf{x}}) = \prod_{i}^{n^2}{p(x_{i}|{\mathbf{x}}_{<i})}
\label{eq:autoregressive-pixelcnn-reformulated}$$
where $i$ denotes the virtual index of the pixel, if the image were flattened into a 1-D array. Rather than training the model on full-sized $n \times n$ images ${\mathbf{x}}$, we instead choose random patches ${\mathbf{y}}$ of size $m \times m$ taken from the images:
$$p({\mathbf{y}}) = \prod_{i}^{m^2}{p(y_{i}|{\mathbf{y}}_{<i})}
\label{eq:autoregressive-patch}$$
Given that our goal is to model the set of images ${\mathbf{X}}$, rather than merely a collection of all patches ${{\mathbf{Y}_{m \times m}}}$, we condition on a normalized coordinate grid ${{\mathbf{G}_{n \times n}}}$ $\in \mathbb{R}^{2\times n \times n}$ that has the same number of coordinate pairs as pixels in the full-sized image. Specifically, ${{\mathbf{G}_{n \times n}}}$ is composed of $n \times n$ evenly spaced coordinate pairs $(i, j)$ within the range $[-1, 1]$. We choose patches ${\mathbf{g}}$ from ${{\mathbf{G}_{n \times n}}}$ corresponding to the given patch ${\mathbf{y}}$. To condition each pixel $y_i$ on its corresponding coordinate pair $g_i$, we make use of the gated spatial conditioning as outlined in [@van2016conditional] arriving at the conditional probability:
$$p({\mathbf{y}}|{\mathbf{g}}) = \prod_{i}^{m^2}{p(y_{i}|{\mathbf{y}}_{<i}, g_{i})}
\label{eq:grid-conditioned-patch}$$
We found such spatial conditioning crucial to imparting coherence to the patches ${\mathbf{y}}$. Without conditioning on ${\mathbf{g}}$, the model may assign a high probability to distinct patches of the generated image, allowing juxtapositions (Fig. \[fig:image-defects\]). However, as the model is trained on patches ${\mathbf{y}}$ of fewer dimensions, the extra spatial information provided by ${\mathbf{g}}$ is not enough to ensure global coherence (Fig. \[fig:image-defects-c\]).
In order to provide this global coherence, we also condition on a latent code, provided by a VAE. Previous treatments [@gulrajani2016pixelvae; @chen2016variational] combinining PixelCNN and VAE have made use of PixelCNN as a decoder. We found that by only training on patches, the VAE was unable to capture global features (data not shown). Instead, we jointly train the two models: VAE on images ${\mathbf{X}}$ and Spatial PixelCNN on patches ${{\mathbf{Y}_{m \times m}}}$.
Specifically, we minimize the sum of an *image loss* and a *patch loss*:
$$\mathcal{L} = \mathcal{L}^x + \mathcal{L}^y
\label{eq:total-loss}$$
The *image loss* $\mathcal{L}^x$ is the typical VAE loss, that of the negative log-likelihood of the images and the KL-divergence of the approximate posterior from the prior:
$$\mathcal{L}^x = -\mathbb{E}_{{\mathbf{x}}\sim {\mathbf{X}},{\mathbf{z}}\sim q({\mathbf{z}}|{\mathbf{x}})}\mathrm{log}p({\mathbf{x}}) + D_{KL}(q({\mathbf{z}}|{\mathbf{x}})||p({\mathbf{z}}))
\label{eq:image-loss}$$
The *patch loss* $\mathcal{L}^y$ is the negative log-likelihood of the patches, given the coordinate grid ${\mathbf{g}}$ and latent ${\mathbf{z}}$:
$$\mathcal{L}^y = -\mathbb{E}_{{\mathbf{y}}\sim {{\mathbf{Y}_{m \times m}}},{\mathbf{z}}\sim q({\mathbf{z}}|{\mathbf{x}})}\mathrm{log}p({\mathbf{y}}|{\mathbf{g}},{\mathbf{z}})
\label{eq:patch-loss}$$
Note that instead of co-training both VAE and Spatial PixelCNN, it is also possible to pre-train the VAE first, and then train the Spatial PixelCNN. This pre-trained VAE typically has a lower KL-divergence than the co-trained approach. Global features extracted from the pre-trained VAE also allow for successful reconstructions, though we do not evaluate its efficacy in this paper.
Generating Images
-----------------
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As our model is based on patches of images, a sliding window is used during image generation. In order to ensure that each pixel being generated has the maximal number of preceding pixels to condition on, the sliding window moves one pixel at a time from left to right, top to bottom (Fig. \[fig:training-patches\]). This is the same ordering defined by the model for pixels to condition on (Eq. \[eq:autoregressive-pixelcnn\]). Only the maximally conditioned pixels are generated from each patch of the sliding window.
One of the unique aspects of our model is its ability to upscale images to a higher resolution, while only being trained on lower resolution images. To accomplish this, we condition on coordinate patches from a larger coordinate grid during the generation process. That is, when we generate a $56 \times 56$ image, we subdivide the grid into $56 \times 56$ evenly spaced steps. Then as we slide the window over the image we wish to generate, we condition on the associated patch from ${{\mathbf{G}_{56 \times 56}}}$.
Experiments
===========
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_original_28x28.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_latent_28x28.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_reconstruction_28x28.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_reconstruction_56x56.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_reconstruction_112x112.png "fig:"){width="1.0\linewidth"}
[1]{} ![ Reconstructions from a Spatial PixelCNN trained on $8 \times 8$ patches. While VAE reconstructions are blurry (b), Spatial PixelCNN reconstructs the input quite convincingly with fine local details at the original $28 \times 28$ resolution (c). The model shows an impressive ability to produce coherent, plausible reconstructions at arbitrarily high resolutions; (d–f) show reconstructions up to resolution $224 \times 224$—an $8\times$ upscaling. Though note the increase in striations as the resolution increases. See Appendix \[app:large-upscaling\] for $20 \times$ and $50 \times$ reconstructions. []{data-label="fig:reconstructions"}](with_vae_coord_8x8_reconstruction_224x224.png "fig:"){width="1.0\linewidth"}
0.15in
[ccc|M[1.75cm]{}M[1.75cm]{}|M[1.75cm]{}M[1.75cm]{}]{} & &\
& & & &\
Patch Size & Coordinates & Bpd & 28x28 & 56x56 & 28x28 & 56x56\
[[\[row:28x28\]]{}]{}$28 \times 28$& & [****]{}0.88 & 0.17 $\pm$ 0.26 & 0.66 $\pm$ 0.16 & 95.6 $\pm$ 1.2 & 70.7 $\pm$ 5.1\
$28 \times 28$ & $\surd$ & 0.89 & [****]{}0.17 $\pm$ 0.27 & [****]{}0.48 $\pm$ 0.29 & [****]{}97.5 $\pm$ 0.7 & [****]{}86.3 $\pm$ 3.3\
[[\[row:20x20\]]{}]{}$20 \times 20$ & & 1.50 & 0.09 $\pm$ 0.25 & [****]{}0.15 $\pm$ 0.19 & 89.6 $\pm$ 2.6 & [****]{}82.1 $\pm$ 3.6\
$20 \times 20$ & $\surd$ & [****]{}1.48 & [****]{}0.18 $\pm$ 0.26 & 0.27 $\pm$ 0.15 & [****]{}93.7 $\pm$ 1.7 & 81.5 $\pm$ 3.5\
[[\[row:16x16\]]{}]{}$16 \times 16$ & & 1.68 & 0.03 $\pm$ 0.24 & 0.02 $\pm$ 0.12 & 87.0 $\pm$ 2.9 & 79.9 $\pm$ 3.7\
$16 \times 16$ & $\surd$ & [****]{}1.65 & [****]{}0.14 $\pm$ 0.25 & [****]{}0.15 $\pm$ 0.18 & [****]{}91.1 $\pm$ 2.3 & [****]{}82.6 $\pm$ 3.7\
[[\[row:12x12\]]{}]{}$12 \times 12$ & & 1.73 & 0.02 $\pm$ 0.24 & 0.02 $\pm$ 0.13 & [****]{}86.4 $\pm$ 3.0 & 80.3 $\pm$ 3.8\
$12 \times 12$ & $\surd$ & [****]{}1.70 & [****]{}0.13 $\pm$ 0.24 & [****]{}0.20 $\pm$ 0.19 & 85.6 $\pm$ 3.5 & [****]{}83.3 $\pm$ 3.5\
[[\[row:4x4\]]{}]{}$ 4 \times 4$ & & 1.52 & 0.05 $\pm$ 0.21 & 0.05 $\pm$ 0.14 & 71.8 $\pm$ 6.0 & 76.3 $\pm$ 4.3\
$ 4 \times 4$ & $\surd$ & [****]{}1.42 & [****]{}0.14 $\pm$ 0.23 & [****]{}0.17 $\pm$ 0.19 & [****]{}79.1 $\pm$ 5.5 & [****]{}86.4 $\pm$ 2.9\
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& &\
& & & &\
Patch Size & Coordinates & Bpd & 28x28 & 56x56 & 28x28 & 56x56\
[[\[row:16x16-vae\]]{}]{}$16 \times 16$ & & $\leq$1.95 & 0.03 $\pm$ 0.23 & 0.03 $\pm$ 0.11 & 87.6 $\pm$ 2.8 & 81.9 $\pm$ 3.5\
[[\[row:16x16-vae-coord\]]{}]{}$16 \times 16$ & $\surd$ & $\leq$1.87 & 0.19 $\pm$ 0.27 & [****]{}0.16 $\pm$ 0.21 & 93.4 $\pm$ 1.6 & 91.3 $\pm$ 2.3\
[[\[row:16x16-vae-coord-large\]]{}]{}$16 \times 16$& $\surd$ & [****]{}$\leq$1.39 & [****]{}0.18 $\pm$ 0.27 & 0.15 $\pm$ 0.21 & [****]{}97.0 $\pm$ 0.8 & 93.2 $\pm$ 1.7\
[[\[row:8x8-vae-coord\]]{}]{}$ 8 \times 8$ & $\surd$ & $\leq$1.67 & [****]{}0.18 $\pm$ 0.27 & 0.13 $\pm$ 0.20 & 95.1 $\pm$ 1.3 & [****]{}94.1 $\pm$ 1.6\
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Here we describe the details for the experiments conducted. This includes model architecture, training hyperparameters, and the specific metrics used to evaluate the model.
Model Details
-------------
We use a modified version of PixelCNN++ [@salimans2017pixelcnn++]. Our model similarly makes use of six blocks of ResNet [@he2016deep] layers. Spatial PixelCNN is conditioned on both a latent vector and a regularized coordinate grid. For each change in dimensions between blocks, the coordinate grid is resampled to the new layer’s dimensions. This resampling is performed as a simple linear interpolation. Given a grid patch defined by $m \times m$ coordinates in the range $[(i_1, j_1), (i_m, j_m)]$, we linearly interpolate the range into $\frac{m}{2} \times \frac{m}{2}$ equal sized steps. This regularization is key to preserving spatial coherence between ResNet blocks.
We additionally make use of ResNet blocks for the encoder and decoder of our VAE. For each ResNet layer of the decoder, we condition on the latent vector ${\mathbf{z}}$.
For most of our MNIST [@lecun1998gradient] experiments we use two ResNet layers for each block, with a convolution filter size of $25$. We term this the *small network*. In order to determine the trade-off between the model’s ability to compress data (as determined by having a smaller expected bits per dimension), versus its ability to effectively upscale images, we also conduct an experiment with a *large network*. This large network utilizes five ResNet layers for each block, with a convolution filter size of $140$. The size of the latent code is fixed at $60$ for both network sizes.
Training Details
----------------
We make use of the Tensorflow [@tensorflow2015-whitepaper] framework for training and evaluating our models. We train using the Adam [@kingma2014adam] optimizer, with a batch size of $128$, and an initial learning rate of of $0.001$ which is annealed using exponential decay at a rate of $0.999995$ every batch. Additionally a dropout rate of $0.5$, along with L2 regularization of $0.0001$ is utilized. We train each model until convergence, as determined by a lack of improvement over one hundred epochs in the model’s bits per dimension.
At testing and image generation time we use the exponential moving average over the model parameters [@polyak1992acceleration], calculated with a decay of $0.9995$ at each parameter update. We report our negative log-likelihood values in bits per dimension. The reported bits per dimension is calculated as the average bits per dimension over all possible patches for the images in the test set.
Evaluation Details
------------------
As previously noted [@theis2015note], the negative log-likelihood may not be a great qualitative measure for evaluating generative models. This can be seen clearly in our experiments, where bits per dimension does not accurately correlate with the ability of the model to generate coherent images (Fig. \[fig:image-defects\]).
**MS-SSIM:** We first assess the diversity of the images generated by utilizing Multiscale Structural Similarity [@wang2003multiscale; @odena2016conditional] (MS-SSIM). To calculate the MS-SSIM for a given model, we begin by randomly sampling $500$ images from the model. We produce an MS-SSIM score for all unique pairs of images. We then report the mean and standard deviation of these scores. The lower the MS-SSIM, the more diverse the images in the set. The ideal MS-SSIM of a given model should closely resemble that of the underlying data, so we also report an MS-SSIM for $500$ images from the MNIST test set for comparison.
While a measure of diversity shows the model does not exhibit mode collapse, it is unable to address the subjective assessment of image quality. Such an assessment should include the ability of the model to accurately reconstruct a given input, as well as ensuring generated samples reflect the underlying data distribution. Thus we devise two additional measures.
**LeNet Accuracy:** As the combination of PixelCNN and VAE allows for conditioning on an interpretable latent representation, we can assess reconstruction accuracy for these models. We take inspiration from the use of the Inception model [@odena2016conditional] for assessing accuracy. As our model is trained on MNIST, we instead train a version of LeNet [@lecun1998gradient], which has demonstrated strong classification ability for this dataset. We reconstruct $1000$ images from the test set using our models, then measure the classification accuracy for these reconstructions. For comparison, we also report the classification accuracy for the original $1000$ images from the test set.
**LeNet Confidence:** In order to assess random samples generated from the model, we devise a simple confidence score inspired by the Inception Score [@salimans2016improved]. As part of the Inception Score measures diversity, we reformulate our metric to only measure the conditional probability of the label, given an image. For a given image, we calculate the softmax of the LeNet logits. The index of the largest value indicates the predicted class. We take the largest value multiplied by one hundred as a confidence measure, indicating how confident the model is in the prediction. We report the mean and standard deviation of this value for $1000$ random samples from each model.
Given that our model additionally shows strong ability to upscale images to higher resolutions, we also report results for generating $56 \times 56$ images. For both classification accuracy and confidence, we first downsample images to $28 \times 28$ before running the LeNet classifier. We note that individually these metrics are imperfect, especially when assessing downsampled images, but when taken in aggregate they yield a more holistic assessment.
Results
=======
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Patch Size & Coordinates & 28x28 & 56x56\
[[\[row:16x16-vae-reconst\]]{}]{}$16 \times 16$ & & 33.2 & 12.1\
[[\[row:16x16-vae-coord-reconst\]]{}]{}$16 \times 16$ & $\surd$ & 97.2 & 87.8\
[[\[row:16x16-vae-large-reconst\]]{}]{}$16 \times 16$& $\surd$ & [****]{}98.2 & 91.4\
[[\[row:8x8-vae-coord-reconst\]]{}]{}$ 8 \times 8$ & $\surd$ & 96.2 & [****]{}95.3\
-0.1in
In this section we detail the results of our experiments. As the model we propose has multiple components, we perform ablation experiments to verify the need for each. When comparing MS-SSIM scores, we consider scores close to those computed for the actual MNIST digits to be better scores. For LeNet Accuracy and Confidence, a higher score is considered better.
Effects of Spatial Conditioning
-------------------------------
We first assess the importance of spatial conditioning on the generative ability of PixelCNN++ (Table \[tab:sample-analysis\]). We train PixelCNN++ on various patch sizes, both with and without spatial conditioning. We note that even when trained on full-sized images, conditioning on a grid of coordinates boosts both MS-SSIM and LeNet Confidence (Table \[tab:sample-analysis\] Row \[row:28x28\]). This implies the addition of coordinates may help capture structure versus a baseline PixelCNN++.
As we sweep across patch sizes, we see the addition of coordinates keeps the MS-SSIM within range of the underlying dataset. Though, we note as patch size decreases, we observe a decrease in confidence scores for the $28\times28$ samples, and associated coherence of the resultant images (Fig. \[fig:image-defects-c\]).
An important observation from the experiments is that the trained bits per dimension value is not a great representation of the quality of random samples generated from the models (Figs. \[fig:image-defects-b\] & \[fig:image-defects-c\]). In the case of training on $4 \times 4$ patches (Table \[tab:sample-analysis\] Row \[row:4x4\]), the model achieves a similar ability to compress the data (as denoted by a low bits per dimension), as training on $20 \times 20$ patches (Table \[tab:sample-analysis\] Row \[row:20x20\]). Though, comparing the LeNet Confidence of $28 \times 28$ generated images, the $20 \times 20$ patches clearly model the underlying dataset more accurately (training on $20 \times 20$ patches achieves a Confidence of $93.7 \pm 1.7$, while training on $4 \times 4$ patches only results in a score of $79.1 \pm 5.5$).
Addition of a Latent Code
-------------------------
We next consider the effects provided by the addition of a latent code. Even without spatial conditioning, the model trained on $16 \times 16$ patches produces higher confidence scores with the addition of a VAE (Table \[tab:sample-analysis\] Row \[row:16x16\] & Table \[tab:vae-sample-analysis\] Row \[row:16x16-vae\]). Though, the model has trouble with scale (Fig. \[fig:image-defects-d\]). This follows our intuition that spatial conditioning is key to capturing regularity across images.
Images from Patches
-------------------
We now examine Spatial PixelCNN conditioned on both coordinates and a latent code. While a direct comparison is difficult, we note that Spatial PixelCNN produces samples qualitatively similar to PixelCNN++ (Fig. \[fig:random-samples\]). Additionally Spatial PixelCNN trained on $8 \times 8$ patches receives a similar Confidence score to the PixelCNN++ baseline when generating $28 \times 28$ images (Table \[tab:sample-analysis\] Row \[row:28x28\] & Table \[tab:vae-sample-analysis\] Row \[row:8x8-vae-coord\]).
An interesting result can be observed when comparing the small network trained on $8 \times 8$ patches versus the small network trained on $16 \times 16$ patches. Given an equivalent network size, training on smaller patches produces superior results (Table \[tab:vae-sample-analysis\] Rows \[row:16x16-vae-coord\] & \[row:8x8-vae-coord\]). Only with the addition of more parameters, through the use of the large network, does training on $16 \times 16$ patches produce a higher confidence and accuracy (Table \[tab:vae-sample-analysis\] Rows \[row:16x16-vae-coord-large\] & \[row:8x8-vae-coord\]) when generating $28 \times 28$ images.
Upscaling Ability
-----------------
Finally we assess the ability for the network to generate high resolution images while only training on a low resolution dataset. Looking at the generated $56 \times 56$ images, the small network trained on $8 \times 8$ patches works exceedingly well for upscaling (Fig. \[fig:random-samples-small\]). It even tends to retain structural detail better than the large network trained on $16 \times 16$ patches (Fig. \[fig:random-samples-large\]). Not only does it surpass the accuracy and confidence scores of the large network (Table \[tab:vae-sample-analysis\] Rows \[row:16x16-vae-coord-large\] & \[row:8x8-vae-coord\] and Table \[tab:reconstruction-analysis\] Rows \[row:16x16-vae-large-reconst\] & \[row:8x8-vae-coord-reconst\]), it only loses a small amount of accuracy and confidence, despite upscaling $2\times$. It even shows remarkable ability for large upscaling factors (Figs. \[fig:concept\], \[fig:reconstructions-112x112\], & \[fig:reconstructions-224x224\]). See Appendix \[app:large-upscaling\] for examples of $20 \times$ and $50 \times$ upscaling.
Discussion & Conclusions
========================
We demonstrated that the addition of coordinates to PixelCNN++ has a clear positive effect on its generative ability. This can be seen even when trained on full-sized images, rather than on patches. We also showed that with the combination of a VAE, Spatial PixelCNN trained on patches is able to not only generate convincing images at the same resolution as the underlying dataset, but also high resolution images, despite only training on a low resolution dataset.
We believe that our approach should allow for training on mixed-sized images by feeding a resampled version of the image to the VAE. This may potentially allow for less preprocessing of images used for training. We leave this open for future research.
Acknowledgments
===============
We thank Jeff Clune for his valuable discussions and support. We also thank the University of Wyoming Advanced Research Computing Center (ARCC) for their assistance and computing resources which enabled us to perform experiments and analyses. Additionally, we appreciate OpenAI making their code for PixelCNN++ available for reference online.
Large Upscaling Factors {#app:large-upscaling}
=======================
Upscaling Comparisons {#app:large-upscaling-comparison}
---------------------
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A Single High-Resolution Example {#app:large-upscaling-example}
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---
abstract: 'We report a detailed *ab initio* study of two superlattice heterostructures, one component of which is a unit cell of CuPt ordered InSb$_{0.5}$As$_{0.5}$. This alloy part of the heterostructures is a topological semimetal. The other component of each system is a semiconductor, zincblende-InSb, and wurtzite-InAs. Both heterostructures are semiconductors. Our theoretical analysis predicts that the variation in the thickness of the InSb layer in InSb/InSb$_{0.5}$As$_{0.5}$ heterostructure renders altered band gaps with different characteristics (*i.e.* direct or indirect). The study holds promise for fabricating heterostructures, in which the modulation of the thickness of the layers changes the number of carrier pockets in these systems.'
author:
- 'Atanu Patra$^{1}$, Monodeep Chakraborty$^{2}$, Anushree Roy$^{1}$'
title: 'Electronic band structure engineering in InAs/InSbAs and InSb/InSbAs superlattice heterostructures'
---
Introduction
============
Low band gap materials, like InSbAs alloys, find potential applications in various infra-red device fabrications [@Pitanti1; @Hoglund1; @Steveler1; @Jiang1; @Wu1]. In addition, due to large lande-g factor, significant spin-orbit coupling strength and small effective mass, these compounds are also used in high-speed [@Ashley; @Sourribes1] and spin-related [@Nilsson] devices. The promising characteristics of InSb$_{x}$As$_{1-x}$/InSb$_{y}$As$_{1-y}$ heterostructures (HSs) in various applications motivated us to investigate electronic band structure of two superlattice HSs. One component of both HSs consists of one unit cell of InSb$_{0.5}$As$_{0.5}$ in CuPt ordering. For one of the HSs, the other component is InSb in zinc-blende (ZB) structure along \[111\] direction. For the second HS, the other component is InAs in wurtzite (WZ) phase along \[0001\] direction. We varied the thickness of InAs or InSb segments. Thus, we have studied band structures of two sets of systems, (InSb)$_n$(ZB)/InSb$_{0.5}$As$_{0.5}$(CuPt) and (InAs)$_m$(WZ)/InSb$_{0.5}$As$_{0.5}$(CuPt), named as H-I and H-II, respectively. Here, $n$ and $m$ are the number of unit cells of InSb and InAs segments in H-I and H-II. The choice of two different phases (i.e. ZB and WZ) of InSb and InAs parts originates from the fact that while growing these HSs (eg., in HS nanowires) using MOCVD or CBE techniques, InSb and InAs components are formed in these phases [@Johansson; @Ercolani1]. Moreover, these two different semiconductor layers result in compressive and tensile strain at the heterointerface with CuPt-InSb$_{0.5}$As$_{0.5}$.
In this article, we demonstrate that the band gap of the HSs can be tuned by the choice of semiconductor segments and their thickness (*i.e.*, $m$ and $n$). Interestingly, for H-I with $n$=1 both direct and indirect band gaps are of equal energy. In addition, the calculated band structure opens a possibility of achieving different number of carrier pockets in these HSs under perturbation.
Methodology
===========
We have carried out first principles calculations with Wien2k, which is an all-electron-full potential code [@Singh1; @Blugel; @Blaha1]. All crystal structures, discussed in this article, were optimized following Ref. [@Patra1]. To study the electronic band structure, we have performed calculations using local density approximation (LDA) and through modified Becke-Johnson exchange potential (mBJLDA) [@Tran1]. The spherical harmonic function inside the muffin-tin spheres was limited by *l*$_{max}$ = $12$, where the muffin-tin radii for In, As and Sb were fixed at $1.9$, $2.1$ and $2.15$ a.u., respectively. In interstitial regions the charge density and potential were defined through *G*$_{max}$ at $14$ Bohr$^{-1}$. The tetrahedron method was employed for Brillouin zone (BZ) integrations within self-consistency cycles [@Blochl1]. The basis set convergence parameter ($R_{MT}^{min}K_{max}$) was set to $8$ for all calculations.
The $k$-mesh for the band structure of CuPt ordered InSb$_{0.5}$As$_{0.5}$ was $14\times14\times14$. The calculations for HSs were carried out with a $k$-mesh of $20\times20\times1$. To obtain high accuracy in our calculations the effect of spin-orbit coupling (SOC) was implemented through a second variational procedure, where states up to 9 Ry above Fermi energy ($E_F$) were included in the basis expansion and the relativistic $p_{1/2}$ corrections were incorporated for the higher lying *p* orbitals.
Results and Discussion
======================
![One unit cell of (a) CuPt ordered InSb$_{0.5}$As$_{0.5}$, (b) H-I (*n*=2) and (c) H-II (*m*=3).[]{data-label="structure"}](fig1.eps)
Structural properties
---------------------
The atomic arrangement in a unit cell of CuPt ordered InSb$_{0.5}$As$_{0.5}$ is shown in Fig. \[structure\] (a). The atomic arrangement in optimized HS unit cells of H-I with *n*=2 and H-II with *m*=3 are shown in Fig. \[structure\] (b) and (c). The optimized structures of other HSs, i.e. with $n$=1,3 (in H-I) and $m$=1,2 (in H-II) are also obtained. A unit cell of CuPt ordered InSb$_{0.5}$As$_{0.5}$ has 12 layers of atoms. A unit cell of a ZB-InSb \[111\] has six layers of atom; whereas, the same for a WZ structure in \[0001\] direction has four layers of atomic arrangement. Thus, the number of layers in H-I are 18 (*n*=1), 24 (*n*=2) and 30 (*n*=3) whereas in H-II, the values are 16 (*m*=1), 20 (*m*=2) and 24 (*m*=3). The optimized lattice constant for CuPt ordered InSb$_{0.5}$As$_{0.5}$ are $ a =b=
4.4586$ Å and $ c = 21.8427$ [Å]{} with the space group $R3m$ (no. 160). Both H-I and H-II take a trigonal structure with the space group $P3m1$ (no. 156). The lattice parameters of H-I (*n*=1–3) and H-II (*m*=1–3) are listed in Table-\[l.c.\].
[c c c c c c c c c c c c c c c c c c c]{} HS & & *a* && *c* &\
& & in [Å]{} && in [Å]{} &\
& *n*\
&1 &4.6077&&33.8598 &\
**H-I** &2 &4.6154&&45.2216 &\
&3 &4.6231&&56.6206 &\
& *m*\
&1 &4.4935&&29.3514 &\
**H-II** &2 &4.4636&&36.4449 &\
&3 &4.3752&&43.9056 &\
\[l.c.\]
Electronic properties
---------------------
We have carried out systemic band structure calculations on two HSs, H-I and H-II with (*n,m*=1–3), using LDA, LDA with SOC, mBJLDA and mBJLDA with SOC schemes to demonsrate the role of the mBJLDA exchange potential and SOC in the band structure. From the calculated band structure, we plot the energy bands along $M$-$\Gamma$-$L$ *k*-paths. Fig. \[band-H-I\] (a)–(d) present the same for H-I (*n*=2). The standard LDA method yields this system as a gapless material (Fig. \[band-H-I\] (a)), However, the implementation of SOC interaction over LDA does not open a gap in the system (Fig. \[band-H-I\] (b)). The mBJLDA exchange potential creates a separation between the bands, present near the E$_F$ (see Fig. \[band-H-I\] (c)). Nonetheless, the band structure still exhibits metallic behavior. Incorporation of SOC over the mBJLDA calculation, as shown in Fig. \[band-H-I\] (d), shows a band gap. Similarly, for H-I with $n$=1,3, the band gap could be obtained only using mBJLDA exchange potential with SOC. Above mentioned other calculation schemes showed the metallic behavior of the systems.
0.3cm ![The band structure of H-I (*n*=2) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJLDA (d) mBJLDA with SOC schemes.[]{data-label="band-H-I"}](HI_2_LDA.eps "fig:") ![The band structure of H-I (*n*=2) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJLDA (d) mBJLDA with SOC schemes.[]{data-label="band-H-I"}](HI_2_LDA_SO.eps "fig:") 0.3cm ![The band structure of H-I (*n*=2) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJLDA (d) mBJLDA with SOC schemes.[]{data-label="band-H-I"}](HI_2_MBJ.eps "fig:") ![The band structure of H-I (*n*=2) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJLDA (d) mBJLDA with SOC schemes.[]{data-label="band-H-I"}](HI_2_MBJ_SO.eps "fig:")
We have performed the calculations on H-II, with $m$=1–3, following the same four-steps procedure as used for H-I. H-II (with $m$=1–3) exhibited the same trend, as observed in the case of H-I. Under LDA, LDA with SOC and mBJLDA schemes, we obtained metallic behavior of this system. Incorporation of SOC over mBJ corrected LDA calculation opens a gap at the $\Gamma$ point. In Fig. \[band-HII\] (a)–(d) we plot the energy bands of H-II with $m$=3, as obtained using above the mentioned schemes.
0.3cm ![The band structure of H-II (*m*=3) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJ and (d) mBJ with SOC schemes.[]{data-label="band-HII"}](fig2a.eps "fig:") ![The band structure of H-II (*m*=3) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJ and (d) mBJ with SOC schemes.[]{data-label="band-HII"}](fig2b.eps "fig:") 0.3cm ![The band structure of H-II (*m*=3) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJ and (d) mBJ with SOC schemes.[]{data-label="band-HII"}](fig2c.eps "fig:") ![The band structure of H-II (*m*=3) using different procedures under (a) LDA, (b) LDA with SOC (c) mBJ and (d) mBJ with SOC schemes.[]{data-label="band-HII"}](fig2d.eps "fig:")
0.3cm ![Fat bands showing the contributions of (a) $s$ and (b) $p$ orbitals in H-I (*n*=2) with LDA. The same of (c) $s$ and (d) $p$ orbitals with mBJLDA+SOC scheme.[]{data-label="fat-HI"}](fat_all.eps "fig:")
We note that the LDA underestimates the separation between different bands to a great extent. It is to be noted that for InSb based materials Sb-*p* states dominate near $E_F$ [@Cardona1]. Fat band analysis provides the contribution of atomic orbitals in the band structure. Fat bands showing the contribution of $s$ and $p$ orbitals in the band structure of H-I with *n*=2 under the LDA scheme are plotted in Fig. \[fat-HI\] (a)–(b), mBJLDA in Fig. \[fat-HI\] (c)–(d) and under mBJLDA+SOC in Fig. \[fat-HI\] (e)–(f). While the upper three panels, (a), (c) and (e), reveal the contribution of $s$ orbitals, the lower three panels (b), (d) and (d), demonstrate the contribution of the $p$ orbitals. We find that under LDA, In/As *s* and Sb *p*-orbitals mostly contribute near $E_F$. The spin-orbit interaction becomes significant for high Z elements, like Sb. Incorporation of SOC in the calculation splits the bands and make the band structure more densely packed. However, this does not remove the contribution from the *s*-states, and it does not open the band gap. Since the mBJ potential corrects the conventional LDA [@Kim1] type of exchange correlations by considering the effect of holes, it creates the proper separation between the levels near E$_F$ [@Tran1]. The mBJLDA exchange potential separates the *s*-like states from the *p*-states near E$_F$ (see Fig. \[fat-HI\] (c)–(d)), which were wrongly mixed up at the LDA level. Under this scheme the Sb-*p* states dominate near $E_F$, as expected [@Cardona1]. When SOC is included over the mBJLDA corrected bands, it provides the necessary symmetry breaking at the $\Gamma$ point, which in turn leads to a correct band structure (Fig. \[fat-HI\] (e)–(f)). We could apply the above arguments to explain the role of mBJLDA exchange potential and SOC in determining correct electronic band structure of other systems under study.
The band structure of InSb$_{0.5}$As$_{0.5}$ with CuPt ordering, calculated using mBJLDA potential, with SOC is shown in Fig. \[Winkler\]. We obtain stated [@Winkler1] topological features like band inversion at the $E_F$ level and the existence of the triple point. A novel topological phase is observed in this system through appearance of the triple point in the band structure. Our calculated band structure is an excellent match with that reported using HSEO6 [@Winkler1]. Here also we find that mBJLDA+SOC scheme is reliable to obtain details of electronic band structure.
![The electronic band structure of InSb$_{0.5}$As$_{0.5}$ using the mBJLDA method along with SOC.[]{data-label="Winkler"}](fig5.eps){width="0.5\columnwidth"}
Next we take a close look at band structures near $\Gamma$ point for HSs under study. Refer to Fig. \[band-layer\] (a)–(c) for H-I with *n*=1–3. In the figure the direct and indirect band to band transitions are shown by green and violet lines, respectively. For H-I with *n*=1,2 four transitions, two direct (shown by green) and two indirect (shown by violet lines) transitions are possible. All direct and indirect gaps are equal. The obtained values of the gaps are 0.143 eV and 0.115 eV for H-I with *n*=1 and $n$=2, respectively. However, the band structure gets modified as the InSb layer thickness is further increased to $n=3$. It has two indirect band gaps of same energy, 0.54 eV. On the contrary, no qualitative change in the nature of the band structure was observed as we vary the InAs layer thickness in H-II (Fig. \[band-layer\] (d)–(f)). The systems, with *m*=1–3, exhibit two indirect band gaps of equal energy. Table-\[bandgap\] lists the gap energies (E$_g$s) of the HSs. H-I shows a decrease of the gap from 0.143 to 0.054 eV for $n=$1–3. However, for H-II with $m$=1–3 the band gap changes only from 0.075 to 0.084 eV. The variation in band gap is an order of magnitude higher in H-I than in H-II with increase in the thickness of the layers.
0.3cm ![\[f2\] Zoomed view of the band structures near $\Gamma$ point for (a)–(c) H-I with *n*=1,2 and 3 (d)–(f) H-II with *m*=1,2 and 3.[]{data-label="band-layer"}](gap_layer.eps "fig:")
[c c c c c c c c c c c c c c c c c c c]{} HS &&E$_g$ & hole &electron &\
&& in eV& pockets &pockets &\
& *n*\
&1 &0.143& 6&1&\
**H-I** &2 &0.115& 7&6&\
&3 &0.054& 1&-&\
& *m*\
&1 &0.075& 1&-&\
**H-II** &2 &0.074& 1&-&\
&3 &0.084& 1&-&\
\[bandgap\]
Doping or external field is often used to shift the E$_F$ in a system [@Zhu1; @Yuan1; @Song1]. A close inspection of the band structures of both HSs near the $\Gamma$ point reveals that a small shift in E$_F$ is expected to generate different number of Fermi pockets. A positive effective mass defines an electron pocket while a hole pocket has a negative effective mass. The curvature of the bands define the effective mass either to be positive or to be negative. Refer to Table-\[bandgap\]. With a shift in E$_F$, six hole pockets (three along equivalent $\Gamma$-$M$ and similarly three along $\Gamma$-$L$ direction) can be obtained in the band structure of H-I with $n$=1. In addition, we find the possibility of having an electron pocket only at the $\Gamma$ point. With increase in the layer thickness, i.e. for $n=$2 the number of hole and electron pockets increased to seven and six respectively. However, the number of carrier pockets reduced to one for $n$=3. With a shift in $E_F$, the band structure of H-II with $m$=1–3 (see Fig. \[band-layer\] (d)–(f)) is expected to exhibit only one hole pocket. Thus, we demonstrate that by varying the InSb layer thickness, one can engineer different carrier pockets (CPs) in H-I. CPs at a low symmetry points are enhances material functionality. For example, materials with large numbers of CPs are ideal for thermoelectric devices [@Rabin1; @Zhou1].
=6.5in
In above we observe that the intricate features of electronic band structures of H-I and H-II are markedly different. To find the possible origin of this difference in evolution characteristic of the electronic band structures of H-I and H-II with layer thickness, we looked into the charge modulation along the $c$ axis of the HSs. As seen through the fat band analysis, Sb *p*-orbitals of these systems have a dominant contribution near $E_F$. We studied the variation of charge on Sb sites along the $c$ axis, as shown in Fig. \[OS\] (a)–(c) and (d)–(f) for H-I and H-II respectively. In H-I, as we change the InSb layer thickness, the number of Sb atoms increases from six to twelve and they are inequivalent (see Fig. \[OS\] (a)-(c)). Thus, the increase in InSb layer thickness in H-I has two consequences. First it shrinks the BZ along the Z-direction and secondly, it increases the number of states within the BZ (due to increase in inequivalent Sb atoms). As a consequence, the band structure significantly changes when $n$ changes from 1 to 3. In H-II, with an increase in the InAs layer thickness, the BZ shrinks. However, the numbers of relevant inequivalent atoms remain three for $m$=1–3 (see Fig. \[OS\] (d)–(f)). Hence, we do not find appreciable change in the band structure for this system.
Conclusion
==========
In summary, we have calculated electronic band structure of two different HSs, one component of which is a topological semimetal of CuPt ordered InSb$_{0.5}$As$_{0.5}$ and other component is a band insulator. The HSs are also semiconductors. We have shown that for InSb/InAs$_{0.5}$Sb$_{0.5}$ HS, the band gap as well as fine features near the Fermi level can be modulated by varying the thickness of the InSb segment. Another important take away from this work is how differently these HSs would respond to a small perturbation owing to the difference in the number of their Fermi pockets.
**Acknowledgements**\
Authors thank Professor Debraj Choudhury, IIT Kharagpur, for valuable discussion. Authors also acknowledge the use of the computing facility from the DST-Fund for Improvement of S&T infrastructure ( phase-II) Project installed in the Department of Physics, IIT Kharagpur, India. AR thanks Department of Science and Technology, Government of India, for financial assistance.
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|
---
abstract: |
Observations of performed with the Westerbork Synthesis Radio Telescope at four frequencies, over the interval 1.4$-$8.4 GHz, have provided us with the first broadband radio spectrum of a ‘quiescent’ stellar mass black hole. The measured mean flux density is of 0.35 mJy, with a spectral index $\alpha= +
0.09\pm0.19$ (such that $\rm S_{\nu}\propto \nu^{\alpha}$). Synchrotron emission from an inhomogeneous partially self-absorbed outflow of plasma accounts for the flat/inverted radio spectrum, in analogy with hard state black hole X-ray binaries, indicating that a steady jet is being produced between a few $10^{-6}$ and a few per cent of the Eddington X-ray luminosity.
author:
- |
E. Gallo$^{1}$, R. P. Fender$^{1}$[^1], R. I. Hynes$^{2}$[^2]\
\
$^{1}$ Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam,\
$~$and Center for High Energy Astrophysics, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands\
$^{2}$ McDonald Observatory and Department of Astronomy, The University of Texas at Austin, 1 University Station C1400, Austin, Texas 78712, USA\
title: The radio spectrum of a quiescent stellar mass black hole
---
2[cm$^2$ ]{} 1[s$^{-1}$ ]{}
2[cm$^2$ ]{} 1[s$^{-1}$ ]{}
Binaries: general – ISM: jets and outflows – Individual: V404 Cyg
Introduction
============
While accreting gas at relatively low rates, black hole candidates in X-ray binary (BHXB) systems are able to power steady, collimated outflows of energy and material, oriented roughly perpendicular to the orbital plane. The jet interpretation of the radio emission from hard state BHXBs came before the collimated structures were actually resolved with Very Long Based Interferometry (VLBI) techniques. In a seminal work, Blandford & Königl (1979) proposed a model to interpret the flat radio spectrum of extragalactic compact radio sources in terms of isothermal, conical outflows, or jets. A jet model for X-ray binaries was later developed by Hjellming & Johnston (1988), in order to explain both the steady radio emission with flat/inverted spectra observed in the hard state of BHXBs, and transient outbursts with optically thin synchrotron spectra. We refer the reader to McClintock & Remillard (2004) and Fender (2004) for comprehensive reviews on X-ray states and radio properties (respectively) of BHXBs. High resolution maps of Cyg X-1 in the hard X-ray state have confirmed the jet interpretation of the flat radio-mm spectrum (Fender 2001), imaging an extended, collimated structure on milliarcsec scale (Stirling 2001). Further indications for the existence of collimated outflows in the hard state of BHXBs come from the stability in the orientation of the electric vector in the radio polarization maps of GX 339$-$4 over a two year period (Corbel 2000). This constant position angle, being the same as the sky position angle of the large-scale, optically thin radio jet powered by GX 339$-$4 after its 2002 outburst (Gallo 2004), clearly indicates a favoured ejection axis in the system. Finally, the optically thick milliarcsec scale jet of the (somewhat peculiar) BH candidate GRS 1915+105 (Dhawan, Mirabel & Rodríguez 2000) in the plateau state (Klein-Wolt 2002) supports the association of hard X-ray states of BHXBs with steady, partially self-absorbed jets.
Having established this association, a natural question arises: what are the required conditions for a steady jet to exist? We wonder especially whether the jet survives in the very low luminosity, *quiescent* X-ray state. While radio emission from BHXBs in the thermal dominant (or high/soft) state is suppressed up to a factor $\sim$50 with respect to the hard state (Fender 1999; Corbel 2001, and references therein), most likely corresponding to the physical disappearance of the jet, little is known about the radio behaviour of quiescent stellar mass BHs, mainly due to sensitivity limitations. Among the very few systems detected in radio is V404 Cygni, which we shall briefly introduce in the next Section.
V404 Cyg (=GS 2023+338)
-----------------------
The X-ray binary system is thought to host a strong BH candidate, with a most probable mass of $\sim$ 12 (Shahbaz 1994), and a low mass K0IV companion star, with orbital period is of 6.5 days, and orbital inclination to the line of sight is of about 56 (Casares & Charles 1994; Shahbaz 1994). Following the decay of the 1989 outburst that led to its discovery (Makino 1989), the system entered a quiescent X-ray state, in which it has remained ever since. The relatively high quiescent X-ray luminosity of V404 Cyg (with an *average* value of about $6 \times 10^{33}\times (\rm D/4 ~kpc)^2$ erg sec$^{-1}$ in the range $0.3-7.0$ keV; Garcia 2001; Kong 2002; Hynes 2004) is possibly related to the long orbital period and surely indicates that the some accretion continues to take place at $\rm L_{X}\simeq
4\times 10^{-6}\rm L_{Edd}$, where $\rm L_{Edd}$ is the Eddington X-ray luminosity (for a 12 BH).
As reported by Hjellming (2000), since (at least) early 1999 the system has been associated with a variable radio source with flux density ranging from 0.1 to 0.8 mJy on time scales of days and it is known to vary at optical (Wagner 1992; Casares 1993; Pavlenko 1996; Hynes 2002; Shahbaz 2003; Zurita 2003) and X-ray wavelengths (Wagner 1994; Kong 2002; Hynes 2004, for a coordinated variability study) as well. Yet no *broadband* radio spectrum of V404 Cyg in quiescence, nor of any other stellar mass BH below $10^{-5} \rm L_{Edd}$, is available in the literature to date (see Corbel 2000 for a 2-frequency radio spectrum of GX 339-4 at $\sim$$ 10^{-5}\rm
L_{Edd}$). Given the quite large degree of uncertainty about the overall structure of the accretion flow in quiescence (e.g. Narayan, Mahadevan & Quataert 1999 for a review), it has even been speculated that the total power output of a quiescent BH could be dominated by a radiatively inefficient outflow (Fender, Gallo & Jonker 2003) rather than by the local dissipation of gravitational energy in the accretion flow. It is therefore of primary importance to establish the nature of radio emission from quiescent BHXBs. In this brief paper we show that the radio properties of closely resemble those of a canonical hard state BH, suggesting that there is no fundamental difference in terms of radio behaviour between the quiescent and the canonical hard X-ray state. A comprehensive study of the spectral energy distribution of in quiescence, from radio to X-rays, will be presented elsewhere (Hynes et al., in preparation).
Radio emission from V404 Cyg
============================
WSRT observations
-----------------
The Westerbork Synthesis Radio Telescope (WSRT) is an aperture synthesis interferometer that consists of a linear array of 14 dish-shaped antennae arranged on a 2.7 km East-West line. was observed by the WSRT at two epochs: [**[i)]{}**]{} on 2001 December 28, start time 05:28 UT (MJD 52271.3), at 4.9 GHz (6 cm) and 8.4 GHz (3 cm), for 8 hours at each frequency; observations were performed with the (old) DCB backend, using 8 channels and 4 polarizations; [**[ii)]{}**]{} on 2002 December 29, start time 06:29 UT (MJD 52637.3), at 1.4 GHz (21 cm), 2.3 GHz (13 cm), 4.9 GHz (6 cm) and 8.4 GHz (3 cm) for a total of 24 hours. Frequency switching between 8.4$-$2.3 GHz and 4.9$-$1.3 GHz was operated every 30 minutes over the two 12 hour runs, resulting in $\sim$ 5.5 hour on the target and $\sim$ 0.5 hour on the calibration sources (3C 286 and 3C 48) at each frequency. During this set of observations, the WSRT was equipped with the DZB backend using eight IVC sub-bands of 20 MHz bandwidth, 64 channels and 4 polarizations. Seven out of the eight sub-bands were employed to reconstruct the images, as the sub-band IVC-IF6 failed to detect any signal other than noise over the whole 24 hour period. The telescope operated in its *max-short* configuration, particularly well suited for observations shorter than a full 12 hour synthesis, and with a minimum baseline of 36 m (see http://www.astron.nl/wsrt/wsrtGuide/ for further details). The data reduction, consisting of editing, calibrating and Fourier transforming the $(u,v)$-data on the image plane, has been performed with the MIRIAD (Multichannel Image Reconstruction Image Analysis and Display) software (Sault & Killeen 1998). The 1.4 and 2.3 GHz data, containing several sources with flux density well above 100 mJy, were self-calibrated in phase.
Results: spectrum and variability
---------------------------------
### 2001 December 28 (MJD 52271.3)
----------------- ------------- ----------------- -------
date $\nu$ (GHz) $S_{\nu}$ (mJy) $S/N$
2001-12-28 4.9 0.49 0.04 12.2
*(MJD 52271.3)* 8.4 0.50 0.20 2.5
2002-12-29 1.4 0.34 0.08 4.2
*(MJD 52637.3)* 2.4 0.33 0.07 4.7
4.9 0.38 0.05 7.6
8.4 0.36 0.15 2.4
----------------- ------------- ----------------- -------
: \[table1\] WSRT observations of .
An unresolved (beam size of $\sim 5.8 \times 3.0$ arcsec$^{2}$ at 8.4 GHz) $\sim$0.50 mJy radio source is detected at both 4.9 and 8.4 GHz, at the position consistent with that of ($\alpha$(J2000) = 20:24:03.78; $\delta$(J2000) = +33:52:03.2; e.g. Downes 2001). Table 1 lists the measured flux densities with errors at each frequency; the corresponding spectral index (hereafter defined as $ \alpha =\Delta~\rm log~S_{\nu} /
\Delta~\rm log~\nu $, such that $\rm S_{\nu}\propto \nu^{\alpha } $) is of 0.040.68; such a large error bar is mainly due to the high noise in the 8.4 GHz map (see Table 1). The signal/noise ratios are too low to measure linearly polarized flux from the source at the expected level of a few per cent, assuming a synchrotron origin for the radio emission (see Section 3).
### 2002 December 29 (MJD 52637.3)
is detected at four frequencies with a mean flux density of 0.35 mJy; flux densities at each frequency are listed in Table 1. The fitted four-frequency spectral index is $\alpha=0.09\pm0.19$. Radio contours as measured at 4.9 GHz are plotted in Figure 1, while Figure \[spectrum\] shows the radio spectra of at two epochs.
Since returning to quiescence, is known to vary on time scales of days, or even shorter, both in radio and in X-rays; such variability is actually detected in our 2002 WSRT observations. The low flux of makes it practically impossible to subtract from the $(u,v)$-data all the other radio sources in the field and generate a reliable light curve of the target. We thus divided each of the two $\sim$ 11-hour data sets on-source in time intervals of $\sim$ 5.5 hours (of which only $\sim$ 2.75 hours on source per frequency, due to the frequency switching) and made maps of each time interval. Significant variability (checked against other bright sources in the field) is detected at 4.9 GHz: the flux density varied from 0.270.07 mJy in the first half of the observation, to 0.470.07 in the second half.
Discussion
==========
As mentioned in the introduction, synchrotron radiation from a relativistic outflow accounts for the observed flat radio spectra of *hard* state BHXBs; we refer the interested reader to more thorough discussions in e.g. Hjellming & Han (1995), Mirabel & Rodríguez (1999) and Fender (2001; 2004). Here we shall note that the *collimated* nature of these outflows is more debated, as it requires direct imaging to be proven. Even though confirmations come from Very Long Based Array (VLBA) observations of Cyg X-1 (Stirling 2001) and GRS 1915+105 (Dhawan 2000; Fuchs 2003) in hard states, failure to image a collimated structure in the hard state of XTE J1118+480 down to a synthesized beam of 0.6$\times$1.0 mas$^{2}$ at 8.4 GHz (Mirabel 2001) may challenge the jet interpretation (Fender 2001). However, apart from GRS 1915+105, which is persistently close to the Eddington rate (see Fender & Belloni 2004 for a review), Cyg X-1 in the hard state displays a 0.1-200 keV luminosity of 2 per cent $\rm L_{Edd}$ (Di Salvo 2001), while XTE J1118+408 was observed at roughly one order of magnitude lower level (Esin 2001). If the jet size scaled as the radiated power, we would expect the jet of XTE J1118+408 to be roughly ten times smaller than that of Cyg X-1 (which is 2$\times$6 mas$^{2}$ at 9 GHz, at about the same distance), and thus still point-like in the VLBA maps presented by Mirabel (2001).
Garcia (2003) have pointed out that long period ($\simgt 1$ day) BHXBs undergoing outbursts tend to be associated with spatially resolved optically thin radio ejections, while short period systems would be associated with unresolved, and hence physically smaller, radio ejections. If a common production mechanism is at work in optically thick and optically thin BHXB jets (Fender, Belloni & Gallo 2004), the above arguments should apply to steady optically thick jets as well, providing an alternative explanation to the unresolved radio emission of XTE J1118+480, with its 4 hour orbital period, the shortest known for a BHXB. It is worth mentioning that, by analogy, a long period system, like , might be expected to have a relatively larger optically thick jet.
A synchrotron emitting outflow in the quiescent state of V404 Cyg
-----------------------------------------------------------------
### Emission mechanism
The WSRT observations of performed on 2002 December 29 provide us with the first broadband (1.4$-$8.4 GHz) radio spectrum of a stellar mass BH candidate below $10^{-5}\rm L_{Edd}$. As we do not have direct evidence (no linear polarization measurement, no especially high brightness temperature, see below) for the synchrotron origin of the radio emission from in quiescence, we must first briefly explore different mechanisms, such as free-free emission from an ionised plasma. The donor in is a K0IV star with most probable mass of 0.7 and temperature around 4300 K (Casares & Charles 1994; Shahbaz 1994), simply too cool to produce any observable free-free radio emission (see Wright & Barlow 1975). Alternatively, the accretion flow onto the compact object may provide the needed mass loss rates and temperatures in order to produce a flat/inverted free-free radio spectrum. In (line- and radiation-driven) disc wind models, global properties such as the total mass loss rate and wind terminal velocity depend mainly on the system luminosity (see Proga & Kallman 2002; Proga, Stone & Drew 1998 and references therein); very high accretion rates are required in order to sustain significant mass loss rates and hence observable wind emission, ruling out a disc wind origin for the observed radio flux from V404 Cyg. However, that mass loss via winds in sub-Eddington, radiatively inefficient accretion flows (ADAFs) may be both dynamically crucial and quite substantial, has been pointed out by Blandford & Begelman (1999). Quataert & Narayan (1999) calculated the spectra of such advection dominated inflows taking into account wind losses, and found that the observations of three quiescent black holes, including V404 Cyg, are actually consistent with at least 90 per cent of the mass originating at large radii to be lost to a wind. Under the rough assumption that models developed for ionised stellar winds (Wright & Barlow 1975, Reynolds 1986; see Dhawan 2000 for an application to the steady jet of GRS 1915+105) might provide an order of magnitude estimate of the mass loss rate even for such ‘advection-driven’ winds, still the required mass loss rate in order to sustain the observed radio emission for a *fully ionised* hydrogen plasma is close the Eddington accretion rate for a 12 BH (assuming a 10 per cent efficiency in converting mass into light). Lower ionisation parameters would further increase the needed mass loss, bringing it to super-Eddington rates. Even taking into account geometrical effects, such as wind collimation and/or clumpiness, the required mass loss rates can not be more than three orders of magnitude below the spherical homogeneous wind, still far too high for a $\simlt 10^{-5}$ Eddington BH to produce any observable radio emission. As free-free emission does not appear to be a viable alternative, we are led to the conclusion that the radio spectrum of in quiescence is likely to be synchrotron in origin. This conclusion is supported by polarization measurements during the second phase (following a bright optically thin event) of the 1989 radio outburst of , when a slow-decay, optically thick component had developed (Han & Hjellming 1992). At this time, after 1989 June 1-3, displayed the same flat/inverted spectrum of the present 2002 WSRT observations, but was at a few mJy level, still high enough to allow the detection of linearly polarized flux, which confirmed the synchrotron nature of the emission. In addition, the roughly constant and similar polarization angles measured at that time, indicated that the averaged magnetic field orientation changed very little, if at all. As entered a quiescent regime following the decay of that outburst (it reached the typical quiescent flux densities about 1 year after the outburst peak), it seems reasonable to assume that the present $\sim$ 0.5 mJy radio emission with flat/inverted spectrum is of the same nature as the few mJy flat/inverted spectrum component detected in 1989, and therefore synchrotron in origin.
A further argument for the *jet* interpretation of this synchrotron emission is the fact that the radio and X-ray fluxes of over the decline of its 1989 outburst *and* at its current quiescent X-ray and radio luminosities, display the same non-linear correlation found to hold in the whole hard state of BHXBs (Corbel 2003; Gallo, Fender & Pooley 2003) and later extended to super-massive nuclei in active nuclei as well (Merloni, Heinz & Di Matteo 2003; Falcke, Körding & Markoff 2004), where there is little doubt about the jet origin of the radio emission.
[ccccc]{} requirement & frequency & &Size &\
& GHz &cm &$\rm R_{\odot}$ & mas\
\
${\rm T_{b}}< 10^{12}$K & 1.4&$\simgt 5 \times 10^{11}$ & $\simgt 7 $ & $\simgt
0.01$\
${\rm L}< c~\Delta t$ &4.9& $\simlt 6 \times
10^{14}$ & $\simlt 8530 $& $\simlt 10$\
### Angular size
The maximum brightness temperature for a galactic incoherent synchrotron source is a (weak) function of the measured spectral index, the upper frequency $\nu_{\rm up}$ of the synchrotron spectrum and the Doppler boosting factor $D$ (e.g. Hughes & Miller 1991). For $\alpha=0.1$ and $\nu_{\rm
up}$=8.4 GHz, as in the case presented here, $T_{\rm b} \simlt 10^{12}\times
D^{1.2}$ K at 1.4 GHz. With an orbital inclination of 56$^{\circ}$, the Doppler boosting factor is likely to vary in the range $D=$1–1.1, calculated for bulk Lorentz factors between 1–1.7. Assuming a distance to V404 Cyg of 4 kpc (Jonker & Nelemans 2004 and references therein), we can thus derive a minimum linear size for the (1.4 GHz) emitting region of $\sim 5 \times
10^{11}$ cm, or $\sim 7~\rm R_{\odot}$. This corresponds to about one fifth of the system orbital separation (Shahbaz 1994). For comparison, the highest [*measured*]{} brightness temperature in a Galactic binary system is of a few $10^{11}$ K at 5 GHz, during a flaring event in Cyg X-3 (Ogley 2001).
Because of limits on the signal propagation speed, the 5.5-hour time scale variability detected at 4.9 GHz gives an upper limit to the linear size $\rm
L$ of the variable region: $\rm L < 5.9 \times 10^{14}$ cm, or about 8530 $\rm
R_{\odot}$. At a distance of 4 kpc, this translates into an angular extent $\theta \simlt 10$ mas at 4.9 GHz (see Table 2). In the framework of standard conical jet models (Blandford & Königl 1979; Hjellming & Johnston 1988; Falcke & Biermann 1996), flux variability could be induced by e.g. the propagation of shocks within the compact outflow. These shocks will not be visible until they reach the point along the outflow where it becomes optically thin at the observing frequency. The actual morphology of the radio source will depend on the ratio between the thickness $\Delta r$ of the region where the variability occurs (the lower the observing frequency, the higher the thickness) and its distance $\rm R$ from the core. If ${\rm R} \gg \Delta r$, we would expect a double radio source with flux ratios depending on Doppler boosting, while if ${\rm R}\simeq \Delta r$, then we would expect to observe a continuous elongated structure. The ratio $\Delta
r/\rm R$ is unknown in the case of and could only be determined by measuring *delays* between different frequencies. For comparison, the average flux rise time in the oscillations of the flat spectrum radio component in GRS 1915+105 is of about a few minutes, while the infrared-radio delays are typically of 15 min, indicating that the variable radio source should not be too distant from the core (Mirabel 1998; Fender 2002, and references therein). Combined with the extended core morphology of both Cyg X-1 (Stirling 2001) and the core of GRS 1915+105 (Dhawan 2000; Fuchs 2003), this suggests that ${\rm R}\simeq \Delta r$ in these two sources, but it is not clear how the outflow properties might scale with the luminosity.
Han & Hjellming (1992), based on the same arguments, were able to constrain the linear size of the optically *thin* ejection associated with the fast-decay phase in the radio light curve of the 1989 outburst: from the 5 min time scale variability measured on 1989 June 1st, and from brightness temperature limits, they derived $\theta\simeq 0.2$ mas. Fluctuations on time scales of tens of minutes were later measured during the slow-decay phase, when an optically thick component had developed, and interpreted as possible hot shocks propagating downstream an underlying compact jet. By analogy, this would appear a reasonable explanation for the 5-hour time scale variability detected in our WSRT observations as well.
Summary
=======
WSRT observations of performed on 2002 December 29 (MJD 52637.3) at four frequencies over the interval 1.4$-$8.4 GHz have provided us with the first broadband radio spectrum of a quiescent (with average $\rm L_X$ of a few $10^{-6}\rm L_{Edd}$) stellar mass BHXB. We measured a mean flux density of 0.35 mJy, and a flat/inverted spectral index $\alpha= 0.09\pm0.19$. WSRT observations performed one year earlier, at 4.9 and 8.4 GHz, resulted in a mean flux density of 0.5 mJy, confirming the relatively stable level of radio emission from on a year time-scale; even though the spectral index was not well constrained at that time, the measured value was consistent with the later one.
Synchrotron emission from an inhomogeneous, optically thick relativistic outflow of plasma seems to be the most likely explanation for the flat radio spectrum, in analogy with hard state BHXBs (Fender 2001). Optically thin free-free emission as an alternative explanation is ruled out on the basis that mass loss rates far too high would be required, either from the companion star or from the inflow of plasma to the accretor. The collimated nature of this outflow remains to be proven; based on brightness temperature arguments and the 5.5-hour time-scale variability detected at 4.9 GHz, we conclude that the angular extent of the radio source is constrained between 0.01 at 1.4 GHz and 10 mas at 4.9 GHz (at a distance of 4 kpc; Jonker & Nelemans 2004). In the context of standard self-absorbed jet models, the flux variability may be due to shocks or clouds propagating in an inhomogeneous jet.
If our interpretation is correct, a compact steady jet is being produced by BHXBs between a few $10^{-6}$ and $\sim$$ 10^{-2}$ times the Eddington luminosity, supporting the notion of quiescence as a low luminosity level of the standard hard state. However, as is the most luminous quiescent BHXB known to date, the existence of a steady jet in this system does not automatically extend to the whole quiescent state of stellar mass BHs. Sensitive radio observations of the nearby, truly quiescent system A0620$-$00 (three orders of magnitude less luminous than in X-rays; e.g. Kong 2002), will hopefully provide an answer about the ubiquity of compact jets from stellar mass black holes with a hard spectrum.
Acknowledgments {#acknowledgments .unnumbered}
===============
EG wishes to thank Raffaella Morganti for her kind assistance in the data reduction, and Alex de Koter for suggestions. RIH is supported by NASA through Hubble Fellowship grant $\#$ HF-01150.01-A awarded by STScI, which is operated by AURA, for NASA, under contract NAS 5-26555. The Westerbork Synthesis Radio Telescope is operated by the ASTRON (Netherlands Foundation for Research in Astronomy) with support from the Netherlands Foundation for Scientific Research NWO.
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[^1]: Present address: School of Physics and Astronomy, University of Southampton, Hampshire SO17 1BJ, United Kingdom
[^2]: Present address: Louisiana State University, Department of Physics and Astronomy, Baton Rouge, LA 70803-4001, USA
|
---
abstract: 'A new class of lattice gas models with trivial interactions but constrained dynamics are introduced. These are proven to exhibit a dynamical glass transition: above a critical density, $\rho_c$, ergodicity is broken due to the appearance of an infinite spanning cluster of jammed particles. The fraction of jammed particles is discontinuous at the transition, while in the unjammed phase dynamical correlation lengths and timescales diverge as $\exp[C(\rho_c-\rho)^{-\mu}]$. Dynamic correlations display two-step relaxation similar to glass-formers and jamming systems.'
author:
- Cristina Toninelli
- Giulio Biroli
- 'Daniel S. Fisher'
title: Jamming Percolation and Glass Transitions in Lattice Models
---
In the majority of liquids dramatic slowing down occurs upon super-cooling. In a rather small temperature window, typically from the melting temperature $T_m$ to about $2 T_m/3$, the viscosity increases by $14$ orders of magnitude and the relaxation becomes complicated: non-exponential and spatially heterogeneous [@DeBenedettiStillinger]. Similar features are observed in soft materials, such as colloidal suspensions and more generally in non-thermal “jamming" systems [@Trappe]. Despite a great deal of effort, these remarkable phenomena, associated with “glass transitions" are still far from understood. Even the most basic issues remain open: is the rapid slowing down due to proximity to a phase transition? Is this putative glass transition static or purely dynamic [@Krauth]? Experimental results make it clear, however, that [*if*]{} an ideal glass transition does exist it should have some peculiar features: the density auto-correlation function $C(t)$ should exhibit a lengthening plateau that, at the transition, extends out to infinite times. Thus, the Edwards-Anderson order parameter, defined as the infinite time limit of $C(t)$, will be discontinuous at the transition. But this discontinuity should be accompanied by a critical divergence of the relaxation time. And, contrary to usual critical slowing down, the relaxation time should diverge exponentially, as in the Vogel-Fulcher law [@DeBenedettiStillinger]. Long-range spatial correlations, if they exist at all, must be very subtle. These unusual properties present major theoretical challenges: whether or not there is a true transition there is no “standard" framework to start from. There are promising results for models on Bethe lattices [@LGM] and for some with long-range interactions [@Bernasconi]. But the quest for [*models with short range interactions and no quenched disorder*]{} that are [*simple*]{} enough to be analyzed and can be shown to have a glass transition, namely a transition with the basic properties discussed above, is still open, in spite of much effort.
In this paper we introduce the first examples of such models [@foot; @Liu]. These are [*kinetically constrained models*]{} (KCMs): stochastic lattice gases with no static interactions, except hard core, but constrained dynamics [@ReviewKLG]. The elementary moves are particle jumps for conservative models and birth/death moves for non conservative models. Whether a move can occur depends on the nearby configuration and is non-zero only if some local constraints are satisfied. These kinetic constraints can radically change the dynamical behavior and typically induce glassy phenomenology[@ReviewKLG; @KA; @Jackle]. For some KCMs the dynamics becomes so slow at high density or low temperature, that they have been conjectured to undergo a true glass transition. The simplest examples are Kob-Andersen models on a square lattice (SKA) [@KA], where particles can move if and only if they have no more than two nearest neighbors both before and after the move. Although the analogous model on a Bethe lattice [@Bethe; @KATBF] does have a jamming transition, we have shown previously in [@KATBF] that the SKA and a broad class of generalizations on hypercubic lattices cannot have ergodicity breaking transitions: in any finite dimension the relaxation time diverges — in many cases in a super-Arrhenius way — only at the close packing density ($\rho=1$).
But this is not the only possible behavior. We here introduce a new class of KCMs that do exhibit a jamming transition at a non-trivial critical density, $\rho_c$ on finite dimensional lattices.\
For simplicity we focus on one of the simplest: a square lattice model [*without*]{} particle conservation; vacancies can loosely be thought of as “free volume“ which need only be conserved on average. At the end, we discuss generalizations to both higher dimensional and conservative models. The stochastic dynamics is as follows: An occupation variable at site $x$ cannot change if $x$ is [*blocked*]{} along [*either*]{} of the diagonal directions, as defined below. Unblocked sites change from occupied to empty and from empty to occupied with rates $(1-\rho)$ and $\rho$, respectively. Thus detailed balance is satisfied with the trivial product measure with density $\rho$. The blocking is determined by the eight fourth-nearest-neighbor sites of $x$. Denote [*pairs*]{} of these the north-east (NE), north-west (NW), south-east (SE), and south-west (SW) pairs as in Fig. \[directed1\] b). Site $x$ is blocked if [*either*]{} at least one of the NE sites [*and*]{} at least one of the SW sites is occupied, [*or*]{} at least one of the SE sites [*and*]{} at least one of the NW sites is occupied. Blocking can thus be along either the NW-SE or the NE-SW diagonals. As the distance to the blocking neighbors resembles a knight’s move in chess, we call this the ”knights model".\
If a site cannot be unblocked even by first emptying with allowed moves an arbitrarily large number of other sites, we call the site [*frozen*]{}. Any finite cluster of particles cannot be frozen: one can always unblock all sites by emptying from the perimeter in (see Fig. \[directed1\]c). A crucial question is whether an infinite spanning cluster of frozen sites exists in infinite systems. We will call this problem [*jamming percolation*]{}; it is akin to bootstrap percolation [@Aizenmann].
\[\]\[\][a)]{} \[\]\[\][$\ell$]{} \[\]\[\] \[\]\[\][$\ell$]{} \[\]\[\][c)]{} \[\]\[\][b)]{}
![a): Sites connected by arrows form one of the graphs on which directed percolation can form frozen clusters: e.g. the set of occupied sites (big dots) shown. b)Site X and its NE,NW,SW,SE pairs of neighbors. c) Portion of an empty octagonal annulus. The interior of the annulus, as any finite region surrounded by a double frame of vacancies, can be eaten away. Whether the vacant region can expand is determined by the three key sites indicated by question marks. If one of these belongs to an occupied DP path in the NE direction which is anchored on two perpendicular DP paths, as shown, it blocks the expansion. Inset: the full octagon. A necessary condition for the octagon not to be expandable of one step in the NW direction is that a DP path spans the dotted rectangle. []{data-label="directed1"}](9f.eps){width="\columnwidth"}
We will show the following results (which can be proved [@TBF2long]): (i) with blocked or periodic boundary conditions on $L\times L$ squares, there exist configurations with system-spanning clusters of frozen sites; (ii) on infinite lattices, below a critical density, $\rho_c$, there are no infinite frozen clusters; while (iii) above $\rho_c$, there is an infinite cluster of frozen particles that occupies a non-zero fraction, $\phi_{\infty}$, of the area; (iv) $\rho_c$ coincides with the critical density for directed site percolation (DP) on a square lattice; (v) $\phi_\infty$ is discontinuous at $\rho_c$; (vi) below $\rho_c$, there is a crossover length $\Xi(\rho)$: squares of size $L<<\Xi$ are very likely to have a frozen cluster, while for $L>>\Xi$, the probability of a frozen cluster falls-off exponentially; (vii) as $\rho$ increases to $\rho_c$, $\Xi$ diverges exponentially rapidly, as $\log \Xi\sim(\rho_c-\rho)^{-\mu}$ with $\mu\cong 0.64$ related to DP exponents; (viii) the relaxation time diverges as $\Xi$ or faster. Thus even though the critical density is the same as for DP, the behavior is completely different.
We first show that both an unfrozen and a frozen phase exist. At sufficiently low densities, the occupied sites will not percolate via connections up to fourth neighbors: the resulting finite clusters can always be unblocked from their perimeters. Thus at low densities, frozen clusters will not occur. In contrast, at high densities, spanning frozen clusters occur. Consider site directed percolation with directed links that connect a site to its two NE (fourth) neighbors as in Fig. \[directed1\] a). Infinite directed paths of occupied sites exist for $\rho\ge p_c^{DP}\cong 0.705$ (the critical threshold for conventional site DP on a square lattice [@reviewDP]). These clusters of sites are frozen since each has at least one occupied fourth-neighbour in both NE and SW directions. Thus, $\rho_c\le p_c^{DP}$. For the above argument it was sufficient to use blocking along just one of the two diagonal directions. But, due to blocking in the perpendicular diagonal direction, typical frozen configurations do [*not*]{} resemble DP clusters: they consist of short DP paths that terminate at each end in a T-junction with a DP path in the perpendicular direction. Thus large regions can be frozen even if they are not spanned by DP clusters.
An explicit construction is instructive. Consider a structure built of DP paths of length, $s$, intersecting at T-junctions as in Fig.\[lower\] a). This structure does not contain any long DP cluster, yet it is frozen. And, crucially, there exists a similar frozen cluster as long as each DP path remains inside a nearby rectangular region of size $s\times s/6$ (see Fig. \[lower\] a). Therefore the probability of the system being frozen is bounded from below by the probability that [*all*]{} these rectangles are spanned. This will be substantial if the DP spanning probability of each such rectangle is very high.
The crucial needed property follows from the anisotropy of critical DP clusters: a cluster of length ${s}$ typically has transverse dimensions of order ${s}^\zeta$ with $\zeta$, (often called $1/z$) the anisotropy exponent, $\zeta\cong 0.63$ [@reviewDP]. Therefore an ${s}\times b{s}$ rectangle with $s$ in the parallel direction can be cut into slices of width ${s}^{\zeta}$ such that for each slice the probability of DP spanning paths is substantial for $\rho\geq p_c^{DP}$. Therefore, even [*at*]{} $p_c^{DP}$, the probability $r_b(s)$ of [*not*]{} having [*any*]{} DP crossing in a large ${s}\times b{s}$ rectangle is $r_b(s) < {\cal O} [\exp(-2Cb {s}^{1-\zeta})]$ (with $C$ a constant).
What happens just below $p_c^{DP}$? The above argument will still hold for rectangles that are are of order the DP parallel correlation length, ${\xi_\parallel}$. This, together with the previously explained construction of a frozen structure, implies that an $L\times L$ square is likely to have a frozen structure built of DP clusters of length ${\xi_\parallel}$ if $r_{1/6}({\xi_\parallel}) L^2/{\xi_\parallel}^2\ll 1$. For $L<\Xi_< \sim
{\xi_\parallel}\exp(C_<{\xi_\parallel}^{1-\zeta})$ — a lower bound on $\Xi$ — squares will thus contain frozen clusters with high probability in spite of the rarity of DP clusters with length larger than ${\xi_\parallel}\ll \Xi$ .
We now need to show that below $p_c^{DP}$, sufficiently large squares are [*unlikely*]{} to contain frozen clusters. Again, this can be done by construction — now of unfrozen regions. Consider an infinite system within which is an octagonal annulus of radius (center to NW) $\ell$ that is completely empty. Whether or not this empty region can be expanded depends crucially on three key sites in each diagonal corner, say, the NW corner, as shown in Fig. \[directed1\] c). If all three key sites are empty or emptyable — i.e. unfrozen — the empty annulus can be expanded along its two adjacent (NNW and WNW) sides. A necessary condition to have a key site frozen is that it belongs to a DP cluster in the NE direction that is anchored at both ends, as in Fig. \[directed1\] c). Furthermore, in order for this anchorage to occur, it is necessary that the NE path spans the rectangular dotted region of size $\ell\times b\ell$ (with $b$ a constant) in Fig 1 c). For $\rho<p_c^{DP}$ DP clusters with length $\ell$ much larger than the DP correlation length, $\xi_\parallel$, are exponentially rare and the rectangle spanning probability is $\sim \exp(-\ell/\xi_\parallel)$ [@reviewDP]. Thus, if $\rho<p_c^{DP}$, the probability that the annulus can be expanded out to infinity by successive expansions is high for $\ell >>\xi_{\parallel}$ [@ReviewKLG; @Aizenmann; @KATBF]. Since the infinite system will contain a non-zero density of these empty regions which can be expanded to unblock the whole system, we conclude that $\rho_c=p_c^{DP}$.
Estimating how rare the unblocking regions are near $\rho_c$, yelds an upper bound on the crossover length $\Xi$. Starting with a small empty nucleus the — small — probability, $\delta$, that it can be expanded out to size $\sim{\xi_\parallel}$ (and hence readily to infinity) is the product of many small terms. This is dominated by $\ell \sim{\xi_\parallel}$: from $r({\xi_\parallel})$, we obtain $\delta >
\exp(-2C_>{\xi_\parallel}^{1-\zeta})$. Since in an $L\times L$ square there are $(L/{\xi_\parallel})^2$ roughly independent positions for such empty nuclei, some are likely to occur if $\delta L^2/{\xi_\parallel}^2$ is not small. Thus we find that $\Xi\le \Xi_>\sim {\xi_\parallel}\exp(C_>{\xi_\parallel}^{1-\zeta})$.\
We have found upper and lower bounds for the crossover length, $\Xi$, of similar form, hence $$\log\Xi \sim {\xi_\parallel}^{1-\zeta} \sim (\rho^c-\rho)^{-\mu} ~~ {\mbox {with}} ~~\mu=(1-\zeta)\nu_\parallel\cong 0.64$$ with $\nu_\parallel \cong 1.73$ the correlation length exponent for DP [@reviewDP].
\[\]\[\][$\ell_0$]{} \[\]\[\][a)]{} \[\]\[\] \[\]\[\][b)]{} ![a) Frozen structure made of DP paths of length $s$ (represented by straight continuous lines) intersecting at T-junctions. The A path anchors one end of the B and C paths, while its ends are anchored by the D and E paths. This anchoring is retained if path A is displaced until the nearby dotted line, even if the B and E paths are similarly displaced no further than their corresponding dotted lines. Thus the structure is frozen if all the rectangles — shown and unshown — formed by the solid and dashed lines are spanned lengthwise by DP paths b) Two sequences of intersecting rectangles, ${\cal{R}}_i$, that when spanned length-wise by directed paths, make the central site, $O$, frozen. For clarity, the rectangles in one direction are shown with dashed lines. []{data-label="lower"}](disc6.eps "fig:"){width=".98\columnwidth"}
We now discuss another peculiarity of this transition: the nature of the frozen clusters implies that the density, $\phi_\infty$, of the infinite frozen cluster jumps at $\rho_c$. To analyze the probability that a site is frozen, consider an occupied site, e.g. the origin, which belongs to a DP cluster that extends to a (small) distance $\ell_0/2$ in both the NE and SW directions: this occurs with some probability $p_0$. Now focus on two infinite sequences of rectangles ${\cal{R}}_i$ of increasing size $\ell_i\times \ell_i/12$ with $\ell_1=\ell_0$, $\ell_{i}=2\ell_{i-2}$ and intersecting as in Fig. \[lower\]. If each of these rectangles is spanned lengthwise by a DP cluster, the origin is frozen. The probabilities of perpendicular DP paths are positively correlated. Thus $\phi_\infty\ge p_0 \prod_i
[1-r_{1/12}(\ell_i)]^2$. Since ${\xi_\parallel}$ is infinite at $\rho_c$, as shown above, $r_b(\ell)$ decays exponentially to zero as $\exp[-Cb\ell^{1-z}]$. Therefore, the infinite product is non-zero, giving a strictly positive bound on $\phi_{\infty}(\rho_c)$ for any $\ell_0$. Thus, in contrast to DP or conventional percolation, the infinite frozen cluster of jamming percolation is “compact" — i.e. with dimension $d$ — at the transition.
To supplement our predictions, we have studied $L\times L$ systems numerically. The distribution of the densities, $\phi_L$, of the frozen clusters shows two peaks with weak size dependence as found at conventional first order transitions. This is consistent with the predicted discontinuous behavior. The probability that there [*exists*]{} a frozen cluster is substantial for $\rho$ fifteen percent below $\rho_c$ even in our largest systems, ($L=1600$): it is thus hard to study the asymptotic critical behavior (see [@Dawson] for an analogous problem in the context of bootstrap percolation). But in a slightly different model one can get closer to the transition [@TBF2long]: these data are consistent with the predicted $\ln\Xi \sim (\rho_c-\rho)^{-\mu}$ with $\mu\cong 0.64$, but the small range of $\ln L$ available makes the uncertainties in $\mu$ large.
Our results on [*jamming percolation*]{} have important implications for the equilibrium dynamics of the knights model. For $\rho<\rho_c$ the fact that infinite frozen clusters do not exist imply that the system is ergodic \[proof can be done as in ([@KATBF] 2.5)\] but dynamical correlations, such as the density auto-correlation function, $C(t)$, display two step temporal relaxation with a long plateau followed eventually by a relaxation (numerical results will be reported elsewhere [@TBF2long]). The relaxation time diverges exponentially near $\rho_c$, at least as $\Xi ^{z}$, with $z\geq 1$: This is reminiscent of the Vogel-Fulcher law found experimentally near glass transitions. Above $\rho_c$, the plateau stretches to infinite times with the Edwards-Anderson order parameter, $q\equiv\lim_{t\rightarrow \infty}C(t)$, discontinuous at the transition. This follows from our results, since $q$ is related to the density of frozen sites.
We have seen that, even though the critical densities are the same, the properties of jamming percolation are strikingly different from the power law behavior of directed percolation. Most of the physics is controlled by relatively short DP clusters joined together at T-junctions. The only role of long DP clusters is to prevent very rare large unfrozen regions from unblocking their surroundings. There is substantial universality in the primary features of the jamming percolation. This extends even to the SKA model which has frozen configurations composed of double-width occupied bars that terminate in T-junctions with similar perpendicular bars, but there is no real transition because very long bars are unlikely. Yet the SKA’s behavior as $\rho\nearrow 1$ is similar to the knights model as $\rho\nearrow \rho_c$ with $1/(1-\rho)$ roughly replacing powers of ${\xi_\parallel}$. Surprisingly, if the square lattice of the SKA is replaced by a particular complicated four-fold coordinated lattice, $\rho_c<1$ and the behavior is similar to the knights model. Thus the local structure matters a lot as in real glasses. Note that cooperative models with a transition, in contrast to those without, display two-step relaxation from the dynamics within blocked regions: this is like beta-relaxation in glasses [@DeBenedettiStillinger]. Models with [*particle-conserving*]{} dynamics behave surprisingly similarly to those without: the nature of the transition (and in some cases the critical density) remain the same because the slowing of the dynamics is dominated by the large clusters of the underlying jamming percolation. Diffusive transport rides on top of this [@TBF2long; @Huse; @Chandler]. In three dimensions, two natural generalizations of our jamming percolation exist: one composed of DP clusters — which should slow down as a double exponential of $(\rho-\rho_c)^{{-\overline{\mu}}}$ — and the other of directed sheet-like structures which will have exponential slowing down like we have found in 2D. The key ingredients are kinetic constraints that enable huge jammed clusters to form out of small objects without these becoming much more common or much larger.
For the future, the connection between our results and the jamming transition found for continuum particle systems [@Trappe] needs exploring. With the hope of increased understanding of the rapid liquid to glass crossover observed experimentally, one should also analyze the effects of constraint-violating processes occurring with a very low rate. For both glasses and granular materials, studying the non-equilibrium effects caused by a quench or by driving forces [@TBF2long] is merited even in the simplest models that exhibit a jamming transition.\
After the completion of this work a new version of a preprint [@Liu] appeared in which other models with a jamming transition are introduced and studied numerically.
We thank J. M. Schwarz, A. J. Liu for discussions. The numerical simulations have been performed on the parallel computer cluster of CEA under grant p576. GB is partially supported by EU contract HPRN-CT-2002-00307 (DYGLAGEMEM).CT by EU contract HPRN-CT-2002-00319(STIPCO) and DSF by the NSF via DMR-0229243.
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abstract: 'The recently developed semiclassical variational Wigner-Kirkwood (VWK) approach is applied to finite nuclei using external potentials and self-consistent mean fields derived from Skyrme interactions and from relativistic mean field theory. VWK consists of the Thomas-Fermi part plus a pure, perturbative $\hbar^2$ correction. In external potentials, VWK passes through the average of the quantal values of the accumulated level density and total energy as a function of the Fermi energy. However, there is a problem of overbinding when the energy per particle is displayed as a function of the particle number. The situation is analyzed comparing spherical and deformed harmonic oscillator potentials. In the self-consistent case, we show for Skyrme forces that VWK binding energies are very close to those obtained from extended Thomas-Fermi functionals of $\hbar^4$ order, pointing to the rapid convergence of the VWK theory. This satisfying result, however, does not cure the overbinding problem, i.e., the semiclassical energies show more binding than they should. This feature is more pronounced in the case of Skyrme forces than with the relativistic mean field approach. However, even in the latter case the shell correction energy for e.g. $^{208}$Pb turns out to be only $\sim -6$ MeV what is about a factor two or three off the generally accepted value. As an ad hoc remedy, increasing the kinetic energy by 2.5%, leads to shell correction energies well acceptable throughout the periodic table. The general importance of the present studies for other finite Fermi systems, self-bound or in external potentials, is pointed out.'
author:
- 'M. Centelles$^1$[^1], P. Schuck$^2$, and X. Viñas$^1$'
title: 'Thomas-Fermi theory for atomic nuclei revisited'
---
Introduction
============
One of the most important problems of finite fermion systems such as nuclei, atoms, helium- and metallic-clusters, quantum dots, etc., is the determination of the ground-state binding energy and the corresponding particle density distributions. In the nuclear case, to overcome the problems encountered when starting from realistic bare nucleon-nucleon forces, approximate and phenomenological schemes have widely been employed. This is the case of the very successful density dependent Hartree-Fock method with Skyrme [@Sky] or Gogny [@Gog] forces in the non-relativistic framework and of the relativistic mean field theory (non-linear $\sigma-\omega$ model) [@SW] in the relativistic formulation.
To investigate how properties of global character vary with the number of nucleons $A$, which is the subject of the present work, semiclassical or statistical techniques are very useful. The best known example is the nuclear mass formula, based on the liquid drop or droplet model [@Myers]. The success of the mass formula in describing binding energies lies in the fact that the quantal effects, i.e. shell corrections, are small as compared with the part of the energy which smoothly varies with $A$. The perturbative treatment of the shell correction energy in finite Fermi systems was elaborated by Strutinsky in the case of nuclei [@Strut]. It was proposed to divide the total quantal ground state energy in two parts: $$E = \tilde{E} + \delta E.
\label{eq1}$$ The by far largest part, $\tilde{E}$, varies smoothly with the number of fermions and is to be associated with the liquid drop energy. It can be calculated from e.g. the Hartree-Fock (HF) approach using the Strutinsky smoothing method [@Strut], which is a well defined mathematical procedure to erase the quantal oscillations in a finite Fermi system. However, this method may in general be more difficult to handle than the solution of the full quantal problem if realistic potentials are used. Thus, the search of alternative methods is an interesting and still partly open problem, as we will see. Semiclassical methods of the Thomas-Fermi (TF) type, which evaluate the smooth part of the energy, have widely been used in atomic, nuclear and metallic clusters physics. These TF methods, like the liquid droplet or Strutinsky calculations, smooth the quantal shell effects and estimate the average part of the HF energy [@BP; @Swia].
The semiclassical methods of the TF type are usually based on the Wigner-Kirkwood (WK) expansion of the density matrix [@WK]. In this approach, the single-particle density $\rho$ and the kinetic energy density $\tau$ are expressed by means of functionals of the one-body single-particle mean field potential $V$. The $\hbar^2$ or $\hbar^4$ corrections to the lowest-order TF term contain gradients of $V$ of second or fourth order that arise from the non-conmutativity between the momentum $\hat{{\bf p}}$ and position $\hat{{\bf R}}$ operators. The $\hbar$ corrections to the pure TF particle or kinetic energy densities are known to diverge at the classical turning point. They are to be considered rather as distributions than as functions [@RS; @KCM], in the sense that only integrated quantities have a real physical meaning. It has been shown that, in the case of a harmonic oscillator potential well, the WK theory including up to $\hbar^4$ corrections is equivalent to the Strutinsky average [@BP].
An important property of the WK expansion of the energy in powers of $\hbar$ concerns its variational content. For a set of non-interacting fermions submitted to an external potential, as for instance harmonic oscillator or Woods-Saxon wells, the variational solution for the particle density which minimizes the semiclassical WK energy at each order of the $\hbar$ expansion, is just the WK expansion of the particle density $\rho$ at the same order in $\hbar$ [@SV; @CVDSE]. The method for solving this variational problem [@SV; @CVDSE], which sorts out properly the different powers of $\hbar$ at each step of the minimization, was called variational Wigner-Kirkwood (VWK) theory. This VWK method has been applied to describe half infinite nuclear matter in the self consistent case using Skyrme forces [@CVDSE] and the relativistic mean field approximation [@ECV]. The formal VWK approach, up to $\hbar^4$ order, was developed in [@CVDSE].
Another related approach widely used for dealing with the semiclassical approximation of the self consistent HF problem, is based on the so-called density functional theory (DFT). The theoretical justification of DFT is formally provided by the Hohenberg and Kohn theorem [@HK]. In the nuclear context it states that the energy of a set of interacting nucleons is a unique functional of the local density alone, that is $$E = \int d \vec{r} \varepsilon \big[\rho (\vec{r})\big].
\label{eq2}$$ which reaches its minimal value when calculated with the exact ground-state density. The ground-state density $\rho(\vec{r})$ is determined by a variational approach of Eq.(\[eq2\]) with the constraint of a fixed number of particles: $$\frac{\delta}{\delta \rho} \int d \vec{r}
\{ \varepsilon \big[ \rho(\vec{r})
\big] - \mu \rho(\vec{r}) \} = 0.
\label{eq3}$$
In spite of the appeal of Eqs. (\[eq2\]) and (\[eq3\]), in general the exact energy functional is unknown and approximate techniques have to be worked out. The most popular and successful semiclassical approach based on DFT and developed together with the use of the Skyrme forces is the extended Thomas-Fermi (ETF) method. There the WK $\hbar$ expansion of the density is inverted to recast the kinetic energy density as a functional $\tau[\rho]$ of the local density and its derivatives [@GV; @BGH]. If the potential part of the energy density is also a known functional of $\rho(\vec{r})$, as it happens for the Skyrme forces, the approximate energy density functional can be minimized to obtain an Euler-Lagrange equation like in Eq.(\[eq3\]). The solution of this equation will provide the ground-state particle density and energy. The quantum shell oscillations are absent in the ETF model, which yields average densities and energies with good accuracy [@BGH; @CPVGB; @PS; @Li; @BB; @CVBS; @CV].
Using the VWK method, we have studied in Refs.[@SV; @CVDSE] the surface energy of a half infinite Fermi gas embedded in an external Woods-Saxon potential well. When $\hbar^2$ corrections are taken into account, the VWK surface energy reproduces the quantal values within 1% and the agreement is almost perfect when $\hbar^4$ corrections are considered. This result indicates that quantal Friedel oscillations have a negligible influence on the nuclear surface energy. We also have solved this problem using ETF (that in the case of the external potential reduces to the use of the ETF kinetic energy density). However, in this case we find discrepancies between ETF and quantal surface energies of 10% and 7% considering $\hbar^2$ and $\hbar^4$ corrections, respectively. This is an indication that ETF results are less well converged than the VWK ones. As it also will be discussed later on, this is mainly due to the fact that VWK properly sorts out the different orders in $\hbar$ which is not the case in ETF.
In Ref.[@CVDSE] we have analyzed the surface energy in self consistent problems using the TF, VWK, and ETF (up to $\hbar^2$ order) semiclassical approaches in comparison with the quantal (HF) results. In this study we considered several Skyrme forces that cover a wide range of effective mass values and incompressibility moduli in bulk matter at saturation. In general the VWK2 surface energies are closer to the quantal values than the TF ones. (We call VWK2 the VWK theory developed up to order $\hbar^2$, similarly VWK4 for the theory up to order $\hbar^4$, and analogously for the ETF theory.) The ETF2 surface energies are systematically smaller than the corresponding quantal values, and their absolute error is always larger than in the VWK2 case. An analysis of VWK and ETF calculations of half infinite nuclear matter in relativistic mean field theory yields the same kind of conclusions and the quality of the VWK2 results is seen to be systematically better than in ETF2 [@ECV]. However, if one compares the situation in the case of self consistent mean fields with respect to the case of the external Woods-Saxon potential well, the agreement between the semiclassical and quantal calculations of the surface energy worsens in the self consistent case, pointing to the fact that in the self consistent problems the semiclassical $\hbar$-expansions are more involved than in the case of an external potential. This aspect will be treated with more detail along this paper.
Summarizing, from the previous discussion it is clear that the VWK and ETF methods exhibit small but significant differences [@SV; @CVDSE; @ECV]. The reason lies in the fact, as already mentioned, that ETF does not properly sort out the different powers in $\hbar$ and that it partially sums terms to all orders in $\hbar$. On the other hand, our previous findings in the study of the surface energy in the self consistent case [@SV; @CVDSE; @ECV] indicate that the splitting of the quantal binding energies into their smooth and fluctuating parts is not so well under control (both in ETF and in VWK) as in the case of an external potential.
The main purpose of this paper is to develop and apply the VWK theory to [*finite*]{} nuclei in the self consistent case using both non-relativistic Skyrme forces and relativistic mean field (RMF) interactions [@SW]. In the next section we present the basics of the VWK method in the non-relativistic case with an alternative derivation to the one used in Refs. [@SV; @CVDSE] to explicitly show the differences between the VWK and ETF methods. In section III we first discuss within WK the external potential case to set the stage for the study of finite nuclei with self consistent mean fields later. We show that for strongly triaxially deformed mean field potentials with absence of any degeneracies, the semiclassical energies are extremely close to the quantal ones. Approaching sphericity in a homothetic way the shell structure becomes more and more apparent. Spherical open shell nuclei are slightly less bound quantally than semiclassically. This gives rise for the binding energy per particle as a function of particle number to the typical quantal arch structure between magic numbers whereas the semiclassical curve is, of course, monotonous. Section IV is devoted to the self consistent problem in finite nuclei. First, in section IV.A, we will show that using Skyrme forces, the VWK2 and ETF4 approaches practically give the same energy along the periodic table and that this fact is independent of the Skyrme interaction chosen. This a priori satisfying feature reveals, however, that the semiclassical approaches VWK2 and ETF4 overbind in excess, since even for the doubly magic nucleus $^{40}$Ca the Hartree-Fock results gives less binding than the semiclassical ones. This is shown and discussed in section IV.B, where also an ad hoc remedy to this problem is proposed. Several technical aspects are discussed in Appendices 1 and 2.
The variational Wigner-Kirkwood theory
======================================
The VWK theory has formally been introduced in Refs. [@SV; @CVDSE; @ECV]. Here we present, for a non-relativistic Skyrme force, a shortcut derivation in order to show explicitly the similarities and differences with the ETF method. For the sake of simplicity we consider symmetric and uncharged nuclei for the moment and, in this section, a Skyrme force with an effective mass equal the physical one. In this case the total energy of a nucleus in the ETF approach up to order $\hbar^2$ (ETF2) is written as [@BGH; @CPVGB]: $$\begin{aligned}
E = \int d \vec{r} \{ \frac{\hbar^2}{2 m} \big[ \frac{3}{5} \big(\frac{3
\pi^2}{2}\big)^{2/3} \rho^{5/3}
+ \frac{1}{36} \frac{(\nabla \rho)^2}{\rho}
+\frac{1}{3} \Delta \rho \big]
+ a_0 \rho^2 + a_3 \rho^{2 + \alpha} + a_{12} \, (\nabla \rho)^2 \},
\label{eq4} \end{aligned}$$ where $a_0= 3 t_0 / 8$, $a_{12}= (9t_1 - 5t_2 - 4t_2 x_2)/64$, $a_3=
t_3 / 16$, and $\alpha$ are the parameters that characterize the Skyrme interaction. The terms inside square brackets correspond to the ETF kinetic energy density $\tau_{ETF}$ up to $\hbar^2$ order. The $\rho^{5/3}$ term is the well-known pure TF contribution, which is of order $\hbar^0$ in the expansion of the kinetic energy density in $\hbar$ powers. The two remaining terms are of $\hbar^2$ order, and the first one is the so-called Weizsäcker term. Clearly, $\tau_{ETF}$ is a functional of the local density where the gradient terms are of second order in $\hbar$.
Starting from Eq.(\[eq4\]) the Euler-Lagrange equation for the local density constrained to give $A$ nucleons reads: $$\begin{aligned}
\frac{\hbar^2}{2 m} \bigg[ \bigg(\frac{3
\pi^2}{2}\bigg)^{2/3} \rho^{2/3}
+ \frac{1}{36} \frac{(\nabla \rho)^2}{\rho^2}
- \frac{1}{18} \frac{\Delta \rho}{\rho} \bigg]
+ 2 a_0 \rho +(2 + \alpha) a_3
\rho^{1 + \alpha} - 2 a_{12} \, \Delta \rho = \mu,
\label{eq5} \end{aligned}$$ where the chemical potential $\mu$ is the Lagrange multiplier that ensures the right normalization of the local density $\rho$. In the ETF method the variational equation (\[eq5\]) is solved numerically, for instance using the imaginary-time step method [@CPVGB]. However, the ETF approximation has some consistency problems with respect to the correct sorting out of powers in $\hbar$ [@CVDSE]. The reason is that the solution of Eq. (\[eq5\]) contains $\hbar$ at all orders due to the fact that the Weizsäcker term in Eq.(\[eq4\]) is of order $\hbar^2$. Actually Eq.(\[eq5\]) has a similar structure as a Schrödinger equation for $\rho$ and thus the density contains $\hbar$ as an essential singularity [@SV].
In order to properly sort out the different powers in $\hbar$ (to second order in the present example) one should split the local density and chemical potential entering in Eq.(\[eq5\]) into their $\hbar^0$ and $\hbar^2$ parts: $$\rho= \rho_0 + \hbar^2 {\rho_2}
\label{eq6}$$ and $$\mu = \mu_0 + \hbar^2 \mu_2.
\label{eq7}$$ Using (\[eq6\]) and (\[eq7\]), the Euler-Lagrange equation (\[eq5\]) can be sorted into $\hbar^0$ and $\hbar^2$ terms. One key point in the VWK theory is that the minimization is performed for each order in the expansion parameter $\hbar^2$ separately since, in principle, $\hbar$ can be considered as an arbitrary parameter (see Refs. [@SV; @CVDSE; @ECV] for a more detailed discussion of this point). Thus from the Euler-Lagrange equation (\[eq5\]) one obtains $$\frac{\hbar^2}{2 m} \big(\frac{3
\pi^2}{2}\big)^{2/3} \rho_0^{2/3}
+ 2 a_0 \rho_0 +
(2 + \alpha) a_3
\rho_0^{1 + \alpha} - 2 a_{12} \, \Delta \rho_0 - \mu_0 = 0
\label{eq8}$$ at TF ($\hbar^0$) order, and $$\begin{aligned}
&& \frac{\hbar^2}{2 m} \big[ \frac{2}{3} \big(\frac{3
\pi^2}{2}\big)^{2/3} \rho_0^{-1/3} {\rho_2}
+ \frac{1}{36} \frac{(\nabla \rho_0)^2}{\rho_0^2}
-\frac{1}{18} \frac{\Delta
\rho_0}{\rho_0}
\big] \nonumber \\
&+& 2 a_0 {\rho_2} +
(2 + \alpha)(1 + \alpha) a_3
\rho_0^{\alpha} {\rho_2} - 2 a_{12} \, \Delta {\rho_2} - \mu_2 = 0
\label{eq9} \end{aligned}$$ for the linearized second order correction. Another important point in the VWK theory is that the TF local density $\rho_0$, i.e. the variational solution of Eq.(\[eq8\]), fulfills the normalization condition: $$\int d\vec{r} \rho_0 = A
\label{eq10}$$ and due to the fact that $$\int d \vec{r} \big(\rho_0 + \hbar^2 {\rho_2} \big) = A,
\label{eq11}$$ we immediately see that the integral over the second-order density vanishes. This condition can be assured in adjusting $\mu_2$.
Now, splitting the total energy $E$ into its $\hbar^0$ and $\hbar^2$ contributions and using Eqs. (\[eq8\]), (\[eq10\]) and (\[eq11\]), one finds that in the VWK approach the energy of a finite nucleus including corrections of order $\hbar^2$ can be written as: $$\begin{aligned}
E = \int d \vec{r} \{ \frac{\hbar^2}{2 m} \big[ \frac{3}{5} \big(\frac{3
\pi^2}{2}\big)^{2/3} \rho_0^{5/3} + \frac{1}{36} \frac{(\nabla
\rho_0)^2}{\rho_0}
+\frac{1}{3} \Delta \rho_0 \big]
+ a_0 \rho_0^2 + a_3
\rho_0^{2 + \alpha} + a_{12} \, (\nabla \rho_0)^2 \}.
\label{eq12} \end{aligned}$$ Thus we arrive at the important result that the total energy up to order $\hbar^2$ is computed using only the lowest-order solution (TF) of the Euler-Lagrange equation. In practice this amounts to take the expression of the total energy as formally given by ETF2 but to compute it with the TF density solution. Consequently, the VWK procedure is consistent with the spirit of perturbation theory, since to calculate the energy at order $\hbar^2$ only requires knowledge of the solution of $\rho$ to the previous ($\hbar^0$) order [@SV; @CVDSE; @ECV]. The integral in Eq. (\[eq12\]) is defined between $r=0$ and the classical turning point $r_t$ where the TF density $\rho_0$ vanishes. The analysis of Eq. (\[eq12\]) near $r_t$ shows that $\rho_0$ behaves as $(r_t-r)^2$ and, as a consequence, the integrand of (\[eq12\]) is always finite in the whole domain of definition.
Of course the procedure can be continued to obtain the fourth order correction, see Ref.[@CVDSE] where this has been worked out in a slightly different way. The fourth order is, however, much more complicated, and necessitates for instance the knowledge of ${\rho_2}$ which may not easily be accessible [@CVDSE]. We remark that ${\rho_2}$ is not needed in VWK2 for the calculation of the energy. This is a consequence of the fact that the [*total*]{} energy is just the quantity that is minimized and then the use of the Euler-Lagrange equations allows to eliminate ${\rho_2}$ in the expansion of the energy. However, the evaluation of other quantities that are not minimized, e.g. kinetic energies or root mean square radii, etc., needs the explicit knowledge of ${\rho_2}$ when computed to $\hbar^2$ order.
It should be pointed out that in the general realistic case with effective mass different from the bare nucleon mass, inclusion of the spin-orbit potential, etc., the VWK2 method follows the same principle as in our schematic example. In practice one can take the corresponding ETF2 expression for the ground-state energy and replace $\rho(\vec{r})$ by its TF solution $\rho_0(\vec{r})$ which is the self consistent solution of the lowest-order variational TF equation. We refer the reader to Eqs. (A1)-(A5) of Ref. [@CEV] for the detailed ETF2 expression of the energy in the case of realistic Skyrme forces.
The same VWK theory can be applied to finite range effective forces such as the Gogny interaction [@Gog] although this case will not be treated explicitly in this paper. For this type of forces the semiclassical single-particle potential is not only position but also momentum dependent because of the finite range [@CVDSE]. Then, in addition to the kinetic and spin-orbit $\hbar^2$ corrections to the energy, there is another $\hbar^2$ contribution coming from the exchange term. Due to the $k$-dependence of the single-particle potential, the effective mass also becomes momentum dependent, which introduces extra terms in the $\hbar^2$ energy not present in the case of local forces as the Skyrme ones. The reader can find in Ref. [@SV2000] a detailed discussion of the ETF approach in the case of a general finite-range effective force. In particular, the kinetic and exchange energy densities in this case are given by Eqs.(39) and (40) of that reference. On the other hand, we also will consider in this paper the VWK approach applied to the relativistic mean field theory for the description of nuclei. The relativistic model automatically contains the finite range, spin-orbit and density dependence of the nucleon-nucleon interaction. The basic relativistic VWK theory up to $\hbar^2$ order has been worked out in Ref. [@ECV] and applied to the analysis of half infinite nuclear matter. In the case of finite nuclei the basic equations to be used for the VWK2 calculations are Eqs.(A7)-(A11) of Ref. [@CEV] together with Eqs.(5.8)-(5.12) of Ref. [@CVBS] computed with the solution of the relativistic TF equations.
The external potential case
===========================
In order to get a deeper insight into the behavior of the semiclassical energies as compared with the quantal ones before the study of finite nuclei with the use of self consistent mean field potentials, we first analyze the simpler problem of a set of non-interacting fermions submitted to an external potential well. In this case the VWK solution up to $\hbar^2$ order is just the WK expansion of the local density [@SV; @CVDSE] as pointed out in the Introduction. We will consider the model problems of harmonic oscillator and Woods-Saxon potentials. The discussion of the harmonic oscillator, apart of being of interest by itself as it is one of the most important model potentials in quantum mechanics, is relevant in different areas of physics beyond the context of atomic nuclei, such as confined electron systems or trapped ultracold fermion gases. A separate study for the harmonic oscillator potential including deformation degrees of freedom and the problem of a cavity with sharp boundaries, where the WK expansion cannot be applied, will be presented in a forthcoming publication [@Leboeuf06].
One important quantity is the number of states (accumulated level density) up to an energy $\varepsilon$, which is defined as [@RS] $$N(\varepsilon) = \int^{\varepsilon}_0 g(\varepsilon') d\varepsilon'.
\label{eq13}$$ The level density $g(\varepsilon)$ is given by $$\begin{aligned}
g(\varepsilon) &=& {\rm Tr}\/ [\delta (\varepsilon - {\hat H})] =
\frac{\partial}{\partial \varepsilon}
{\cal{L}}^{-1}_{\beta \to \varepsilon} \bigg[ \frac{2}{(2 \pi \hbar)^3} \int
\int
\frac{C^{\beta}(\vec{r},\vec{p})}{\beta} d\vec{r} d\vec{p} \bigg] \nonumber \\
&=& \frac{2}{(2 \pi \hbar)^3} \int \int
\frac{\partial f_{\varepsilon}(\vec{r},\vec{p})}{\partial \varepsilon}
d\vec{r} d\vec{p}
\label{eq14} \end{aligned}$$ where ${\cal{L}}^{-1}_{\beta \to \varepsilon}$ is the inverse Laplace transform and the factor 2 takes into account spin degeneracy. We use the notation $C^{\beta}(\vec{r},\vec{p})$ for the Wigner transform of the single-particle propagator $\hat{C}^{\beta}= \exp{(- \beta
\hat{H})}$, and $f_{\varepsilon} (\vec{r},\vec{p})$ is the corresponding Wigner function whose semiclassical expansion up to order $\hbar^2$ reads [@RS] $$\begin{aligned}
f_{\varepsilon}(\vec{r},\vec{p}) &=& \Theta(\varepsilon - H_{\rm w}) -
\frac{\hbar^2}{8m} \Delta V \delta' (\varepsilon - H_{\rm w}) \nonumber \\
&+& \frac{\hbar^2}{24m} \big[ (\nabla V)^2 + \frac{1}{m} (\vec{p}
\cdot \nabla)^2
V \big] \delta'' (\varepsilon - H_{\rm w}),
\label{eq15} \end{aligned}$$ where $H_{\rm w}$ is the classical mean field Hamiltonian (Wigner transform of ${\hat H}$).
Inserting Eq.(\[eq14\]) into (\[eq13\]) one obtains the accumulated level density from the Wigner function as $$N(\varepsilon) =
\frac{2}{(2 \pi \hbar)^3} \int \int
f_{\varepsilon}(\vec{r},\vec{p})d\vec{r} d\vec{p}.
\label{eq16}$$ In the same way the energy of a set of fermions in a potential well filled up to the Fermi energy $\varepsilon$ can be expressed as $$E(\varepsilon) = \int^{\varepsilon}_0 \varepsilon'
g(\varepsilon') d\varepsilon' =
\frac{2}{(2 \pi \hbar)^3} \int \int
f_{\varepsilon}(\vec{r},\vec{p})H_{\rm w} d\vec{r} d\vec{p} .
\label{eq17}$$
To simplify the calculation of $N(\varepsilon)$ and $E(\varepsilon)$, it is helpful to realize that $H_{\rm w}$ is the natural variable for $f_{\varepsilon}(\vec{r},\vec{p})$. In particular, the classical spherical harmonic oscillator (HO) Hamiltonian $H_{\rm w} = p^2/2m + m
\omega^2 r^2/2 = P^2 + Q^2$ can be seen as the square of a radial component $\sqrt{P^2 + Q^2}$ in polar coordinates, with a polar angle $\theta = \arctan (P/Q)$. In a similar way, for a more general potential with spherical symmetry, radial and polar angle coordinates can be defined in phase space by $$\sqrt{\tilde{H}_{\rm w}} = \sqrt{H_{\rm w} - V(0)},
\label{eq18}$$ where $V(0)$ is the bottom of the potential, and $$\frac{p^2}{2m} = \tilde{H}_{\rm w} \sin^2 \theta, \qquad
V(\vec{r}) - V(0) = \tilde{H}_{\rm w} \cos^2 \theta .
\label{eq19}$$ This allows switching from the variables $(r,p)$ to the new ones $(H_{\rm w},\theta)$ in the integrals over phase space. An advantage of this procedure is that one automatically circumvents the divergency problems usually encountered at the classical turning point when the $\hbar^2$ corrections are taken into account. We will use this method to obtain the results for the accumulated level density and energy as a function of $\varepsilon$ for an external Woods-Saxon potential that we will discuss later in this Section.
In the case of an external potential of HO type the integration of Eqs.(\[eq16\]) and (\[eq17\]) can be done analytically. The semiclassical expressions of the accumulated level density and energy read $$N_{\rm WK}(\varepsilon) = \frac{1}{3} \bigg(\frac{\varepsilon}{\hbar
\omega}\bigg)^3
- \frac{1}{4} \frac{\varepsilon}{\hbar \omega}
\label{eq20}$$ and $$E_{\rm WK}(\varepsilon) = \bigg[ \frac{1}{4}
\bigg(\frac{\varepsilon}{\hbar \omega}\bigg)^4
- \frac{1}{8} \bigg(\frac{\varepsilon}{\hbar \omega}\bigg)^2
- \frac{17}{960} \bigg] \hbar \omega,
\label{eq21}$$ respectively, where the contribution $-17\hbar \omega/960$ in the last equation comes from the $\hbar^4$ correction. Notice that in a HO potential there is no $\hbar^4$ correction in $N_{\rm WK}$ [@BP].
For the HO potential the quantal level density can also be obtained analytically [@BB; @BJ] and reads: $$g(\varepsilon) = \frac{1}{\hbar \omega}
\bigg[ \bigg(\frac{\varepsilon}{\hbar
\omega}\bigg)^2 - \frac{1}{4} \bigg]
\bigg( 1 + 2 \sum_{M=1}^{\infty} (-1)^M \cos \bigg( 2 \pi M
\frac{\varepsilon}{\hbar \omega} \bigg) \bigg),
\label{eq22}$$ which is seen to split into a part that smoothly varies with $\varepsilon$ and a fluctuating part. The smooth part is equal to the semiclassical WK expansion of the level density up to $\hbar^2$ (as already mentioned, the contributions of higher order in $\hbar$ vanish for the HO potential). The fluctuating part corresponds to the shell correction and contains all the quantal effects not included in the WK expansion. The quantal expressions for the accumulated level density and energy can easily be calculated starting from Eq.(\[eq22\]): $$\begin{aligned}
N(\varepsilon) &=& N_{\rm WK}(\varepsilon) +
2 \sum_{M=1}^{\infty} (-1)^M \bigg[\frac{1}{4 \pi^2
M^2}\frac{\varepsilon}{\hbar \omega}
\cos \bigg(2 \pi M \frac{\varepsilon}{\hbar \omega} \bigg) \nonumber \\
&+& \bigg( \frac{1}{2 \pi M} \bigg(\frac{\varepsilon}{\hbar \omega}\bigg)^2 -
\frac{1}{4 \pi^3 M^3} -
\frac{1}{8 \pi M}\bigg) \sin \bigg(2 \pi M \frac{\varepsilon}{\hbar \omega}
\bigg) \bigg] \label{eq23} \end{aligned}$$ and $$\begin{aligned}
E(\varepsilon)
&=& E_{\rm WK}(\varepsilon)
+ 2 \sum_{M=1}^{\infty} (-1)^M \bigg[ \bigg( \frac{3}{4 \pi^2 M^2}
\bigg(\frac{\varepsilon}{\hbar \omega}\bigg)^2 - \frac{3}{8 \pi^4 M^4} -
\frac{1}{16 \pi^2 M^2} \bigg) \cos \bigg(2 \pi M \frac{\varepsilon}{\hbar
\omega} \bigg) \nonumber \\
&+& \bigg( \frac{1}{2 \pi M} \bigg(\frac{\varepsilon}{\hbar \omega}\bigg)^3 -
\frac{3}{4 \pi^3M^3}\frac{\varepsilon}{\hbar \omega} -
\frac{1}{8 \pi M} \frac{\varepsilon}{\hbar \omega}\bigg) \sin \bigg( 2 \pi M
\frac{\varepsilon}{\hbar \omega} \bigg) \bigg] \hbar \omega.
\label{eq24} \end{aligned}$$ Therefore, in this simple model the separation of the total energy in a smooth (liquid drop like) part $\tilde{E}$ and a shell correction part $\delta E$, like in Eq.(\[eq1\]), is obtained analytically.
The upper panel of Fig.1 displays the accumulated level density $N(\varepsilon)$ for a set of fermions in a fixed [*spherical*]{} HO potential calculated semiclassically and quantally, as a function of the Fermi energy $\varepsilon$ divided by $\hbar \omega$. The quantal result exhibits discontinuities at each major shell ($N= 2$, 8, 20, 40, 70, and 112 in the figure) and is represented by a staircase function formed by horizontal and vertical lines which fluctuate around the smooth value of $N(\varepsilon)$. The latter is provided by the WK value given by Eq.(\[eq20\]) and is represented by the solid curve of the upper panel of Fig.1. In the same panel we display the oscillatory part of $N(\varepsilon)$ (dashed curve), i.e., the quantal minus the semiclassical values, which contains the fluctuations due to the shell effects. One sees that the quantal part of the accumulated level density oscillates around zero.
The lower panel of Fig.1 displays the quantal and semiclassical WK values of the total energy $E(\varepsilon)/\hbar \omega$ for the spherical HO potential by the staircase and solid curves, respectively. In the same lower panel, the shell energy, i.e. the difference between the quantal and semiclassical values, is represented by the dashed line. Again it can be seen that the shell energy fluctuates around zero and that the semiclasical WK estimate of $E(\varepsilon)/\hbar \omega$ averages the quantal values. As it has been pointed out in the Introduction, the WK approach to $E(\varepsilon)$ including $\hbar^4$ corrections coincides with the Strutinsky average in the HO potential [@BP].
We have performed the same kind of analysis for the more realistic potential well of Woods-Saxon (WS) type used in Ref. [@JBB]: $V(r)=V_0/[1+\exp{(\frac{r-R}{a})}]$ with the values $V_0=-44$ MeV, $a=0.67$ fm and $R=1.27 A^{1/3}$ fm. We have computed quantally and semiclassically (with pure TF and with WK up to $\hbar^2$ order) the accumulated level density and energy of neutrons (spin degeneracy is assumed) in the above WS potential with a size corresponding to a nucleus of $A=208$ nucleons. The calculated $N(\varepsilon)$ and $E(\varepsilon)$ are displayed as a function of the Fermi energy $\varepsilon$ in the upper and lower panels of Fig.2, respectively. Again the staircase and solid curves correspond to the quantal and WK results, respectively, whereas the dashed lines are now the pure TF values. As in the case of the HO potential, the WK estimate of the smooth parts of $N(\varepsilon)$ and $E(\varepsilon)$ passes well through the corresponding staircase functions and averages the quantal accumulated level density and energy.
For the WS potential the equivalence between the semiclassical WK expansion and the Strutinsky average cannot be established analytically. It has been checked numerically that both methods, with high accuracy, give the same value for the energy, at least in the case where the chemical potential is sufficiently negative [@RS; @CVDSE; @CPVGB; @JBB]. However, the situation may be different when the Fermi energy is close to zero. In this case, the semiclassical WK and Strutinsky level densities start to deviate from one another when $\varepsilon$ approaches zero. The WK level density, which includes $\hbar^2$ corrections shows a ${\varepsilon}^{-1/2}$ divergency at $\varepsilon=0$ for a finite potential as the WS one, whereas the Strutinsky averaged level density only has a strongly pronounced, but finite, maximum [@S]. In Refs. [@NWD; @VKLNSW] it was concluded that the divergency of the WK level density for $\varepsilon \to 0$ is unphysical and preference should be given to the Strutinsky smoothed level density. However, we would like to recall that WK quantities have to be understood in the sense of distributions [@RS; @KCM]. Therefore, a diverging WK level density should not be taken literally and only used under integrals. For example, in the upper and lower panels of Fig.2 one sees that the accumulated semiclassical level density $N(\varepsilon)$ and the total energy $E(\varepsilon)$ are well behaved and accurately average the corresponding quantal values even for $\varepsilon \to 0$. The TF accumulated level density and energy show similar tendencies to those exhibited by the WK results. However, the TF average of the quantal values is less good than the one obtained at the WK level. This fact demonstrates the importance of the $\hbar$-corrections in the Wigner function (\[eq15\]) to obtain the correct average of the quantal results.
Usually the various quantities like energy, kinetic energy, etc. for a system containing a fixed number of particles $N$ are not displayed as a function of the chemical potential $\mu$ \[given by $N=N(\mu)$\], but rather as functions of the particle number. For example, having the energy $E(\mu)$ and the accumulated level density $N(\mu)$ as functions of the chemical potential $\mu$ we can consider the inversion $\mu=\mu(N)$ and then obtain the energy as a function of the particle number $N$, i.e $E=E(N)$. The $N$ dependence can be studied for a fixed external potential. More realistically the potential well will change with the number of particles, as e.g. the HO potential with $\hbar \omega \simeq 41 A^{-1/3}$ MeV or the WS potential of Ref. [@BB]. Below we will consider both cases: the most simple case of a fixed potential well and the case where the potential changes with the particle number.
If the potential well has degenerate levels, the inversion $\mu=\mu(N)$ is not unambiguous in the quantal case, because the chemical potential is the same for various values of the particle number $N$. This is for example the case for the spherical HO. To get around this problem one can consider the spherical HO as the limit of a triaxially deformed HO in the limit of zero deformation. In the triaxial case each level has only spin-isospin degeneracy. However, for the purpose of our reasoning we here can disregard spin and isospin. Then in the infinitesimal triaxially deformed HO all levels can be occupied by only “one nucleon”. In the case of sphericity a major shell with HO principal quantum number $n$ has a degeneracy $D(n)$ and the functions $N(\mu)$ and $E(\mu)$ are sharp staircase functions, whereas for very small triaxial deformation the vertical jumps become slightly tilted and resolved in $D(n)$ minuscule staircases. In that case one then always has a definite number of particles for definite values of $\mu$ and perfectly can find $\mu=\mu(N)$ unambigously. Therefore also $E(N)$ is well defined. In the limit of zero deformation this leads to the uniform filling prescription of a degenerate shell at sphericity.
With these preliminaries in mind, we show in the upper and lower panels of Fig.3 the energy per particle as a function of the particle number for (i) a strongly deformed HO with frequencies $$\omega_x= \sigma^{-1/3} \delta^{-1/2} \omega_0 , \qquad
\omega_y= \sigma^{-1/3} \delta^{1/2} \omega_0 , \qquad
\omega_z= \sigma^{2/3} \omega_0 ,
\label{eq25}$$ taking the values $\omega_x=0.460 \, \omega_0$, $\omega_y= 1.111 \,
\omega_0$, and $\omega_z= 1.954 \, \omega_0$, and (ii) a spherical HO in the sense explained above. The HO depends as usual on particle number through $\hbar \omega_0 = 41
A^{-1/3}$ MeV and deformation is such that volume is conserved ($\omega_x \omega_y \omega_z= \omega_0^3$). In the upper panel of Fig.3 we see that in the deformed potential [@BB] the quantal values (dots) practically coincide with the WK values (solid line) and in any case WK perfectly averages the quantal values. On the other hand, in the spherical case there is a surprise in the sense that the WK-values do not pass, as a function of the particle number, through the average of the quantal values: there are much more values above the WK-line than below and also the deviations above the WK-line are stronger than below. This means that WK overbinds with respect to the true average except at magicity.
In the light of the fact that for the separate curves $E(\varepsilon)$ and $N(\varepsilon)$ (see Fig.1) the semiclassical values perfectly average the quantal ones also in the spherical case, the global overbinding of WK as a function of the particle number may appear puzzling. The effect is, however, known [@Leb]. One can indeed show that an average over $\varepsilon$ (or $\mu$) of the fluctuating part in (\[eq24\]) yields zero, whereas when expressed as a function of $N$ the fluctuating part shows a non-vanishing average, i.e. $\langle
\, \delta E(\mu) \, \rangle_{\mu} = 0$ but $\langle \, \delta E(N)
\, \rangle_N \ne 0$, where the brackets $\langle \, \dots \, \rangle_{\mu,N}$ indicate averages over $\mu$ or $N$, respectively. This feature can also be understood schematically from a different aspect in the following way. Suppose we consider a HO potential of fixed size with very small triaxial deformation, i.e. we consider the uniform filling prescription at sphericity. In a given shell the total quantal energy increases linearly with the number of nucleons in the shell. On the other hand, on average the total energy, according to Eqs. (\[eq20\]) and (\[eq21\]), increases at the TF level as $E_{tot} \propto N^{4/3}$. This situation is detailed in Table 1 for the $n$=4 shell of a spherical HO potential of fixed size, which contains the $1g$, $2d$ and $3s$ levels. There we display the quantal and semiclassical (TF and WK including $\hbar^2$ corrections) chemical potentials and energies obtained in filling uniformly the shell assuming spin degeneracy (the values are expressed in units of $\hbar \omega$). The semiclassical chemical potentials are obtained inverting Eq.(\[eq20\]) to find the corresponding value of $\mu$ which in turn is used in (\[eq21\]) to calculate the semiclassical energies. The quantal chemical potential in each spherical shell of the HO potential is given by $\mu/\hbar \omega =n+3/2$. The number of particles and energy in the $n$=4 shell are given by $$N= \sum_{n=0}^{3} D(n) +2m = 40 + 2m
\label{eq24a}$$ and $$\frac{E}{\hbar \omega} = \sum_{n=0}^{3} D(n)\bigg(n + \frac{3}{2}\bigg)
+2m \bigg(4 +
\frac{3}{2}\bigg) = 150 + 2m \bigg(4 + \frac{3}{2}\bigg) ,
\label{eq24b}$$ where $D(n)=(n+1)(n+2)$ is the degeneracy of a shell including spin and $m=1,2,3, \ldots$ is the number of pairs (spin up and spin down) added to fill up the $n$=4 shell. From Table 1 we see that the TF energies always overbind the quantal values and the same is true for the WK ones, except close to magicity where the $n$=4 shell is empty or completely full.
The shell correction, i.e. the difference between quantal and semiclassical energies, is displayed in Fig.4 as a function of the number of particles in the shell. From this figure it is clear that the TF approach is far from averaging the quantal values and that in the WK case the average is much improved. If the $\hbar^4$ corrections are added, in this case of a fixed HO potential, the energy is shifted down by a constant amount of 17/960 (in $\hbar \omega$ units) according to Eq.(\[eq21\]), but it cannot be distinguished from the $\hbar^2$-corrected result on the scale of the figure. Therefore, the $\hbar^4$ corrections are very small as compared with the $\hbar^2$-ones demonstrating again the rapid convergence of the WK series. Thus, the situation for a fixed HO potential is similar to that found in the more general case of a size dependent HO potential as it can be seen comparing the lower panel of Fig.3 with Fig.4.
The lack of averaging in the energies found in the spherical potential is due to the large degeneracy of the HO shells. If each shell is broken into $D(n)$ small pieces, as in the case of strong triaxiality, they are bound to stay close to the average which varies, as already mentioned, as $\sim N^{4/3}$. In the case of sphericity this $N^{4/3}$ behaviour is, as demonstrated in Table 1, quantally replaced by straight line segments, each segment corresponding to a major shell. Two segments join at a magic number with a characteristic overbinding which is relatively small. In between two magic points the quantal straight line passes most of the time [*above*]{} the concave semiclassical curve. This scenario can further be clarified by the following investigation. The fact that for strong triaxiality quantal and semiclassical calculations almost agree can be understood because in that case there do not exist degeneracies (besides some special cases where the axis ratios are formed by rational numbers [@BJ]). Therefore, the quantal level density is also practically smooth, and it almost coincides with the semiclassical result.
In Fig.5 we show this, displaying the energy per particle as a function of triaxial deformation for a HO well. To have a single deformation parameter $d$ for the representation, in this figure we have chosen the frequencies of the deformed HO according to $$\sigma=1 + d \sqrt{3} , \qquad
\delta=1 + \vert d \vert \sqrt{2}
\label{eq25a}$$ in Eq. (\[eq25\]). We see that for a mid-shell configuration (spin degeneracy) of $N=92$ fermions, the semiclassical and quantal values practically agree, up to very small fluctuations, down to quite low deformations. The quantal energy suddenly raises when approaching sphericity. For real nuclei this means that binding energy is lost at sphericity. The only slight exception to this scenario is for deformation $\sim$ 0.6 where the frequency ratio is close to $\omega_x: \omega_y: \omega_z \sim 1:2:3$. We, therefore, see that in forcing open shell nuclei to be spherical one loses a lot of binding energy. As a matter of fact, this loss of binding energy starts immediately off magic numbers and increases towards mid shell fillings. This explains why the semiclassical curve mostly overbinds as a function of the shell filling. In the general case with mass number dependent potentials, these considerations are slightly more complicated but the reasoning which leads to the underbinding of quantal results for energies per particle keeping open shell nuclei spherical is essentially the same. This is one of the explanations of the fact that the shell corrections as a function of the particle number, do not oscillate around zero but show a finite average value. Above we already have mentioned that this can also be seen from the fact that as function of the particle number the fluctuating part in Eq.(\[eq24\]) does not average to zero [@Leb]. Below we will find that the same situation prevails in the case of self consistent mean fields. We also would like to mention that similar features as those discussed above in connection with deformation and degeneracy of the single-particle levels have been found by Pomorski [@pomorski04] using the Strutinsky smearing method applied in energy space and in particle number space.
The above considerations only apply to spherical nuclei. In reality the force which holds semimagic nuclei, e.g. tin isotopes, spherical is the magicity of the protons which resists to deformation. For deformed nuclei the situation is different and needs separate investigation. In the lower panel of Fig. 3 we show what happens when the shape of the potential is free and the energy minimised for each particle number with respect to deformation. The absolute minimum of the quantal calculation is obtained allowing triaxial deformation in Eq.(\[eq25\]). Now the arches are strongly flattened and in between magic numbers the energies per particle lie practically on the semiclassical curve. Magic nuclei appear as exceptional points and a particle number average will be close to the semiclassical result. Notice again that we are now comparing absolute minima both in the semiclassical case (where they occur exclusively at sphericity) and in the quantal one (where they are deformed, besides around magicity). From this point of view, the close agreement between quantal and semiclassical results is very satisfying and it likely is a generic feature valid also for other types of mean field potentials. For real nuclei, deformed as well as spherical situations can happen. If both proton and neutron numbers correspond to open shell situations, nuclei are in their majority deformed, whereas if either the proton or neutron number is magic, nuclei usually are spherical, as it happens for instance for the chain of Sn isotopes. Because of the numerical complexity of the deformed case, we only will concentrate on spherical nuclei in the remainder of the paper. More detailed investigations of the deformed situation will be presented elsewhere [@Leboeuf06].
The fact that the semiclassical results are not going through the average of the quantal results as a function of $A$ is somewhat annoying from the practical point of view, since we cannot judge whether the semiclassical results are converged to the right value or not. In the case of external potentials the answer to this question is easy to find: we take a fixed potential and look at the WK results as a function of the chemical potential $\mu$. We know that in this case the semiclassical results should pass through the average of the quantal ones (see e.g. Fig.2).
The self consistent potential case
==================================
Finite nuclei
-------------
For spherical nuclei described self consistently through an effective interaction, the scenario for the energy per particle as a function of the mass number stays qualitatively the same as for the external potential case. Again, the typical arch structure with the values at magicity barely undershooting the semiclassical line (see lower panel of Fig.3) appears. In the upper panel of Fig.6 we present self consistent calculations of the shell energy per particle, which is defined as $E_{\rm shell}/A =
(E_{\rm HF} - E_{\rm semicl})/A$, as a function of the mass number, for the TF, VWK2, and ETF4 semiclassical approaches using the T6 force [@T6]. This Skyrme interaction has an effective nucleon mass $m^*$ equal to the bare one $m$. In the calculations shown in Fig.6, the Coulomb repulsion among protons and the spin-orbit force have been switched off and only hypothetical spherical symmetric nuclei with $N=Z$ are considered. Later we will study realistic nuclei, but for the moment we want to avoid that these more subtle effects contaminate the comparison of the semiclassical results with the quantal ones.
The same type of calculation is presented in the lower panel of Fig.6, now performed using the very different Skyrme force SV [@Sk5] that has no density-dependent part ($t_3=0$) and for which the effective mass is $m^*/m$=0.38 in nuclear matter. In the case of the VWK2 calculation we encounter the same pattern as with the T6 force. However, the predictions of the TF calculation are at variance with the case of the T6 interaction: for T6 they overbind, whereas for SV they underbind. This change in the behaviour of the TF solution is largely due to the different values of the effective masses of the two forces, and we have documented this fact already in earlier publications [@CVDSE; @CVBML]. Very satisfactorily, the deviation of the VWK2 results from the HF ones only depends very little on the particular properties of the effective interaction.
We have to remark the important feature that, as can be seen in Fig.6, the shell energies per particle calculated with the VWK2 and ETF4 approaches are very close. We have tested a whole series of Skyrme forces and did not find exceptions to this fact. As we will see below, the inclusion of Coulomb and spin-orbit forces does not change this rule essentially. Thus, with the VWK2 calculation one is able to obtain energies of an equivalent quality to the by far more complicated ETF4 approach which requires sophisticated techniques for the self consistent numerical solution of the variational equations.
Several additional comments should, however, still be made. Above we argued that ETF is inconsistent in sorting out the powers in $\hbar$ and that consequently it converges less well than VWK. We also gave arguments backed by explicit examples that VWK should converge faster than ETF and that the $\hbar^4$-contribution to VWK should practically be negligible. This was, however, for an external potential case. Perhaps the external potentials are particularly difficult cases for ETF (for instance one has to solve a non-linear differential equation even in the external potential case, whereas the WK expression can be used as is) and its convergence properties are better in the completely self consistent case. Thus, it could be that both the VWK2 (besides a small VWK4 correction as in the external potential case) and the ETF4 results are converged to the same and definite semiclassical value for nuclear binding energies. However, at this point this is a speculation. It may be that VWK2 and ETF4 coincide without both really having reached complete convergence to the actual semiclassical average. For instance, it cannot be excluded that VWK4 could, in the self consistent case, yield a contribution which is sensitively more important than in the external potential case. This remark should be kept in mind when we discuss the results more closely below.
The uncertainty of the situation also comes from the fact that, as discussed above, we do not have a precise criterion in the self consistent case what the semiclassical binding energies as a function of the particle number should be, besides that they should coincide with a Strutinsky self consistent calculation. The latter is, however, also slightly uncertain, because the plateau condition is difficult to satisfy [@BB; @Pom] and up to date only a few self consistent Strutinsky calculations with Skyrme forces exist in the literature [@BQ; @BQ1]. However, in any case the general agreement of VWK2 with ETF4 is quite remarkable and adds more confidence to the semiclassical results, in spite of the fact that problems are not all resolved as we will further discuss below.
In order to give additional backing to what is just outlined with more studies, we now show on the upper panels of Figs.7 and 8 the energy and the shell correction per nucleon, respectively, in a realistic case for nuclei along the valley of stability calculated with the SkM$^{*}$ interaction including the Coulomb and spin-orbit forces. Most of the $\beta$-stable nuclei displayed in Figs.7 and 8 were also used in Ref. [@Pom], and they are spherical according to the finite range droplet model (FRDM) [@FRDM1]. We have also considered some other additional nuclei which, according to the FRDM, are also $\beta$-stable [@FRDM1] and spherical except for a few of them [@FRDM2]. Also in this realistic case we can observe the very close agreement between the VWK2 and ETF4 methods. We also display the results of the ETF2 calculation and notice that it is much less converged. The VWK2 approach to the relativistic mean field theory has also been worked out [@ECV] and we present the results for the same sample of nuclei along the valley of stability using the realistic, accurately calibrated parameter set NL3 [@NL3] on the lower panels of Figs.7 and 8. In the relativistic case the ETF4 corrections have not been elaborated and we do not have the corresponding results available. We have found the relativistic ETF2 results to be, as in the non-relativistic case, not so well converged as VWK2 and we do not show them.
In Fig.8 we quite clearly see a deficiency of the semiclassical energies, already present in the preceding figures: there is too much binding, even keeping in mind that the semiclassical binding energies for spherical nuclei have a natural tendency to be stronger than the average of the quantal values as discussed in section III. This drawback is particularly pronounced in the case of Skyrme forces (we checked this with multiple Skyrme interactions). Also in the relativistic case the semiclassical results give too much binding, in spite of the fact that the doubly magic nuclei are (slightly) more bound quantally than semiclassically, what is not the case for Skyrme forces, see Table 2. This situation clearly is unphysical. We will comment further on this in the next subsection. Let us point out that in our prescription the shell correction has been taken as $E_{\rm shell} = E_{\rm HF} - E_{\rm
semicl}$, which in principle is different from the often employed prescription where one takes $E_{\rm shell} = \sum \varepsilon_i -
\sum \varepsilon_i \tilde {n}_i$ as the difference of the sum of the quantal single-particle energies and the Strutinsky averaged sum. However, due to the Strutinsky energy theorem [@RS] the predictions of both procedures should agree if the considered semiclassical approaches reproduce well the Strutinsky averaged value.
Concluding this subsection, we can say that the semiclassical limit of the energy per particle of finite nuclei based on Skyrme or relativistic mean field theories has been established on the VWK2 level. A quite intriguing coincidence between the VWK2 and ETF4 methods has been found. The significance of this fact is not entirely clear and will be discussed further in the next section. The ETF4 method exists since long whereas VWK2 is new. Apart from the discussed conceptual differences in the rigorous power counting in the $\hbar$ expansion, the VWK2 method, see Eq.(\[eq12\]), has the advantage that the convergence is faster and that the final formulas for the calculation $E/A$ are very simple (only the solution of the zeroth-order TF variational equation is needed!). The overbinding of TF and ETF calculations of nuclei has been recognized in many studies since years ago (see e.g. Refs. [@BGH; @CPVGB; @PS; @Li; @BB; @CVBS; @CV; @Kumar; @BCKT]), and it is also very much present in atomic physics calculations [@March]. We have shown that the problem persists even in the more refined ETF4 and VWK2 appoaches, and we will turn to it in more detail now.
The overbinding problem
-----------------------
In the last section we have seen that the scenario of the arch structure in the energy per particle remains in the self consistent case qualitatively the same as in the external potential case. However, we remarked a deficiency in the self consistent case which becomes apparent when having a close look at the realistic cases presented in Figs.7 and 8. In Table 2 we present the quantal and VWK2 energies for some magic nuclei calculated with the SkM$^*$ force and with the NL3 parameter set of the relativistic theory. We see, that in particular Skyrme forces overbind semiclassically, as it is also seen in the upper panels of Figs.7 and 8. The fact that even doubly magic nuclei like $^{40}$Ca are more bound semiclassically than quantally is clearly incorrect, even though we should be aware of the fact that we are dealing with small differences of large numbers. In any case, taking self consistent Strutinsky calculations as reference [@BQ; @BQ1], $^{40}$Ca and $^{208}$Pb are less bound, when averaged, than quantally. This failure of the semiclassical approach is disappointing with respect to the external potential case where, as a function of energy (or chemical potential), we are used to the fact that the VWK2 method gives extremely precise average quantities, like level densities, energies, etc., as has been demonstrated in the past with many examples [@RS; @BP; @SV; @CVDSE] (see also Figs.1 and 2 of this paper).
Let us try to find some reason for this deficiency and eventually a cure. What is different between the external and self consistent potential cases? The only convincing difference we can imagine is the fact that in the external potential case the density is a functional of an external [*fixed*]{} potential $V$, i.e. $\rho=\rho[V]$ which is expanded in powers of $\hbar$, that is in gradients of $V$. In the self consistent case the potential itself is a functional of the density and therefore also the potential has then to be expanded in a power series of $\hbar$ (we did not explicitly proceed in this way, but implicitly that is what it amounts to in the VWK method). This double $\hbar$-expansion is very likely one of the reasons for the deterioration of the results with respect to the external potential case. The WK expansion of the density matrix is a local expansion in terms of distributions functions what probably does too much harm to the self consistent potential. Some global features should be kept, even for the average potential. For example, if we were given a self consistently averaged Strutinsky HF potential, we would believe from our past experience that when it is taken as an external potential in the evaluation of the semiclassical WK-HF energy this should give very precisely the true Strutinsky averaged value of $E/A$. Of course, this would be an extremely laborious detour. One can think of employing an approximate substitute of the Strutinsky potential. A possibility is to take the self consistent potential evaluated in the ETF2 approach, instead. Indeed, as we mentioned earlier, in the ETF2 approach the density contains powers in $\hbar$ which are partially resummed to all orders in the self consistent calculation [@SV]. Therefore, the corresponding single-particle potential, which is a well behaved smooth average potential (see Fig.12), also contains some global properties.
We will apply this strategy to obtain another semiclassical estimate of the HF energy. To this end we first run a self-consistent ETF2 calculation with the T6 force. Next, we take the computed ETF2 mean field potential, including its spin-orbit and Coulomb contributions, as if it were an [*external*]{} potential and with it we perform a WK2 calculation to obtain the Skyrme energy, by using the WK expressions for particle and kinetic energy densities including $\hbar^2$ corrections [@RS]. In this procedure, which clearly differs from the VWK method, some divergences arise in the evaluation of some $\hbar^2$ contributions (see Appendix 1 for the treatment of the divergence of the term $( \nabla \rho_{\rm WK} )^2$ in the Skyrme energy density). To circumvent this technical difficulty, a finite temperature WK calculation is performed [@BGH; @VG] which is extrapolated to $T=0$. The details of this method to estimate the semiclassical HF energy are described in Appendix 2. The results obtained for the shell correction per particle calculated for $\beta$-stable nuclei are shown in Fig.9 (curve labelled by “temperature extrapolation”). We see that there is a substantial improvement over the VWK2 and ETF4 results. The cure is not 100% though and there remains the fact that $^{40}$Ca is slightly more bound semiclassically than quantally. However, globally, the average $A$-dependence of $E/A$ is now quite acceptable in particular towards the heavier nuclei. For example the shell correction for $^{208}$Pb is now $\sim 20$ MeV, well in line with the value reported from Strutinsky calculations with the Gogny D1S and RMF NL3 effective nuclear interactions [@Pom]. We agree that the procedure is ad hoc; however, it helps to shed some light on the situation.
It is interesting to note that the WK-HF results in Fig.9 obtained with the ETF2 potential can almost perfectly be reproduced within the VWK2 method with a fudge factor on the VWK2 kinetic energy in the following way: in Eq.(\[eq12\]) we replace the factor 1/36 in front of the Weizsäcker term by 1.26/36 in order to increase the kinetic energy, i.e. to decrease the binding. The same result can be obtained with ETF2 by replacing 1/36 by 1.8/36 (modifications of the value of the coefficient of the Weizsäcker term have been studied in the literature since a long time ago, see e.g. Ref.[@KT]). We show these results for the T6 force in the upper panel of Fig.10 and find that all these different prescriptions practically lead to shell correction values on top of one another. The reason for this very close agreement, found using different prescriptions to estimate the shell corrections, is at present unknown, but it is a surprising and interesting feature.
Still we would like to deepen somewhat the discussion of the situation and of the possible reasons for the failure of VWK2 (and ETF4) to correctly reproduce the average. As mentioned above, the implicit expansion of the average mean field in powers of $\hbar$ may be the main direct reason. However, the very short range character of the nuclear force may reinforce the problem at least in the non relativistic case. For Skyrme forces the zero range character entails an unphysical shape of the self consistent TF potential. This can best be studied in half infinite nuclear matter where the self consistent TF density can be obtained in an analytic way by quadratures in the case of Skyrme forces [@CVDSE; @VCDS]. The TF density $\rho_0(z)$ and the corresponding single-particle potential for a Skyrme force with $m^* = m$ $$V(z) = 2 a_0 \rho_0(z) +
(2 + \alpha) a_3 {\rho_0(z)}^{1 + \alpha}
- 2 a_{12} \, \rho''_0(z),
\label{eq35}$$ are displayed in Fig.11. The TF density close to the turning point, chosen at $z=0$, behaves like $\lim_{z \to 0} \rho_0 \approx z^2$ and the single-particle potential $V(z)$ reaches the classical turning point at $z=0$ with zero slope. This feature is not very well seen in Fig.11 because it turns out that the bending into the horizontal tangent only happens extremly close to the classical turning point. On the other hand, such pathological behaviour is absent in the relativistic mean field approach, since the forces are of finite range. In this case the TF potential has a WS like shape and is continuous in whole space. This is shown in Fig.12 where we display the TF neutron self consistent potential for $^{208}$Pb obtained with the NL3 parameter set. We see that this potential has a very acceptable shape, not much different from the usual phenomenological WS potentials with about a 2 fm wide fall off width. Also the derivative of this potential is in no way anywhere more pronounced than the one corresponding to phenomenological potentials (see Fig.12). From this fact we understand that, with respect to the non-relativistic case of Skyrme forces, the semiclassical results are considerably better in the relativistic case (see Table 2 and Figs.8 and 9). At least practically all doubly magic nuclei are more bound quantally than semiclassically. However, for example for $^{208}$Pb the shell energy turns out to be $E_{shell} \approx 6$ MeV, a value which is about a factor 3 times too small with regard to commonly accepted values [@Pom; @BQ; @BQ1]. Again this problem may be attributed to this double expansion in gradients of the mean field potential and the potential itself and, as already mentioned, it cannot be excluded that the $\hbar$-expansion converges in the self consistent case more slowly than in the the external potential case. At any rate, contrary to the situation with the Skyrme forces, in the relativistic case, as already stated, the TF mean field potentials are perfectly smooth and well behaved and can, therefore, not be incriminated. At the present moment it is unclear how to remedy this situation, other than by prescriptions such as the ones presented above. However, even this must still be refined in order to become entirely realistic.
A last comment may be in order at this point. We remark in Fig.10 that for $^{208}$Pb the shell correction predicted by the ad-hoc methods is quite acceptable. However, there is a continuous deterioration of the situation towards lighter nuclei. Such a deterioration can in fact also be seen in Fig.8 for the SkM$^*$ force and in Fig.9 for the T6 force, whereas in the relativistic case (lower panel of Fig.8) the predictions for light nuclei are more robust. For instance, both SkM\* and T6 yield (wrongly) a positive shell correction energy of about 12 MeV for $^{40}$Ca in VWK2, while NL3 at least predicts a negative value of $-2$ MeV for this nucleus (see Table 2). One possible explanation for this different behaviour could be the different treatment of the spin-orbit potential in both cases. In the non-relativistic case one should realize that the spin-orbit is not only expanded in powers of $\hbar$ but in addition one assumes the smallness of the coupling constant and only the lowest order term is taken into account. In reality the spin-orbit is a matrix problem as recognized by Frisk [@Frisk] and then no expansion in the coupling constant is needed. To our knowledge, the validity in the expansion of the coupling constant has never been checked. On the other hand in the relativistic case such an expansion is absent, and the coupling of the spin-orbit is treated at all the orders even in the semiclassical approach [@CVBS]. It is an open hypothesis whether this difference can explain the different behaviour in the upper and lower panels of Fig.8. The spin-orbit potential yields a surface contribution and this could point to the fact why in one case things deteriorate towards lower mass nuclei whereas in the other not. More studies on this issue are needed.
On the above grounds, it seemed interesting to us to also apply a fudge factor to the kinetic energy in the relativistic case. With the very small coefficient 1.025 we obtain the results shown in the lower panel of Fig.10. Now the shell energy of $^{40}$Ca is 5.6 MeV and the one of $^{208}$Pb is 15 MeV. Both results are compatible with previously known values [@Pom; @BQ; @BQ1]. We therefore have now at hand, at least in the relativistic case, an ad hoc procedure which yields reasonable shell energies throughout the periodic table. This very small correction needed in the relativistic case may hint to the point that there the $\hbar^4$-corrections could cure the overbinding problem with no need of a fudge factor. However, for $^{208}$Pb still $\sim 10$ MeV overbinding in the total energy occurs semiclassically whereas for an external potential the $\hbar^4$-corrections to the energy are typically $\sim 1$ MeV only [@RS].
The conclusion of our study therefore is that the semiclassical method based on the asymptotic expansion of the Wigner-Kirkwood type is in the self consistent case more fragile, i.e. inaccurate, than in the external potential case. This failure in the case of finite nuclei is in agreement with our earlier studies on the surface energies [@SV; @CVDSE; @ECV] which also turn out to be much more accurate in the external potential case than in the self consistent one. These findings are, however, sensitively more pronounced in the case of Skyrme forces than in the case of relativistic mean field theory.
Conclusions
===========
In this paper we took up again the old problem of the Thomas-Fermi approach to nuclei with incorporation of $\hbar$-corrections. As we have pointed out in earlier work [@SV], the well established ETF scheme, lacking a correct sorting out of powers in $\hbar$, may show unnecessary slow convergence properties. We therefore established a rigorous order by order $\hbar$ expansion of the self consistent nuclear mean field problem which we named variational-Wigner Kirkwood (VWK) theory. We here apply it for the first time to finite nuclei in realistic self consistent mean fields at order $\hbar^2$ (VWK2), supposing that the $\hbar^4$-corrections be very small, similarly to what is documented for the external potential case since several decades [@RS].
One essential finding of our investigation is that practically for all Skyrme forces VWK2 yields binding energies per particle very close to fourth order ETF functional theory (ETF4). This result is good and bad at the same time. The good point is that the results from ETF4 can be reproduced with the much simpler VWK2 approach and that the agreement gives further credit to the correctness of the semiclassical values. The bad side is that it is known since long [@BGH; @CPVGB; @PS; @Li; @BB; @CVBS; @CV; @Kumar; @BCKT] that ETF even at order $\hbar^4$ produces E/A values with too much binding yielding for instance for doubly magic nuclei (e.g. $^{40}$Ca or $^{208}$Pb) values which are lower in energy than the ones obtained from quantal Hartree-Fock calculations. Evidently this overbinding problem is then also present in VWK2.
We advanced several arguments in regard to the overbinding problem such as the zero range character of the Skyrme forces, leading to unphysical shapes of the self consistent Thomas-Fermi mean field potential, and/or an insufficient treatment of the spin-orbit potential. Those arguments are backed by the fact that in the relativistic RMF, with finite range meson exchange potentials, the situation is considerably better. Indeed in that case at least practically all of the doubly magic nuclei are more bound quantally than semiclassically. This could stem from the fact that there the Thomas-Fermi mean field potential is perfectly smooth resembling very much a realistic Woods-Saxon type of potential. Also the spin-orbit is treated properly. However, the shell corrections, e.g. for $^{208}$Pb, are with relativistic VWK2 (no ETF4 exists in that case) still roughly a factor of three too small. This remaining failure could have as origin that in the self consistent case, contrary to the external potential case, the mean field is itself a functional of the density and has to undergo an $\hbar$-expansion (this remark is also true in the non-relativistic case). Evidently also missing $\hbar^4$-corrections can be invoked. One has to keep in mind, however, that for heavy nuclei $\hbar^4$-corrections are typically of order 1 MeV in the external potential case, whereas even in RMF semiclassical energies are about 10 MeV overbound. Still, a slower convergence of the $\hbar$-expansion in the self consistent case is not to be excluded.
Since $\hbar^4$-terms enormously complicate the theory and the numerical treatment, we refrained from studying this here, and to gain further insight we rather investigated whether the situation can be improved by ad hoc prescriptions. We report on several possibilities, where a fudge factor of 1.025 on the kinetic energy in the relativistic case gives the most satisfying results. Indeed, we show in the lower panel of Fig.10 the corresponding shell energies for spherical nuclei as a function of mass number which we believe are quite realistic throughout. Probably, if in the non-relativistic case finite range forces were used, the situation also would improve there and an approach like in Ref.[@Swia] [*including*]{} $\hbar^2$-corrections could be undertaken. However, even the Thomas-Fermi solution with the Gogny force shows pathologies, since it still contains zero range pieces. We also should mention that for simplicity our studies were done almost exclusively for spherical nuclei.
A result on the side, obtained in the external potential case, was that for spherical nuclei the semiclassical binding energies per particle as a function of particle number do [*not*]{} pass through the average of the quantal results. Rather the semiclassical curve (see Fig.3) shows more binding than the average. This fact, though not unknown [@Leb], has not been mentioned much in the past. This natural tendency of the semiclassical results to give in the spherical case (for the deformed one, see Fig.3 and remarks towards the end of Section III) more binding than the average should, however, not be confused with the overbinding problem encountered in the self consistent semiclassical approach as discussed above and in section IV.B. This is an additional and erroneous binding contribution contained in the present semiclassical expressions which brings in the self consistent case semiclassical binding energies below the ones of doubly magic nuclei, in contradiction with results from self consistent Strutinsky calculations where this is not the case and which should be the gauge for semiclassical results.
Our studies may have relevance not only for nuclear systems but also for atomic physics calculations and for all other inhomogeneous and/or self-bound Fermi systems like $^3$He drops, trapped cold atoms, metallic clusters, quantum dots, etc., where the application of statistical Thomas-Fermi methods is particularly helpful and valuable.
Acknowledgments
===============
The authors are indebted to R.K. Bhaduri, O. Bohigas, P. Leboeuf, and W.J. Swiatecki for useful comments and informations. Especially we thank P. Leboeuf for pointing out to us that the fluctuating part of the energy in (\[eq24\]) contains a non-vanishing contribution when averaged over particle number and that this feature is worked out in Ref. [@Leb]. This work has been partially supported by the IN2P3-CICYT collaboration. Two of us (X.V. and M.C.) also acknowledge financial support from Grants No. FIS2005-03142 from MEC (Spain) and FEDER, and No. 2005SGR-00343 from Generalitat de Catalunya.
Appendix 1
==========
As is well known the WK $\hbar$-expansion of the particle density involves divergent terms at the classical turning point of the kind $\propto (\mu - V)^{-n/2}$ where $n$ is an [*odd integer and positive*]{} number, $V(\vec{r})$ the external mean field potential, and $\mu$ the chemical potential. It is well documented in the literature [@RS; @KCM] how to deal with this divergent part and to extract the non divergent contribution of the corresponding integral. Actually, we gave in Section 3 a recipe for solving this problem in the case of an external potential. Another way consists in writing $$(\mu - V)^{-n/2} \propto \bigg (\frac{\partial}{\partial \mu} \bigg)^{n'}
(\mu - V)^{-1/2} ,
\label{eqA1}$$ where $n'= (n-1)/2$ and $(\mu - V)^{-1/2}$ is an integrable divergency, and the differentation can then be done after the integration has been performed.
A case which is not so well studied is the one of $\propto (\mu -
V)^{-2n'}$, i.e. terms with integer powers of $\mu - V$ in the denominator. Such terms arise for instance when some powers of the density have to be integrated. We will show here how to extract the finite part of such integrals. In particular in the Skyrme energy computed at the WK-$\hbar^2$ level we have to find the integral of the following expression (see Eq.(\[eq34\]) of Appendix 2): $$( \nabla \rho_{\rm WK} )^2 = I_0 + I_2
\label{eqA2}$$ with $$I_0 = C^2 \bigg[ \frac{9}{4} (\mu - V) \big(\nabla V)^2 + \frac{3
\hbar^2} {16 m} \nabla\big(\Delta V\big) \nabla V \bigg]
\label{eqA3}$$ the non-diverging part and $$I_2 = C^2 \frac{3 \hbar^2}{64 m} \bigg[ \frac{\nabla V \nabla
\big(\nabla V\big)^2 + 2 \Delta V \big( \nabla V \big)^2}{\mu - V}
+ \frac{3}{2} \frac{\big(\nabla V \big)^2}{(\mu - V)^2} \bigg]
\label{eqA4}$$ the diverging one ($C$ is a constant). Without loss of generality we can assume here the spherically symmetric case and then we can write for the integral of (\[eqA4\]) $i = i_1 + i_2$ with $$i_1 = B_1 \int^{r_c}_0 dr r^2 \frac{G_1(r)}{\mu - V}
\label{eqA5}$$ $$i_2 = B_2 \int^{r_c}_0 dr r^2 \frac{G_2(r)}{(\mu - V)^2}
\label{eqA6}$$ where $B_1, B_2$ and $G_1$, $G_2$ are well defined constants and functions, respectively.
A judicious and frequently used strategy to isolate the finite contribution is to integrate by parts and disregard the diverging integrated piece. We have $$\frac{\partial}{\partial r} \frac{1}{\mu - V} = \frac{V'}{(\mu - V) ^2}
\, ; \qquad V'= \frac{\partial V}{\partial r} .
\label{eqA7}$$ Then we can write for the integral of $I_2$: $$i_1 + i_2 = \int^{r_c}_0 dr \frac{B_1 {\tilde{G}}_1 - B_2 {{\tilde{G}}_2}'}
{\mu - V} = i ,
\label{eqA8}$$ where ${\tilde{G}}_1= r^2 G_1$ and ${\tilde{G}}_2= r^2 G_2/V'$. We perform one further partial integration and write, with $x=r/r_c$, $$\frac{\partial}{\partial x} \ln (1 - V/\mu ) =
- \frac{1}{\mu - V} \,
\frac{\partial V}{\partial x} .$$ Again neglecting the integrated part one obtains $$i = r_c \int^1_0 dx \frac{\partial F}{\partial x} \, \ln (1 - V/\mu) ,
\label{eqA9}$$ where $$F(x) = \big[B_1 {\tilde{G}}_1 - B_2 {{\tilde{G}}_2}' \big]
\frac{1}{\partial V/\partial x} .
\label{eqA10}$$ For example, for a Woods-Saxon potential the integral in Eq.(\[eqA9\]) can directly be performed numerically. In the case of a harmonic oscillator potential $V=a r^2$ the integral can be performed analytically: $$i = \gamma \big[ \ln 2 - \frac{4}{3} \big],
\label{eqA11}$$ with $$\gamma = \frac{3}{16} \frac{\hbar^2}{m} A^2 a^2 r^3_c.$$
The above result is the value which is obtained looking up integral tables. We now want to check the value of $i$ with a second independent method. This can be done writing the integral $i$ at finite temperature (see Appendix 2), where it is not divergent, and evaluating it as a function of $T$. At the end an extrapolation to $T
=0$ is performed. This method is also well documented in the literature [@BGH] and will be briefly discussed in Appendix 2 for the sake of completeness. However, the extrapolation process needs some care and usually the final number will only be precise to a couple of percent. Fig.13 displays, as a function of temperature, the value of $\int (\nabla \rho)^2 d \vec{r}$ obtained with a HO potential (upper panel) for $A=40$ and a WS potential (lower panel) for $A=90$. As it is discussed in Appendix 2, the linear behaviour of this integral with $T^2$ breaks down about $T=1$ MeV (as shown by the curves that bend upwards in Fig.13), and the extrapolation to $T=0$ MeV (dashed lines of Fig.13) is needed. We find that the extrapolated values are 1.789 fm$^{-5}$ and 2.494 fm$^{-5}$ for the HO and WS potentials respectively. The values calculated with (\[eqA9\]) give 1.867 fm$^{-5}$ (HO) and 2.383 fm$^{-5}$ (WS), which correspond to relative differences of 4.2% and 7.4%, respectively. Such errors are to be expected in the extrapolation procedure.
Appendix 2
==========
In this Appendix we present the details of the method which has been used to obtain the results displayed by the semiclassical “temperature extrapolation” curve in Fig. 9 of Sect. IV.B. It is based on a calculation of the WK-HF energy including $\hbar^2$ corrections that is built on top of a previously computed smooth mean field potential. The smooth potential, including the spin-orbit ($\vec{W}$) and Coulomb ($V_{\rm Coul}$) contributions, is generated in a self consistent ETF2 calculation with the Skyrme T6 interaction. It is then used as input for the WK calculation, where it is treated as an external potential. To circumvent the divergence problems (see Appendix 1) in some $\hbar^2$ terms of the WK-HF energy functional, we compute the energy at finite temperature and take its limit (numerically) when $T \to 0$. This has been shown in the past [@VG] to be a very efficient procedure to overcome the divergences.
The expression of the WK energy in the case of a Skyrme force with $m^* = m$, including Coulomb and spin-orbit contributions, reads $$\begin{aligned}
E_{{\rm WK}} &=& \int d \vec{r} \bigg\{ \frac{\hbar^2}{2 m_n} \bigg[ \frac{3}{5}
\big(\frac{3\pi^2}{2}\big)^{2/3} \rho_{{\rm WK},n}^{5/3} + \frac{1}{36}
\frac{(\nabla
\rho_{{\rm WK},n})^2}{\rho_{{\rm WK},n}}+ \frac{1}{3} \Delta \rho_{{\rm WK},n} \bigg] \nonumber
\\ &+& \frac{\hbar^2}{2 m_p} \bigg[ \frac{3}{5}
\big(\frac{3\pi^2}{2}\big)^{2/3} \rho_{{\rm WK},p}^{5/3} +
\frac{1}{36}\frac{(\nabla \rho_{{\rm WK},p})^2}{\rho_{{\rm WK},p}}+ \frac{1}{3} \Delta
\rho_{{\rm WK},p} \bigg] \nonumber \\
&+& \frac{1}{2}t_0 \bigg[ \big( 1 + \frac{x_0}{2} \big) \rho_{{\rm WK}}^2 -
\big( x_0 + \frac{1}{2} \big) \big( \rho_{{\rm WK},n}^2 + \rho_{{\rm WK},p}^2 \big) \bigg]
\nonumber \\
&+& \frac{1}{12}t_3 \rho_{{\rm WK}}^{\alpha}
\bigg[ \big( 1 + \frac{x_3}{2} \big) \rho_{{\rm WK}}^2 -
\big( x_3 + \frac{1}{2} \big) \big( \rho_{{\rm WK},n}^2 + \rho_{{\rm WK},p}^2 \big) \bigg]
\nonumber \\
&+& \frac{1}{16} \bigg[ 3t_1 \big( 1 + \frac{x_1}{2} \big) - t_2 \big( 1 +
\frac{x_2}{2} \big) \bigg] \big( \nabla \rho_{{\rm WK}} \big)^2 \nonumber \\
&-& \frac{1}{16} \bigg[ 3t_1 \big( x_1 + \frac{1}{2} \big) - t_2 \big( x_2 +
\frac{1}{2} \big) \bigg] \bigg[ \big( \nabla \rho_{{\rm WK},n} \big)^2 +
\big( \nabla \rho_{{\rm WK},p} \big)^2 \bigg] \nonumber \\
&+& \frac{1}{2} W_0 \bigg[ \vec{J}_{{\rm WK}} \cdot {\nabla \rho_{{\rm WK}}}
+\vec{J}_{{\rm WK},n} \cdot {\nabla \rho_{{\rm WK},n}}
+ \vec{J}_{{\rm WK},p} \cdot {\nabla \rho_{{\rm WK},p}} \bigg]
+ {\cal{H}}_{\rm Coul}
\bigg\}.
\label{eq34} \end{aligned}$$ In this equation $\rho_{{\rm WK},q}$ ($q= n,p$) is the WK neutron or proton density including $\hbar^2$ corrections [@RS], and $\rho_{{\rm WK}}=\rho_{{\rm WK},n}+\rho_{{\rm WK},p}$. The neutron or proton semiclassical spin-current density up to $\hbar^2$ order is given by [@GV; @BGH] $$\vec{J}_{{\rm WK},q}= - \frac{2m_q}{\hbar^2} \rho^{0}_{{\rm WK},q}
\vec{W}_q ,
\label{eq34a}$$ where $\rho^{0}_{{\rm WK},q}$ is the TF ($\hbar^0$ part) of the WK neutron or proton densities, $\vec{W}_q$ is the (external ETF2) spin-orbit potential, and $\vec{J}_{{\rm WK}}=\vec{J}_{{\rm WK},n} + \vec{J}_{{\rm WK},p}$. The semiclassical Coulomb energy density appearing in Eq.(\[eq34\]) is computed as $${\cal{H}}_{\rm Coul} = \frac{1}{2} e^2 \rho_{{\rm WK},p}
V_{\rm Coul}
- \frac{3}{4}\bigg(\frac{3}{\pi}\bigg)^{1/3}
e^2 \rho_{{\rm WK},p}^{4/3} \,,
\label{eq34b}$$ where $V_{\rm Coul}$ is the Coulomb potential provided by the self consistent ETF2 calculation.
At a finite temperature the relevant thermodynamical potential which has to be minimized is the free energy $F$, instead of the energy $E$. The free energy, the energy, and the entropy $S$ are related through $$F = E - TS .
\label{eqB1}$$ In the WK approach $E$ is given by Eq.(\[eq34\]) and the particle and kinetic energy densities at finite temperature, for (external ETF2) nuclear $V_q$ and spin-orbit $W_q$ potentials, for each kind of particle, read as [@BGH; @VG] $$\begin{aligned}
\rho_{{\rm WK},q}^T &=& \frac{1}{2 \pi^2} \bigg( \frac{2mT}{\hbar^2}
\bigg)^{3/2}
\bigg\{ J_{1/2} (\eta_q) + \frac{\hbar^2}{48 m} \bigg[ \frac{\Delta V_q}{T^2}
J_{-3/2}(\eta_q) + \frac{3}{4} \frac{(\nabla V_q)^2}{T^3} J_{-5/2}(\eta_q)
\bigg] \nonumber \\
&+& \frac{m W_q^2}{2 \hbar^2 T}J_{-1/2}(\eta_q) \bigg\}
\label{eqB2} \end{aligned}$$ and $$\begin{aligned}
\tau_{{\rm WK},q}^T
&=& \frac{1}{2 \pi^2} \bigg( \frac{2mT}{\hbar^2} \bigg)^{5/2}
\bigg\{ J_{3/2} (\eta_q) - \frac{\hbar^2}{48 m} \bigg[5 \frac{\Delta V_q}{T^3}
J_{-1/2}(\eta_q) + \frac{9}{4} \frac{(\nabla V_q)^2}{T^3} J_{-3/2}(\eta_q)
\bigg] \nonumber \\
&+& \frac{5 m W_q^2}{2 \hbar^2 T}J_{1/2}(\eta_q) \bigg\},
\label{eqB3} \end{aligned}$$ where $J_{\nu}(\eta_q)$ are the so-called Fermi integrals $$J_{\nu}(\eta_q) = \int^{\infty}_{0} dx \, \frac{x^{\nu}}{1 +
\exp(x-\eta_q)} \label{eqB4}$$ and $\eta_q=(\mu_q - V_q)/T$ is the fugacity parameter. In particular, the free energy $F$ for a free Fermi gas moving in single-particle and spin-orbit potentials is given by $$\begin{aligned}
F_{{\rm free},q} &=& \mu_q A_q - \frac{1}{2 \pi^2} \bigg(
\frac{2mT}{\hbar^2} \bigg)^{3/2} \int
\bigg\{ d \vec{r} \frac{2}{3} T J_{3/2} (\eta_q) \nonumber \\
&-& \frac{\hbar^2}{24 m} \bigg[
\frac{\Delta V_q}{T}
J_{-1/2}(\eta_q) + \frac{1}{4} \frac{(\nabla V_q)^2}{T^2} J_{-3/2}(\eta_q)
\bigg] + \frac{m W_q^2}{\hbar^2} J_{1/2}(\eta_q) \bigg\} ,
\label{eqB5} \end{aligned}$$ where $\mu_q$ and $A_q$ are the chemical potential and the particle number of each kind of nucleon. The final expression of the total free energy contains in addition to (\[eqB5\]) the interacting potential part of the Skyrme-WK energy (\[eq34\]). The contributions of the powers of the particle density and its gradients in Eq. (\[eq34\]) are expanded in a Taylor series starting from the expression (\[eqB2\]) and only the linear terms in $\hbar^2$ are retained.
The ETF2 potentials used in Eqs.(\[eqB2\]), (\[eqB3\]) and (\[eqB5\]) are obtained at zero temperature, i.e., it is assumed that they are temperature independent. It should be pointed out that, even in a fixed external potential, the system starts to evaporate nucleons as soon as the finite temperature appears [@VG] and one should resort to e.g. a subtraction procedure [@S87] to keep the integrated magnitudes finite and independent of the size of the box in which they are calculated. However, as far as we are interested in the $T \to 0$ limit of the free energy, we can safely neglect the effects from evaporated nucleons because they are negligible below $T \simeq
2$ MeV [@BQ]. In such conditions one can consider the low-temperature expansion [@BT] of the free energy and parametrize it below $T=2$ MeV as $F(T)=E(T=0) - a(T=0) \, T^2$. However, it is to be noted that the integrals of some terms of the interacting WK free energy, namely the ones coming from $(\nabla V_q)^2$ and from the exchange Coulomb potential, which show a logarithmic divergence at zero temperature, start to depart from the linear behaviour with $T^2$ below $T=1$ MeV and bend upwards of the linear curve, similarly to what happens in Fig.13 where the particular term $\int (\nabla \rho)^2 d \vec{r}$ is plotted as a function of $T^2$. Thus, we have estimated the WK energy by extrapolating the linear region in $T^2$ between $T=2$ and $T=1$ MeV. As an example, we display in Fig.14 the results of this procedure for the nuclei $^{40}$Ca and $^{90}$Zr calculated with the Skyrme T6 force for which we find $E \simeq -329.3$ and $-766.8$ MeV, respectively.
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N $\mu$(QM) $\mu$(TF) $\mu$(WK) E(QM) E(TF) E(WK)
---- ----------- ----------- ----------- ------- --------- ---------
42 5.5 5.013 5.063 161. 157.919 161.092
44 5.5 5.092 5.141 172. 168.024 171.296
46 5.5 5.168 5.216 183. 178.284 181.653
48 5.5 5.241 5.289 194. 188.693 192.159
50 5.5 5.313 5.360 205. 199.248 202.809
52 5.5 5.383 5.430 216. 209.945 213.599
54 5.5 5.451 5.497 227. 220.780 224.526
56 5.5 5.518 5.563 238. 231.750 235.587
58 5.5 5.583 5.628 249. 242.851 246.778
60 5.5 5.646 5.690 260. 254.080 258.096
62 5.5 5.708 5.752 271. 265.434 269.539
64 5.5 5.769 5.812 282. 276.912 281.103
66 5.5 5.828 5.871 293. 288.510 292.787
68 5.5 5.887 5.929 304. 300.225 304.588
70 5.5 5.943 5.986 315. 312.056 316.504
: \[tab1\] Chemical potential $\mu$ calculated quantally and semiclassically with the TF and WK approaches \[by inversion of Eq.(\[eq20\])\] as a function of the accumulated number of fermions occupying the $n=4$ shell in a spherical HO potential. Within this shell the quantal energy is computed adding a quantum $11 \hbar \omega / 2$ per fermion to the background energy obtained filling the previous HO shells. The semiclassical energies are obtained from Eq.(\[eq21\]). Both $\mu$ and $E$ are in $\hbar
\omega$ units. Spin degeneracy of each level is assumed.
SkM$^*$ SkM$^*$ T6 T6 T6 NL3 NL3
-- ----- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -- --
A HF VWK HF VWK VWK-T H VWK
16 $-$7.081 $-$7.309 $-$7.026 $-$7.348 $-$6.934 $-$7.282 $-$6.965
40 $-$8.126 $-$8.447 $-$8.141 $-$8.428 $-$8.243 $-$8.315 $-$8.265
48 $-$8.387 $-$8.565 $-$8.332 $-$8.596 $-$8.336 $-$8.461 $-$8.463
90 $-$8.502 $-$8.659 $-$8.514 $-$8.719 $-$8.527 $-$8.603 $-$8.641
208 $-$7.779 $-$7.777 $-$7.786 $-$7.828 $-$7.701 $-$7.845 $-$7.817
: \[tab2\] Total energies per nucleon of some magic nuclei obtained with the SkM$^*$ and T6 Skyrme interactions and with the NL3 relativistic mean field parameter set in several approaches. The Coulomb and spin-orbit forces are included in the calculations. The column VWK-T refers to the empirical “temperature extrapolation” method described in Appendix 2.
![(Color online) \[figure1\] Accumulated level density (upper panel) and total energy (lower panel) with degeneracy 2 for a [*fixed*]{} spherical harmonic oscillator potential as a function of the Fermi energy $\varepsilon$. Staircase, solid, and dashed lines correspond to the quantal, semiclassical (WK with $\hbar^4$ corrections), and shell correction (quantal minus semiclassical) values, respectively. ](vwkf01_leveld_ho.eps){width="10.5cm"}
![(Color online) \[figure2\] Accumulated level density (upper panel) and total energy (lower panel) with degeneracy 2 for a [*fixed*]{} spherical Woods-Saxon potential as a function of the Fermi energy $\varepsilon$. Staircase lines correspond to the quantal values, while solid and dashed lines correspond to the semiclassical WK with $\hbar^2$ corrections and TF results, respectively.](vwkf02_leveld_ws.eps){width="9.5cm"}
![(Color online) \[figure3\] Upper panel: quantal (dots) and WK (solid line) energy per particle in a strongly triaxially deformed size-dependent harmonic oscillator potential as a function of the number of particles. Lower panel: the same as in the upper panel but for a spherical size-dependent harmonic oscillator potential (notice that the semiclassical WK curves are different in the deformed and spherical cases). The squares depict the quantal energies per particle in the case that the deformation of the harmonic oscillator potential is optimized leading to maximal binding. Notice the close agreement with the semiclassical curve obtained for the spherical harmonic oscillator.](vwkf03_eon_ho.eps){width="11.5cm"}
![(Color online) \[figure4\] WK2 (solid line) and TF (dashed line) shell energies, per particle, defined as the difference between the quantal and semiclassical energies (in $\hbar \omega$ units) filling the $n=4$ shell of a fixed spherical harmonic oscillator potential as a function of the number of the particles in the shell. The $\hbar^4$-corrections are indistinguishable from the solid VWK2-line.](vwkf04_shell_ho.eps){width="10.5cm"}
![(Color online) \[figure5\] Quantal (solid line) and WK (dashed line) values of the energy per particle of a set of 92 fermions submitted to a triaxilly deformed HO potential as a function of the deformation $d$ \[Eqs.(\[eq25\]) and (\[eq25a\])\]. Spin degeneracy is included.](vwkf05_ldho3def92.eps){width="12.0cm"}
![(Color online) \[figure6\] Upper panel: Shell correction $E_{\rm shell} = E_{\rm HF} - E_{\rm
semicl}$ in the TF, ETF4, and VWK2 approaches for symmetric uncharged nuclei without spin-orbit force as a function of the mass number $A$ calculated with the Skyrme force T6 [@T6]. Lower panel: the same for the Skyrme force SV [@Sk5].](vwkf06_shell_T6_symm.eps){width="13cm"}
![(Color online) \[figure6\] Upper panel: Shell correction $E_{\rm shell} = E_{\rm HF} - E_{\rm
semicl}$ in the TF, ETF4, and VWK2 approaches for symmetric uncharged nuclei without spin-orbit force as a function of the mass number $A$ calculated with the Skyrme force T6 [@T6]. Lower panel: the same for the Skyrme force SV [@Sk5].](vwkf06_shell_SV_symm.eps){width="13cm"}
![(Color online) \[figure11\] Energy per nucleon of $\beta$-stable spherical nuclei along the periodic table calculated with the non-relativistic Skyrme force SkM$^*$ (upper panel) and with relativistic mean field with parameter set NL3 (lower panel).](vwkf07_eoa_SKM.eps){width="13.cm"}
![(Color online) \[figure11\] Energy per nucleon of $\beta$-stable spherical nuclei along the periodic table calculated with the non-relativistic Skyrme force SkM$^*$ (upper panel) and with relativistic mean field with parameter set NL3 (lower panel).](vwkf07_eoa_NL3.eps){width="13.cm"}
![(Color online) \[figure13\] Shell correction at VWK2 level of $\beta$-stable spherical nuclei as a function of the mass number $A$ calculated with the Skyrme force SkM$^*$ (upper panel) and using relativistic mean field with parameter set NL3 (lower panel).](vwkf08_shell_SKM.eps){width="13cm"}
![(Color online) \[figure13\] Shell correction at VWK2 level of $\beta$-stable spherical nuclei as a function of the mass number $A$ calculated with the Skyrme force SkM$^*$ (upper panel) and using relativistic mean field with parameter set NL3 (lower panel).](vwkf08_shell_NL3.eps){width="13cm"}
![(Color online) \[figure15\] Shell energies per nucleon along the $\beta$-stability line computed with the Skyrme force T6 in several approaches, including the empirical temperature extrapolation method described in Appendix 2.](vwkf09_shell_T6_with_T-extrapol.eps){width="12.cm"}
![(Color online) \[figure16\] Upper panel: shell energies per nucleon along the $\beta$-stability line for spherical nuclei computed with the Skyrme force T6 using the VWK2 approach and the different empirical methods described in the text. Lower panel: same for the RMF parameter set NL3. In this case the doubly magic, non $\beta$-stable nuclei $^{100}$Sn and $^{132}$Sn have been added for more complete information.](vwkf10_shell_T6_with_T-extrapol_and_Fudge.eps){width="13cm"}
![(Color online) \[figure16\] Upper panel: shell energies per nucleon along the $\beta$-stability line for spherical nuclei computed with the Skyrme force T6 using the VWK2 approach and the different empirical methods described in the text. Lower panel: same for the RMF parameter set NL3. In this case the doubly magic, non $\beta$-stable nuclei $^{100}$Sn and $^{132}$Sn have been added for more complete information.](vwkf10_shell_NL3_with_Fudge.eps){width="13cm"}
![\[figure20\] Thomas-Fermi density (upper panel) and single-particle potential (lower panel) profiles in half infinite symmetric nuclear matter obtained using the T6 [@T6] force. The fact that the potential enters the classical turning point with a horizontal tangent cannot be seen on the scale of the graph.](vwkf11_sinm.eps){width="11.cm"}
![(Color online) \[figure21\] Neutron single-particle potential and its derivative for the nucleus $^{208}$Pb obtained with the T6 Skyrme interaction in ETF2 and with the relativistic NL3 parameter set in TF approximation.](vwkf12_Pb_pot_T6_NL3.eps){width="11.cm"}
![(Color online) \[figure23\] Extrapolation to $T=0$ of the WK value of $\int (\nabla \rho)^2 d
\vec{r}$ obtained from HO (upper panel) and WS (lower panel) potentials.](vwkf13_grad2ft.eps){width="11.cm"}
![(Color online) \[figure24\] Extrapolation to $T=0$ of the Skyrme energy ($E$) and free energy ($F$) corresponding to the T6 force obtained with the thermal WK method on top of the self consistent ETF2 potential for $^{40}$Ca (upper panel) and $^{90}$Zr (lower panel).](vwkf14_etf2.eps){width="11.cm"}
[^1]: [*E-mail address:*]{} mario@ecm.ub.es
|
---
abstract: '[The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. We prove this conjecture in the minuscule case.]{}'
address:
- |
Mathematisches Institut der Universität Bonn\
Endenicher Allee 60\
53115 Bonn, Germany\
email: rapoport@math.uni-bonn.de
- |
Institut für Experimentelle Mathematik\
Universität Duisburg-Essen, Campus Essen\
Ellernstra[ß]{}e 29\
45326 Essen, Germany\
email: ulrich.terstiege@uni-due.de
- |
Department of Mathematics\
Columbia University\
New York, NY 10027, USA\
email: wzhang@math.columbia.edu
author:
- Michael Rapoport
- Ulrich Terstiege
- Wei Zhang
title: On the Arithmetic Fundamental Lemma in the minuscule case
---
[^1]
Introduction
============
In this introduction, we first formulate (a variant of) the fundamental lemma conjecture (FL) of Jacquet-Rallis [@JR] and the arithmetic fundamental lemma conjecture (AFL) of the third author [@Z]. Then we state our main result, which is a confirmation of the second conjecture in arbitrary dimension under restrictive conditions.
Let $p$ be an odd prime. Let $F$ be a finite extension of ${{\mathbb {Q}}}_p$, with ring of integers ${{\mathcal {O}}}_F$, uniformizer $\pi$ and residue field $k$ with $q$ elements. Let $E$ be an unramified quadratic extension, with ring of integers ${{\mathcal {O}}}_E$, and residue field $k'$. We denote the non-trivial element in ${\rm Gal}(E/F)$ by $\sigma$ or by $a\mapsto \bar a$. Also, we denote by $\eta=\eta_{E/F}$ the quadratic character of $F^\times$ corresponding to $E/F$.
Let $n\geq 1$. Let $v=(0, 0,\ldots,0,1)\in F^n$. We denote by $F^{n-1}$ the subspace of vectors in $F^n$ with trivial last entry. We have a canonical inclusion $GL_{n-1}\hookrightarrow GL_n$ of algebraic groups over $F$. An element $g\in GL_n(E)$ is called regular semi-simple (with respect to the action of $GL_{n-1}(E)$ by conjugation) if both the vectors $(g^iv)_{i=0,\ldots,n-1}$ and the vectors $(^t\!vg^i)_{i=0,\ldots,n-1}$ are linearly independent. This property is equivalent to the condition that the stabilizer ${\rm Stab}_{GL_{n-1}}(g)$ is trivial, and that the orbit of $g$ under $GL_{n-1}$ is Zariski closed in $GL_n$, cf. [@RS Theorem 6.1][^2]. To $g\in GL_n(E)$ we associate the following numerical invariants: the coefficients of the characteristic polynomial ${\rm char}_g(T)\in E[T]$, and the $n-1$ elements $^t\!vg^iv\in E$ for $ i=1,\ldots,n-1$. Then two regular semi-simple elements are conjugate under an element of $GL_{n-1}(E)$ if and only if they have the same invariants, cf. [@Z].
Let $$S_n(F)=\{ s\in GL_n(E)\mid s\sigma(s)=1\} .$$ Then $GL_{n-1}(F)$ acts on $S_n(F)$, and two elements in $S_n(F)$ which are regular semi-simple (as elements of $GL_n(E)$) are conjugate under $GL_{n-1}(E)$ if and only if they are conjugate under $GL_{n-1}(F)$.
Let $J\in {\rm Herm}_{n-1}(E/F)$ be a hermitian matrix of size $n-1$. It defines a hermitian form on $E^{n-1}$. We obtain a hermitian form $J\oplus 1$ of size $n$, which corresponds to extending the hermitian form to $E^n$ by adding an orthogonal vector $u$ of length $1$. We obtain an inclusion of unitary groups $$U(J)(F)\hookrightarrow U(J\oplus 1)(F) ,$$ and therefore an action of $U(J)(F)$ on $U(J\oplus 1)(F)$ by conjugation. We consider $U(J\oplus 1)(F)$ as a subset of $GL_n(E)$ in the obvious way by sending $u$ to $v$, and $E^{n-1}$ to the subspace of vectors with trivial last entry. We call an element $g\in U(J\oplus 1)(F)$ regular semi-simple if it is regular semi-simple as an element of $GL_n(E)$. Two regular semi-simple elements $\gamma\in S_n(F)$ and $g\in U(J\oplus 1)(F)$ are said to match if they are conjugate under $GL_{n-1}(E)$ (when considered as elements of $GL_n(E)$), or, equivalently, if they have the same invariants. This property only depends on the orbits of $\gamma$ under $GL_{n-1}(F)$, resp. $g$ under $U(J)(F)$. This matching condition defines a bijection between orbit spaces [@Z], Lemma 2.3, $$\big[U(J_0\oplus 1)(F)_{\rm rs}\big] \sqcup \big[U(J_1\oplus 1)(F)_{\rm rs}\big]\simeq \big[S_n(F)_{\rm rs}\big] .$$ Here $J_0$ denotes the split hermitian form, and $J_1$ the non-split hermitian form, i.e., the discriminant of $J_0$ has even valuation, and the discriminant of $J_1$ has odd valuation.
For $\gamma\in S_n(F)_{\rm rs}$, and $f\in C^\infty_c(S_n(F))$, consider the weighted orbital integral $$O(\gamma, f)=\int_{GL_{n-1}(F)} f(h^{-1}\gamma h)\eta({\rm det}\,h)dh ,$$ where we normalize the measure so that $GL_{n-1}({{\mathcal {O}}}_F)$ has measure $1$. Similarly, for any $g\in U(J_0\oplus 1)(F)_{\rm rs}$, and $f\in C^\infty_c(U(J_0\oplus 1)(F))$, we form the orbital integral $$O(g, f)=\int_{U(J_0)(F)} f(h^{-1}g h)dh ,$$ where we normalize the measure so that the stabilizer $K'$ of a self-dual lattice $\Lambda'$ in $E^{n-1}$ has measure $1$. Let $K$ be the stabilizer of the self-dual lattice $\Lambda=\Lambda'\oplus {{\mathcal {O}}}_Eu$.
The FL is now the following statement (for the “Lie algebra” version see [@JR]).
For $\gamma\in S_n(F)_{\rm rs}$, $$\begin{aligned} O(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=\begin{cases} {\omega(\gamma)O(g, 1_{K}) \text{\rm \, if $\gamma$ matches
$g\in U(J_0\oplus 1)(F)_{\rm rs}$ }} ,\\
\\
{0 \quad\quad\quad\quad\quad\text{\rm \, if $\gamma$ matches no $g\in U(J_0\oplus 1)(F)_{\rm rs}$ . }}
\end{cases}
\end{aligned}$$
Here the sign $\omega(\gamma)$ is given by $$\omega(\gamma)=(-1)^{v({\rm det}(\gamma^iv)_{i=0,\ldots, n-1})} .$$ Both orbital integrals appearing in the conjecture count certain ${{\mathcal {O}}}_E$-lattices in $E^n$. Let $L=L_g$ be the lattice generated by the vectors $u, gu,\ldots g^{n-1}u$, where we recall the vector $u$ of length one from above. Then the first clause of the above identity can be written as $$\omega(\gamma)\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} =\omega(\gamma)\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda, \Lambda^*=\Lambda\}}1.$$ Here $\tau$ is the antilinear involution on $E^n$, depending on $g$, which sends $g^iu$ to $g^{-i}u$ for $i=0,\ldots, n-1$. Also, for any lattice $\Lambda$, we denote by $\Lambda^*$ the lattice of elements of $E^n$ which pair integrally with all elements of $\Lambda$ ([*dual lattice*]{}).
The equal characteristic analogue of FL was proved by Z. Yun, for $p>n$; J. Gordon deduced FL in the $p$-adic case, for $p$ large enough (but unspecified), cf. [@GY].
Now we come to the AFL conjecture. For $\gamma\in S_n(F)_{\rm rs}$, and $f\in C^\infty_c(S_n(F))$, and $s\in {{\mathbb {C}}}$, let $$O(\gamma, f, s)=\int_{GL_{n-1}(F)} f(h^{-1}\gamma h)\eta({\rm det}\, h)|{\rm det} h|^s dh ,$$ and introduce $$O'(\gamma, 1_{S_n({{\mathcal {O}}}_F)})= \frac{d}{ds}O(\gamma, 1_{S_n({{\mathcal {O}}}_F)},s)_{\big| s=0} .$$ Then the conjecture is as follows.
\[AFL\] For $\gamma\in S_n(F)_{\rm rs}$ which matches $g\in U(J_1\oplus 1)(F)_{\rm rs}$, $$O'(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=-\omega(\gamma)\big\langle\Delta({{\mathcal {N}}}_{n-1}), ({\rm id}\times g)\Delta ({{\mathcal {N}}}_{n-1}) \big\rangle .$$
On the RHS appears the arithmetic intersection product of two formal subschemes inside the formal scheme ${{\mathcal {N}}}_{n-1}\times_{{\rm Spf}\, {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}_n$. Here ${{\mathcal {N}}}_n$ denotes the moduli space over the ring of integers in the completion of the maximal unramified extension $\breve F$ of $F$ of formal ${{\mathcal {O}}}_F$-modules of height $n$ with ${{\mathcal {O}}}_E$-action of signature $(1, n-1)$ and with principal polarization compatible with the involution $\sigma$ on ${{\mathcal {O}}}_E$. (${{\mathcal {N}}}_n$ is a special case of an [*RZ-space*]{} [@RZ].) Similarly for ${{\mathcal {N}}}_{n-1}$, which is naturally embedded in ${{\mathcal {N}}}_n$. The element $g\in U(J_1\oplus 1)(F)$ acts on ${{\mathcal {N}}}_n$ in a natural way. Then $\Delta({{\mathcal {N}}}_{n-1})$ and $ ({\rm id}\times g)\Delta ({{\mathcal {N}}}_{n-1}) $ are two formal schemes of formal dimension $n-1$, contained in the formal scheme ${{\mathcal {N}}}_{n-1}\times_{{\rm Spf}\, {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}_n$ of formal dimension $2(n-1)$, i.e., we are in a situation of middle dimension intersection. We refer to [@Z] for the precise definition of ${{\mathcal {N}}}_n$, and for the definition of the intersection product, and the proof of the fact that the RHS is a finite quantity (cf. also §§2–4 below).
In the following, we fix $n\geq 2$ and denote ${{\mathcal {N}}}_n$ simply by ${{\mathcal {N}}}$, and ${{\mathcal {N}}}_{n-1}$ by ${{\mathcal {M}}}$.
As before, the LHS can be expressed in a combinatorial way, as $$\omega(\gamma){\rm log}\, q\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} \ell(\Lambda/L) .$$ In the case that the intersection of the formal schemes $\Delta({{\mathcal {M}}})$ and $({\rm id}\times g)\Delta ({{\mathcal {M}}})$ is proper, i.e., is a set of isolated points, and using the Bruhat-Tits stratification of ${{\mathcal {N}}}_{\rm red}$ [@VW], the RHS can also be written as a sum over lattices, as $$-\omega(\gamma){\rm log}\, q \sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda, \pi\Lambda\subset\Lambda^*\subset\Lambda\}}{\rm mult}(\Lambda) .$$ Here the number ${\rm mult}(\Lambda)$ is the intersection multiplicity of $\Delta({{\mathcal {M}}})$ and $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ [*along the stratum ${{\mathcal {V}}}(\Lambda)^o$*]{}.
We note that this conjecture holds true for $n=2, n=3$, by results of the third author [@Z]. In these cases the intersection appearing above is automatically proper.
We now come to the description of the results of this paper, which are valid for any $n$ but with strong restrictions on $g$. Let $$({{\mathbb {Z}}}^n)_+=\{(r_1,\ldots,r_n)\in {{\mathbb {Z}}}^n\mid r_1\geq\ldots\geq r_n\}.$$ Let ${\rm inv}(g)=(r_1,\ldots,r_n)\in ({{\mathbb {Z}}}^n)_+$ be the unique element such that $L_g^*$ has a basis $e_1,\ldots, e_n$ such that $\pi^{r_1}e_1,\ldots, \pi^{r_n}e_n$ is a basis of $L_g$. Note that $r_n=0$, and that $\sum_i r_i$ is odd. It turns out that the ‘bigger’ ${\rm inv}(g)$ is, the more difficult it is to prove the identity in AFL. From this point of view we treat here the simplest non-trivial case.
\[mainthm\] Let $g\in U(J_1\oplus 1)(F)_{\rm rs}$.
\(i) The underlying reduced scheme of the intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ has a stratification by Deligne-Lusztig varieties (we refer to §\[fpinstratum\] for the precise description of which Deligne-Lusztig varieties can occur).
[From now on assume that ${\rm inv}(g)$ is minuscule, i.e., that ${\rm inv}(g)=(1^{(m)}, 0^{(n-m)})$, for some $m\geq 1$. Then:]{}
\(ii) The intersection of $\Delta({{\mathcal {M}}})$ and $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ is proper. Furthermore, the arithmetic intersection product $\big\langle\Delta({{\mathcal {M}}}), ({\rm id}\times g)\Delta ({{\mathcal {M}}}) \big\rangle$ is equal to $${\rm log}\, q \sum_{x\in (\Delta({{\mathcal {M}}})\cap({\rm id}\times g)\Delta ({{\mathcal {M}}}))(\bar k)} \ell\big({{{\mathcal {O}}}_{\Delta({{\mathcal {M}}})\cap ({\rm id}\times g)\Delta({{\mathcal {M}}}), x}}\big) ,$$ i.e., there are no higher Tor-terms.
\(iii) The intersection of $\Delta({{\mathcal {M}}})$ and $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ is concentrated in the special fiber, i.e., the uniformizer $\pi$ annihilates its structure sheaf.
\(iv) The AFL identity holds, provided that $n\leq 2p-2$. Furthermore, in this case the lengths of the local rings appearing in (ii) are all identical.
Assertion (i) is proved in section 6, and (ii) follows from Propositions \[scheme\] and \[fixpointstr\]. Assertion (iii) follows from Theorem \[speciald\], and assertion (iv) follows from Propositions \[cardform\], \[derivform\], and \[length\]. That the lengths of all local rings are identical follows from our explicit determination of these lengths, although we think that there should be an [*a priori*]{} proof, without the restriction on $p$.
In fact, we will prove the assertions above only in the case when $F={{\mathbb {Q}}}_p$, because in this case we can refer to [@V] and [@VW] for the structure of ${{\mathcal {N}}}_n$, and also to [@KR2] for some global results. However, there is no doubt that the results should generalize to arbitrary $p$-adic fields.
There is a fundamental difference between the seemingly very similar combinatorial descriptions of both sides in the FL and in the AFL. Whereas in the FL there is a rather simple criterion to decide when both sides of the identity are non-zero, the corresponding question for the AFL seems very subtle in general. However, in the case of a minuscule element $g$, we give a simple criterion in terms of the induced automorphism of the $k'$-vector space $L_g^*/L_g$ to decide when the two sides of the AFL identity are non-zero, cf. §\[theminusculecase\].
There is some relation between the AFL problem and the problem of intersecting [*special divisors*]{} considered in [@KR]. Indeed, the intersection $\Delta({{\mathcal {M}}})\cap({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ is contained in the intersection of the special divisors (in the sense of [@KR]) ${{\mathcal {Z}}}(g^iu)$, for $i=0,\ldots, n-1$. Then point (iii) of Theorem \[mainthm\] is a consequence of the following theorem, which is of independent interest.
\[secondmain\] Consider the intersection of special divisors ${{\mathcal {Z}}}(x_1),\ldots{{\mathcal {Z}}}(x_n)$ on ${{\mathcal {N}}}_n$, where the fundamental matrix (in the sense of [@KR]) is equivalent to the diagonal matrix ${\rm diag}(\pi^{(m)}, 1^{(n-m)})$. Then this intersection is concentrated in the special fiber, and is in fact equal to a closed Bruhat-Tits stratum of type $m$ of $({{\mathcal {N}}}_{n})_ {\rm red}$.
Again, we prove this only in the case $F={{\mathbb {Q}}}_p$. We view this theorem as a confirmation of the following conjecture in a special case.
Consider the intersection of special divisors ${{\mathcal {Z}}}(x_1),\ldots,{{\mathcal {Z}}}(x_n)$ on ${{\mathcal {N}}}_n$, where the fundamental matrix is equivalent to the diagonal matrix ${\rm diag}(\pi^{r_1}, \pi^{r_2}, \ldots,\pi^{r_n})$ with $r_1\geq r_2\geq\ldots\geq r_n$. Then $\pi^{r_1}$ annihilates the structure sheaf of ${{\mathcal {Z}}}({x_1})\cap{{\mathcal {Z}}}({x_2})\ldots\cap{{\mathcal {Z}}}({x_n})$.
The lay-out of the paper is as follows. In sections 2 and 3 we recall some facts about the formal moduli spaces ${{\mathcal {N}}}_n$ and the geometry of their underlying schemes. In section 4 we explain the intersection product appearing on the RHS of Conjecture \[AFL\]. In sections 5 and 6 we address the problem of determining the underlying point set of the intersection. More precisely, we write in section 5 this intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ as a disjoint union over Bruhat-Tits strata of certain fixed point sets in each stratum. The determination of the individual fixed point sets then becomes a problem in Deligne-Lusztig theory that is discussed in section 6. In particular, we give a criterion for when this fixed point set is finite. In section 7 we explain the statements of the FL and the AFL, and show that these conjectures can be interpreted as elementary counting expressions of lattices, as mentioned above. In the rest of the paper we concentrate on the minuscule case. In section 8 we determine the cardinality of the intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ and calculate the LHS of Conjecture \[AFL\], which turn out to be amusing combinatorial exercises. In section 9 we reduce the calculation of the length of the local ring at each point of this intersection to Theorem \[speciald\], alias Theorem \[secondmain\] above, and Theorem \[idealk\]. These theorems are then proved in sections 10 and 11. Here the main tool is Zink’s theory of displays of formal groups.
We conclude this introduction with a few remarks and questions. One remark is that we find it striking that the intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ may be a discrete point set, but not consist entirely of [*superspecial*]{} points. This is in contrast to what occurs, e.g., in [@KR], or [@Z]. A question that seems very interesting to us is to clarify the relationship between the regular semi-simplicity of an element $g\in U(J_1\oplus 1)(F)$ and the finiteness of the intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$: it is easy to see that there are regular semi-simple elements $g$ such that the length of this intersection is not finite. It would be very interesting to characterize those regular semi-simple elements with corresponding proper intersection, in analogy with the corresponding characterization in [@KR] of the cases when the intersection of special divisors is finite. For instance, if $g$ is not regular semi-simple, is the intersection $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ of infinite length?
We thank J.-B. Bost, B. Gross, X. He, S. Kudla, G. Lusztig, P. Scholze, J.-L. Waldspurger, X. Zhu, and Th. Zink for helpful discussions. We also thank the referee for his/her careful reading of the paper.
Parts of this work were done during research stays of the first author at Harvard University, of the first two authors at the Erwin Schrödinger Institute in Vienna, and of the third author at the Morningside Center of Mathematics and MSC of Tsinghua University at Beijing. We thank these institutions for their hospitality.
[**Notation**]{} Throughout the paper, we make the blanket assumption that $p$ is odd. We fix an algebraic closure $\bar k={{\mathbb {F}}}$ of $k$, and denote by $\sigma$ the relative Frobenius in ${\rm Gal}({{\mathbb {F}}}/k)$. We identify $k'$ with the quadratic extension of $k$ in ${{\mathbb {F}}}$.
The set up {#setup}
==========
Fix $n\geq 1$. In the following, ${{\mathcal {N}}}_n$ is the formal moduli space of $p$-divisible groups of unitary type of signature $(1,n-1)$, that parametrizes tuples $(X, \iota, \lambda, \rho)$, where the quasi-isogeny $\rho$ is of height zero, cf. [@KR]. Here is what we mean.
Let $\breve F$ be the completion of the maximal unramified extension of $F$, with ring of integers ${{\mathcal {O}}}_{\breve F}$ and residue field $\bar k$. We denote by ${\rm Nilp}={\rm Nilp}_{{{\mathcal {O}}}_{\breve F}}$ the category of ${{\mathcal {O}}}_{\breve F}$-schemes such that locally $\pi$ is a nilpotent element in the structure sheaf. We consider triples $(X, \iota, \lambda)$ where $X$ is a formal ${{\mathcal {O}}}_F$-module of height $2n$, and $\iota: {{\mathcal {O}}}_E\longrightarrow {\rm End}(X)$ is an action of ${{\mathcal {O}}}_E$ on $X$ with Kottwitz condition of signature $(1, n-1)$, and where $\lambda$ is a principal polarization whose associated Rosati involution induces the automorphism $\sigma$ on ${{\mathcal {O}}}_E$. There is a unique such triple $({{\mathbb {X}}}, \iota, \lambda)$ over $\bar k$ such that ${{\mathbb {X}}}$ is supersingular, up to ${{\mathcal {O}}}_E$-linear isogeny preserving the polarizations up to a scalar. We also write ${{\mathbb {X}}}_n$ when we want to stress the dependence on $n$. Then ${{\mathcal {N}}}_n$ represents the functor which to $S\in {\rm Nilp}$ associates the set of isomorphism classes of quadruples $(X, \iota, \lambda, \rho)$, where $(X, \iota, \lambda)$ is a triple as above over $S$, and where $\rho$ is an ${{\mathcal {O}}}_E$-linear quasi-isogeny $\rho: X\times_S\bar S\to{{\mathbb {X}}}_n\times_{{\rm Spec}\, \bar k} \bar S$ of height zero, which carries the polarization on ${{\mathbb {X}}}_n$ into one which differs locally by an element in ${{\mathcal {O}}}_F^\times$ from $\lambda\times_S\bar S$. Here $\bar S=S\times_{{\rm Spec}\, {{\mathcal {O}}}_{\breve F}} {\rm Spec}\, \bar k$.
The functor ${{\mathcal {N}}}_1$ is representable by ${\rm Spf}\, {{\mathcal {O}}}_{\breve F}$, with universal object $(Y, \iota_0, \lambda_0)$. We denote by $(\overline{Y}, \bar \iota_0, \bar \lambda_0)$ the same formal ${{\mathcal {O}}}_F$-module as $Y$, but where $\bar \iota_0$ is obtained from $\iota_0 $ by pre-composing with $\sigma$. We may (and will) assume that for the framing objects for ${{\mathcal {N}}}_{n-1}$, resp. ${{\mathcal {N}}}_n$ we have the relation $${{\mathbb {X}}}_n={{\mathbb {X}}}_{n-1}\times (\overline{Y}\times_{{\rm Spec}\, {{\mathcal {O}}}_{\breve F}} {\rm Spec}\, \bar k) .$$
For fixed $n\geq 2$, we abbreviate ${{\mathcal {N}}}_{n-1}$ into ${{\mathcal {M}}}$ and ${{\mathcal {N}}}_{n}$ into ${{\mathcal {N}}}$. We define the embedding $$\label{delta}
\delta: {{\mathcal {M}}}\hookrightarrow{{\mathcal {N}}},$$ via $$\delta\big((X, \iota, \lambda, \rho)\big)= (X\times \overline{Y}, \iota\times \bar \iota_0, \lambda\times \bar\lambda_0, \rho\times {\rm id}).$$
Let $$G=G_n=\{g\in {\rm End}_{E}^0({{\mathbb {X}}}_n)\mid \quad gg^\dag=1\}.$$ Here $\dag$ is the Rosati involution induced by $\lambda$. Then $G$ acts on ${{\mathcal {N}}}_n$, by changing $\rho$ into $g\circ \rho$.
The Bruhat-Tits stratification
==============================
We recall some basic structure of the reduced part of the formal schemes of the last section, especially the Bruhat-Tits stratification, comp. [@KR; @V; @VW]. This applies both to ${{\mathcal {M}}}={{\mathcal {N}}}_{n-1}$ and to ${{\mathcal {N}}}={{\mathcal {N}}}_n$. Let us explain the case of ${{\mathcal {M}}}$.
We identify $E$ with the invariants of $\sigma^2$ in $\breve{F}$. Let $C_{n-1}$ be the hermitian space of dimension $n-1$ with hermitian form isomorphic to ${\rm diag}(1, \ldots ,1, p)$ (this differs by the factor $p$ from the form in [@V]). Recall the concept of a [*vertex lattice*]{} in $C_{n-1}$: this is a lattice $\Lambda$ with $\pi\Lambda\subset\Lambda^*\subset\Lambda$, cf. [@KR]. Here, as elsewhere in the paper, $\Lambda^*$ denotes the dual lattice, consisting of elements in the ambient vector space which pair integrally with all elements of $\Lambda$. The [*type*]{} of $\Lambda$ is the dimension of the $k'$-vector space $\Lambda/\Lambda^*$.
We denote by $\tau$ the automorphism ${\rm id}\otimes\sigma^2$ of $C_{n-1}\otimes_E {\breve F}$. We extend the hermitian form on $C_{n-1}$ to a sesqui-linear form on $C_{n-1}\otimes_E {\breve F}$ by $$(x\otimes c, y\otimes c')=c\sigma(c')\cdot (x, y),\ x, y\in C_{n-1};\ c, c'\in {\breve F} .$$ The set ${{\mathcal {M}}}({{\mathbb {F}}})$ can be identified with the set of lattices $A\subset
C_{n-1}\otimes_{E} {\breve F}$ such that $$A^\ast\subset^1 A\subset \pi^{-1}A^\ast ,$$ where the notation “$\subset^1$” means that the quotient $A/A^\ast $ is a $\bar k$-vector space of dimension $1$.
Recall [@KR] that to a lattice $A\in {{\mathcal {M}}}({{\mathbb {F}}})$, there is associated a vertex lattice $\Lambda=\Lambda (A)$ in $C_{n-1}$ via the following rule: $$\Lambda(A)\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}=\sum_{0}^{d}\tau^i A \text { is } \tau \text{-stable, for some } d.$$ Then $\Lambda(A)\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}$ is the smallest $\tau$-invariant lattice containing $A$. Dually, $\Lambda(A)^*\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}$ is the largest $\tau$-invariant lattice contained in $A^*$. The [*type*]{} of $A$ is the integer $t= t(A)=2d+1$, where $d$ is minimal. Equivalently, it is the type $t(\Lambda)$ of $\Lambda=\Lambda(A)$.
For a given vertex lattice $\Lambda$, the lattices $A$ with $\Lambda=\Lambda(A)$ form the open Bruhat-Tits stratum ${{\mathcal {V}}}_\Lambda({{\mathbb {F}}})^o$ associated to $\Lambda$. The closed Bruhat-Tits stratum associated to $\Lambda$ is given by $${{\mathcal {V}}}_\Lambda({{\mathbb {F}}})=\{ A\in\mathcal M({{\mathbb {F}}})\mid A\subset\Lambda\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}\} .$$ It turns out that these strata are in fact the ${{\mathbb {F}}}$-points of algebraic subvarieties of ${{\mathcal {M}}}_{\rm red}$, cf. [@VW]. More precisely, for any vertex $\Lambda$, ${{\mathcal {V}}}_\Lambda({{\mathbb {F}}}) $ is the set of ${{\mathbb {F}}}$-points of a closed irreducible subvariety ${{\mathcal {V}}}_\Lambda$ of ${{\mathcal {M}}}_{\rm red}$ which is smooth of dimension $\frac{1}{2}(t(\Lambda)-1)$, the inclusions $ {{\mathcal {V}}}_{\Lambda'}({{\mathbb {F}}})\subset {{\mathcal {V}}}_\Lambda({{\mathbb {F}}})$ for $\Lambda'\subset \Lambda$ are induced by closed embeddings of algebraic varieties over ${{\mathbb {F}}}$, and the open stratum ${{\mathcal {V}}}_\Lambda({{\mathbb {F}}})^o$ is the set of ${{\mathbb {F}}}$-points of the open subvariety of ${{\mathcal {V}}}_\Lambda$ obtained by removing all ${{\mathcal {V}}}_{\Lambda'}$ for $\Lambda'\subsetneq \Lambda$.
Let $C_{n}=C_{n-1}\oplus E u$ with $(u,u)=1$. We again extend the pairing to a sesqui-linear pairing on $C_n\otimes_E {\breve F}$. Then the preceding explanations apply to ${{\mathcal {N}}}$ instead of ${{\mathcal {M}}}$, and in particular $${{\mathcal {N}}}({{\mathbb {F}}})=\{B \mid B \text{ a lattice in } C_n\otimes_E{\breve F} \text{ with } B^\ast\subset^1 B\subset \pi^{-1}B^\ast\} ,$$ and again there is a Bruhat-Tits stratification, this time parametrized by vertex lattices in $C_n$. The relation between ${{\mathcal {M}}}$ and ${{\mathcal {N}}}$ is given by the following lemma.
\[deltaonpoints\] The injection $\delta: \mathcal M({{\mathbb {F}}})\to \mathcal N({{\mathbb {F}}})$ induced on ${{\mathbb {F}}}$-points by is given by $A\mapsto B=A\oplus {{\mathcal {O}}}_{\breve F}u$. Furthermore, $$\Lambda(B)=\Lambda(A)\oplus \mathcal O_E u .$$ In particular, the types of $B$ and $A$ are the same.
The first assertions follow easily from the identification of ${{\mathcal {N}}}(\bar k)$ in terms of lattices. The last assertion is obvious since $u$ is a unimodular vector.
The morphism $\delta$ is compatible with the Bruhat-Tits stratifications of ${{\mathcal {M}}}_{\rm red}$, resp. ${{\mathcal {N}}}_{\rm red}$, in the sense that the stratum of ${{\mathcal {M}}}_{\rm red}$ corresponding to the vertex lattice $\Lambda$ in $C_{n-1}$ is mapped to the stratum of ${{\mathcal {N}}}_{\rm red}$ corresponding to the vertex lattice $\Lambda\oplus \mathcal O_E u$ in $C_n$.
An intersection problem
=======================
The morphism $\delta$ induces a closed embedding of formal schemes, $$\Delta: {{\mathcal {M}}}\to {{\mathcal {M}}}\times_{{\rm Spf}\, {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}},$$ with components ${\rm id}_{{\mathcal {M}}}$ and $\delta$. Let $g\in G$. Then $g$ induces an automorphism $g: {{\mathcal {N}}}\to{{\mathcal {N}}}$. We denote by ${{\mathcal {N}}}^g$ the fixed point locus, defined to be the intersection in ${{\mathcal {N}}}\times_{{\rm Spf} {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}$, $${{\mathcal {N}}}^g=\Delta_{{\mathcal {N}}}\cap\Gamma_g .$$ Here $\Delta_{{\mathcal {N}}}\subset {{\mathcal {N}}}\times_{{\rm Spf} {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}$ is the diagonal of ${{\mathcal {N}}}$, and $\Gamma_g$ is the graph of $g$.
An element $g\in G$ is called regular semi-simple, if the matrix in ${ M} _n(E)$ $$(g^iu, g^ku), i=0,\ldots, n-1; k=0,\ldots n-1\ ,$$ is non-singular. Equivalently, the vectors $g^iu, i=0,\ldots,n-1, $ form a basis of $C_n$.
Here we have identified the group $G=G_n$ with the unitary group of $C_n$ as explained in §2.2 of [@Z].
This definition coincides with the definition of regular semi-simplicity in the introduction. Indeed, we may identify $C_n$ with the hermitian space $(E^n, J)$ for $J=J_1\oplus 1$ such that $u$ is mapped to $v=(0,0,\ldots,0,1)$. If the vectors $g^iv, i=0,\ldots,n-1, $ form a basis of $C_n$, then so do the vectors $^tg^iv=J^{-1}\bar g^i Ju, i=0,\ldots,n-1$ since $Ju=u$. Hence $^tvg^{i}, i=0,\ldots,n-1,$ also form a basis of $E^n$.
\[scheme\] (i) There is an equality of formal schemes over ${\rm Spf}\, {{\mathcal {O}}}_{\breve F}$ $$\delta({{\mathcal {M}}})\cap{{\mathcal {N}}}^g=\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}}) .$$
\(ii) If $g$ is regular semi-simple, then this formal scheme is a scheme (i.e., any ideal of definition is nilpotent) with underlying reduced subscheme proper over ${\rm Spec}\, {{\mathbb {F}}}$.
\(iii) If $g$ is regular semi-simple and $\big(\delta({{\mathcal {M}}})\cap{{\mathcal {N}}}^g\big)({{\mathbb {F}}})$ is finite, then $${{\mathcal {O}}}_{\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})}={{\mathcal {O}}}_{\Delta({{\mathcal {M}}})}\otimes^{{\mathbb {L}}}{{\mathcal {O}}}_{({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})} ,$$ i.e., the sheaf on the LHS represents the object on the RHS in the derived category.
The first assertion follows by checking the equality on $S$-valued points, for $S\in {\rm Nilp}$, where it is a tautology. For the second assertion, we refer to [@Z], Lemma 2.8.
For the third assertion, first note that if the intersection has a finite number of points, it is an artinian scheme. Now both $\Delta({{\mathcal {M}}})$ and ${{\mathcal {M}}}\times_{{\rm Spf}\, {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}$ are regular formal schemes of dimension $n-1$, resp. $2(n-1)$, and therefore locally $\Delta({{\mathcal {M}}})$ is the intersection of $n-1$ regular divisors in ${{\mathcal {M}}}\times_{{\rm Spf}\, {{\mathcal {O}}}_{\breve F}}{{\mathcal {N}}}$. The same applies to $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$. Hence, if $\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ is discrete, the intersection of the $2(n-1)$ regular divisors is proper. Hence there are no higher Tor-terms, and the assertion follows, comp. [@T].
The hypothesis $F={{\mathbb {Q}}}_p$ for (ii) and (iii) is made because the proof of [@Z], Lemma 2.8 makes use of global methods. In fact, the proof uses a globalization of the special divisors ${{\mathcal {Z}}}(x)$ of [@KR]. The assertions should be true for arbitrary $F$.
It follows from (ii) that the Euler-Poincaré characteristic of ${{\mathcal {O}}}_{\Delta({{\mathcal {M}}})}\otimes^{{\mathbb {L}}}{{\mathcal {O}}}_{({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})}$ is finite. The [*arithmetic intersection number*]{} is defined to be $$\label{defofarint}
\big\langle\Delta({{\mathcal {M}}}), ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})\big\rangle=\chi( {{\mathcal {O}}}_{\Delta({{\mathcal {M}}})}\otimes^{{\mathbb {L}}}{{\mathcal {O}}}_{({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})}\big) {\rm log}\, q .$$
In case the intersection $\delta({{\mathcal {M}}})\cap{{\mathcal {N}}}^g$ is discrete, it follows that locally at a point of intersection the fixed point locus ${{\mathcal {N}}}^g$ is purely formally one-dimensional: indeed, in this case the formal scheme ${{\mathcal {N}}}^g$ intersects properly the formal divisor ${{\mathcal {M}}}$ of ${{\mathcal {N}}}$.
In case the intersection $\delta({{\mathcal {M}}})\cap{{\mathcal {N}}}^g$ is discrete, its underlying set is stratified by the Bruhat-Tits stratification of ${{\mathcal {M}}}$. We define in this case for a vertex lattice $\Lambda$ in $C_{n-1}$ $$\label{defmult}
{\rm mult}(\Lambda)=\sum_{x\in{{\mathcal {V}}}(\Lambda)^o({{\mathbb {F}}})} \ell\big({{{\mathcal {O}}}_{\Delta({{\mathcal {M}}})\cap ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}}), x}}\big) .$$ The total arithmetic intersection number is in this case given by a finite sum $$\big\langle\Delta({{\mathcal {M}}}), ({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})\big\rangle={\rm log}\, q \sum_{\Lambda} {\rm mult}(\Lambda) .$$
Our next task will be to analyze which vertex lattices $\Lambda$ contribute effectively to this sum, and to understand the set of points in ${{\mathcal {V}}}(\Lambda)^o\cap{{\mathcal {N}}}^g$.
Description of $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)^{}({{\mathbb {F}}})$
======================================================================================
Let $g\in G$. It is clear that $${{\mathcal {N}}}^g({{\mathbb {F}}})=\{B\in {{\mathcal {N}}}({{\mathbb {F}}})\mid g(B)\subset B\}=\{B\in {{\mathcal {N}}}({{\mathbb {F}}})\mid g(B)=B\}.$$
If $B\in {{\mathcal {V}}}_\Lambda({{\mathbb {F}}})^o$ is stable under $g$, then $g(\Lambda)=\Lambda$. In particular, if ${{\mathcal {N}}}^g({{\mathbb {F}}})\neq\emptyset$, then the characteristic polynomial ${\rm char}_g(T)$ of $g$ has integral coefficients, i.e., ${\rm char}_g(T)\in {{\mathcal {O}}}_E[T]$.
Obvious, since $g$ and $\tau$ commute.
To a regular semi-simple element $g\in G$ we associate the ${{\mathcal {O}}}_E$-lattice $L_g$ in $C_n$ generated by $g^iu$ (an ${{\mathcal {O}}}_E$-lattice, since $u,
g u,\ldots, g^{n-1}u$ form a basis of $C_n$).
\[squeeze\] Let $g\in G$ be regular semisimple. Then for any $B=A\oplus {{\mathcal {O}}}_{\breve F}u$ stable under $g$, we have $$L_g \subset
\Lambda(B)^\ast\subset \Lambda(B)\subset L_g^\ast.$$ In particular, the invariants $(g^iu,g^{k}u)$ take values in $\mathcal O_E$.
Conversely, if $B\in {{\mathcal {N}}}({{\mathbb {F}}})$ contains $L_{g}$, then $B$ is of the form $B=A\oplus {{\mathcal {O}}}_{\breve F}u$, for a unique $A\in {{\mathcal {M}}}({{\mathbb {F}}})$.
Recall that $\Lambda(B)^{\ast}$ is the largest $\tau$-invariant lattice contained in $B^\ast$. Since $B$ is of the form $B=A\oplus {{\mathcal {O}}}_{\breve F}u$, it follows that $u\in B^*$. Since $gB=B$, we also have $gB^*=B^*$. Hence $g^iu\in B^*$ for all $i\geq 0$. Hence $L_g\subset B^*$, and therefore $L_g\subset \Lambda(B)^*$ by the maximality of $\Lambda(B)^*$. The other inclusion is obtained by taking duals.
For the converse, note that the inclusion $L_g\subset B^*$ implies that $u\in B^*$. Since $u$ is a unimodular vector, setting $A=u^\perp$, we obtain $B=A\oplus {{\mathcal {O}}}_{\breve F}u$.
By definition, $L=L_g$ is a $g$-cyclic lattice, i.e., there exists $v\in L$ such that $L$ is generated over ${{\mathcal {O}}}_E$ by $\{ g^iv\mid 0\leq i\leq n-1\}$.
\[cycl\] Let $g\in G$ and let $L$ be a $g$-cyclic lattice with $gL=L$. Then $L^*$ is also a $g$-cyclic lattice with $gL^*=L^*$.
Let $L$ be generated by $v_i=g^iv$, where $i=0,\ldots,n-1$. Then the $v_i$ form a basis of $C_n$. Let $v'_i$ be the dual basis, i.e., $$(v_i, v'_j)=\delta_{ij} .$$ Then $L^*$ is the ${{\mathcal {O}}}_E$-span of $\{ v'_i\mid i=0,\ldots,n-1\}$. Let $v'=v'_0$. We claim that $v'$ is a cyclic generator of $L^*$, i.e., that the elements $w_i=g^iv'$ for $i=0,\ldots, n-1$ generate $L^*$ as an ${{\mathcal {O}}}_E$-module. It is clear that $w_i\in L^*$ for all $i$.
[**Claim**]{}: [*$v'_j-w_j$ is a ${{\mathcal {O}}}_E$-linear combination of $w_0,\ldots,w_{j-1}$, or equivalently, $v'_j-w_j$ is a ${{\mathcal {O}}}_E$-linear combination of $v'_0,\ldots,v'_{j-1}$ (i.e., the matrix representing the base change from $v'_i$ to $w_i$ is a unipotent upper triangular matrix with integral entries).* ]{}
It is clear that this claim implies the lemma. Now for any vector $w$, we have $w=\sum\nolimits_i(w, v_i)v'_i$. Hence the claim is equivalent to $$\label{inn}
(v'_j-w_j, v_i)=\begin{cases} \in {{\mathcal {O}}}_E&\text
{if $i<j$,}\\
0&\text{if $i\geq j$.}
\end{cases}$$ Now for $i\geq j$, we have $$(w_j, v_i)=(g^jv', g^iv)=(v', g^{i-j}v)=\delta_{i j}=(v'_j, v_i) .$$ This proves the second clause in (\[inn\]). The first clause is trivial since $v'_j-w_j\in L^*$.
Let $L$ be a lattice in $C_n$ with $L\subset L^*$. Then set $${\rm Vert}(L)=\{\Lambda\mid \Lambda \text{ vertex lattice with } L\subset \Lambda^*\subset\Lambda\subset L^* \} ,$$ cf. [@KR]. If $g\in G$ with $gL=L$, then $g$ acts on ${\rm Vert}(L)$, and we set $${\rm Vert}^g(L)=\{\Lambda\in{\rm Vert}(L)\mid g\Lambda=\Lambda \} .$$
Note that by Lemma \[squeeze\], the assumption on $L$ is satisfied for $L=L_g$, if $\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g\neq\emptyset$. We may summarize Lemma \[squeeze\] as follows.
\[stratCZ\] (i) If $g$ is regular semi-simple, there is an equality of sets $$(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})=\{B\in {{\mathcal {N}}}({{\mathbb {F}}})\mid {L_g}\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}\subset B^*\subset B\subset {L_g^*}\otimes_{{{\mathcal {O}}}_E}{{\mathcal {O}}}_{\breve F}, gB=B\} .$$
\(ii) There is an equality of $\bar k$-varieties $$\big(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g\big)_{\rm red}= \bigcup_{\Lambda\in {\rm Vert}^g(L_g)}\big({{\mathcal {V}}}(\Lambda)^o\big)^{g_\Lambda} .$$
Here (ii) makes use of Proposition \[scheme\], and the algebraicity of the Bruhat-Tits stratification. The action of $g$ on ${{\mathcal {V}}}(\Lambda)^o$ is induced by the automorphism $g_\Lambda$ on $\Lambda/\Lambda^*$ induced by $g$.
Fixed point set in a stratum {#fpinstratum}
============================
We next analyze the fixed point variety of $g_\Lambda$ on ${{\mathcal {V}}}(\Lambda)^o$. Since ${{\mathcal {V}}}(\Lambda)^o$ is a Deligne-Lusztig variety [@VW], this can be considered as a general question on Deligne-Lusztig varieties (called “DL-varieties” below for brevity). Accordingly, we use notation that is standard in this context, e.g., [@DL].
\[fixpointstr\] Let $\Lambda$ be a vertex lattice in $C_n$ with $g\Lambda=\Lambda$, and denote by $\bar g=g_\Lambda\in {\rm U}(V)({{\mathbb {F}}}_p)$ the automorphism of the hermitian space $V=V_\Lambda=\Lambda/\Lambda^*$ over $k'$ induced by $g_\Lambda$.
\(i) If $\big({{\mathcal {V}}}(\Lambda)^o\big)^{\bar g}$ is non-empty, then $g_\Lambda$ is semi-simple and contained in a Coxeter type maximal torus.
\(ii) If $\big({{\mathcal {V}}}(\Lambda)^o\big)^{\bar g}$ is a non-empty finite set, then $g_\Lambda$ is a regular elliptic element contained in a Coxeter type maximal torus. Furthermore, in this case the cardinality of $\big({{\mathcal {V}}}(\Lambda)^o\big)^{\bar g}$ is given by the type of $\Lambda$.
This follows from the following lemmas on DL-varieties.
We first recall that an element $w$ in the Weyl group $W$ is called [*elliptic*]{}, if the following equivalent properties are satisfied:
1. The torus $T_w$ of type $w$ is elliptic, i.e., $T_w/Z$ is anisotropic (i.e., $X^*(T_w/Z)^F=(0)$).
2. $T_w$ is not contained in a proper $F$-stable parabolic subgroup.
3. $1$ is not an eigenvalue of $w\cdot F_*$.
4. The $F$-conjugacy class of $w$ contains no element in a proper $F$-stable parabolic subgroup of $W$.
Here $F_*$ denotes the action of Frobenius on $X_*(T),$ where $T$ is a maximal torus contained in a Borel subgroup (if $G$ is split, then $F_*$ is trivial; for a unitary group in $n$ variables with standard basis for the hermitian space, $F_*$ acts through the longest element in $S_n$). Note that any Coxeter element in the sense of Lusztig [@Lu1] is elliptic.
(The equivalence of (i) and (iii) follows from [@C], Proposition 3.2.2. The equivalence of (iii) and (iv) follows from [@H], Lemma 7.2. The equivalence of (i) and (ii) is easy[^3].)
The DL-varieties appearing in [@VW] are associated to unitary groups in an odd number of variables and standard Coxeter elements. More precisely, let $V$ be a hermitian vector space over ${{\mathbb {F}}}_{q^2}$ of dimension $n=2d+1$. We choose the basis $e_1,\ldots, e_n$ in such a way that under the hermitian pairing $e_i$ pairs trivially with $e_j$, unless $i+j=n+1$, and we identify $W$ with the symmetric group $S_n$. Then the DL-variety of interest is associated to the cyclic permutation $w=(d+1, d+2,\ldots, n)$. The DL-varieties associated to different Coxeter elements all differ at most by a power of Frobenius [@Lu1], Prop. 1.10; in particular, they are all universally homeomorphic. The DL-variety $X_w$ associated to the Coxeter element $w=(1, 2,\ldots,d+1)$ is the variety of complete flags ${{\mathcal {F}}}_\bullet$ such that $${{\mathcal {F}}}_{n-i}^\perp\subset {{\mathcal {F}}}_{i+1}, \, {{\mathcal {F}}}_{n-i}^\perp\neq {{\mathcal {F}}}_i,\, (1\leq i\leq d);\,\, {{\mathcal {F}}}_i={{\mathcal {F}}}_{n-i}^\perp,\, (d+1\leq i\leq n-1) .$$ Let $\tau=\sigma^2$. Then $X_w$ can also be identified with the variety of complete selfdual flags ${{\mathcal {F}}}_\bullet$ of $V$ such that $${{\mathcal {F}}}_i+\tau({{\mathcal {F}}}_i)={{\mathcal {F}}}_{i+1} , \, i=1,\ldots,d .$$ In other words, $X_w$ is the variety of complete isotropic flags ${{\mathcal {F}}}_1\subset{{\mathcal {F}}}_2\subset\ldots\subset{{\mathcal {F}}}_d$ of $V$ such that $${{\mathcal {F}}}_1\neq \tau({{\mathcal {F}}}_1)\subset {{\mathcal {F}}}_2, {{\mathcal {F}}}_2\neq \tau({{\mathcal {F}}}_2)\subset {{\mathcal {F}}}_3,\ldots, {{\mathcal {F}}}_{d-1}\neq \tau({{\mathcal {F}}}_{d-1})\subset {{\mathcal {F}}}_d, {{\mathcal {F}}}_d\neq \tau ({{\mathcal {F}}}_d) .$$ Hence we can identify $X_w$ with the set of $\ell\in{{\mathbb {P}}}(V)$ such that $$\big(\ell, \ell\big)=\big(\ell, \tau(\ell)\big)=\ldots =\big(\ell, \tau^{d-1}(\ell)\big)=0;\, \big(\ell, \tau^{d}(\ell)\big)\neq 0 .$$ This DL-variety is defined over ${{\mathbb {F}}}_{q^2}$.
[(Lusztig [@Lu2 5.9])]{}\[semisimple\] Let $X_w$ be a DL-variety, where $w$ is elliptic and of minimal length in its $F$-conjugacy class. Let $s\in G({{\mathbb {F}}}_q)$. If the fixed point set $X_w^s$ is non-empty, then $s$ is semi-simple.
(Lusztig) In the case of a unitary group in an odd number of variables and the standard Coxeter element, this can be easily seen as follows. In this case, as explained above, $X_w$ can be viewed as a subset of projective space, by associating to a complete flag its one-dimensional component $\ell\subset V\otimes_{{{\mathbb {F}}}_{q^2}}{{\mathbb {F}}}$. Now assume that $\ell$ is fixed under $s$. Then so are $\tau(\ell), \tau^2(\ell),\ldots$. But if $\ell\in X_w$, then $\ell, \tau(\ell), \ldots, \tau^{n-1}(\ell)$ form a basis of $V\otimes_{{{\mathbb {F}}}_{q^2}}{{\mathbb {F}}}$, cf. [@Lu3], Prop. 26, (i). Hence $s$ is a diagonal element wrt this basis.
Let $X_w$ be a DL-variety, and let $s\in G({{\mathbb {F}}}_q)$ be a semi-simple element. Then the fixed point set $X_w^s$ is non-empty if and only if $s$ is conjugate under $G({{\mathbb {F}}}_q)$ to an element in $T({{\mathbb {F}}}_q)$ for a maximal torus $T$ of type $w$.
This follows immediately from the formula (4.7.1) for $X_w^s$ in [@DL], Prop. 4.7.
We know from [@DL] that the fixed point set is a finite disjoint sum of DL-varieties, for various groups and various Weyl group elements. Let us spell out which DL-varieties occur in the case of interest to us, namely the unitary group of odd size $n=2d+1$, and when $w=(1, 2,\ldots,d+1)$ is the Coxeter element as above. Now in this case the maximal torus $T$ of type $w$ is given by $$T({{\mathbb {F}}}_q)={\rm Ker \, \big(Nm}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^n}}: {{{\mathbb {F}}}^\times_{q^{2n}}}\to{{{\mathbb {F}}}^\times _{q^n}}\big).$$ We may identify the hermitian space $V$ with ${{\mathbb {F}}}_{q^{2n}}$, equipped with the hermitian form $(x, y)\mapsto {\rm Tr}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^2}}(\sigma^n(x) y)$. Now $s\in T({{\mathbb {F}}}_q)$. Hence $s$ generates a subfield ${{\mathbb {F}}}_q(s)$ of ${{\mathbb {F}}}_{q^{2n}}$. Let ${{\mathbb {F}}}_q(s)={{\mathbb {F}}}_{q^h}$. Then $h\vert 2n$. If $h$ is odd, then the norm equation for $s$ gives $s^2=1$, hence $h=1$ and $s=\pm 1$, and $s$ acts trivially on $X_w$. If $h=2k$ is even, then $k\vert n$. In this case, we may identify the hermitian space $V$ with ${{\mathbb {F}}}_{q^{2n}}$, equipped with the hermitian form $(x, y)\mapsto {\rm Tr}_{{{\mathbb {F}}}_{q^{2k}}/{{\mathbb {F}}}_{q^2}}\big({\rm Tr}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^{2k}}}(\sigma^n(x) y)\big) .$ Then the centralizer $Z^0(s)$ can be identified with $$Z^0(s)={\rm Res}_{{{\mathbb {F}}}_{q^k}/{{\mathbb {F}}}_q}(U_h) ,$$ where $U_h$ is the unitary group for the hermitian form ${\rm Tr}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^{2k}}}(\sigma^n(x) y)$ on ${{\mathbb {F}}}_{q^{2n}}$, and the maximal torus can be identified with the restriction of scalars of the maximal torus ${\rm Ker\, Nm}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^n}}$ of $U_h$. In this case the corresponding DL-variety is simply the DL-variety of dimension $\frac{1}{2}(\frac{n}{k}-1)$ associated to the Coxeter torus in a unitary group of odd size $\frac{n}{k}$ over ${{\mathbb {F}}}_{q^k}$. And the fixed point set $X_w^s$ is a disjoint sum of isomorphic copies of this DL-variety.
Let $w$ be elliptic, and $s\in G({{\mathbb {F}}}_q)$ semi-simple. If $X_w^s$ has only finitely many elements, then $s$ is regular, and conversely.
We use the formula (4.7.2) for $X_w^s$ in [@DL], Prop. 4.7., which presents $X_w^s$ as a disjoint union of varieties which are DL-varieties for $Z^0(s)$, of the form $X_{T'\subset B'}$. However, $T'$ is of the same type as $T$, hence is elliptic. On the other hand, if the fixed point set is finite, then ${\rm dim}\, X_{T'\subset B'}=0$. This implies that $T=Z^0(s)$, which is precisely the claim. The converse is obvious, because a regular element has only finitely many fixed points in the flag variety.
\[cardcentr\] Let $s\in G({{\mathbb {F}}}_q)$ be regular and contained in a maximal torus $T$ of type $w$. Then the number of fixed points of $s$ in $X_w$ is equal to the cardinality of the $F$-centralizer of $w$ in $W$.
We use the formula (4.7.1) in [@DL], Prop. 4.7. It shows that the cardinality of $X_w^s$ is equal to the cardinality of $N({{\mathbb {F}}}_q)/T({{\mathbb {F}}}_q)$, where $N$ denotes the normalizer of $T$. However $N({{\mathbb {F}}}_q)/T({{\mathbb {F}}}_q)$ can be identified with the fixed points under the action of Frobenius on $N/T$. After identifying $N/T$ with $W$, this action is via $x\mapsto wF(x)w^{-1}$. Hence the fixed points are identified with the $F$-centralizer of $w$.
\[cardfix\] Let $G$ be the unitary group in an odd number $n$ of variables. Then
\(i) The $F$-centralizer of a Coxeter element $w$ has $n$ elements.
\(ii) Let $s\in T({{\mathbb {F}}}_q)$ be a regular element in a Coxeter torus. Then all points in $X_w^s$ are conjugate under ${\rm Gal}({{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^2})$, and in fact, this Galois group acts simply transitively on the fixed points.
By Lemma \[cardcentr\], the second assertion implies the first one, since ${\rm Gal}({{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^2})$ has $n$ elements. Now we may identify $T$ with ${\rm Ker}({\rm Nm}_{{{\mathbb {F}}}_{q^{2n}}/{{\mathbb {F}}}_{q^n}}) $, and the hermitian space $V$ with ${{\mathbb {F}}}_{q^{2n}}$, cf. above. Then the set of fixed points of a regular $s\in T({{\mathbb {F}}}_q)$ in ${{\mathbb {P}}}(V)$ is just the set of eigenlines of ${{\mathbb {F}}}_{q^{2n}}^\times$ in $V\otimes_{{{\mathbb {F}}}_{q^2}}{{\mathbb {F}}}$. These all lie in $X_w$, and this implies the assertion.
At this point all statements of Proposition \[fixpointstr\] are proved. We also note the following consequence.
Let $\Lambda\in{\rm Vert}^g(L)$ such that $\big({{\mathcal {V}}}(\Lambda)^o\big)^{g_\Lambda}$ is finite. Then there is no $\Lambda'\in {\rm Vert}^g(L)$, with $\Lambda'$ strictly contained in $\Lambda$.
Indeed, $\Lambda'$ would correspond to a proper parabolic in ${\rm U}(V_{\Lambda})({{\mathbb {F}}}_p)$; but $ g_\Lambda$ is not contained in a proper parabolic by Proposition \[fixpointstr\], and hence cannot fix $\Lambda'^*/\Lambda^*$.
Statement of the AFL
====================
Let $C'_n$ be a hermitian space of dimension $n$ with discriminant of *even* valuation, and equipped with a vector $u$ of norm one. We fix a self-dual lattice $L_0$ in $C'_n$ such that $u\in L_0$. We denote by $K$ the stabilizer of $L_0$ in $U(C_n')(F)$. We define, for $g\in U(C_n')(F)$ regular semi-simple, $$\label{orbintu}
O(g,1_K)=\int_{U(u^\perp)(F)}1_K(h^{-1} g h)dh ,$$ where the Haar measure is normalized by ${\rm vol}\big(K\cap {U(u^\perp)(F)}\big)=1$. Here $u^\perp$ denotes the orthogonal complement of $u$ in $C_n'$.
We now denote by $C$ either $C_n$ or $C_n'$. For $g\in U(C)(F)$ regular semi-simple, we denote by $L=L_g$ the lattice in $C$ generated by $g^i u,i=0,1,\ldots,n-1$. We define an involution $\tau$ on $C=L_g\otimes_{{{\mathcal {O}}}_E} E$ (depending on $g$) by requiring that $(a \cdot g^iu)^\tau= \bar a\cdot g^{-i}u$ for $a\in E$ and $i=0,1,\ldots,n-1$.
\[lem Cn’\] Let $g\in U(C) (F)$ be regular semisimple. Then $$O(g,1_K)=\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda, \Lambda^*=\Lambda\}}1.$$
The orbital integral counts the number of self-dual lattices $\lambda$ in $u^\perp$ such that $\lambda\oplus {{\mathcal {O}}}_E u$ is fixed by $g$. To show the equality, it suffices to show that any lattice $\Lambda$ occuring on the RHS splits as a direct sum $\lambda\oplus {{\mathcal {O}}}_E u$ for a self-dual lattice $\lambda$. But since $(u,u)=1$ it follows that $\Lambda=(\Lambda\cap u^\perp)\oplus{{\mathcal {O}}}_E u$, where $\lambda=\Lambda\cap u^\perp$ is self-dual.
Now let $S_n$ be the variety over $F$ whose $F$-points are $$S_n(F)=\{s\in GL_n(E)\mid s\bar s=1.\}$$ In fact, $S_n$ is defined over ${{\mathcal {O}}}_F$. For $\gamma\in S_n(F)$, recall that its invariants are the characteristic polynomial ${\rm char}_\gamma(T)\in E[T]$ and the $n-1$ elements $v \gamma^i\, ^t\! v,i=1,2,\ldots,n-1$ of $E$, for $v$ the row vector $(0,\ldots,0,1)$.
For $\gamma\in S_n(F)$ regular semi-simple and $s\in{{\mathbb {C}}}$, we consider $$O(\gamma, 1_{S_n({{\mathcal {O}}}_F)},s)=\int_{GL_{n-1}(F)}1_{S_n({{\mathcal {O}}}_F)}(h^{-1}\gamma h)\eta({\rm det}\,h)|{\rm det}\, h|^sdh ,$$ where the Haar measure on $GL_{n-1}(F)$ is normalized by ${\rm vol}(GL_{n-1}({{\mathcal {O}}}_F))=1$. This is a polynomial in ${{\mathbb {Z}}}[q^s, q^{-s}]$, comp. Lemma \[lem Mi\] below.
We will simply denote the value at $s=0$ by $O(\gamma, 1_{S_n({{\mathcal {O}}}_F)})$; it is given by$$\label{orbintS}
O(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=\int_{GL_{n-1}(F)}1_{S_n({{\mathcal {O}}}_F)}(h^{-1}\gamma h)\eta({\rm det}\, h)dh.$$ We also introduce the first derivative at $s=0$: $$\label{orbintder}
O'(\gamma, 1_{S_n({{\mathcal {O}}}_F)})= \frac{d}{ds}O(\gamma, 1_{S_n({{\mathcal {O}}}_F)},s)_{\big| s=0} .$$
For a regular semisimple $\gamma\in S_n(F)$, we define $\ell(\gamma)=v({\rm det}(\gamma^i v))\in {{\mathbb {Z}}}$, where $(\gamma^i v)$ is the matrix $(v,\gamma v,\ldots,\gamma^{n-1} v)$. And we define a sign $$\omega(\gamma)=(-1)^{\ell(\gamma)} \in \{\pm 1\}.$$
Now let $g\in U(C)(F)$ match $\gamma$, i.e., have the same invariants as $\gamma$. Then, with $L=L_g$, we define the set of lattices in $C$ $$M=\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}$$ and its subsets indexed by $i\in {{\mathbb {Z}}}$, $$M_i=\{\Lambda\mid\Lambda\in M,\ell(\Lambda/L)=i\}.$$ Here $\ell(\Lambda/L)$ is the length of the ${{\mathcal {O}}}_E$-module $\Lambda/L$.
\[lem Mi\] Let $GL_{n-1}(F)_i$ be the open subset of $GL_{n-1}(F)$ consisting of $h$ with $v({\rm det}\, h)=i$. Then $$\int_{GL_{n-1}(F)_i}1_{S_n({{\mathcal {O}}}_F)}(h^{-1}\gamma h)dh =| M_{i-\ell(\gamma)}|.$$
Consider the row vector space $F^{n}$, with $F^{n-1}$ as a natural subspace (vectors with zero last entry). We also consider $v=(0,\ldots,0,1) $ as an vector in $F^{n}$. Consider the set of lattices $$\mathfrak{m}:=\{\lambda\subset F^{n-1}\mid \gamma(\Lambda)=\Lambda, \text{ where } \Lambda=(\lambda\otimes {{\mathcal {O}}}_E)\oplus{{\mathcal {O}}}_E v\}$$ and the subsets $$\mathfrak{m}_i:=\{\lambda\in
\mathfrak{m}\mid \ell(\lambda/\lambda_0)=i\} ,\quad \lambda_0={{\mathcal {O}}}_F^{n-1}.$$ Here the length is defined by $\ell(\lambda/{{\mathcal {O}}}_F^{n-1}):=\ell(\lambda/A)-\ell({{\mathcal {O}}}_F^{n-1}/A)$ for any lattice $A\subset \lambda\cap {{\mathcal {O}}}_F^{n-1}$. It is obvious that the LHS in Lemma \[lem Mi\] is given by the cardinality $|\mathfrak{m}_i|$.
Denote by $\sigma$ the Galois conjugation on $E^n$. Define a hermitian form on $E^n$ by requiring that $$(\gamma^i v,\gamma^j v):=v \gamma^{i-j}\,^t\!v.$$ Let ${{\mathcal {L}}}={{\mathcal {L}}}_\gamma$ be the ${{\mathcal {O}}}_E$-lattice in $E^n$ generated by $\gamma^i v, i=0,1,\ldots,n-1$. Denote by ${{\mathcal {L}}}^\ast$ the dual of ${{\mathcal {L}}}$, i.e., $${{\mathcal {L}}}^\ast=\{ x\in E^n\mid (x, {{\mathcal {L}}})\subset {{\mathcal {O}}}_E \} .$$ Now we introduce the set of lattices $$\mathfrak{m}':=\{\Lambda\subset E^{n}\mid {{\mathcal {L}}}\subset \Lambda\subset {{\mathcal {L}}}^\ast, \gamma \Lambda=\Lambda, \Lambda^\sigma=\Lambda\} ,$$ and $$\mathfrak{m}'_i:=\{\Lambda\in
\mathfrak{m}'\mid \ell(\Lambda/{{\mathcal {L}}})=i\} .$$ We claim that the map $\lambda\mapsto \Lambda:=(\lambda\otimes {{\mathcal {O}}}_E)\oplus{{\mathcal {O}}}_Ev$ defines a bijection between $\mathfrak{m}$ and $ \mathfrak{m}'$. First of all, such $\Lambda$ do lie in $ \mathfrak{m}'$. Indeed, we only need to verify that $\Lambda\subset {{\mathcal {L}}}^\ast$ or, equivalently, $(\Lambda,\gamma^i v)\in {{\mathcal {O}}}_E$ for all $i$. This follows from $\gamma\Lambda=\Lambda$ and $(\Lambda, v)\in {{\mathcal {O}}}_E$. Now we only need to show the surjectivity of the map. Similarly to the unitary case, any $\Lambda\in \mathfrak{m}'$ is a direct sum $ (\Lambda\cap E^{n-1})\oplus{{\mathcal {O}}}_E v$. Obviously $ \Lambda\cap E^{n-1}$ is also invariant under the Galois conjugation on $E^{n-1}$. So we may find a lattice $\lambda\subset F^{n-1}$ such that $ \lambda\otimes {{\mathcal {O}}}_E=\Lambda\cap E^{n-1}$. This proves the surjectivity.
We claim that the set $\mathfrak{m}_i$ is sent to $\mathfrak{m}'_{i-\ell(\gamma)}$. Clearly we have $\ell(\lambda/\lambda_0)=\ell(\Lambda/{{\mathcal {O}}}_E^n)$ under this map. Hence the claim follows, since the length of ${{\mathcal {L}}}$ over the image of ${{\mathcal {O}}}_E^{n}$ is obviously given by $\ell(\gamma)$.
To finish the proof, we need to exhibit a bijection from $ \mathfrak{m}'$ to $M$ that sends $\mathfrak{m}'_i$ to $M_{i}$. Since $(g^iu,u)=v\gamma^i\, ^t\!v$ for all $i$, the map $\gamma^iv\mapsto g^i u$ defines an isometry between ${{\mathcal {L}}}={{\mathcal {L}}}_\gamma$ and $L=L_g$. Moreover, the involution $\sigma$ on $E^n$ maps $\gamma^i v$ to $\bar \gamma^i v=\gamma^{-i}v$. Therefore $\sigma$ transfers to the involution $\tau$ on $L\otimes E$. Clearly this map sends $\mathfrak{m}'_i$ to $M_{i},$ since it sends ${{\mathcal {L}}}$ to $L$.
\[cor lattice counting\] Let $\gamma\in S_n(F)$ be regular semisimple and match $g\in U(C)(F)$.
- If $C=C_n'$, then $$O(\gamma,1_{S_n({{\mathcal {O}}}_F)})=\omega(\gamma)\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} .$$
- If $C=C_n$, then $O(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=0$ and $$O'(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=-\omega(\gamma){\rm log}\, q\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} \ell(\Lambda/L) .$$
By Lemma \[lem Mi\], we have $$O(\gamma,1_{S_n({{\mathcal {O}}}_F)},s)=\sum_{i\in {{\mathbb {Z}}}}(-1)^i |M_{i-\ell(\gamma)}|q^{-is}.$$ Or equivalently $$O(\gamma,1_{S_n({{\mathcal {O}}}_F)},s)=(-1)^{\ell(\gamma)}\sum_{i\in {{\mathbb {Z}}}}(-1)^i |M_{i}|q^{-(i+\ell(\gamma))s}.$$ This shows that $$O(\gamma,1_{S_n({{\mathcal {O}}}_F)})=(-1)^{\ell(\gamma)}\sum_{i\in {{\mathbb {Z}}}}(-1)^i |M_{i}|.$$ In particular, if we set $C=C_n'$, the first identity is proved.
Now let $C=C_n$. The map $\Lambda \mapsto \Lambda^\ast$ defines an involution on $M$ and sends $M_{i}$ to $M_{r-i}$ where $r$ is the length of $L^\ast/L$, which is odd. This shows that $O(\gamma, 1_{S_n({{\mathcal {O}}}_F)})=0$. We now take the first derivative $$\begin{aligned}
O'(\gamma,1_{S_n({{\mathcal {O}}}_F)},0)&=-(-1)^{\ell(\gamma)}{\rm log}\, q \sum_{i\in {{\mathbb {Z}}}}(-1)^i |M_{i}|(i+\ell(\gamma))\\&=-\omega(\gamma) {\rm log}\, q \sum_{i\in {{\mathbb {Z}}}}(-1)^i i|M_{i}|.\end{aligned}$$ This completes the proof.
Using Lemma \[lem Cn’\] and Corollary \[cor lattice counting\] above, the statement of the FL (cf. Introduction) is the following identity for $g\in U(C_n')(F)$ regular semi-simple: $$\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} = \sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda, \Lambda^*=\Lambda\}}1.$$
Now, in the special case that the intersection of $\Delta({{\mathcal {M}}})$ and $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ is discrete, the statement of the AFL (cf. Introduction) is as follows.
Let $g\in U(C_n)(F)$ be regular semi-simple. Assume that $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is finite. Then $$\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda, \pi\Lambda\subset\Lambda^*\subset\Lambda\}}{\rm mult}(\Lambda)=-\sum_{\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}} (-1)^{\ell(\Lambda/L)} \ell(\Lambda/L) .$$ Here the number ${\rm mult}(\Lambda)$ is the intersection multiplicity of $\Delta({{\mathcal {M}}})$ and $({\rm id}_{{\mathcal {M}}}\times g)\Delta({{\mathcal {M}}})$ [*along the stratum ${{\mathcal {V}}}(\Lambda)^o$*]{}, cf. .
The minuscule case {#theminusculecase}
==================
In this section we assume that $V=L^\ast/L$ is killed by $\pi$. We thus consider it as a vector space over $k'={{\mathbb {F}}}_{q^2}$, the residue field of ${{\mathcal {O}}}_E$. Denote by $r$ its dimension. Then $r$ is an odd integer between $1$ and $n-1$. Then the hermitian form on $L$ naturally induces a non-degenerate hermitian form on $V$. (It is obtained as follows: For $\bar x, \bar y \in V$ with representatives $x,y\in L^*$, the value $(\bar x, \bar y)_V$ of the hermitian form on $V$ is the image modulo $(\pi)$ of $\pi \cdot (x,y)\in {{\mathcal {O}}}_E$, where $(\ , \ )$ denotes the form on $L\otimes E$.) We denote the corresponding unitary group by $U(V)$ and consider it as an algebraic group defined over ${{\mathbb {F}}}_q$. As $g$ defines automorphisms of both $L$ and $L^\ast$, it induces an automorphism $\bar g\in U(V)$. Then via the map $\Lambda \mapsto \Lambda^*/L $ the set of vertices $\Lambda\in {\rm Vert}^g(L)$ is in natural bijection with the set of $\bar g$-invariant $k'$-subspaces $W$ of $V$ such that $W$ is totally isotropic with respect to the hermitian form on $V$. We thus define ${\rm Vert}^{\bar g}(V)$ to be the set of such $W$. And we write ${{\mathcal {V}}}_W$ for the closed Bruhat-Tits stratum ${{\mathcal {V}}}_\Lambda$ that corresponds to $\Lambda\in {\rm Vert}^g(L)$ in the sense of Vollaard’s paper [@V], cf. also [@VW], and we call [*type*]{} of $W$ the type of $\Lambda$, i.e., the dimension of $W^\perp/W$. The open stratum ${{\mathcal {V}}}_W^\circ$ can then be identified with the Deligne-Lusztig variety associated to a Coxeter torus of $U(W^\perp/W)$, cf. [@V].
We will consider the characteristic polynomial $P_{\bar g|V}(T):={\rm det}(T-\bar g|V)\in k'[T]$ of degree $r$. Since, by Lemma \[cycl\], $L^\ast$ is $g$-cyclic, $V$ is $\bar g$-cyclic. This is equivalent with the regularity of $\bar g$ as an endomorphism of $V$. In particular, its characteristic polynomial is equal to its minimal polynomial. Let $$\begin{aligned}
\label{prodirr}
P_{\bar g}=\prod_{i=1}^{\ell} P_i^{a_i} \end{aligned}$$ be the decomposition into irreducible monic polynomials.
If $P(T)=T^d+b_1T^{d-1}+\ldots+b_d \in k'[T],b_d\neq 0$, we set $$P^\ast(T)= \overline{b}_{d}^{-1}T^d \overline{P}(T^{-1}) ,$$ where the bar denotes the Galois conjugate on $k'$. Since $\bar g\in U(V)$, we have $P_{\bar g}=P^\ast_{\bar g}=\prod_i P_i^{\ast a_i}$, and hence we have an involution $\tau$ of $\{1,2,\ldots,\ell\}$ such that $P_{i}^\ast=P_{\tau(i)}$ and $a_{\tau(i)}=a_i$, cf. [@AG]. Note that since $V$ has odd dimension, the degree of $P_{\bar g}$ is odd, and hence there exists at least one index $i$ with $\tau(i)=i$ and such that $a_i$ is odd.
\[cardform\] (i) The set $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is non-empty if and only if there exists a unique $i_0\in \{1,2,\ldots,\ell\}$ such that $\tau(i_0)=i_0$ and such that $a_{i_0}$ is odd. Then the set $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is finite.
\(ii) If $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is non-empty (hence finite), these points lie on some strata ${{\mathcal {V}}}^\circ_W$, all of the same type ${\rm deg}\, P_{i_0}$ for the unique $i_0$ in part $(i)$. And the cardinality of $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is given by $$\prod_{\{i,j\},j=\tau(i)\neq i} (1+a_i)\cdot {\rm deg}\,P_{i_0}.$$
If $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is non-empty, then there exists $W\in {\rm Vert}^{\bar g}(V)$. Then $W^\perp$ is $\bar g$-invariant and the hermitian form allows us to identify $W$ with the dual of $V/W^\perp$. This yields a filtration $$0\subset W\subset W^\perp \subset V$$and a decomposition $$\begin{aligned}
\label{decomp}
P_{\bar g|V}=P_{\bar g|W}\cdot P_{\bar g|W^\perp/W}\cdot P_{\bar g|V/W^\perp},\quad\end{aligned}$$ with the property $$\begin{aligned}
\label{propdecomp}
P_{\bar g|W^\perp/W}=P_{\bar g|W^\perp/W}^\ast,\,\, P_{\bar g|W}=P^\ast_{\bar g|V/W^\perp}.\end{aligned}$$ Note that the fixed point set ${{\mathcal {V}}}_W^{\circ,\bar g}$ is non-empty if and only if $\bar g|(W^\perp/W)$ lies in a Coxeter torus, i.e., (because we are dealing here with a unitary group in an [*odd*]{} number of variables), if and only if $\bar g|(W^\perp/W)$ generates inside ${\rm End}(W^\perp/W)$ a subfield of ${{\mathbb {F}}}_{q^{2r}}$. Since $\bar g$ is a regular endomorphism, so is the induced endomorphism on $W^\perp/W$. Hence the fixed point set ${{\mathcal {V}}}_W^{\circ,\bar g}$ is non-empty if and only if $P_{\bar g|W^\perp/W}$ is an irreducible polynomial. This irreducible polynomial has to be of the form $P_{i_0}$ with $\tau(i_0)=i_0$. Moreover if $P_i|P_{\bar g|W}$, then $P_i^\ast|P_{\bar g|V/W^\perp}$. This shows that $a_{i_0}$ is odd and that for every $j\neq i_0$, either $\tau(j)\neq j$ or $\tau(j)=j$ and $a_j$ is even. This shows the “only if" part of $(i)$. Moreover, the type of $W$, i.e., the dimension of $W^\perp/W$, is equal to the degree of $P_{i_0}$.
We now assume that there exists a unique $i_0$ such that $\tau(i_0)=i_0$, and with $a_{i_0}$ odd. To show the “if" part of $(i)$ and part $(ii)$, it suffices to prove the formula of cardinality and that the strata ${{\mathcal {V}}}^o_W$ have the desired type.
The decomposition (\[prodirr\]) induces a decomposition as a direct sum of generalized eigenspaces $$V=\bigoplus_{i=1}^\ell V_i,\quad V_i:={\rm Ker}\, P_i^{a_i}(\bar g).$$
For $v_i\in V_i,v_j\in V_j$, there is some non-zero constant $c$ such that (cf. [@AG]) $$0=\langle P_i^{a_i}(\bar g)v_i,v_j \rangle=c\langle v_i, P^{\ast a_i}_i(\bar g^{-1}) v_j \rangle=c\langle v_i, P^{\ast a_i}_{i}(\bar g) \bar g^{-s_i}v_j \rangle ,$$ where $s_i=a_i{\rm deg}\, P_i$. Then we have two cases:
- If $\tau(i)=i$, by the above equation we see that $V_{i}$ is orthogonal to $\oplus_{j\neq i}V_j$ and the restriction of the Hermitian form to $V_{i}$ is non-degenerate.
- If $\tau(i)=j\neq i$, then $V_i\oplus V_j$ is orthogonal to $V_k,k\neq i,j$. And the restriction of the hermitian form to $V_i\oplus V_j$ is non-degenerate, both $V_i$ and $V_j$ being totally isotropic subspaces.
Consider the decomposition $$W=\bigoplus_{i}W_i,\quad W_i:=W\cap V_i.$$ Then each $W_i$ is invariant under $\bar g$ and totally isotropic in each $V_i$. By the regularity of $\bar g$, we may list all $\bar g$-invariant subspaces in $V_{i}$: for each $m=0,1,\ldots,a_{i}$: there is precisely one invariant subspace (denoted by $V_{i,m}$) of dimension $m\cdot {\rm deg}\, P_{i}$ and these exhaust all invariant subspaces of $V_i$. Moreover $V_{i,m}={\rm Ker}\, P_i^m(\bar g)$. Let now $i=i_0$. The proof of the “only if" part of $(i)$ shows that $W_{i_0}=V_{i_0,\frac{a_{i_0}-1}{2}}$. We also know that $W':=\bigoplus_{i\neq i_0}W_i$ must be maximal totally isotropic in $V':=\bigoplus_{i\neq i_0}W_i$. Now suppose that $i\neq i_0$. We have two cases.
- If $\tau(i)=i$, $W_{i}$ must be a maximal totally isotropic subspace of $V_i$. Hence $W_i=V_{i,a_{i}/2}$ is unique (note that $a_i$ must be even by the assumption of the uniqueness of $i_0$).
- If $\tau(i)=j\neq i$, then $W_i\oplus W_j$ must be a maximal totally isotropic subspace of $V_i\oplus V_j$. Therefore $W_j$ is uniquely determined by $W_i$ and we can take $W_i=V_{i,m}$ for $m=0,1,\ldots, a_i$. We thus have precisely $a_{i}+1=a_{j}+1$ number of choices.
In summary we have shown that the cardinality of the set of $W$ with ${{\mathcal {V}}}_W^{o,\bar g}$ non-empty is $$\prod_{\{i,j\},j=\tau(i)\neq i}(1+ a_i)\cdot \prod_{i\neq i_0,\tau(i)=i}1.$$
Moreover, this also shows that the type of all such ${{\mathcal {V}}}_W^{o}$ is the same, namely ${\rm deg}\, P_{i_0}$. And by Lemma \[cardfix\], for each $W$, the cardinality of ${{\mathcal {V}}}_W^{o,\bar g}$ is precisely ${\rm dim}\, W^\perp/W={\rm deg}\, P_{i_0}$. We conclude that the cardinality of $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is $$\sum_{W\in {\rm Vert}^{\bar g}(V)}|{{\mathcal {V}}}_W^{o,\bar g}|=\prod_{\{i,j\},j=\tau(i)\neq i}(1+ a_i)\cdot {\rm deg}\,P_{i_0}.$$
We now calculate the derivative of the orbital integral using Lemma \[lem Cn’\].
\[derivform\] Let $g$ be as above. Let $\gamma\in S_n(F)$ match $g$. Then $O'(\gamma,1_{S_n({{\mathcal {O}}}_F)})=0$ unless there is a unique $i_0$ such that $\tau(i_0)=i_0$ and with $a_{i_0}$ odd, in which case
$$O'(\gamma,1_{S_n({{\mathcal {O}}}_F)})=-\omega(\gamma) {\rm log}\,q\prod_{\{i,j\},j=\tau(i)\neq i} (1+a_i)\cdot {\rm deg}\, P_{i_0}\cdot \frac{a_{i_0}+1}{2}.$$
By mapping $\Lambda$ to $\Lambda/L$, the set of lattices $$\{\Lambda\mid L\subset \Lambda\subset L^*, g\Lambda=\Lambda,\Lambda^\tau=\Lambda\}$$ is in bijective correspondence with the set of subspaces $${{\mathcal {W}}}:=\{W\mid W\subset V, \bar g W=W,W^{\bar\tau}=W\} ,$$ where $\bar \tau$ is the involution on $V$ induced by the restriction of the involution $\tau$ to $L^\ast$. We need to describe this involution. By definition, the involution $\tau$ on $C$ has the property that $$(x^\tau,y^\tau )=(y,x),\quad (g\tau)^2=1$$ The induced involution $\bar \tau$ on $V$ inherits the same properties. In particular, for a polynomial $P\in k'[T]$, we have $$P(\bar g) \bar \tau=\bar \tau \bar P(\bar g^{-1}).$$ In particular, $\bar \tau$ maps $V_{i,m}={\rm Ker}\, P_{i}^m(\bar g)$ to $V_{\tau(i),m}$ and this is the reason we use the same notation $\tau$ to denote the involution on the index set $\{1,2,\ldots,\ell\}$. And clearly we have $\ell(\Lambda/L)={\rm dim}_{k'} \Lambda/L$.
We consider the decomposition $V=\bigoplus_i V_i$ and $W=\bigoplus_i W_i$, where $W_i=W\cap V_{i}$.
First we assume there is a unique $i=i_0$ such that $\tau(i)=i$ and with $a_i$ odd. According to $W_{i_0}=W\cap V_{i_0}$, we write ${{\mathcal {W}}}=\coprod_{m=0}^{a_{i_0}} {{\mathcal {W}}}_m$ as a disjoint union where ${{\mathcal {W}}}_m$ consists of $W\in {{\mathcal {W}}}$ such that $W_{i_0}= V_{i_0,m}$. Since $V_{i_0,m}$ is invariant under the involution $\bar \tau$, it is clear that the map $W\mapsto W\oplus V_{i_0,m}$ defines a bijection between ${{\mathcal {W}}}_0$ and ${{\mathcal {W}}}_m$. Therefore we may write the sum of Lemma \[lem Cn’\] $$\sum_{W\in{{\mathcal {W}}}}(-1)^{{\rm dim}\, W}{\rm dim}\, W=\sum_{W\in{{\mathcal {W}}}_0}(-1)^{{\rm dim}\, W} \sum_{m=0}^{a_{i_0}} (-1)^{{\rm dim}\,V_{i_0,m}}({\rm dim}\, W+{\rm dim} V_{i_0,m}).$$ Note that $a_{i_0}\cdot {\rm deg}\, P_{i_0}$ is odd. Therefore the inner sum simplifies to $$-\frac{a_{i_0}+1}{2}{\rm deg}\, P_{i_0} ,$$ which is independent of $W\in {{\mathcal {W}}}_0$.
We now compute the sum $\sum_{W\in{{\mathcal {W}}}_0}(-1)^{{\rm dim}\, W} $. As before we still use the notation $V'=\bigoplus_{i\neq i_0}V_i.$ Note that ${{\mathcal {W}}}_0=\{ W\subset V' \mid W\in {{\mathcal {W}}}\}$. Then for $W\in{{\mathcal {W}}}_0$ we have the decomposition $W=\bigoplus_{i\neq i_0}W_i$. Similar to the proof of the previous proposition, we have two cases for $i\neq i_0$:
- If $\tau(i)=i$, then $a_i$ is even and there are $a_i+1$ choices of $W_{i}=V_{i,m}$ for $m=0,1,\ldots,a_i$.
- If $\tau(i)=j\neq i$, then $W_j=\bar \tau W_i$ and there are $a_i+1$ choices of $W_{i}=V_{i,m}$ for $m=0,1,\ldots,a_i=a_j$.
Hence $\sum_{W\in{{\mathcal {W}}}_0}(-1)^{{\rm dim}\, W}$ is equal to the product of $$\sum_{m=0}^{a_i}(-1)^{2{\rm dim} V_{i,m}} =a_i+1$$ for each pair $(i,j)$ with $j=\tau(i)\neq i$, and $$\sum_{m=0}^{a_i}(-1)^{{\rm dim} V_{i,m}}=1$$ for $i\neq i_0$ with $\tau(i)=i$ (and since $a_i$ is even). This proves the formula when there is a unique $i$ such that $\tau(i)=i$ and $a_i$ odd.
Now suppose that there are at least two such $i$’s. We claim that then $O'(\gamma,1_{S_n({{\mathcal {O}}}_F)})=0$. The same argument as above shows that $\sum_{W\in{{\mathcal {W}}}_0}(-1)^{{\rm dim}\, W}$ is a product which has a factor of the form $$\sum_{m=0}^{a_i}(-1)^{{\rm dim} V_{i,m}}=0 ,$$ when $i\neq i_0$ with $\tau(i)=i$ and $a_i$ odd! This shows that $O'(\gamma,1_{S_n({{\mathcal {O}}}_F)})=0$ and hence completes the proof.
Calculation of length
=====================
We continue to assume $\pi\cdot (L^*/L)=(0)$, and keep all other notation from the previous section. In particular, we assume that $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ is non-empty.
A point of $(\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ corresponds to a lattice $B$ in $C_n\otimes_E{\breve F}$, occurring in a chain of inclusions of lattices, $$L\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset \Lambda^*\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset B^*\subset^1B\subset \Lambda\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset L^*\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F} ,$$ or also to a subspace $U$ of $V\otimes_{k'} \bar k$, occurring in a chain of inclusions of vector spaces over $\bar k$, all of which are invariant under $\bar g$, $$\label{chain}
(0)\subset W\otimes_{k'}\bar k\subset U^\perp\subset^1U\subset W^\perp\otimes_{k'}\bar k\subset V\otimes_{k'} \bar k .$$
Let $\lambda$ be the eigenvalue of $\bar g|(U/U^{\perp})$. Then by the regularity of $\bar g|U$, there exists a unique Jordan block to $\lambda$ in $U$. The size of this Jordan block is of the form $c+1$, where $c$ is the size of the Jordan block of $\bar g|U^\perp$. The size of the Jordan block of $\bar g$ to $\lambda$ is equal to the exponent $a_{i_0}$ with which the irreducible polynomial $P=P_{i_0}$ occurs in $P_{\bar g}$ and is equal to $a_{i_0}=2c+1$. To see this, consider the decomposition $\prod P_i^{a_i}$ of the characteristic polynomial of $\bar{g}$ into irreducible factors. Over $\bar{k}$ only $\lambda$ is a zero of $P_{i_0}$. Since $P_{i_0}$ is an irreducible polynomial over a finite field, $P_{i_0}$ has only simple zeros in $\bar{k}$. Since the minimal polynomial of $\bar{g}$ equals the characteristic polynomial of $\bar{g}$ it follows that the size of the (unique) Jordan block of the eigenvalue $\lambda$ is the multiplicity of the zero $\lambda$ in the characteristic polynomial, and this is $a_{i_0}$. Now we use the chain of inclusions and the formulas and to conclude that $a_{i_0}=2c+1$.
\[length\] Assume $F={{\mathbb {Q}}}_p$. Suppose that $n\leq2p-2$. The length of the local ring of $\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g$ at the point $[B]\in (\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g)({{\mathbb {F}}})$ corresponding to $B$ is equal to $c+1=\frac{1}{2}(a_{i_0}+1)$.
We note that this proposition, together with Propositions \[cardform\] and \[derivform\], proves assertion (iv) of Theorem \[mainthm\].
We first determine the tangent space of $({{\mathcal {M}}}\cap{{\mathcal {N}}}^g)\otimes{{\mathbb {F}}}$ at the point $[B]$ corresponding to $B$.
\[tangspace\] The tangent space of $({{\mathcal {M}}}\cap{{\mathcal {N}}}^g)\otimes{{\mathbb {F}}}$ at $[B]$ is a one-dimensional subspace of the tangent space of ${{\mathcal {M}}}\otimes{{\mathbb {F}}}$ if $c\geq 1$. If $c=0$, the tangent space is trivial.
For this, we have to first recall how one associates the lattice $B$ to a point $(X, \iota, \lambda, \rho)$ of ${{\mathcal {N}}}_n({{\mathbb {F}}})$. Let $M(X)$ be the Dieudonné module of $X$. Then $\iota$ induces a ${{\mathbb {Z}}}/2$-grading of $M(X)$, and $\lambda$ induces an alternating form $\langle\, , \, \rangle$ on $M(X)$. Furthermore, $\rho$ induces an identification of $M(X)_0\otimes _{{{\mathcal {O}}}_{\breve F}}{\breve F}$ with $C_n\otimes_E{\breve F}$ such that ${\rm id}_{C_n}\otimes \sigma^2$ corresponds to $\pi V^{-2}$, and the extended form on $C_n\otimes_E{\breve F}$ to $x, y\mapsto
\pi^{-1}\delta^{-1}\langle x , \pi V^{-1} y \rangle$. Here $\delta$ is a fixed element of ${{\mathcal {O}}}_E^\times$ with $\bar \delta=-\delta$. We then have $B=M(X)_0$, and $B^*=\pi V^{-1}M(X)_1$.
Now the tangent space is given by ${\rm Hom}_{[B]}({\rm Spec}\, {{\mathbb {F}}}[\epsilon], {{\mathcal {N}}}_n)$, where the index $[B]$ indicates that only morphisms are considered whose image point is $[B]$. By Grothendieck-Messing theory, the tangent space ${{\mathcal {T}}}_{{{\mathcal {N}}}_{n}, [B]}$ is equal to $${\rm Hom}_{[B]}({\rm Spec}\, {{\mathbb {F}}}[\epsilon], {{\mathcal {N}}}_n)={\rm Hom}_{0, {\rm isot}}(VM(X)/\pi M(X), M(X)/VM(X)) ,$$ where the index indicates that only homomorphisms are considered which respect the ${{\mathbb {Z}}}/2$-grading and the alternating form. Hence we have $$\begin{aligned}
{{\mathcal {T}}}_{{{\mathcal {N}}}_{n}, [B]}=& \ {\rm Hom}(VM(X)_0/\pi M(X)_1, M(X)_1/VM(X)_0)\\
\simeq & \ {\rm Hom}(M(X)_0/\pi V^{-1}M(X)_1, V^{-1}M(X)_1/M(X)_0)\\
\simeq&\ {\rm Hom}(B/B^*, B^*/\pi B).
\end{aligned}$$ Similarly, if $[B]\in {{\mathcal {N}}}^g$, the tangent space to ${{\mathcal {N}}}^g$ at $[B]$ is given by ${\rm Hom}_g(B/B^*, B^*/\pi B)$, where the index indicates that only $g$-equivariant homomorphisms are considered. And if $[B]\in {{\mathcal {M}}}$, then $$B=A\oplus {{\mathcal {O}}}_{\breve F}u \text { and } B^*=A^*\oplus {{\mathcal {O}}}_{\breve F}u ,$$ and $$B/B^*=A/A^* \text { and } B^*/\pi B=A^*/\pi A\oplus {{\mathbb {F}}}\bar u .$$ Then the tangent space to $\delta({{\mathcal {M}}})$ at $[B]$ is equal to the subspace of ${\rm Hom}(B/B^*, B^*/\pi B)$ consisting of homomorphisms whose image is contained in $A^*/\pi A$, and the tangent space to $\delta({{\mathcal {M}}})\cap{{\mathcal {N}}}^g$ at $[B]$ is given by the subspace of ${\rm Hom}_g(B/B^*, B^*/\pi B)$ of elements which factor through $A^*/\pi A$. In other words, this tangent space is identified with the intersection of the eigenspace $(B^*/\pi B)(\lambda)$ to $\lambda$ in $B^*/\pi B$ with $A^*/\pi A$.
We now show that this intersection, denoted by ${{\mathcal {T}}}$, has dimension one if $c\geq 1$ and zero if $c=0$. Since $(B,B^\ast)\subset {{\mathcal {O}}}_{\breve F}$ and $B^\ast\subset B$, we have an induced sesqui-linear pairing on $B^\ast/\pi B$ valued in ${{\mathbb {F}}}$. We still denote this pairing by $(\cdot,\cdot)$. As $u\in B^\ast$, we may denote by $\bar u$ its image in the ${{\mathbb {F}}}$-vector space $B^\ast/\pi B$. The map $\bar b \mapsto (\bar b,\bar u)$ defines an ${{\mathbb {F}}}$-linear functional denoted by $\ell_u$ on $B^\ast/\pi B$. Using this, we may identify $A^\ast/\pi A$ with the kernel of $\ell_u$. If $\bar b\in {{\mathcal {T}}}$, it is an eigenvector of $\bar g$ with eigenvalue $\lambda$, and $(\bar b, \bar u)=0$. Since $\bar b$ is eigenvector of $\bar g$ we have $$\bar g^{-1}\bar b=\lambda^{-1}\bar b ,$$ and hence $$(\bar g^{-1}\bar b,u)=\lambda^{-1}(\bar b,\bar u)=0 ,\,\text{ i.e., }\, (\bar b,\bar g \bar u)=0 .$$ Similarly, $(\bar b,\bar g^i\bar u)=0$ for all $i\in {{\mathbb {Z}}}$. This implies that $(b,L)=0 \mod \pi$, where $b\in B^\ast $ is any lifting of $\bar b\in B^\ast/\pi B$. Equivalently we have $(b/\pi,L)\in {{\mathcal {O}}}_{\breve F}$, and hence $b\in \pi L^\ast\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}$. We note the following sequence of inclusions, $$\pi B\subset \pi L^\ast\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset L\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset \Lambda^*\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F}\subset B^*\subset^1B .$$ We have proved that ${{\mathcal {T}}}$ is a subspace of $X:=(\pi L^\ast\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F})/\pi B$. And, in fact, ${{\mathcal {T}}}$ is precisely the $\lambda$-eigenspace in $X$. Now $X$ is obviously isomorphic to $Y:=(L^\ast\otimes_{{{\mathcal {O}}}_E} {{\mathcal {O}}}_{\breve F})/ B$ as $\bar g$-modules, and hence is a $\bar g$-cyclic ${{\mathbb {F}}}$-vector space. It is easy to see that the $\lambda$-eigenspace $Y(\lambda)$ of $Y$ is one-dimensional when $c\geq 1$ and zero if $c=0$.
It follows from the preceding lemma that the completed local ring $R=\widehat{{{\mathcal {O}}}}_{{{\mathcal {M}}}\cap {{\mathcal {N}}}^g, [B]}$ is an ${{\mathcal {O}}}_{\breve F}$-algebra of the form $$R={{\mathcal {O}}}_{\breve F}[[t]]/I \text{\, if $c\geq 1$, \, resp. $R={{\mathcal {O}}}_{\breve F}/I$\,
if $c=0$},$$ where $I$ is an ideal in ${{\mathcal {O}}}_{\breve F}[[t]]$, resp. in ${{\mathcal {O}}}_{\breve F}$. Therefore Proposition \[length\] follows from the following proposition.
\[ideal\] Assume $F={{\mathbb {Q}}}_p$. Suppose that $n\leq 2p-2$. Then $I=(\pi, t^{c+1}) . $
The fact that $\pi\in I$ follows from the relation to the special divisors of [@KR]. Recall that to any non-zero element $x\in C_n$, there is associated the special divisor ${{\mathcal {Z}}}(x)$ of ${{\mathcal {N}}}_n$, cf. [@KR], Lemma 3.9. It is the closed formal subscheme of ${{\mathcal {N}}}$ with $S$-valued points $$\begin{aligned}
\{(X,\iota, \lambda, \rho)\mid \text{ the composed quasi-homomorphism } &\overline{Y}\times_{{{\mathcal {O}}}_{\breve F}}\bar S\stackrel{x}{\longrightarrow}{{{\mathbb {X}}}}\times_{{{\mathbb {F}}}}
\bar{S}\stackrel{\varrho_X^{-1}}{\longrightarrow}X\times_S \bar{S}
\\
\text{ lifts to an ${{\mathcal {O}}}_E$-linear homomorphism }&\overline{Y}\times_{{{\mathcal {O}}}_{\breve F}}S\to X\}.
\end{aligned}$$ Here we have identified $C_n$ with ${{\mathrm{Hom}}}_{{{\mathcal {O}}}_E}(\overline Y \times_{{{\mathcal {O}}}_{\breve F}}{{\mathbb {F}}}, {{\mathbb {X}}})\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$ as explained in [@KR], Lemma 3.9. The elements of ${{\mathrm{Hom}}}_{{{\mathcal {O}}}_E}(\overline Y \times_{{{\mathcal {O}}}_{\breve F}}{{\mathbb {F}}}, {{\mathbb {X}}})\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$ are called [*special homomorphisms*]{}, cf. [@KR]. The special divisor ${{\mathcal {Z}}}(x)$ is a relative divisor, with set of ${{\mathbb {F}}}$-points equal to $${{\mathcal {Z}}}(x)({{\mathbb {F}}})=\{ B\in {{\mathcal {N}}}_n({{\mathbb {F}}})\mid x\in B^* \} .$$ Similarly, if ${\bf x}=[x_1,\ldots, x_m]\in (C_n)^m$, then ${{\mathcal {Z}}}({\bf x})={{\mathcal {Z}}}(x_1)\cap\ldots\cap {{\mathcal {Z}}}(x_m)$ has ${{\mathbb {F}}}$-points equal to $${{\mathcal {Z}}}({\bf x})({{\mathbb {F}}})=\{ B\in {{\mathcal {N}}}_n({{\mathbb {F}}})\mid \{x_1,\ldots, x_m\}\subset B^* \} .$$ Now let $g\in G_n$ be regular semi-simple. Then $$\label{reltosp}
\delta({{\mathcal {M}}})\cap {{\mathcal {N}}}^g\subset {{\mathcal {Z}}}(u, gu, \ldots,g^{n-1}u) .$$ Indeed, $\delta ({{\mathcal {M}}})$ can be identified with ${{\mathcal {Z}}}(u)$, comp. [@KR], Lemma 5.2. Hence the assertion follows by the $g$-invariance of the LHS in .
Note that the fundamental matrix of $(u, gu,\ldots, g^{n-1}u)$ in the sense of [@KR] is equivalent to the diagonal matrix $\pi^{{\rm inv}(g)}$. Therefore we may apply the following theorem.
\[speciald\] Assume $F={{\mathbb {Q}}}_p$. Let ${\bf x}=[x_1,\ldots, x_n]\in (C_n)^n$ with fundamental matrix $T({\bf x})$ equivalent to $\pi^\mu$, where $\mu=(1^{(m)}, 0^{(n-m)})$ is minuscule. Then $${{\mathcal {Z}}}(x_1, x_2,\ldots,x_n)\subset {{\mathcal {N}}}_n\otimes_{{{{\mathcal {O}}}_{\breve F}}}\bar k .$$
The proof is given in §\[Proof1\]. Assuming this theorem, we may write $$\label{equalk}
R={{\mathbb {F}}}[[t]]/\bar I ,$$ with an ideal $\bar I\subset {{\mathbb {F}}}[[t]]$. Note that at this point, the case $c=0$ is proved completely (in particular, in this case the restriction $n\leq 2p-2$ is not needed). The general case follows from the following theorem which together with Theorem \[speciald\] implies Propositions \[ideal\] and \[length\].
\[idealk\] Assume $F={{\mathbb {Q}}}_p$. If $n\leq 2p-2$, the ideal $\bar I\subset {{\mathbb {F}}}[[t]]$ equals $(t^{c+1})$.
The proof is given in §\[proofidealk\].
Assume $F={{\mathbb {Q}}}_p$. Let $n\leq 2p-2$. Assume that $g$ is regular semisimple and that $L_g^\ast/L_g$ is killed by $\pi$. Then the contribution of $\Lambda\in {\rm Vert}^g(L_g)$ to the intersection multiplicity, if non-zero, is equal to $${\rm mult}(\Lambda)={\rm deg} \, P_{i_0} \frac{a_{i_0}+1}{2}.$$
Of course, we are using here the notation of Proposition \[cardform\]. Now all assertions of Theorem \[mainthm\] are proved.
Proof of Theorem \[speciald\] {#Proof1}
=============================
In this section, we assume $F={{\mathbb {Q}}}_p$. Accordingly, we write ${{\mathbb {Q}}}_{p^2}$ for $E$, and $W$ for ${{\mathcal {O}}}_{\breve F}$. In the terminology of [@KR], we will prove the following theorem.
\[specialdQ\] Let $j_1,\ldots,j_n$ be special homomorphisms such that the corresponding fundamental matrix $T(j_1,\ldots,j_n)$ is equivalent to a diagonal matrix of the form ${\rm diag}(p,\ldots,p, 1,\ldots,1)$ (where $p$ occurs say $m$ times and $1$ occurs $n-m$ times). Let ${{\mathcal {Z}}}=\bigcap_{i=1,\ldots,n}{{\mathcal {Z}}}(j_i)\subseteq \mathcal N$. Then ${{\mathcal {Z}}}$ is an integral scheme. In particular, $p\cdot \mathcal{O}_{{{\mathcal {Z}}}}=0$. In fact, ${{\mathcal {Z}}}$ is equal to ${{\mathcal {V}}}(\Lambda)$ for some vertex lattice $\Lambda$ in $C_n$ of type $m$.
\[DLmod\] We point out that this theorem gives a modular interpretation of the closure ${{\mathcal {V}}}(\Lambda)$ of the Deligne-Lusztig variety ${{\mathcal {V}}}(\Lambda)^o$. Here $\Lambda=\langle x_1,\ldots, x_n\rangle^*$ is the dual of the lattice generated by the elements $x_1,\ldots, x_n$ of $C_n$ corresponding to $j_1,\ldots,j_n$.
First we remark that we may replace $n$ by $m$, cf. [@KR], proof of Lemma 5.2. Hence we may assume that $T(j_1,\ldots,j_n)$ is equivalent to the diagonal matrix ${\rm diag}(p,\ldots,p)$.
We use the following simple fact.
\[pg0\] Let ${{\mathcal {O}}}$ be a complete discrete valuation ring, with uniformizer $\pi$ and algebraically closed residue field $k$. Let $\mathcal Y$ be a (formal) scheme locally (formally) of finite type over Spf $ {{\mathcal {O}}}$ and such that its special fiber $\mathcal{Y}_k$ is regular. Suppose that there does not exist a ${{\mathcal {O}}}/(\pi^2)$-valued point of $\mathcal Y$. Then $\pi\cdot \mathcal{O}_{\mathcal{Y}}=0$ .
Suppose the claim is false. Then there exists a $k$-valued point $x$ of ${{\mathcal {Y}}}$ such that $\pi\neq 0$ in $\mathcal O_{\mathcal Y, x}$. We show that under this assumption there is an ${{\mathcal {O}}}/(\pi^2)$-valued point of ${{\mathcal {Y}}}$ with underlying $k$-valued point $x$. Locally around $x$, the (formal) scheme $\mathcal Y$ is a closed (formal) subscheme of a (formal) scheme $\mathcal X$ which is locally of finite type over ${{\mathcal {O}}}$ and smooth over ${{\mathcal {O}}}$. Let $R=\widehat{\mathcal O}_{\mathcal X, x}$. We may identify $R$ with ${{\mathcal {O}}}$ or with $ {{\mathcal {O}}}[\![x_1,\ldots,x_{N}]\!] $ for some $N\geq 1$. In the first case there is nothing to do, so we assume the second case. Let $I$ be the ideal of $\mathcal Y$ in $R$. Thus we assume that $\pi\not\in I$. It is enough to construct an ${{\mathcal {O}}}$-linear homomorphism $\psi: R/I\rightarrow {{\mathcal {O}}}/(\pi^2)$. Let $\mathfrak m$ be the maximal ideal of $R$ and let $\mathfrak m'$ be the maximal ideal of $R/(\pi)$. Let $l=\dim \mathcal{X}_k-\dim \mathcal Y_k$. Since $\mathcal Y_k$ is regular, we find $l$ distinct elements $q_{1},\ldots,q_{l}\in I$ such that the images of $q_{1},\ldots,q_{l}$ in $R/(\pi)$ generate the ideal of $\mathcal Y_k$ in $R/(\pi)$ and such that the images of $q_{1},\ldots,q_{l}$ in $\mathfrak m' /{\mathfrak m'}^ 2$ are linearly independent. We extend the $q_i$ to a system of generators $q_1,\ldots,q_r$ of $I$. For $i\leq l$, let $y_i=q_i$. We find elements $y_{l+1},\ldots,y_{N}\in \mathfrak m$ such that the images of $y_1,\ldots,y_{N}$ in $\mathfrak m' /{\mathfrak m'}^ 2$ form a basis of $\mathfrak m' /{\mathfrak m'}^ 2$. Thus ${{\mathcal {O}}}[\![x_1,\ldots,x_{N}]\!] = {{\mathcal {O}}}[\![y_1,\ldots,y_{N}]\!] $. Now we consider the ${{\mathcal {O}}}$-linear homomorphism $\phi: {{\mathcal {O}}}[\![y_1,\ldots,y_{N}]\!] \rightarrow {{\mathcal {O}}}$ given by $y_i \mapsto \pi^2$ for all $i$.
[**Claim** ]{}[*The image of the ideal $I$ under $\phi$ is $(\pi^2)$.*]{}
The ideal $\phi(I)$ is generated by $\phi(q_1),\ldots,\phi(q_r)$. By definition we have $\phi(q_i)=\pi^2$ for $i\leq l$. Now assume that $i > l$. Then $q_{i}=\sum_{k\leq l} c_kq_k + \pi\cdot z$ for suitable elements $c_1,\ldots,c_l,z\in R$ (depending on $i$). Since we assume $\pi\not\in I$, it follows that $z$ is not a unit, hence $z\in \mathfrak m = (\pi,y_1,\ldots,y_{N})$. Hence $\phi(z)\in (\pi)$ and $\phi(q_i)\in (\pi^2)$. This confirms the claim.
Using the claim it follows that $\phi$ induces a ${{\mathcal {O}}}$-linear homomorphism $\psi: R/I\rightarrow {{\mathcal {O}}}/(\pi^2)$ yielding an ${{\mathcal {O}}}/(\pi^2)$-valued point of $\mathcal Y$. But this contradicts our assumption. Hence $\pi\cdot \mathcal{O}_{\mathcal{Y}}=0$.
Taking into account Grothendieck’s infinitesimal characterization of smoothness, the previous lemma gives a purely infinitesimal *sufficient condition* for a (formal) ${{\mathcal {O}}}$-scheme to be a (formal) $k$-scheme.
We will prove Theorem \[specialdQ\] by showing in Proposition \[liftW\] and Corollary \[regu\] that ${{\mathcal {Z}}}$ satisfies the hypotheses of the previous lemma.
For any (formal) $W$-scheme $S$, we denote by $S_p$ its special fiber.
\[liftW\] Let $j_1,\ldots,j_n$ and ${{\mathcal {Z}}}$ be as in the theorem. Then ${{\mathcal {Z}}}$ does not have a $W/(p^2)$-valued point.
We may assume that $j_1,\ldots,j_n$ all have valuation $1$ and are all perpendicular to each other. We also first assume that $n>1$. Suppose there was a $W/(p^2)$-valued point $\varrho$ of ${{\mathcal {Z}}}$. Let $M$ be the Dieudonné module of the underlying ${{\mathbb {F}}}$-valued point. Let $M_{W/(p^2)}=M\otimes_W W/(p^2)$. We obtain a Hodge filtration $\mathcal{F}\hookrightarrow M_{W/(p^2)}$ corresponding to $\varrho$, and lifting the Hodge filtration of the underlying ${{\mathbb {F}}}$-valued point. From the $\mathbb Z_{p^2}$-action we get a decomposition $\mathcal{F}=\mathcal{F}_0\oplus\mathcal{F}_1$, where $\mathcal{F}_0 $ is free of rank of rank $n-1$ and $\mathcal{F}_1$ is free of rank $1$. Let $x_i=j_i(\overline{1}_0)\in M_0$, where we are using the notation of [@KR]. We denote the image of $x_i$ in $M_{W/(p^2)}$ by $\overline{x}_i$. Then it follows that $\overline{x}_i\in \mathcal{F}_0$. Let $\overline{f}_1,\ldots,\overline{f}_{n-1}$ be a basis of $\mathcal{F}_0$. Let $f_i\in M_0$ be a lift of $\overline{f}_i$, and choose $f_n\in M_0$ such that $f_1,\ldots,f_n$ is a basis of $M_0$. Let $\widehat{x}_i$ be the image of $x_i$ in the span of $f_1,\ldots,f_{n-1}$ (viewed as a quotient of $M_0$). Then $\widehat{x}_1,\ldots,\widehat{x}_n$ are linearly dependent, i.e., $\sum_{i}c_i\widehat{x}_i=0$ for suitable $c_i\in W$, which are not all zero. We may assume that the valuation of $c_n$ is minimal among the valuations of the $c_i$. Dividing by $-c_n$ we may therefore assume that $\widehat{x}_n=\sum_{i<n}c_i\widehat{x}_i$. Therefore $x_n=\sum_{i<n}c_i{x_i}+cf_n$ for some $c\in W$. Since the image of $x_n-\sum_{i<n}c_i{x_i}=cf_n$ in $M_{W/(p^2)}$ lies in $\mathcal{F}_0 $, it follows that $c$ is divisible by $p^2$.
Now for any $i \neq n$ we have $0=\{x_n,x_i\}=c_i\{x_i,x_i\}+c\{f_n,x_i\}=c_ip+c\{f_n,x_i\}$. Here $\{ , \}$ is the hermitian form on $C=(M_0\otimes {{\mathbb {Q}}})^{V^{-1}F}$ as in §3. Since $x_i\in M_0^*$, it follows that $\{f_n,x_i\}$ is integral. Since further $c$ is divisible by $p^2$, it follows that $c_i$ is divisible by $p$. It follows that $x_n/p\in M_0^* $ so that $j_n$ has valuation bigger than $1$, a contradiction which completes the proof in the case $n>1$.
Finally, we observe that this reasoning also works for $n=1$, since it shows that in this case $x_n=cf_n$, where $c$ is divisible by $p^2$ so that $j_n$ cannot have valuation $1$.
\[supergen\] Let $n$ be odd. Let $x$ be a ${{\mathbb {F}}}$-valued point of $\mathcal{N}$. The following conditions are equivalent.
\(i) $x$ lies on only one irreducible component of $\mathcal N_{\rm red}$.
\(ii) No special cycle of valuation $0$ passes through $x$.
\(iii) The Dieudonné module modulo $p$ of $x$ is of type $\mathbb B(n)$, in the sense of [@VW], §3 (cf. also the beginning of the proof of Theorem \[prep\] below).
We remark that (ii) and (iii) both imply that $n$ is odd. A point satisfying (ii) is called *super-general*.
The equivalence of (ii) and (iii) follows from [@BW], Proposition 3.6 and Lemma 4.1.
Next we prove the equivalence of (i) and (ii). The point $x$ lies on two irreducible components if and only if there are two vertex lattices $\Lambda_1$ and $\Lambda_1'$ of type $n$ in $C_n$ such that $x$ is a ${{\mathbb {F}}}$-valued point of the corresponding irreducible components $\mathcal{V}(\Lambda_1)$ and $\mathcal{V}(\Lambda_1')$. This is equivalent to the statement that $x\in \mathcal{V}(\Lambda)({{\mathbb {F}}})$ for some vertex lattice $\Lambda$ of type $t<n$. (Given $\Lambda_1$ and $\Lambda_1'$ define $\Lambda$ as $\Lambda_1\cap \Lambda_1'$.) We claim that for any vertex lattice $\Lambda$ of type $t<n$ there is a special homomorphism $j$ of valuation $0$ with $ \mathcal{V}(\Lambda)\subseteq
{{\mathcal {Z}}}(j)$. To see this, note that $\Lambda$ has an orthogonal basis $e_1,\ldots,e_n$ such that $\{e_i,e_i\}=1/p$ for $i\leq t$ and $\{e_i,e_i\}=1$ for $i> t$. Let $j$ be the special homomorphism with $j(\bar{1}_0)=e_{t+1}$. Then for the hermitian form $h(\,,\,)$ on the space ${{\mathbb {V}}}$ of special homomorphisms, we have $h(j,j)=1$ and $y=j(\bar{1}_0)\in \Lambda^*$. This shows the claim, which implies $(ii)\!\!\!\implies\!\! \!(i)$. For the reverse implication, assume that $\Lambda$ is a vertex lattice such that $\Lambda^*$ contains a vector $y$ with $\{y, y\}=1$. Then $\Lambda$ cannot be of type $n$. This shows that ${{\mathcal {V}}}(\Lambda)$ cannot be contained in special divisor of valuation $0$.
\[prep\] Let $n\geq 3$. Let $x$ be a super-general ${{\mathbb {F}}}$-valued point of $\mathcal{N}$. Then the following statements hold.
\(i) For any special homomorphism $j$ with $x\in {{\mathcal {Z}}}(j)({{\mathbb {F}}}) $ and $x\not\in {{\mathcal {Z}}}(j/p)({{\mathbb {F}}}), $ the special fiber ${{\mathcal {Z}}}(j)_p$ is regular at $x$.
\(ii) Let $j_1,\ldots,j_n$ be a basis of the ${{\mathbb {Z}}}_{p^2}$-module of special homomorphisms $j$ with $x\in {{\mathcal {Z}}}(j)({{\mathbb {F}}})$. Then the intersection $\bigcap{{\mathcal {Z}}}(j_i)_p$ is regular at $x$.
It is enough to show the claims of the theorem in $\widehat{\mathcal{O}}_{\mathcal{N}_p,x}$ instead of $\mathcal{O}_{\mathcal{N}_p,x}$. Let $(X, \iota, \lambda)$ be the $p$-divisible group with its $\mathbb Z_{p^2}$-action and its $p$-principal polarization corresponding to $x$, and let $M$ be its Dieudonné module. Let $M_p=M_{p,0}\oplus M_{p,1}$ be the reduction mod $p$ of $M.$ Since we assume that no special cycle of valuation $0$ passes through $x$, it follows that $M_p$ is isomorphic to $\mathbb{B}(n)$ and that $n$ is odd. Here we are using the notation of [@VW], §3.1. This means that we find bases $\overline{e}_1,\ldots,\overline{e}_n$ of $M_{p,0}$ and $\overline{f}_1,\ldots,\overline{f}_n$ of $M_{p,1}$ such that $V( \overline{f}_i)=(-1)^i\overline{e}_{i+1}$ for $i<n$, $V(\overline{e}_n)=\overline{f}_1 $, $F(\overline{f}_i)=(-1)^i\overline{e}_{i-1}$ for $i \geq 3$, $F(\overline{f}_2)=-\overline{e}_1$, $F(\overline{e}_1)=\overline{f}_n$ and for the induced alternating form we have $\langle \overline{e}_i, \overline{f}_j \rangle= \varepsilon_i\delta_{ij}$, where $\varepsilon_i=1$ for $i=1$ and $\varepsilon_i=-1$ for $i>1$.
We find lifts $e_i\in M_0$ of $ \overline{e}_i$ and lifts $f_i\in M_1$ of $ \overline{f}_i$ such that still $\langle{e_i},{f_j} \rangle= \varepsilon_i\delta_{ij}$. Denote by $T$ the $W$-span of $e_1,f_2,f_3,\ldots,f_n$ and by $L$ the $W$-span of $f_1,e_2,e_3,\ldots,e_n$. Then $$M=L\oplus T, \ \ \ VM=L\oplus pT.$$ Let $h_1=e_1,h_2=f_2,\ldots,h_n=f_n,h_{n+1}=f_1,h_{n+2}=e_2,\ldots,h_{2n}=e_n.$ Define the matrix $(\alpha_{ij})$ by $$Fh_j=\sum_i \alpha_{ij}h_i \text{ for } j=1,\ldots,n, \\$$ $$V^{-1}h_j=\sum_i \alpha_{ij}h_i \text{ for } j=n+1,\ldots,2n.$$ Since we know the action of $F$ resp. $V$ on the $\overline{e}_i$ and $\overline{f}_i$, we can conclude that $V^{-1}(f_1)=e_n + \sum_{i<n}x_ie_i+pe$ for suitable $x_i\in W$ and $e\in M_0$. Similarly for $i\geq 2$ we have $ V^{-1}(e_i)=(-1)^{i-1}f_{i-1}+y_if_n+pg_i$ for suitable $y_i\in W$ and $g_i\in M_1$.
Thus $(\alpha_{ij})$ is of the form $$(\alpha_{ij})=
\left(
\begin{array}{cccccc|ccccccccccccccccc}
&-1&&&&&x_1&&&&\\
&&&&&&&&1&&& \\
&&&&&&&&&-1&&\\
&&&&&&&&&&\ddots \\
&&&&&&&&&&&-1 \\
&&&&&&&&&&&&1 \\
1&&&&&&&y_2&y_3&y_4&\ldots&y_{n-1}&y_n\\
\hline
&&&&&&&-1&&&&\\
&&-1&&&&x_2&&&&\\
&&&1&&&x_3&&&&\\
&&&&\ddots&&\vdots&&&&\\
&&&&&-1&x_{n-1}&&&&&\\
&&&&&&1&&&&&\\
\end{array}
\right) +pD,$$ where $D$ has entries in $W$ and also maps $M_0$ to $M_1$ and $M_1$ to $M_0$ . (The vertical and horizontal lines divide the first matrix into four $n\times n$ matrices, and only non-zero entries are displayed.) It follows (see [@Zi], p. 48) that the universal deformation of $X$ over ${{\mathbb {F}}}[\![ t_{11},\ldots,t_{nn}]\!]$ corresponds to the display $(L\oplus T)\otimes W({{\mathbb {F}}}[\![t_{11},\ldots,t_{nn} ]\!]) $ with matrix $(\alpha_{ij})^{\rm univ}$ (wrt. the basis $h_1,\ldots, h_{2n}$ and with entries in $W({{\mathbb {F}}}[\![t_{11},\ldots,t_{nn} ]\!])$ given by $$(\alpha_{ij})^{\rm univ}=
\begin{pmatrix}
1&&&[t_{11}]&\hdots&[t_{1n}] \\
&\ddots&&\vdots&\ddots&\vdots \\
&&1&[t_{n1}]&\hdots&[t_{nn}] \\
&&&1&& \\
&&&&\ddots& \\
&&&&&1 \\
\end{pmatrix} \cdot
(\alpha_{ij}).$$ Here $[t]$ denotes the Teichmüller representative of $t$. Now let $A^{'}=W[\![t_{11},\ldots,t_{nn}]\!]$ and let $R^{'}={{\mathbb {F}}}[\![t_{11},\ldots,t_{nn}]\!]$. We extend the Frobenius $\sigma $ on $W$ to $A^{'}$ by setting $\sigma(t_{ij})=t_{ij}^p.$ Let $R$ be the completed universal deformation ring (in the special fiber) of $X$, together with its ${{\mathbb {Z}}}_{p^2}$-action and its $p$-principal polarization. Then $R$ is a quotient of $R^{'}$ by an ideal $J$. Using the fact that $(\alpha_{ij})^{\rm univ}$ has to respect the ${{\mathbb {Z}}}/2$-grading, it is easy to see that the ideal describing the deformation of the ${{\mathbb {Z}}}_{p^2}$-action is $(t_{11},t_{ij})_{i,j\neq 1}$. Using this, it is easy to see that $J=\big ((t_{11},t_{ij})_{i,j\neq 1}, (t_{1i}-t_{i1})_{i\leq n}\big)$. (Compare also [@G], p. 231.) Thus we may identify $R$ with the ring ${{\mathbb {F}}}[\![t_{2},\ldots,t_{n}]\!]$, where $t_i$ corresponds to the image of $t_{1i}$ in $R^{'}/J$. We also define $A=W[\![t_{2},\ldots,t_{n}]\!]$. For any $m \in {{\mathbb {N}}}$, denote by $\mathfrak{a}_m$ resp. $\mathfrak{r}_m$ the ideal in $A$ resp. in $R$ generated by the monomials $t_2^{a_2}\cdot \ldots \cdot t_n^{a_n}$, where $a_i\geq 0$ and $\sum a_i = m$. Hence $\mathfrak{r}_m=\mathfrak m^m, $ where $\mathfrak m$ denotes the maximal ideal of $R$. Let $A_m=A/\mathfrak{a}_m$ and $R_m=R/\mathfrak{r}_m$. Then $A^{'}$ is a frame for $R^{'}$, resp. $A$ is a frame for $R$, resp. $A_m$ is a frame for $R_m$. (See [@Zi2] for the definition of frames.)
For an $A^{'}$-$R^{'}$-window $(M^{'}, M^{'}_1, \Phi^{'},\Phi^{'}_1)$, let $M_1^{'^{\sigma}}=A^{'}\otimes_{A^{'}, \sigma}M_1^{'}$ and denote by $\Psi^{'}: M_1^{'^{\sigma}} \rightarrow M^{'}$ the linearization of $\Phi^{'}_1$. It is an isomorphism of $A^{'}$-modules. Denote by $\alpha^{'}:M_1^{'} \rightarrow M_1^{'^{\sigma}}$ the composition of the inclusion map $M_1^{'} \hookrightarrow M^{'} $ followed by $\Psi^{'^{-1}}$. In this way, the category of formal $p$-divisible groups over $R^{'}$ becomes equivalent to the category of pairs $(M_1^{'}, \alpha^{'})$ consisting of a free $A^{'}$-module of finite rank and an $A^{'}$-linear injective homomorphism $\alpha^{'}:M_1^{'} \rightarrow M_1^{'^{\sigma}}$ such that Coker $\alpha^{'}$ is a free $R^{'}$-module, and satisfying the [*nilpotence condition*]{} [@Zi]. Since we will only consider deformations of formal $p$-divisible groups, the nilpotence condition will be fulfilled automatically, and we will ignore it, comp. also [@KR], section 8. A corresponding description holds for the category of formal $p$-divisible groups over $R$ resp. $R_m$.
In the sequel we are using notation that is customary in Zink’s theory. The notation $M_1$ conflicts with its usage when taking the degree-$1$-component of $M$ under the ${{\mathbb {Z}}}/2$-grading. Henceforward we will write $M^1$ for the degree-$1$-component.
Let $(\beta_{ij})^{\rm univ}$ be the matrix over $A^{'}$ which is obtained from $(\alpha_{ij})^{\rm univ}$ by replacing the $[t_i]$ by $t_i$ and by multiplying the last $n$ rows by $p$. We consider the $A^{'}$-$R^{'}$-window $(M^{'},M^{'}_1, \Phi^{'},\Phi^{'}_1)$ given by $
M^{'}=M\otimes A^{'}, \ M_1^{'}=VM\otimes A^{'}, \
\Phi^{'}=(\beta_{ij})^{\rm univ}\sigma, \ \ \Phi^{'}_1=\frac{1}{p}\cdot \Phi^{'},
$ where the matrix of $\Phi^{'}$ is described in the basis $h_1,\ldots,h_{2n}$. The corresponding display is the universal display described above (easy to see using the procedure described on p.2 of [@Zi2]). Hence $(M^{'},M^{'}_1, \Phi^{'},\Phi^{'}_1)$ is the universal window. Using this and the form of the ideal $J$ given above, one checks that the map $\alpha: M_1 \rightarrow M_1^{\sigma}$ corresponding to the $A$-$R$ window of the universal defomation of $(X, \iota, \lambda)$ (which is the base change of $(M_1^{'}, \alpha^{'})$) can be written as follows (using the bases $pe_1,pf_2,..,pf_n,$ $ f_1,e_2,\ldots, e_n$ resp. $p(1\otimes e_1),p(1\otimes f_2),\ldots,p(1\otimes f_n),1\otimes f_1, 1\otimes e_2,\ldots,1\otimes e_n$ ) $$\tilde \alpha=
\left(
\begin{array}{cccccc|ccccccccccccccccc}
&-py_3&py_4&\hdots&-py_n&p&-t_n+y_2+\sum_{i\geq 3}(-1)^{i-1}y_it_{i-1}&&&&\\
-p&&&&&&&t_2&t_3&\hdots&t_{n-1}&x_1+t_n \\
&&&&&&&-1&&&&x_2\\
&&&&&&&&1&&&x_3 \\
&&&&&&&&&\ddots&&\vdots \\
&&&&&&&&&&-1&x_n \\
\hline
&&&&&&&&&&&1 \\
&&&&&&-1&&&&\\
&p&&&&&-t_2&&&&\\
&&-p&&&&t_3&&&&\\
&&&\ddots&&&\vdots&&&&\\
&&&&p&&-t_{n-1}&&&&&\\
\end{array}
\right) +B.$$ Here $B=\begin{pmatrix}
B_{11}&B_{12}\\
B_{21}&B_{22}\\
\end{pmatrix}
$, where $B_{ij}$ has size $n\times n$, and $B_{11}$ and $ B_{21}$ have entries in $p^2A$ and $B_{12}$ and $B_{22}$ have entries in $pA$. Rewriting this in the bases $pe_1,e_2,..,e_n,$ $ f_1,pf_2,\ldots, pf_n$ resp. $p(1\otimes e_1),1\otimes e_2,\ldots,1\otimes e_n,1\otimes f_1, p(1\otimes f_2),\ldots,p(1\otimes f_n)$ we obtain a block matrix $
\alpha=\begin{pmatrix}
&U\\
\tilde{U}
\end{pmatrix},
$ where $U$ is of the form $$U=
\left(
\begin{array}{ccccccccccccccc}
-t_n+y_2+\sum_{i\geq 3}(-1)^{i-1}y_it_{i-1}&-py_3&py_4&\hdots&-py_n&p\\
-1&&&&\\
-t_2&p&&&\\
t_3&&-p&&\\
\vdots&&&\ddots&\\
-t_{n-1}&&&&p\\
\end{array}
\right) +pB_U,$$ where $B_U$ has entries in $A$ and, in the last $n-1$ rows, even has entries in $pA$, and $\tilde{U}$ is of the form $$\tilde{U}=
\left(
\begin{array}{ccccccccccccccc}
&&&&&1\\
-p&t_2&t_3&\hdots&t_{n-1}&x_1+t_n\\
&-1&&&&x_2\\
&&1&&&-x_3\\
&&&\ddots&&\vdots\\
&&&&-1&x_{n-1}\\
\end{array}
\right) +pB_{\tilde{U}},$$ where $B_{\tilde{U}}$ has entries in $A$ and in the first row even has entries in $pA$.
The corresponding universal $p$-divisible groups over $R_m$ correspond to the pairs $(M_1(m), \alpha(m))$ obtained by base change from $(M_1, \alpha)$.
Consider the $p$-divisible group $\overline{\mathbb{Y}}$ with its Dieudonné module $\overline{\mathbb{M}}=W\overline{1}_0\oplus W\overline{1}_1.$ Let $\overline{M}_1=W\overline{1}_0\oplus W p\overline{1}_1$ and let $n_0=\overline{1}_0$ and $n_1=p\overline{1}_1$. Then $\overline{\mathbb{Y}}$ corresponds to the pair $(\overline{M}_{1}, \beta)$ where $\beta(n_0)=1\otimes n_1$ and $\beta(n_1)=-p\otimes n_0$. By base change $W \rightarrow A$ resp. $W \rightarrow A_m$ we obtain pairs $({\overline{M}_{1}}_{R}, \beta)$ resp. $({\overline{M}_{1}}_{R_m}, \beta)$ corresponding to the constant $p$-divisible group $\overline{\mathbb{Y}}$ over $R$ resp. $R_m$. We denote the matrix of $\beta$ by $S$, hence $$S=
\begin{pmatrix}
0&-p\\
1&0\\
\end{pmatrix}.$$
Now let $j$ be as in the statement of the theorem, i.e. $x\in {{\mathcal {Z}}}(j)({{\mathbb {F}}})$ but $x\notin {{\mathcal {Z}}}(j/p)({{\mathbb {F}}})$. We want to investigate the ideal in $R$ describing the maximal deformation of the homomorphism $j$, and its image in $R_m$. We will determine explicitly the image of this ideal in $R_p.$
The map $j$ corresponds to a map $ j(1):\overline{M}_{1} \rightarrow M_1(1) $ such that the following diagram commutes, $$\xymatrix{ \overline{M}_{1} \ar[d]_{j(1)} \ar[r]^{\beta}& {\overline{M}}_{1}^{\sigma}
\ar[d]^{\sigma(j(1))} \\M_1(1)\ar[r]_{\alpha(1)} & M_1(1)^{\sigma} .}$$ Then $j$ lifts over $R_m$ if and only if there is a lift $j(m)$ of $j(1)$ such that the following diagram commutes,
$$\xymatrix{ {\overline{M}_{1}}_{R_m} \ar[d]_{j(m)} \ar[r]^{\beta}& {{\overline{M}}^{\sigma}_{1 {R_m}}}
\ar[d]^{\sigma(j(m))} \\M_1(m)\ar[r]_{\alpha(m)} & M_1(m)^{\sigma} .}$$
We write $j(\overline{1}_0)=a_1 \cdot pe_1+a_2 \cdot e_2+\ldots+a_n\cdot e_n.$ We also write $j(1)=(X(1), Y(1)).$ Then $X(1)$ can be written in the above basis as $$X(1)=
\begin{pmatrix}
a_1&0 \\
\vdots&\vdots\\
a_n&0\\
\end{pmatrix}.$$ Similarly we write $j(p\overline{1}_1)=b_1 \cdot f_1+b_2 \cdot pf_2+\ldots+b_n\cdot pf_n$ and $$Y(1)=
\begin{pmatrix}
0&b_1 \\
\vdots&\vdots\\
0&b_n\\
\end{pmatrix}.$$
Since $j$ commutes with the Frobenius operator, we have $j(p\overline{1}_1)=FjF^{-1}(p\overline{1}_1)=Fj(\overline{1}_0)$. Using the matrix $(\alpha_{ij})$ we see that $$b_1=-pa_2^{\sigma}+p^2s_1,\, b_i=(-1)^{i}a_{i+1}^{\sigma}+ps_i \text{ for } 2\leq i\leq n-1,\text{ and } b_n=a_1^{\sigma}+\sum\nolimits_{i\geq 2}a_i^{\sigma}y_i+ps_n,$$ for suitable elements $s_i\in W.$
Similarly, exploiting the relation $j(\overline{1}_0)=FjF^{-1}(\overline{1}_0)=-\frac{1}{p}Fj(p\overline{1}_1)$, we obtain the system of equations $$a_1=-b_1^{\sigma}x_1/p-b_2^{\sigma}/p+r_1, a_i=-b_1^{\sigma}x_i+(-1)^{i}b_{i+1}^{\sigma}+pr_i \text{ for } 2\leq i\leq n-1,\text{ and }a_n=b_1^{\sigma}+pr_n,$$ for suitable elements $r_i\in W$. The equations $b_1=-pa_2^{\sigma}+p^2s_1$ and $a_1=-b_1^{\sigma}x_1/p-b_2^{\sigma}/p+r_1$ show that $b_1$ and $b_2$ are divisible by $p$. However, not all $b_i$ are divisible by $p$. Indeed, if they were, then, because of $b_i=(-1)^{i}a_{i+1}^{\sigma}+pr_i$ for $2\leq i\leq n-1$, the elements $a_3,\ldots,a_n$ would also be divisible by $p$. Using $a_2=-b_1^{\sigma}x_2+b_{3}^{\sigma}+pr_2$, we see that then also $a_2$ would be divisible by $p$ and, using $b_n=a_1^{\sigma}+\sum_{i\geq 2}a_i^{\sigma}y_i+ps_n$, it finally would follow that also $a_1$ is divisible by $p$. However, this would contradict our assumption that $x\in {{\mathcal {Z}}}(j)({{\mathbb {F}}})\setminus {{\mathcal {Z}}}(j/p)({{\mathbb {F}}})$.
We are looking for liftings $X(m)$ of $X(1)$ and $Y(m)$ of $Y(1)$ over $A_m$ such that $$\label{displayrec}
UY(m)= \sigma (X(m))S \ \text{ and } \ \tilde{U}X(m)= \sigma (Y(m))S .$$ Suppose $m=p^l$, where $l\geq1$, and suppose we have found liftings $X(p^{l-1})$ and $Y(p^{l-1})$ satisfying . For any choice of liftings $X(p^{l})$ and $Y(p^{l})$ of $X(p^{l-1})$ and $Y(p^{l-1})$, the matrices $\sigma(X(p^{l})),$ resp. $\sigma(Y(p^{l}))$ are equal to $\sigma(X(p^{l-1}))$ resp. $\sigma(Y(p^{l-1}))$, interpreted as matrices over $A_{p^l}$. Hence there are liftings $X(p^{l})$ and $Y(p^{l})$ satisfying if and only if the matrices $$U^{-1}\sigma (X(p^{l-1}))S \ \text{ and } \ \tilde{U}^{-1} \sigma (Y(p^{l-1}))S$$ are integral, and in this case $$X(p^l)= \tilde{U}^{-1} \sigma (Y(p^{l-1}))S \ \text{ and } \ Y(p^l)= U^{-1}\sigma (X(p^{l-1}))S.$$ Define now inductively matrices $X_{{{\mathbb {Q}}}}(p^l)$ and $Y_{{{\mathbb {Q}}}}(p^l)$ over $A_{p^l}\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$ as follows: $X_{{{\mathbb {Q}}}}(1)=X(1)$ and $Y_{{{\mathbb {Q}}}}(1)=Y(1)$ and $$X_{{{\mathbb {Q}}}}(p^{l+1})= \tilde{U}^{-1} \sigma (Y(p^{l}))S \ \text{ and } \ Y_{{{\mathbb {Q}}}}(p^{l+1})=U^{-1}\sigma (X(p^{l}))S.$$ (Again $\sigma (X_{{{\mathbb {Q}}}}(p^l))$ and $\sigma (Y_{{{\mathbb {Q}}}}(p^l))$ are well defined over $A_{p^{l+1}}\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$.) It is easy to see that $Y_{{{\mathbb {Q}}}}(p)$ is integral.
Let $\overline{b}_i $ denote the image of $b_i$ in ${{\mathbb {F}}}$. Let $D = \{i\geq 3\mid \overline{b}_i\neq 0\}$. (This set is not empty as we saw above.) Using the form of the matrix $\tilde{U}$ we easily see that $X_{{{\mathbb {Q}}}}(p)$ is of the form $$X_{{{\mathbb {Q}}}}(p)=
\begin{pmatrix}
\frac{1}{p}\sum_{{i}\in D} (-1)^ib_{i}^{\sigma}t_{i-1} & 0\\
0&0\\
\vdots&\vdots\\
0&0\\
\end{pmatrix}+A(p),$$ where $A(p)$ is integral. We claim that the equation of ${{\mathcal {Z}}}(j)_p$ in $R/\mathfrak m^p$ is $\sum_{i\in D} (-1)^i\overline{b}_{i}^{\sigma}t_{i-1} = 0$. Let $t=\sum_{i\in D} (-1)^i\overline{b}_{i}^{\sigma}t_{i-1}$ and lift $t$ to an element $\tilde t\in A$ using Teichmüller lifts of the coefficients of $t$. Then we claim that $\tilde A:=A/((t_2,\ldots,t_n)^p+\tilde t\cdot A)$ is a frame for $R/(\mathfrak m^p+t\cdot R)$. Since $\tilde A$ is isomorphic to $W[\![X_1,\ldots,X_{n-2} ]\!]/(X_1,\ldots,X_{n-2})^p$, it is torsion free as an abelian group. Let $\sigma $ be the endomorphism on $\tilde A$ which extends the Frobenius on $W$ by sending (the images of) the $t_i$ to $0$. Then $\sigma$ induces the Frobenius on $R/(\mathfrak m^p+t\cdot R)$. The ideal $p\cdot \tilde A$ is in an obvious way equipped with a pd-structure. Using this frame, the same calculation as above shows that $j$ lifts over $R/(\mathfrak m^p+t\cdot R)$. Since ${{\mathcal {Z}}}(j)_p$ is a divisor ([@KR], Proposition 3.5), it follows that we can write the equation for ${{\mathcal {Z}}}(j)_p$ in $R/\mathfrak m^p$ in the form $t\cdot s=0$. We have to show that $s$ is a unit. Assume $s$ is not a unit. Then it follows that $j$ lifts over $R/\mathfrak m^2$. For this ring we have the obvious frame $A/(t_2,\ldots,t_n)^2$. Again the same calculation as above shows that $j$ does not lift over $R/\mathfrak m^2$ because $\sum_{{i}\in D} (-1)^ib_{i}^{\sigma}t_{i-1}$ is not divisible by $p$ in $A/((t_2,\ldots,t_n)^2$. Thus the equation of ${{\mathcal {Z}}}(j)_p$ in $R/\mathfrak m^p$ is indeed $\sum_{i\in D} (-1)^i\overline{b}_{i}^{\sigma}t_{i-1} = 0$. Claim i) of the theorem follows.
Now we come to claim ii). Let $\mathfrak n$ be the maximal ideal in $\widehat{\mathcal O}_{{{\mathcal {Z}}}, x}$. We need to show ${\rm dim}\,\mathfrak n/\mathfrak n^2={\rm dim}\, {{\mathcal {Z}}}_p$. Since $x$ is super-general, it lies on a unique irreducible component of $\mathcal N_{\rm red}$, of the form $\mathcal V(\Lambda)$, where $\Lambda$ is a vertex lattice of type $n$, cf. [@KR], §4. Furthermore, $x\in {{\mathcal {Z}}}(j)({{\mathbb {F}}})$ if and only if $j(\bar 1_0)\in \Lambda^*=p\Lambda$. By Lemma \[supergen\], ${{\mathrm{ord}}}_p (h(j_i,j_i))\geq 1$ for all $i$, and by the results of [@KR], §4, the dimension of ${{\mathcal {Z}}}(j_i)_{\rm red}$ is $(n-1)/2$ at $x$ for all $i$. Hence ${{\mathcal {Z}}}_{\rm red}=\mathcal V(\Lambda)$ locally at $x$, and has dimension $(n-1)/2$. We will show that ${\rm dim}\,\mathfrak n/\mathfrak n^2=(n-1)/2$, which will prove that ${{\mathcal {Z}}}_p={{\mathcal {Z}}}_{\rm red}$ at $x$, and will finish the proof.
We saw above that the equation of ${{\mathcal {Z}}}(j)$ in $\mathfrak m/\mathfrak m^2$ is a linear equation of the form $$\sum_{i\geq 1} (-1)^i\overline{b}_i(j)^{\sigma}t_{i-1} = 0,$$ where the coefficients $ \overline{b}_i(j)$ arise by expressing $j(p\bar 1_1)$ in terms of a specific basis of the ${{\mathbb {F}}}$-vector space $VM^0/pVM^0$ with $\overline{b}_1(j)=\overline{b}_2(j)=0$. We have to see that the rank of this system of linear equations, as $j(\bar 1_0)$ varies through $p\Lambda$, is equal to $(n-1)/2$.
However, as $j$ varies, the elements $j(p\bar 1_1)$ generate the $W$-lattice $pV(\Lambda\otimes_{{{\mathbb {Z}}}_{p^2}} W)$ inside $VM^0$, and the dimension of $pV(\Lambda\otimes_{{{\mathbb {Z}}}_{p^2}} W)/p VM^0$ is equal to $(n-1)/2$.
\[regu\] Let $j_1,\ldots,j_n$ and ${{\mathcal {Z}}}$ be as in Theorem \[specialdQ\]. Then the special fiber ${{\mathcal {Z}}}_p$ of ${{\mathcal {Z}}}$ is regular.
We use induction on $m$ (notation as in Theorem \[specialdQ\]). We observe that $m$ is always odd. For $m=1$ there is nothing to do, since for $n=1$ we have $\mathcal{N}_p\cong {{\mathrm{Spec}}}\, {{\mathbb {F}}}$. If $x$ is super-general, the assertion follows from Theorem \[prep\].
Now assume that $x$ is not super-general. By Lemma \[supergen\], there is a special cycle ${{\mathcal {Z}}}(j_0)$ of valuation $0$ passing through $x$. We consider the ${{\mathbb {Z}}}_{p^2}$-submodule $J$ of $\mathbb V$ generated by $j_0,j_1,\ldots,j_n$ and we define ${{\mathcal {Z}}}(J)=\cap_{i=0,\ldots,n}{{\mathcal {Z}}}(j_i)$. (Recall that $\mathbb V $ is the ${{\mathbb {Q}}}_{p^2}$-space of special homomorphisms, with hermitian form $h(\,,\,)$.)
[**Claim**]{} [*There is an orthogonal ${{\mathbb {Z}}}_{p^2}$-basis $b_1,\ldots,b_n $ of $J$ with $h( b_i,b_i ) \in \{1,p\}$ for all $i$.*]{}
We denote by $U$ the ${{\mathbb {Z}}}_{p^2}$-submodule of $\mathbb V$ generated by $j_1,\ldots,j_n$, so that $U\subseteq J \subseteq \mathbb V$. Both $U$ and $J$ are free ${{\mathbb {Z}}}_{p^2}$-modules of rank $n$. Let $U^{\vee}$ (resp. $J^{\vee}$) be the set of $j\in \mathbb V$ with $h(j,c)\in {{\mathbb {Z}}}_{p^2}$ for all $c\in U$ (resp. all $c\in J$). It follows that $pU^{\vee}=U$. Let $c_1,\ldots,c_n$ be an orthogonal basis of $J$, and denote by $\alpha_i$ the valuation of $h(c_i,c_i)$. Assume now that $\alpha_i\geq 2$ for some $i$. Then it follows that $p^{-1}c_i\in pJ^{\vee}\subseteq pU^{\vee}=U$. Hence $c_i\in pU\subseteq pJ$. But an element of $pJ$ cannot be a member of a basis of $J$. Hence all $c_i$ have valuation $0$ or $1$. Thus the claim follows.
The number of $b_i$ which have valuation $0$ is positive because there is an element of valuation $0$ in $J$ (e.g. $j_0$). Hence by the induction hypothesis ${{\mathcal {Z}}}(J)_p$ is regular. We know that the dimension of ${{\mathcal {Z}}}_p$ is the dimension of the supersingular locus of $\mathcal{N}_p$. Therefore the dimension of ${{\mathcal {Z}}}(J)_p={{\mathcal {Z}}}_p \cap {{\mathcal {Z}}}(j_0)_p$ is smaller than the dimension of ${{\mathcal {Z}}}_p$. Since ${{\mathcal {Z}}}(J)_p$ is regular, it follows that ${{\mathcal {Z}}}_p$ is regular at $x$.
Consider the isogeny $\alpha: \overline{{{\mathbb {Y}}}}^n\to X$ defined by $(j_1,\ldots,j_n)$. The kernel of $\alpha$ is a finite flat group scheme $G$ of rank $p^m$, of type $(p, p,\ldots, p)$ and equipped with an action of ${{\mathbb {F}}}_{p^2}$. As Zink pointed out, if $m=1$, such a group scheme can only exist over a base $Y$ with $p\cdot{{\mathcal {O}}}_Y= 0$. (He uses Oort-Tate theory to show this.) We do not know whether Theorem \[specialdQ\] can be seen from this angle in the general case.
Proof of Theorem \[idealk\] {#proofidealk}
===========================
Choose a $W$-basis of $B$ as follows. Choose $e_0, e_1 \ldots, e_c \in B$ such that $e_1,\ldots,e_c$ project to vectors in $B^*/pB$ and $e_0$ is in $B\setminus B^*$, and such that the images of these vectors in $B/L$ span the Jordan block relative to the eigenvalue $\lambda$ of $\bar{g}$ in $U/U^\perp$. Next let $l$ be the minimal integer $\geq 0$ such that $g^lu\in pB$. For $l>i \geq 0$ denote by $e_{c+1+i}$ the element $g^iu\in L$. Finally, we complete this to a basis by lifting vectors which project to Jordan blocks other than $\lambda$. These last vectors we call $e_{c+l+1},\ldots,e_{n-1}$. We therefore obtain the following identities modulo $L$, $$\label{basis1}
ge_0\equiv\lambda e_0+e_1, ge_1\equiv \lambda e_1+e_2,\ldots,ge_{c-1}\equiv\lambda e_{c-1}+e_c, ge_c\equiv\lambda e_c .$$ If $m>c+l$ and $e_m,\ldots,e_{m'}$ give rise to a Jordan block of $g$ in $B/L$ to an eigenvalue $\mu$, then $$\label{basis2}
ge_m\equiv\mu e_m+e_{m+1},\ldots,ge_{m'-1}\equiv\mu e_{m'-1}+e_{m'}, ge_{m'}\equiv\mu e_{m'} .$$ By perhaps changing the $e_i$ by adding a suitable element of $L$, we may (and will) assume that these congruences also hold modulo $pB$.
The vectors $e_i$ form a $W$-basis of $M_0=B$, where $M$ is the Dieudonné module of $X$, the $p$-divisible group belonging to $B$. Let $f_0,\ldots,f_{n-1}$ be a basis of $M_1$ such that $\langle e_i,f_j\rangle=\delta_{ij}$. Denote by $T$ the $W$-span of $e_0,f_1,f_2,\ldots,f_{n-1}$ and by $L'$ the $W$-span of $f_0,e_1,e_2,\ldots,e_{n-1}$. (We only write $L'$ instead of the usual notion $L$ since the letter $L$ is already used.) Then $$M=L'\oplus T, \ \ \ VM=L'\oplus pT.$$ Let $h_1=e_0,h_2=f_1,\ldots,h_n=f_{n-1},h_{n+1}=f_0,h_{n+2}=e_1,\ldots,h_{2n}=e_{n-1}.$ Define the matrix $(\alpha_{ij})$ by $$Fh_j=\sum_i \alpha_{ij}h_i \text{ for } j=1,\ldots,n, \\$$ $$V^{-1}h_j=\sum_i \alpha_{ij}h_i \text{ for } j=n+1,\ldots,2n.$$ It follows (see [@Zi], p. 48) that the universal deformation of $X$ over ${{\mathbb {F}}}[\![ t_{11},\ldots,t_{nn}]\!]$ corresponds to the display $(L'\oplus T)\otimes W({{\mathbb {F}}}[\![t_{11},\ldots,t_{nn} ]\!]) $ with matrix $(\alpha_{ij})^{\text{univ}}$ (wrt. the basis $h_1,\ldots,h_{2n}$ and with entries in $W({{\mathbb {F}}}[\![t_{11},\ldots,t_{nn} ]\!])$ given by $$(\alpha_{ij})^{\text{univ}}=
\begin{pmatrix}
1&&&[t_{11}]&\hdots&[t_{1n}] \\
&\ddots&&\vdots&\ddots&\vdots \\
&&1&[t_{n1}]&\hdots&[t_{nn}] \\
&&&1&& \\
&&&&\ddots& \\
&&&&&1 \\
\end{pmatrix} \cdot
(\alpha_{ij}).$$ Here the $[t_{i j}]$ denote the Teichmüller representatives of the $t_{i j}$. Now let $A^{'}=W[\![t_{11},\ldots,t_{nn}]\!]$ and let $R^{'}={{\mathbb {F}}}[\![t_{11},\ldots,t_{nn}]\!]$. We extend the Frobenius $\sigma $ on $W$ to $A^{'}$ putting $\sigma(t_{ij})=t_{ij}^p.$ Let $R$ be the completed universal deformation ring (in the special fiber) of $X$ together with the ${{\mathbb {Z}}}_{p^2}$-action and the $p$-principal polarization. Then $R$ is a quotient of $R^{'}$ by an ideal $\mathfrak J$. Using the fact that $(\alpha_{ij})^{\text{univ}}$ has to respect the ${{\mathbb {Z}}}/2$ grading, it is easy to see that the ideal describing the deformation of the ${{\mathbb {Z}}}_{p^2}$-action is $(t_{11},t_{ij})_{i,j\neq 1}$. Using this, it is easy to see that $\mathfrak J=((t_{11},t_{ij})_{i,j\neq 1}, (t_{1i}-t_{i1})_{i\leq n})$. (Compare also [@G], p. 231.) Thus we may identify $R$ with the ring ${{\mathbb {F}}}[\![t_{1},\ldots,t_{n-1}]\!]$, where $t_i$ corresponds to the image of $t_{1i+1}$ in $R^{'}/\mathfrak J$. We also define $A=W[\![t_{1},\ldots,t_{n-1}]\!]$.
Let $(\beta_{ij})^{\rm univ}$ be the matrix over $A^{'}$ which is obtained from $(\alpha_{ij})^{\rm univ}$ by replacing the $[t_i]$ by $t_i$ and by multiplying the last $n$ rows by $p$. We consider the $A^{'}$ - $R^{'}$ window $(M^{'},M^{'}_1, \Phi^{'})$ given by $
M^{'}=M\otimes A^{'}, \ M_1^{'}=VM\otimes A^{'}, \
\Phi^{'}=(\beta_{ij})^{\text{univ}}\sigma, \ \,
$ where the matrix of $\Phi^{'}$ is described in the basis $h_1,\ldots,h_{2n}$. (We consider the $h_i$ as elements in $M^{'}$, and they form a basis of $M^{'}$; similarly $ph_1,\ldots,ph_n,h_{n+1},\ldots,h_{2n}$ form a basis of $M_1^{'}$.) The corresponding display is the universal display described above (easy to see using the procedure described on p.2 of [@Zi2]). Hence we call $(M^{'},M^{'}_1, \Phi^{'})$ the universal window.
For an element $f=\sum a_{k_1,\ldots,k_{n-1}}t_1^{k_1}\cdots t_{n-1}^{k_{n-1}}\in R$ we denote by $\tilde f\in A$ the element $\tilde f=\sum \tilde a_{k_1,\ldots,k_{n-1}}t_1^{k_1}\cdots t_{n-1}^{k_{n-1}}$, where $ \tilde a_{k_1,\ldots,k_{n-1}}$ is the Teichmüller lift of $a_{k_1,\ldots,k_{n-1}}$. Thus $\tilde f$ is a lift of $f$.
Let $\mathfrak m$ be the maximal ideal of $R$. In the sequel, we call an ideal $J\subseteq \mathfrak m$ of $R$ [*admissible*]{} if $R/J$ is isomorphic to $ {{\mathbb {F}}}[T]/(T^l)$ for some $l$ with $1\leq l\leq p$. (In particular, $J$ contains $\mathfrak m^p$.)
Let $J$ be admissible. We now construct a frame for $R/J$. If $l=1$ (i.e. $J=\mathfrak m$) then there is nothing to do since $W$ is a frame for ${{\mathbb {F}}}$. Thus we may assume that $l\geq 2$. Let $\mathfrak m_{J}$ be the maximal ideal of $R/J$. The map $R\rightarrow R/J$ induces a surjective linear map of ${{\mathbb {F}}}$-vector spaces $$\phi: \mathfrak m/ \mathfrak m ^2\rightarrow \mathfrak m_{J}/ \mathfrak m_{J}^2.$$ Since $R/J\cong {{\mathbb {F}}}[T]/(T^l)$, the dimension of $\mathfrak m_{J}/ \mathfrak m_{J}^2$ is $1$. Let $\bar X_1\in \mathfrak m/ \mathfrak m^2$ be an element that is not in the kernel of $\phi$. We can extend $\bar X_1$ to a basis $\bar X_1,\ldots,\bar X_{n-1}$ of $\mathfrak m/ \mathfrak m^2$ such that $\bar X_2,\ldots,\bar X_{n-1}$ are in the kernel of $\phi$. Let $X_1\in \mathfrak m$ be any lift of $\bar X_1$. We find lifts $X_2,\ldots, X_{n-1}$ of $\bar X_2,\ldots,\bar X_{n-1}$ which are all contained in $J$. It follows that $J=(X_1^l,X_2,\ldots,X_{n-1})$. Let $\tilde X_i\in A$ be the lifts of the $X_i$ as explained above. Then it follows that $R={{\mathbb {F}}}[\![X_1,\ldots, X_{n-1}]\!]$ and $A=W[\![\tilde X_1,\ldots, \tilde X_{n-1}]\!]$. Define now $\tilde J= ((\tilde X_1)^l,\tilde X_2,\ldots,\tilde X_{n-1})$. Then $(A/\tilde J)/(p)=R/J$, and $A/\tilde J$ is torsion free as an abelian group. The endomorphism on $A$ extending the Frobenius on $W$ by sending $t_i\mapsto t_i^p$ induces an endomorphism on $A/\tilde J$ sending the images of the $t_i$ to $0$ since we assume that $l\leq p$ and hence $(t_1,\ldots,t_{n-1})^p=(\tilde X_1,\ldots,\tilde X_{n-1})^p\subseteq (\tilde X_1^p,\tilde X_2,\ldots,\tilde X_{n-1}) \subseteq \tilde J$. Since furthermore the ideal $p\cdot A/\tilde J$ in $A/\tilde J$ is obviously equipped with a pd-structure, it follows that indeed $A/\tilde J$ is a frame for $R/J$.
Let $I$ be the ideal of $\mathcal{M}\cap \mathcal{Z}(g)$ in $R$. By Lemma \[tangspace\], $R/I\cong {{\mathbb {F}}}[T]/(T^l)$ for some $l\geq 1$. Thus, the ideal $I+ \mathfrak m^p$ is the smallest admissible ideal $J$ such that ${{\mathrm{Spec}}}(R/J) \subseteq \mathcal{M}\cap \mathcal{Z}(g)$.
Let $J\subseteq R$ be an admissible ideal such that $I+ \mathfrak m^p\subseteq J$.
By base change from the universal window $(M^{'},M^{'}_1, \Phi^{'})$ we obtain a window $(M^{(J)},M^{(J)}_1, \Phi^{(J)})$ over $R/J$. We have a ${{\mathbb {Z}}}/2$-grading[^4] $M^{(J)}={M^{(J),0}}\oplus {M^{(J),1}}$ and $M^{(J)}_1=M^{(J),0}_1\oplus M^{(J),1}_1$. We denote by $G^{(J)}$ the matrix of $g$ wrt. the basis of $M^{(J)}$ coming from the above basis of $M'$, and by $G_1^{(J)}$ the matrix of $g$ wrt. the basis of $M_1^{(J)}$ coming from the above basis of $M_1'$. Denote by $\tilde{\Phi}$ the matrix of $\Phi$ wrt. this basis of $M^{(J)}$. Then $G^{(J)}$ and $G_1^{(J)}$ are integral. Since $g$ commutes with $\Phi$, we have $$G^{(J)}=\tilde{\Phi}\sigma(G^{(J)})\tilde{\Phi}^{-1}=\tilde{\Phi}\sigma(G^{(\mathfrak m)})\tilde{\Phi}^{-1}.$$ Here we use that since $\mathfrak m^p\subseteq J$, we have $\sigma(G^{(J)})=\sigma(G^{(\mathfrak m)})$, where we view $G^{(\mathfrak m)}$ (i.e. the matrix of $g$ over $W$ corresponding to the ${{\mathbb {F}}}$-valued point) as a matrix with entries in $A/\tilde J$. Similarly, $$G_1^{(J)}=\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}\tilde{\Phi}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}
\sigma(G_1^{(J)})\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}\tilde{\Phi}^{-1}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}.$$ Again we observe that $\sigma(G_1^{(J)})=\sigma(G_1^{(\mathfrak m)})$. Let $$\tau=\begin{pmatrix}
0&t_1&\hdots&t_{n-1}\\
t_1&0&\hdots&0\\
\vdots&\vdots&&\vdots \\
t_{n-1}&0&\hdots&0 \\
\end{pmatrix} .$$ Then $$\label{Phi}
\tilde{\Phi}=\begin{pmatrix}
1_n&\tau\\
0&1_n
\end{pmatrix}
(\alpha_{ij})\begin{pmatrix}
1_n\\
&p\cdot1_n
\end{pmatrix}.$$
If $S\neq 0$ is a quotient of $R$ by an admissible ideal and if $(M^S,M_1^S,\Phi^S)$ denotes the corresponding window obtained by base change from the universal one, let $g^S: M^S\otimes {{\mathbb {Q}}}\rightarrow M^S \otimes {{\mathbb {Q}}}$ be the map which is induced by the map $g^{(\mathfrak m^p)}:M^{(\mathfrak m^p)}\otimes {{\mathbb {Q}}}\rightarrow M^{(\mathfrak m^p)}\otimes {{\mathbb {Q}}}$ which in turn lifts $g$ and commutes with $\Phi^{(\mathfrak m^p)}$. (Here $(M^{(\mathfrak m^p)}, M_1^{(\mathfrak m^p)}, \Phi^{(\mathfrak m^p)})$ is the window over $R/\mathfrak m^p$ obtained by base change from the universal one, where we use the obvious frame $A/(t_1,...,t_{n-1})^p$ for $R/\mathfrak m^p$.) The map $g^{(\mathfrak m^p)}$ is given by the matrix $G^{(\mathfrak m^p)}$ of $g^{(\mathfrak m^p)}$ with respect to the above basis of $M^{(\mathfrak m^p)}\otimes {{\mathbb {Q}}}$, i.e. $G^{(\mathfrak m^p)}=\tilde{\Phi}\sigma(G^{(\mathfrak m^p)})\tilde{\Phi}^{-1}=\tilde{\Phi}\sigma(G^{(\mathfrak m)})\tilde{\Phi}^{-1}$, where again by abuse of notation we write $G^{(\mathfrak m)}$ for an arbitrary lift of $G^{(\mathfrak m)}$ over $A/(t_1,...,t_{n-1})^p$ and $\tilde{\Phi}$ is the matrix of $\Phi^{(\mathfrak m^p)}$ wrt. the above basis of $M^{(\mathfrak m^p)}\otimes {{\mathbb {Q}}}$. We claim that $g$ lifts over $S$ if and only if $g^S$ maps $M^{S,0}$ into $M^{S,0}$ and $M_1^{S,0}$ into $M_1^{S,0}$. It is obvious that these conditions are necessary. Suppose they are fulfilled.
Since $g$ is unitary, $\langle x,y\rangle=\langle gx,gy\rangle$ for all $x,y\in M$. In other words, for the adjoint $g^{\dagger}$ of $g$ we have $g^{\dagger}=g^{-1}$. It is obvious (from the above formulas for the matrices of the lifts of $g$) that the map $p^{2}g$ lifts to a map $\tilde{g}_1$ over $R/\mathfrak m^p$. By rigidity $\tilde{g}_1^{\dagger}=p^{4}\tilde{g}^{-1}_1$. Let $\langle, \rangle_{p}$ be the alternating form on $M^{(\mathfrak m^p)}\otimes_{{{\mathbb {Z}}}} {{\mathbb {Q}}}$. Then it follows that $\langle p^{-2}\tilde{g}x,y \rangle_{p}=\langle x,(p^{-2}\tilde{g})^{-1}y \rangle_{p}$. Suppose now that $g^S$ maps $M^{S,0}$ into $M^{S,0}$ and $M_1^{S,0}$ into $M_1^{S,0}$. Denoting by $\langle, \rangle_S$ the alternating form on $M^S$, this means that $\langle g^Se_i, f_j \rangle_S$ is integral for all $i,j $ and $\langle g^Se_i, f_1 \rangle_S$ is an integral multiple of $p$ for all $i$. This implies that $\langle e_i, (g^S)^{-1}f_j \rangle_S$ is integral for all $i,j $ and $\langle e_i, (g^S)^{-1}f_1 \rangle_S$ is an integral multiple of $p$ for all $i$. Since the determinant of $g$ (restricted to an endomorphism of $M^1\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$) is a unit in $W$, also the determinant of $g^S$ (restricted to an endomorphism of ${M^{S,1}}\otimes_{{{\mathbb {Z}}}}{{\mathbb {Q}}}$) is a unit in $S$. Hence it follows that also $\langle e_i, (g^S)f_j \rangle_S$ integral for all $i,j $ and $\langle e_i, (g^S)f_1 \rangle_S$ is an integral multiple of $p$ for all $i$. This shows that $g^S$ maps $M^{S,1}$ into $M^{S,1}$ and $M_1^{S,1}$ into $M_1^{S,1}$, confirming the claim.
Let $J\subseteq R$ again be an admissible ideal such that $I+ \mathfrak m^p\subseteq J$. We compute: $$\begin{aligned}
G_1^{(J)}&=\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}\tilde{\Phi}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}
\sigma\big(G_1^{(\mathfrak m)}\big)\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}\tilde{\Phi}^{-1}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}\\
&=\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}\tilde{\Phi}
\sigma\big(G^{(\mathfrak m)}\big)\tilde{\Phi}^{-1}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}.
\end{aligned}$$ Using that $G^{(\mathfrak m)}=(\alpha_{ij})\begin{pmatrix}
1_n& 0\\
0&p1_n
\end{pmatrix}\sigma\big(G^{(\mathfrak m)}\big)\begin{pmatrix}
1_n& 0\\
0&p1_n
\end{pmatrix}^{-1}(\alpha_{ij})^{-1} $ and the equation , we obtain
$$G_1^{(J)}=
\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}
\begin{pmatrix}
1_n&\tau\\
0&1_n
\end{pmatrix}
G^{(\mathfrak m)}\begin{pmatrix}
1_n&-\tau\\
0&1_n
\end{pmatrix}\begin{pmatrix}
p^{-1}1_n\\
&1_n
\end{pmatrix}^{-1}.$$ This matrix respects the ${{\mathbb {Z}}}/2$-grading and it is integral since $I+\mathfrak m^p\subseteq J$. We consider the block matrix of $g=g^{R/\mathfrak m}$ according to the ${{\mathbb {Z}}}/2$-grading of $M$, and denote by $H$ the ‘left upper block’ describing the endomorphism of $M^0$ induced by $g$ wrt the basis $e_0,\ldots,e_{n-1}$. Let $H^{(J)}$ the upper left block of the matrix obtained from $G^{(J)}$ by base change to the basis $e_0,\ldots,e_{n-1}, f_0,\ldots,f_{n-1}$. Similarly, let $H_1^{(J)}$ be the upper left block of the matrix obtained from $G_1^{(J)}$ by base change to the basis $pe_0,e_1\ldots,e_{n-1}, f_0,pf_1,\ldots,pf_{n-1}$. Then $H^{(J)}_1$ is given by
$$H^{(J)}_1=
\begin{pmatrix}
p^{-1}\\
&1_{n-1}
\end{pmatrix}
\begin{pmatrix}
1&t_1&\ldots&t_{n-1}\\
&1\\
&&\ddots \\
&&&1
\end{pmatrix}
H
\begin{pmatrix}
1&-t_1&\ldots&-t_{n-1}\\
&1\\
&&\ddots \\
&&&1
\end{pmatrix}
\begin{pmatrix}
p^{-1}\\
&1_{n-1}
\end{pmatrix}^{-1}.$$ Note that we only want to consider deformations which factor through $\delta(\mathcal{M})$. In terms of the parameters $t_1,\ldots,t_{n-1}$, this condition just says that $t_{c+1}=0$.
The matrix $H$ is up to multiples of $p$ given by the description of the action of $g$ on the $e_i$ in the beginning of the proof.
We want to find the conditions that the matrix $H^{(J)}_1$ is integral. For this it is enough to check when the entries in the first line are integral (the coefficients of $pe_0$ in the images of the basis vectors). Using the equations (and calculating modulo integral elements, i.e., modulo elements of $R$), the first $c+l$ entries are $$\begin{aligned}
\lambda+t_1, (-t_1^2+t_2)/p, (-t_1t_2+t_3)/p,\ldots,(-t_1t_{c-1}+t_c)/p,(-t_1t_c)/p, \\
\big(-(\lambda+t_1)t_{c+1}+t_{c+2}\big)/p,\ldots, \big(-(\lambda+t_1)t_{c+l-1}+t_{c+l}\big)/p, \big(-(\lambda+t_1)t_{c+l}\big)/p .
\end{aligned}$$ Using $t_{c+1}=0$, this shows that in $R/J$ $$t_1t_i=t_{i+1},\, \forall i\leq c-1,t_1t_c=0, \text{ and } t_{c+2}=\ldots=t_{c+l}=0 .$$ If $e_m,\ldots,e_{m'}$ span another Jordan block (mod $L$) to an eigenvalue $\mu$, then by equations (\[basis1\]) and (\[basis2\]), the entries with index between $m$ and $m'$ are (modulo integral elements) $$\begin{aligned}
\big(-\lambda t_{m}-t_1 t_{m}+\mu t_{m}+ t_{m+1}\big)/p,\ldots, \big(-\lambda t_{m'-1}-t_1 t_{m'-1}+\mu t_{m'-1}+ t_{m'}\big)/p, t_{m'}(-\lambda+\mu-t_1)/p .\end{aligned}$$ Since $\lambda\neq \mu$ (modulo $p$), the expression $(-\lambda+\mu-t_1)$ is a unit, hence we obtain $t_{m'}=0$ in $R/J$. Inductively we obtain that $t_m=\ldots=t_{m'-1}=t_{m'}=0$ in $R/J$. Therefore $$\big(t_1t_i-t_{i+1} \text{ for } i\leq c-1, t_1t_c, t_i \ \text{for } i\geq c+1\big)\subseteq J.$$ Note that there do not occur further conditions from the integrality of $$H^{(J)}=
\begin{pmatrix}
1&t_1&\ldots&t_{n-1}\\
&1\\
&&\ddots \\
&&&1
\end{pmatrix}
H
\begin{pmatrix}
1&-t_1&\ldots&-t_{n-1}\\
&1\\
&&\ddots \\
&&&1
\end{pmatrix}.$$
We claim that $J_0:=\big(t_1t_i-t_{i+1} \text{ for } i\leq c-1, t_1t_c, t_i \ \text{for } i\geq c+1\big)+\mathfrak m^p=I+m^p$. Obviously $J_0$ is admissible. If we choose $S=R/J_0$ then the same calculation as above shows that $g$ lifts over $S$. Since $t_{c+1}\in J_0$ it follows that ${{\mathrm{Spec}}}(S)\subseteq \mathcal{M}\cap \mathcal{Z}(g)$. It follows that indeed $I+m^p=J_0$.
Since $2c+1\leq n$ and since we assume $n \leq 2p-2$, we have $c+1< p$. Hence the claim of Theorem \[idealk\] follows from the next lemma.
Let $r>s>0$ be integers. Consider the ideal $J$ in ${{\mathbb {F}}}[\![X_1,\ldots,X_n]\!]$, where $$J=(X_1^s, X_2,\ldots,X_n) .$$ Let $I$ be an ideal in ${{\mathbb {F}}}[\![X_1,\ldots,X_n]\!]$ such that $$I+\mathfrak m^r=J,$$ where $\mathfrak m$ denotes the maximal ideal. Then $I=J$.
We proceed in several steps.
*Step 1.* Consider the projection ${{\mathbb {F}}}[\![X_1,\ldots,X_{n}]\!]\to{{\mathbb {F}}}[\![X_2,\ldots,X_{n}]\!]$, obtained by dividing out by $(X_1)$. Let $\bar I,$ resp. $ \bar J,$ be the image of $I$, resp. $J$, and let $\bar{ \mathfrak m}$ be the maximal ideal of ${{\mathbb {F}}}[\![X_2,\ldots,X_{n}]\!]$. Then $\bar I=\bar J=\bar{\mathfrak m}$ by Nakayama’s Lemma.
*Step 2.* Let $b\in J$. Then $b$ is congruent modulo $I$ to an element of $\mathfrak m^r$. Writing this latter element as a sum of monomials in $X_1,\ldots, X_n$, and using step 1, we see that $b$ is congruent modulo $I$ to an element in the ideal $(X_1^r)$. Hence it suffices to prove that $X_1^r\in I$.
*Step 3.* We claim that in fact $X_1^s\in I$. We will show that $X_1^s\in I+\mathfrak m^{kr}$ for all $k$, which will prove the claim.
We proceed by induction on $k$, the case $k=1$ holding true by hypothesis. Assume that $X_1^s\in I+\mathfrak m^{kr}$. Hence we are assuming that $X_1^s$ is congruent modulo $I$ to an element of $\mathfrak m^{kr}$. Writing this element of $\mathfrak m^{kr}$ as a sum of monomials in $X_1,\ldots,X_n$, we subdivide this sum into
- a sum of monomials, where the exponent of $X_1$ is $\geq s$,
- a sum of monomials, where the exponent of $X_1$ is $<s$.
In the first sum, we extract the factor $X_1^s$; since $kr>s$, the remainder lies in $\mathfrak m$. Bringing this first sum to the left hand side, we see that this expression differs from $X_1^s$ by a unit. Hence we may disregard the first sum.
In the second sum, the total degree in $X_2,\ldots,X_n$ of each monomial is strictly larger than $kr-s$, i.e., is at least $ (k-1)r+2$. Now $X_2,\ldots,X_n$ are congruent modulo $I$ to elements in $\mathfrak m^r$ and we may replace each $X_2,\ldots,X_n$ by an element in $\mathfrak m^r$. Then each summand lies in $\mathfrak m^{((k-1)r+2)r}\subset \mathfrak m^{(k+1)r}$, which concludes the induction step.
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[^1]: Research of Rapoport and Terstiege partially supported by SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties" of the DFG. Research of Zhang partially supported by NSF grant DMS 1204365.
[^2]: In [@RS], the Lie algebra version is considered. But it is easy to deduce the group version from the Lie algebra version. Moreover, what is called “regular semi-simple" here is called “regular" in [@RS].
[^3]: We thank X. He for pointing out these references.
[^4]: As in the previous section, we now write the grading index as an upper index, to avoid a conflict of notation with Zink’s theory.
|
---
abstract: 'Using an extension of the well-known evaluation symmetry, a new Cauchy-type identity for Macdonald polynomials is proved. After taking the classical limit this yields a new ${\mathfrak{sl}_{3}}$ generalisation of the famous Selberg integral. Closely related results obtained in this paper are an ${\mathfrak{sl}_{3}}$-analogue of the Askey–Habsieger–Kadell $q$-Selberg integral and an extension of the $q$-Selberg integral to a transformation between $q$-integrals of different dimensions.'
address: 'School of Mathematics and Physics, The University of Queensland, QLD 4072, Australia'
author:
- 'S. Ole Warnaar'
title: 'The ${\mathfrak{sl}_{3}}$ Selberg integral'
---
[^1]
Introduction
============
Let ${\mathfrak{g}}$ be a simple Lie algebra of rank $n$, with simple roots, fundamental weights and Chevalley generators given by $\alpha_i$, $\Lambda_i$ and $e_i,f_i,h_i$ for $1\leq i\leq n$. The roots of ${\mathfrak{g}}$ are normalised such that the maximal root $\theta$ has length $\sqrt{2}$, i.e., ${(\theta,\theta)}=2$, where ${(\cdot\,,\cdot)}$ is the standard bilinear symmetric form on the dual of the Cartan subalgebra.
Let $V_{{\lambda}}$ and $V_{\mu}$ be highest weight modules of ${\mathfrak{g}}$ with highest weights ${\lambda}$ and $\mu$, and denote by $\text{Sing}_{{\lambda},\mu}[\nu]$ the space of singular vectors of weight $\nu$ in $V_{{\lambda}}\otimes V_{\mu}$: $$\text{Sing}_{{\lambda},\mu}[\nu]=\bigl\{v\in V_{{\lambda}}\otimes V_{\mu}:
h_i v=\nu(h_i) v,~e_i v=0,~1\leq i\leq n\bigr\}.$$
For fixed nonnegative integers $k_1,\dots,k_n$ assign $k:=k_1+\cdots+k_n$ integration variables $t_1,\dots,t_k$ to ${\mathfrak{g}}$ by attaching the $k_i$ variables $$t_{1+k_1+\cdots+k_{i-1}},\dots,t_{k_1+\cdots+k_i}$$ to the simple root $\alpha_i$. In other words, the first $k_1$ integration variables are attached to $\alpha_1$, the second $k_2$ to $\alpha_2$ and so on. By a mild abuse of notation, also set $$\alpha_{t_j}=\alpha_i \quad\text{if $k_1+\cdots+k_{i-1}<j\leq k_1+\cdots+k_i$}.$$
Exploiting the connection between Knizhnik–Zamolodchikov equations and hypergeometric integrals, see e.g., [@EFK03; @SV91; @Varchenko03], Mukhin and Varchenko [@MV00] conjectured in 2000 that if the space $$\text{Sing}_{{\lambda},\mu}\Bigl[{\lambda}+\mu-\sum_{i=1}^n k_i \alpha_i\Bigr]$$ is one-dimensional, then there exists a real integration domain $\Gamma$ such that a closed-form evaluation exists (in terms of products of ratios of Gamma functions) for the ${\mathfrak{g}}$ Selberg integral $$\label{Sg}
{\int\limits}_{\Gamma}\;
\biggl[\;\:\prod_{i=1}^k t_i^{-{({\lambda},\alpha_{t_i})}}
(1-t_i)^{-{(\mu,\alpha_{t_i})}}
\prod_{1\leq i<j\leq k}{\lvertt_i-t_j\rvert}^{{(\alpha_{t_i},\alpha_{t_j})}}
\biggr]^{\gamma}
\operatorname{d\!}t_1\cdots\operatorname{d\!}t_k.$$
For ${\mathfrak{g}}={\mathfrak{sl}_{2}}$ the evaluation of is well-known and corresponds to the celebrated Selberg integral [@Selberg44; @Mehta04; @FW08]: $$\begin{gathered}
\label{Selberg}
{\int\limits}_{0<t_1<\dots<t_k<1}\,
\prod_{i=1}^k t_i^{\alpha-1}(1-t_i)^{\beta-1}
\prod_{1\le i < j\le k} {\lvertt_i-t_j\rvert}^{2\gamma}\,
\operatorname{d\!}t_1\cdots\operatorname{d\!}t_k \\
=\prod_{i=0}^{k-1} \frac{\Gamma (\alpha+i\gamma)
\Gamma(\beta+i\gamma)\Gamma((i+1)\gamma)}
{\Gamma(\alpha+\beta+(i+k-1)\gamma)\Gamma(\gamma)},\end{gathered}$$ where $${\textup{Re}}(\alpha)>0,~{\textup{Re}}(\beta)>0,~{\textup{Re}}(\gamma)>
-\min\{1/k,{\textup{Re}}(\alpha)/(k-1),{\textup{Re}}(\beta)/(k-1)\}.$$ The prospect that generalisations of this extremely important integral exist for all simple Lie algebras has led to much recent progress in evaluating hypergeometric integrals, see e.g., [@FW08; @Iguri09; @MT05; @TV03; @Varchenko08; @W08; @W08b; @W09].
In [@TV03] Tarasov and Varchenko obtained an evaluation of for ${\mathfrak{g}}={\mathfrak{sl}_{3}}$, ${\lambda}={\lambda}_1\Lambda_1+{\lambda}_2\Lambda_2$, $\mu=\mu_2\Lambda_2$ and $k_1\leq k_2$ as follows.
\[thmTV\] For $0\leq k_1\leq k_2$ let $t=(t_1,\dots,t_{k_1})$, $s=(s_1,\dots,s_{k_2})$, and let $\alpha_1,\alpha_2,\beta_2,\gamma\in{\mathbb C}$ such that ${\textup{Re}}(\alpha_1),{\textup{Re}}(\alpha_2),{\textup{Re}}(\beta_2)>0$ and ${\lvert\gamma\rvert}$ is sufficiently small. Then $$\begin{aligned}
\label{STV}
{\int\limits}_{C^{k_1,k_2}_{\gamma}[0,1]}
&\prod_{i=1}^{k_1} t_i^{\alpha_1-1}
\prod_{i=1}^{k_2} s_i^{\alpha_2-1} (1-s_i)^{\beta_2-1} \\[-2mm]
&\times \prod_{1\leq i<j\leq k_1} {\lvertt_i-t_j\rvert}^{2\gamma}
\prod_{1\leq i<j\leq k_2} {\lverts_i-s_j\rvert}^{2\gamma} \:
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}\, {\lvertt_i-s_j\rvert}^{-\gamma} \;
\operatorname{d\!}t \operatorname{d\!}s \notag \\[2mm]
&\qquad=
\prod_{i=0}^{k_1-1}
\frac{\Gamma(\alpha_1+i\gamma)
\Gamma(1+(i-k_2)\gamma)\Gamma((i+1)\gamma)}
{\Gamma(\alpha_1+1+(i+k_1-k_2-1)\gamma)
\Gamma(\gamma)} \notag \\
&\qquad\quad\times\prod_{i=0}^{k_2-1}
\frac{\Gamma(\alpha_2+i\gamma)
\Gamma(\beta_2+i\gamma)\Gamma((i+1)\gamma)}
{\Gamma(\alpha_2+\beta_2+(i+k_2-k_1-1)\gamma)\Gamma(\gamma)} \notag \\
&\qquad\quad\times\prod_{i=0}^{k_1-1}
\frac{\Gamma(\alpha_1+\alpha_2+(i-1)\gamma)}
{\Gamma(\alpha_1+\alpha_2+\beta_2+(i+k_2-2)\gamma)} \notag \\
&\qquad\quad\times\prod_{i=0}^{k_1-1}
\frac{\Gamma(\alpha_2+\beta_2+(i+k_2-k_1-1)\gamma)}
{\Gamma(\alpha_2+(i+k_2-k_1)\gamma)}, \notag\end{aligned}$$ where $\operatorname{d\!}t=\operatorname{d\!}t_1\cdots\operatorname{d\!}t_{k_1}$ and $\operatorname{d\!}s=\operatorname{d\!}s_1\cdots\operatorname{d\!}s_{k_2}$.
In the above, $C^{k_1,k_2}_{\gamma}[0,1]$ is a somewhat complicated integration chain defined in on page . Since $$C_{\gamma}^{0,k}[0,1]
=\{(s_1,\dots,s_k)\in{\mathbb R}^k:~0<s_1<\cdots<s_k<1\}$$ the Tarasov–Varchenko integral simplifies to the Selberg integral when $(k_1,k_2)=(0,k)$.
In [@W08; @W09] the present author developed a method for proving Selberg-type integrals using Macdonald polynomials. This resulted in an evaluation of for ${\mathfrak{g}}={\mathfrak{sl}_{n}}$ where ${\lambda}=\sum_i {\lambda}_i\Lambda_i$, $\mu=\mu_n\Lambda_n$ and $k_1\leq k_2\leq \dots\leq k_n$, generalising the Selberg and Tarasov–Varchenko integrals. In this paper we again employ the theory of Macdonald polynomials to establish the following Cauchy-type identity. For ${\lambda}$ and $\mu$ partitions (and not, as above, weights of ${\mathfrak{g}}$) let ${\mathsf{P}}_{{\lambda}}$ be a suitably normalised Macdonald polynomial. Furthermore, let $(a)_n$ be a $q$-shifted factorial and $(a)_{{\lambda}}$ a generalised $q$-shifted factorial. (For precise definitions of all of the above, see Section \[SecDN\].)
\[thmCauchy\] Let $X=\{x_1,\dots,x_n\}$ and $Y=\{y_1,\dots,y_m\}$. Then $$\begin{gathered}
\sum_{{\lambda},\mu}
t^{{\lvert{\lambda}\rvert}-n{\lvert\mu\rvert}}
{\mathsf{P}}_{{\lambda}}(X){\mathsf{P}}_{\mu}(Y)\,
(at^{m-1})_{{\lambda}}(qt^n/a)_{\mu}
\prod_{i=1}^n \prod_{j=1}^m
\frac{(a t^{j-i-1})_{{\lambda}_i-\mu_j}}
{(a t^{j-i})_{{\lambda}_i-\mu_j}} \\
=\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(tx_i)_{\infty}}
\prod_{j=1}^m\frac{(qy_j/a)_{\infty}}{(y_j)_{\infty}}
\prod_{i=1}^n \prod_{j=1}^m \frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}}.\end{gathered}$$
For $m=0$ ($n=0$) the above identity reduces to the $q$-binomial theorem for Macdonald polynomials in $X$ ($Y$). Theorem \[thmCauchy\] may thus be viewed as two coupled, multidimensional $q$-binomial theorems. In the special case $(X,Y,a,q,t)\mapsto (X/q,qY,-q^2,q^2,q^2)$ the theorem simplifies to Kawanaka’s $q$-Cauchy identity for Schur functions [@Kawanaka99] (with the proviso that Kawanaka’s description of the summand is significantly more involved).
After a limiting procedure, which turns the sums over ${\lambda}$ and $\mu$ into integrals, Theorem \[thmCauchy\] becomes a new evaluation of the Selberg integral for ${\mathfrak{g}}={\mathfrak{sl}_{3}}$ as follows.
\[thmsl3\] Let $t=(t_1,\dots,t_{k_1})$, $s=(s_1,\dots,s_{k_2})$ and let $\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma\in{\mathbb C}$ such that ${\textup{Re}}(\alpha_1),{\textup{Re}}(\alpha_2),{\textup{Re}}(\beta_1),{\textup{Re}}(\beta_2)>0$, ${\lvert\gamma\rvert}$ is sufficiently small, $$\beta_1+(i-k_2-1)\gamma\not\in{\mathbb Z}\quad\text{for $1\leq i\leq \min\{k_1,k_2\}$}$$ and $$\beta_1+\beta_2=\gamma+1.$$ Then $$\begin{aligned}
\label{sl3new}
{\int\limits}_{C_{\beta_1,\gamma}^{k_1,k_2}[0,1]}
&\prod_{i=1}^{k_1} t_i^{\alpha_1-1}(1-t_i)^{\beta_1-1}
\prod_{i=1}^{k_2} s_i^{\alpha_2-1} (1-s_i)^{\beta_2-1} \\[-2mm]
&\times \prod_{1\leq i<j\leq k_1} {\lvertt_i-t_j\rvert}^{2\gamma}
\prod_{1\leq i<j\leq k_2} {\lverts_i-s_j\rvert}^{2\gamma} \:
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}\, {\lvertt_i-s_j\rvert}^{-\gamma} \;
\operatorname{d\!}t \operatorname{d\!}s \notag \\[2mm]
&\qquad=\prod_{i=0}^{k_1-1}\frac{\Gamma(\alpha_1+i\gamma)
\Gamma(\beta_1+(i-k_2)\gamma)\Gamma((i+1)\gamma)}
{\Gamma(\alpha_1+\beta_1+(i+k_1-k_2-1)\gamma)\Gamma(\gamma)} \notag \\
&\qquad\quad\times \prod_{i=0}^{k_2-1}
\frac{\Gamma(\alpha_2+i\gamma)\Gamma(\beta_2+i\gamma)\Gamma((i+1)\gamma)}
{\Gamma(\alpha_2+\beta_2+(i+k_2-k_1-1)\gamma)
\Gamma(\gamma)} \notag \\
&\qquad\quad\times\prod_{i=0}^{k_1-1}
\frac{\Gamma(\alpha_1+\alpha_2+(i-1)\gamma)}
{\Gamma(\alpha_1+\alpha_2+(i+k_2-1)\gamma)}, \notag \end{aligned}$$ where $\operatorname{d\!}t=\operatorname{d\!}t_1\cdots\operatorname{d\!}t_{k_1}$, $\operatorname{d\!}s=\operatorname{d\!}s_1\cdots\operatorname{d\!}s_{k_2}$ and $C_{\beta,\gamma}^{k_1,k_2}[0,1]$ the integration chain defined in on page .
Since $$C_{\beta,\gamma}^{k,0}[0,1]
=\{(t_1,\dots,t_k)\in{\mathbb R}^k:~0<t_1<\cdots<t_k<1\},$$ the integral again contains the Selberg integral as special case. Unlike , however, exhibits ${\mathbb Z}_2$ symmetry thanks to $$\label{CCsymm}
C_{\beta_2,\gamma}^{k_2,k_1}[0,1]
=C_{\beta_1,\gamma}^{k_1,k_2}[0,1]
\prod_{i=0}^{k_1-1}\frac{\Gamma(\beta_1+i\gamma)}
{\Gamma(\beta_1+(i-k_2)\gamma)}
\prod_{i=0}^{k_2-1}\frac{\Gamma(\beta_2+(i-k_1)\gamma)}
{\Gamma(\beta_2+i\gamma)}$$ for $\beta_1+\beta_2=\gamma+1$, and $$\prod_{i=0}^{k_1-1}
\frac{\Gamma(\alpha_1+\alpha_2+(i-1)\gamma)}
{\Gamma(\alpha_1+\alpha_2+(i+k_2-1)\gamma)}=
\prod_{i=0}^{k_2-1}
\frac{\Gamma(\alpha_1+\alpha_2+(i-1)\gamma)}
{\Gamma(\alpha_1+\alpha_2+(i+k_1-1)\gamma)}.$$
If we specialise $\beta_2=\gamma$ in and $(\beta_1,\beta_2)=(1,\gamma)$ in then the respective products over gamma functions on the right coincide. Since also $$\label{chainid}
C_{1,\gamma}^{k_1,k_2}[0,1]=C_{\gamma}^{k_1,k_2}[0,1]$$ (see Section \[Secchains\] for more details) the two ${\mathfrak{sl}_{3}}$ integrals are indeed identical for this particular specialisation.
Macdonald Polynomials
=====================
Definitions and notation {#SecDN}
------------------------
Let ${\lambda}=({\lambda}_1,{\lambda}_2,\dots)$ be a partition, i.e., ${\lambda}_1\geq {\lambda}_2\geq \dots$ with finitely many ${\lambda}_i$ unequal to zero. The length and weight of ${\lambda}$, denoted by $l({\lambda})$ and ${\lvert{\lambda}\rvert}$, are the number and sum of the nonzero ${\lambda}_i$, respectively. Two partitions that differ only in their string of zeros are identified, and the unique partition of length (and weight) $0$ is itself denoted by $0$. The multiplicity of the part $i$ in the partition ${\lambda}$ is denoted by $m_i=m_i({\lambda})$, and occasionally we will write ${\lambda}=(1^{m_1} 2^{m_2} \dots)$.
We identify a partition with its diagram or Ferrers graph, defined by the set of points in $(i,j)\in {\mathbb Z}^2$ such that $1\leq j\leq {\lambda}_i$. The conjugate ${\lambda}'$ of ${\lambda}$ is the partition obtained by reflecting the diagram of ${\lambda}$ in the main diagonal, so that, in particular, $m_i({\lambda})={\lambda}_i'-{\lambda}_{i+1}'$. The statistic $n({\lambda})$ is given by $$n({\lambda})=\sum_{i\geq 1} (i-1){\lambda}_i=
\sum_{i\geq 1}\binom{{\lambda}_i'}{2}.$$
The dominance partial order on the set of partitions of $N$ is defined by ${\lambda}\geq \mu$ if ${\lambda}_1+\cdots+{\lambda}_i\geq \mu_1+\cdots+\mu_i$ for all $i\geq 1$. If ${\lambda}\geq \mu$ and ${\lambda}\neq\mu$ then ${\lambda}>\mu$.
If ${\lambda}$ and $\mu$ are partitions then $\mu\subseteq{\lambda}$ if (the diagram of) $\mu$ is contained in (the diagram of) ${\lambda}$, i.e., $\mu_i\leq{\lambda}_i$ for all $i\geq 1$.
For $s=(i,j)\in\lambda$ the integers $a(s)$, $a'(s)$, $l(s)$ and $l'(s)$, known as the arm-length, arm-colength, leg-length and leg-colength of $s$, are defined as $$\begin{aligned}
a(s)&={\lambda}_i-j, & a'(s)&=j-1, \\
l(s)&={\lambda}'_j-i, & l'(s)&=i-1.\end{aligned}$$ Note that $n({\lambda})=\sum_{s\in{\lambda}}l(s)$. Using the above we define the generalised hook-length polynomials $c_{{\lambda}}$ and $c'_{{\lambda}}$ as $$\begin{aligned}
c_{{\lambda}}=c_{{\lambda}}(q,t)&:=\prod_{s\in{\lambda}}\bigl(1-q^{a(s)}t^{l(s)+1}\bigr), \\
c'_{{\lambda}}=c'_{{\lambda}}(q,t)&:=\prod_{s\in{\lambda}}\bigl(1-q^{a(s)+1}t^{l(s)}\bigr).\end{aligned}$$
The ordinary $q$-shifted factorial are given by $$(a)_{\infty}=(a;q)_{\infty}:=\prod_{i=0}^{\infty}(1-aq^i)$$ and $$(b)_z=(a;q)_z:=\frac{(b)_{\infty}}{(bq^z)_{\infty}}.$$ Note in particular that for $N$ a positive integer $(b)_N=(1-b)(1-bq)\cdots(1-bq^{N-1})$, and $1/(q)_{-N}=0$. Also note that $c'_{(k)}=(q)_k$. The $q$-shifted factorials can be generalised to allow for a partition as indexing set: $$(b)_{{\lambda}}=(b;q,t)_{{\lambda}}:=\prod_{s\in{\lambda}}\bigl(1-b q^{a'(s)}t^{-l'(s)}\bigr)
=\prod_{i=1}^{l({\lambda})}(bt^{1-i})_{{\lambda}_i}.$$ With this notation,
\[ccp\] $$\begin{aligned}
c_{{\lambda}}&=(t^n)_{{\lambda}}
\prod_{1\leq i<j\leq n}\frac{(t^{j-i})_{{\lambda}_i-{\lambda}_j}}
{(t^{j-i+1})_{{\lambda}_i-{\lambda}_j}}, \\
c'_{{\lambda}}&=(qt^{n-1})_{{\lambda}}
\prod_{1\leq i<j\leq n}\frac{(qt^{j-i-1})_{{\lambda}_i-{\lambda}_j}}
{(qt^{j-i})_{{\lambda}_i-{\lambda}_j}},\end{aligned}$$
where $n$ is an arbitrary integer such that $n\geq l({\lambda})$. We also introduce the usual condensed notation $$(a_1,\dots,a_k)_N=(a_1)_N\cdots (a_k)_N$$ and likewise for $q$-shifted factorials indexed by partitions.
Macdonald polynomials
---------------------
Let ${\mathfrak{S}}_n$ denote the symmetric group, and $\Lambda_n={\mathbb Z}[x_1,\dots,x_n]^{{\mathfrak{S}}_n}$ the ring of symmetric polynomials in $n$ independent variables.
For $X=\{x_1,\dots,x_n\}$ and ${\lambda}=({\lambda}_1,\dots,{\lambda}_n)$ a partition of length at most $n$ the monomial symmetric function $m_{{\lambda}}(X)$ is defined as $$m_{{\lambda}}(X)=\sum_{\alpha}x_1^{\alpha_1}\cdots x_n^{\alpha_n},$$ where the sum is over all distinct permutations $\alpha=(\alpha_1,\dots,\alpha_n)$ of ${\lambda}$. If $l({\lambda})>n$ then $m_{{\lambda}}(X):=0$. The monomial symmetric functions $m_{{\lambda}}(X)$ for $l({\lambda})\leq n$ form a ${\mathbb Z}$-basis of $\Lambda_n$.
A ${\mathbb Q}$-basis of $\Lambda_n$ is given by the power-sum symmetric functions $p_{{\lambda}}(X)$, defined as $$p_r(X)=\sum_{i=1}^n x_i^r$$ for $r\geq 0$ and $p_{{\lambda}}(X)=p_{{\lambda}_1}(X)\cdots p_{{\lambda}_n}(X)$. The power-sum symmetric functions may be used to define an extremely powerful notational tool in symmetric-function theory, known as plethystic or $\lambda$-ring notation, see [@Haglund08; @Lascoux03]. First we define the plethystic bracket by $$f[x_1+\cdots+x_n]=f(x_1,\dots,x_n)$$ where $f$ is a symmetric function. More simply we just write $$f[X]=f(X)$$ where on the left we assume the additive notation for sets (or alphabets), i.e., $X=x_1+\cdots+x_n$ and on the right the more conventional $X=\{x_1,\dots,x_n\}$. With this notation $f[X+Y]$ takes on the obvious meaning of the symmetric function $f$ acting on the disjoint union of the alphabets $X$ and $Y$. Plethystic notation also allows for the definition of symmetric functions acting on differences $X-Y$ of alphabets, or for symmetric functions acting on such alphabets as $(X-Y)/(1-t)$, see e.g., [@Lascoux03]. In this paper we repeatedly need this last alphabet when both $X$ and $Y$ contain a single letter, say $a$ and $b$, respectively. We may then take as definition $$p_r\biggl[\frac{a-b}{1-t}\biggr]=\frac{a^r-b^r}{1-t^r},$$ and extend this by linearity to any symmetric function. Note in particular that $$f\biggl[\frac{1-t^n}{1-t}\biggr]=f(t^{n-1},\dots,t,1)=:f({\langle 0\rangle})$$ corresponds to the so-called principal specialisation, where more generally, $${\langle {\lambda}\rangle}={\langle {\lambda}\rangle}_n:=
(q^{{\lambda}_1}t^{n-1},q^{{\lambda}_2}t^{n-2},\dots,q^{{\lambda}_n}t^0),$$ for $l({\lambda})\leq n$.
After this digression we turn to the definition of the Macdonald polynomials and to some of its basic properties [@Macdonald88; @Macdonald95]. First we define the scalar product ${\langle\cdot\,,\cdot\rangle}$ on symmetric functions by $$\langle p_{{\lambda}},p_{\mu}\rangle=
\delta_{{\lambda}\mu} z_{{\lambda}} \prod_{i=1}^n
\frac{1-q^{{\lambda}_i}}{1-t^{{\lambda}_i}},$$ where $z_{{\lambda}}=\prod_{i\geq 1} m_i! \: i^{m_i}$ and $m_i=m_i({\lambda})$. If we denote the ring of symmetric functions in $n$ variables over the field ${\mathbb F}={\mathbb Q}(q,t)$ of rational functions in $q$ and $t$ by $\Lambda_{n,{\mathbb F}}$, then the Macdonald polynomial $P_{{\lambda}}(X)=P_{{\lambda}}(X;q,t)$ is the unique symmetric polynomial in $\Lambda_{n,{\mathbb F}}$ such that: $$P_{{\lambda}}(X)=m_{{\lambda}}(X)+\sum_{\mu<{\lambda}}
u_{{\lambda}\mu} m_{\mu}(X)$$ (where $u_{{\lambda}\mu}=u_{{\lambda}\mu}(q,t)$) and $$\langle P_{{\lambda}},P_{\mu} \rangle
=0\quad \text{if$\quad{\lambda}\neq\mu$.}$$ The Macdonald polynomials $P_{{\lambda}}(X)$ with $l({\lambda})\leq n$ form an ${\mathbb F}$-basis of $\Lambda_{n,{\mathbb F}}$. If $l({\lambda})>n$ then $P_{{\lambda}}(X):=0$. From the definition it follows that $P_{{\lambda}}(X)$ for $l({\lambda})\leq n$ is homogeneous of (total) degree ${\lvert{\lambda}\rvert}$; $P_{{\lambda}}(zX)=z^{{\lvert{\lambda}\rvert}} P_{{\lambda}}(X)$. A second Macdonald polynomial $Q_{{\lambda}}(X)=Q_{{\lambda}}(X;q,t)$ is defined as $$Q_{{\lambda}}(X)=b_{{\lambda}} P_{{\lambda}}(X),$$ where $b_{{\lambda}}=b_{{\lambda}}(q,t):=c_{{\lambda}}/c'_{{\lambda}}$. Then $${\langleP_{{\lambda}},Q_{\mu}\rangle}=\delta_{{\lambda}\mu}.$$ This last result may equivalently be stated as the Cauchy identity $$\sum_{{\lambda}} P_{{\lambda}}(X)Q_{{\lambda}}(Y)=
\prod_{i,j=1}^n \frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}}.$$
We also need the skew Macdonald polynomials $P_{{\lambda}/\mu}(X)$ and $Q_{{\lambda}/\mu}(X)$ given by $$\begin{aligned}
P_{{\lambda}}[X+Y]&=\sum_{{\lambda}}P_{{\lambda}/\mu}[X]P_{\mu}[Y] \\
Q_{{\lambda}}[X+Y]&=\sum_{{\lambda}}Q_{{\lambda}/\mu}[X]Q_{\mu}[Y],\end{aligned}$$ so that $P_{{\lambda}/0}(X)=P_{{\lambda}}(X)$ and $Q_{{\lambda}/\mu}(X)=b_{{\lambda}}b^{-1}_{\mu}P_{{\lambda}/\mu}(X)$. Equivalently, $$Q_{{\lambda}/\mu}(X)=\sum_{\nu}f_{\mu\nu}^{{\lambda}} Q_{\nu}(X),$$ where $f_{\mu\nu}^{{\lambda}}=f_{\mu\nu}^{{\lambda}}$ are the $q,t$-Littlewood–Richardson coefficients: $$P_{\mu}(X)P_{\nu}(X)=\sum_{{\lambda}}f_{\mu\nu}^{{\lambda}} P_{{\lambda}}(X).$$ From the homogeneity of the Macdonald polynomial it immediately follows that $f_{\mu\nu}^{{\lambda}}(q,t)=0$ if ${\lvert{\lambda}\rvert}\neq{\lvert\mu\rvert}+{\lvert\nu\rvert}$. It may also be shown that $f_{\mu\nu}^{{\lambda}}(q,t)=0$ if $\mu,\nu\not\subseteq{\lambda}$, so that $P_{{\lambda}/\mu}(X)$ vanishes if $\mu\not\subseteq{\lambda}$.
To conclude this section we introduce normalisations of the Macdonald polynomials convenient for dealing with basic hypergeometric series with Macdonald polynomial argument:
$$\begin{aligned}
{\mathsf{P}}_{{\lambda}/\mu}(X)&=t^{n({\lambda})-n(\mu)}
\frac{c'_{\mu}}{c'_{{\lambda}}}\, P_{{\lambda}/\mu}(X) \\[2mm]
{\mathsf{Q}}_{{\lambda}/\mu}(X)&=t^{n(\mu)-n({\lambda})}
\frac{c'_{{\lambda}}}{c'_{\mu}}\, Q_{{\lambda}/\mu}(X).
\label{skewQP}\end{aligned}$$
Note that $$\label{norm}
{\mathsf{Q}}_{{\lambda}/\mu}(X)
=t^{2n(\mu)-2n({\lambda})} \frac{c_{{\lambda}}c'_{{\lambda}}}{c_{\mu}c'_{\mu}}\,
{\mathsf{P}}_{{\lambda}/\mu}(X).$$ If we also normalise the $q,t$-Littlewood–Richardson coefficients as $${\mathsf{f}}^{{\lambda}}_{\mu\nu}=
t^{n(\mu)+n(\nu)-n({\lambda})} \frac{c'_{{\lambda}}}{c'_{\mu}c'_{\nu}}\,
f^{{\lambda}}_{\mu\nu},$$ then all of the preceding formulae have perfect analogues:
$$\begin{aligned}
{\mathsf{P}}_{{\lambda}}[X+Y]&=\sum_{\mu} {\mathsf{P}}_{{\lambda}/\mu}[Y] {\mathsf{P}}_{\mu}[X], \\
{\mathsf{Q}}_{{\lambda}}[X+Y]&=\sum_{\mu} {\mathsf{Q}}_{{\lambda}/\mu}[Y] {\mathsf{Q}}_{\mu}[X],
\label{QQQ}\end{aligned}$$
$$\label{CXY}
\sum_{{\lambda}} {\mathsf{P}}_{{\lambda}}(X){\mathsf{Q}}_{{\lambda}}(Y)=
\prod_{i,j=1}^n \frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}},$$
$$\label{QfQ}
{\mathsf{Q}}_{{\lambda}/\mu}(X)=\sum_{\nu}{\mathsf{f}}_{\mu\nu}^{{\lambda}}{\mathsf{Q}}_{\nu}(X)$$
and $$\label{PPfP}
{\mathsf{P}}_{\mu}(X){\mathsf{P}}_{\nu}(X)=\sum_{{\lambda}}{\mathsf{f}}_{\mu\nu}^{{\lambda}}{\mathsf{P}}_{{\lambda}}(X).$$
Generalised evaluation symmetry
-------------------------------
One of the many striking results in Macdonald polynomial theory — first proved in unpublished work by Koornwinder — is the evaluation symmetry $$\label{TK}
\frac{{\mathsf{P}}_{{\lambda}}({\langle \mu\rangle})}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}
=\frac{{\mathsf{P}}_{\mu}({\langle {\lambda}\rangle})}{{\mathsf{P}}_{\mu}({\langle 0\rangle})},$$ where ${\lambda}$ and $\mu$ are partitions of length at most $n$. As we shall see in Section \[Secproof\], a simple generalisation of this result is the key to proving Theorem \[thmCauchy\]. Before stating this generalisation we put in plethystic notation as $${\mathsf{P}}_{{\lambda}}\biggl[\frac{1-t^n}{1-t}\biggr]
{\mathsf{P}}_{\mu}\bigl[{\langle {\lambda}\rangle}\bigr]
={\mathsf{P}}_{\mu}\biggl[\frac{1-t^n}{1-t}\biggr]
{\mathsf{P}}_{{\lambda}}\bigl[{\langle \mu\rangle}\bigr],$$ where $$f[{\langle {\lambda}\rangle}]=f\bigl[q^{{\lambda}_1}t^{n-1}+\cdots+q^{{\lambda}_n}t^0\bigr]=
f\bigl(q^{{\lambda}_1}t^{n-1},\dots,q^{{\lambda}_n}t^0\bigr)=f({\langle {\lambda}\rangle}).$$
For ${\lambda}$ and $\mu$ partitions of length at most $n$, $$\label{gd}
{\mathsf{P}}_{{\lambda}}\biggl[\frac{1-at^n}{1-t}\biggr]
{\mathsf{P}}_{\mu}\biggl[a{\langle {\lambda}\rangle}+\frac{1-a}{1-t}\biggr]
={\mathsf{P}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
{\mathsf{P}}_{{\lambda}}\biggl[a{\langle \mu\rangle}+\frac{1-a}{1-t}\biggr].$$
Both sides are polynomials in $a$ of degree ${\lvert{\lambda}\rvert}+{\lvert\mu\rvert}$ with coefficients in ${\mathbb Q}(q,t)$. It thus suffices to verify for $a=t^p$, where $p$ ranges over the nonnegative integers. We now write ${\langle {\lambda}\rangle}={\langle {\lambda}\rangle}_n$ and use that $$f\biggl[a{\langle {\lambda}\rangle}_n+\frac{1-a}{1-t}\biggr]\bigg|_{a=t^p}=
f\bigl[{\langle {\lambda}\rangle}_{n+p}\bigr],$$ where, since $l({\lambda})\leq n$, $$f\bigl[{\langle {\lambda}\rangle}_{n+p}\bigr]=
f\bigl(q^{{\lambda}_1}t^{n+p-1},\dots,q^{{\lambda}_n}t^p,t^{p-1},\dots,t^0\bigr).$$ As a result we obtain $${\mathsf{P}}_{{\lambda}}\bigl({\langle 0\rangle}_{n+p}\bigr) {\mathsf{P}}_{\mu}\bigl({\langle {\lambda}\rangle}_{n+p}\bigr)=
{\mathsf{P}}_{\mu}\bigl({\langle 0\rangle}_{n+p}\bigr) {\mathsf{P}}_{{\lambda}}\bigl({\langle \mu\rangle}_{n+p}\bigr)$$ which follows from ordinary evaluation symmetry for Macdonald polynomials on $(n+p)$-letter alphabets.
The generalised evaluation symmetry can also be stated without resorting to plethystic notation as a symmetry for skew Macdonald polynomials.
\[PropII\] $$\label{dual}
(at^n)_{{\lambda}} \sum_{\nu} (a)_{\nu} {\mathsf{Q}}_{\mu/\nu}(a{\langle {\lambda}\rangle})
=(at^n)_{\mu} \sum_{\nu} (a)_{\nu} {\mathsf{Q}}_{{\lambda}/\nu}(a{\langle \mu\rangle}).$$
When $a=1$ both sums vanish unless $\nu=0$. Thanks to the principal specialisation formula [@Macdonald95 page 337] $$\label{tn}
{\mathsf{Q}}_{{\lambda}}({\langle 0\rangle})=(t^n)_{{\lambda}}$$ the $a=1$ case of thus corresponds to in the equivalent form $$\frac{{\mathsf{Q}}_{{\lambda}}({\langle \mu\rangle})}{{\mathsf{Q}}_{{\lambda}}({\langle 0\rangle})}
=\frac{{\mathsf{Q}}_{\mu}({\langle {\lambda}\rangle})}{{\mathsf{Q}}_{\mu}({\langle 0\rangle})}.$$
By changing normalisation we may replace $({\mathsf{P}}_{{\lambda}},{\mathsf{P}}_{\mu})$ in by $({\mathsf{Q}}_{{\lambda}},{\mathsf{Q}}_{\mu})$. Using [@Macdonald95 page 338] $$\label{bla}
(a)_{{\lambda}}={\mathsf{Q}}_{{\lambda}}\biggl[\frac{1-a}{1-t}\biggr]$$ (for $a=t^n$ this is ) and this gives rise to $$(at^n)_{{\lambda}}\sum_{\nu}
{\mathsf{Q}}_{\mu/\nu}\bigl[a{\langle {\lambda}\rangle}\bigr]
{\mathsf{Q}}_{\nu}\biggl[\frac{1-a}{1-t}\biggr]
=(at^n)_{\mu}\sum_{\nu}
{\mathsf{Q}}_{{\lambda}/\nu}\bigl[a{\langle \mu\rangle}\bigr]
{\mathsf{Q}}_{\nu}\biggl[\frac{1-a}{1-t}\biggr].$$ Once again using and dispensing with the remaining plethystic brackets yields .
${\mathfrak{sl}_{n}}$ basic hypergeometric series
-------------------------------------------------
Before we deal with the most important application of the generalised evaluation symmetry — the proof of Theorem \[thmCauchy\] — we will show how it implies a multivariable generalisation of Heine’s transformation formula.
Let $$\tau_{{\lambda}}=\tau_{{\lambda}}(q,t):=(-1)^{{\lvert{\lambda}\rvert}}q^{n({\lambda}')}t^{-n({\lambda})}$$ and $X=\{x_1,\dots,x_n\}$. Then the ${\mathfrak{sl}_{n}}$ basic hypergeometric series $_r\Phi_s$ is defined as $$\label{Phi}
{_r\Phi_s}\biggl[\genfrac{}{}{0pt}{}{a_1,\dots,a_r}
{b_1,\dots,b_s};X\biggr]
=\sum_{{\lambda}} \frac{(a_1,\dots,a_r)_{{\lambda}}}{(b_1,\dots,b_s)_{{\lambda}}}\,
\tau_{{\lambda}}^{s-r+1}\, {\mathsf{P}}_{{\lambda}}(X).$$ For $n=1$ this is in accordance with the standard definition of single-variable basic hypergeometric series ${_r\phi_s}$ as may be found in [@AAR99; @GR04]: $$\begin{aligned}
{_r\Phi_s}\biggl[\genfrac{}{}{0pt}{}{a_1,\dots,a_r}
{b_1,\dots,b_s};\{z\}\biggr]
&=\sum_{k=0}^{\infty} \frac{(a_1,\dots,a_r)_k}{(q,b_1,\dots,b_s)_k}\,
\Bigl((-1)^k q^{\binom{k}{2}}\Bigr)^{s-r+1}\, z^k \\
&={_r\phi_s}\biggl[\genfrac{}{}{0pt}{}{a_1,\dots,a_r}
{b_1,\dots,b_s};z\biggr],\end{aligned}$$ where in the second line the $q$-dependence of the $_r\phi_s$ series has been suppressed.
\[thmPhitrafo\] Let $X=\{x_1,\dots,x_n\}$ and $Y=\{y_1,\dots,y_m\}$. Then $$\begin{gathered}
{_{m+1}\Phi_m}\biggl[\genfrac{}{}{0pt}{}{a,ay_1/t,\dots,ay_m/t}
{ay_1,\dots,ay_m};X\biggr] \\
=\biggl(\:\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(x_i)_{\infty}}
\biggr)\biggl(\:\prod_{j=1}^m \frac{(y_j)_{\infty}}{(ay_j)_{\infty}}
\biggr)\,
{_{n+1}\Phi_n}\biggl[\genfrac{}{}{0pt}{}{a,ax_1/t,\dots,ax_n/t}
{ax_1,\dots,ax_n};Y\biggr].\end{gathered}$$
For $m=0$ this is the $q$-binomial theorem for Macdonald polynomials [@Kaneko96; @Macdonald] $$\label{qbt}
{_1\Phi_0}\biggl[\genfrac{}{}{0pt}{}{a}{\text{--}};X\biggr]
=\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(x_i)_{\infty}}$$ and for $m=n=1$ it is Heine’s $_2\phi_1$ transformation formula [@GR04 Equation (III.2)] $${_2\phi_1}\biggl[\genfrac{}{}{0pt}{}{a,ay/t}{ay};x\biggr]
=\frac{(y,ax)_{\infty}}{(x,ay)_{\infty}}\:
{_2\phi_1}\biggl[\genfrac{}{}{0pt}{}{a,ax/t}{ax};y\biggr].$$
First assume that $m=n$, multiply by ${\mathsf{P}}_{{\lambda}}(X){\mathsf{P}}_{\mu}(Y)$ and sum over ${\lambda}$ and $\mu$ to get $$\begin{gathered}
\label{interm}
\sum_{\nu,\mu,{\lambda}} (a)_{\nu} (at^n)_{{\lambda}}
{\mathsf{Q}}_{\mu/\nu}(a{\langle {\lambda}\rangle}) {\mathsf{P}}_{{\lambda}}(X) {\mathsf{P}}_{\mu}(Y) \\
=\sum_{\nu,\mu,{\lambda}} (a)_{\nu} (at^n)_{\mu}
{\mathsf{Q}}_{{\lambda}/\nu}(a{\langle \mu\rangle}) {\mathsf{P}}_{{\lambda}}(X) {\mathsf{P}}_{\mu}(Y).\end{gathered}$$ If we multiply by ${\mathsf{P}}_{\nu}(Y)$ and sum over $\nu$ then and permit this $\nu$-sum to be carried out explicitly on both sides. As a result we obtain the skew Cauchy identity (see also [@Macdonald95 page 352]) $$\label{Cauchy}
\sum_{{\lambda}}{\mathsf{P}}_{{\lambda}}(X){\mathsf{Q}}_{{\lambda}/\mu}(Y)=
{\mathsf{P}}_{\mu}(X) \prod_{i,j=1}^n
\frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}}.$$ Applying this to we can perform the sum over $\mu$ on the left and the sum over ${\lambda}$ on the right, leading to $$\begin{gathered}
\sum_{\nu,{\lambda}} (a)_{\nu} (at^n)_{{\lambda}}
{\mathsf{P}}_{{\lambda}}(X){\mathsf{P}}_{\nu}(Y)\prod_{i,j=1}^n
\frac{(aty_i{\langle {\lambda}\rangle}_j)_{\infty}}{(ay_i{\langle {\lambda}\rangle}_j)_{\infty}} \\
=\sum_{\nu,\mu} (a)_{\nu} (at^n)_{\mu}
{\mathsf{P}}_{\mu}(Y){\mathsf{P}}_{\nu}(X)\prod_{i,j=1}^n
\frac{(atx_i{\langle \mu\rangle}_j)_{\infty}}{(ax_i{\langle \mu\rangle}_j)_{\infty}}.\end{gathered}$$ Using the $q$-binomial theorem to perform both sums over $\nu$ gives $$\begin{gathered}
\sum_{{\lambda}} (at^n)_{{\lambda}}
{\mathsf{P}}_{{\lambda}}(X) \prod_{i=1}^n \frac{(ay_i)_{\infty}}{(y_i)_{\infty}}
\prod_{i,j=1}^n
\frac{(aty_i{\langle {\lambda}\rangle}_j)_{\infty}}{(ay_i{\langle {\lambda}\rangle}_j)_{\infty}} \\
=\sum_{\mu} (at^n)_{\mu}
{\mathsf{P}}_{\mu}(Y) \prod_{i=1}^n \frac{(ax_i)_{\infty}}{(x_i)_{\infty}}
\prod_{i,j=1}^n
\frac{(atx_i{\langle \mu\rangle}_j)_{\infty}}{(ax_i{\langle \mu\rangle}_j)_{\infty}}.\end{gathered}$$ Simplifying the products and replacing $a\mapsto at^{-n}$ completes the proof of the theorem for $m=n$.
The general $m,n$ case trivially follows from $m=n$; assuming without loss of generality that $m\leq n$ we set $y_{m+1},\dots,y_n=0$ and use that $${\mathsf{P}}_{{\lambda}}(y_1,\dots,y_m,\underbrace{0,\dots,0}_{n-m})=
\begin{cases}
{\mathsf{P}}_{{\lambda}}(y_1,\dots,y_m)
& \text{if $l({\lambda})\leq m$}, \\[1mm]
0
& \text{if $l({\lambda})>m$}.
\end{cases}\qedhere$$
Proof of Theorem \[thmCauchy\] {#Secproof}
------------------------------
Using the generalised evaluation symmetry to prove Theorem \[thmCauchy\] is much more difficult than the proof of Theorem \[thmPhitrafo\], and we proceed by first proving an identity for skew Macdonald polynomials.
\[thmPQ\] For ${\lambda}$ and $\mu$ partitions of length at most $n$, $$\begin{gathered}
\sum_{\nu} t^{-{\lvert\nu\rvert}}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
{\mathsf{Q}}_{{\lambda}/\nu}\biggl[\frac{1-q/at}{1-t}\biggr] \\
=t^{-n{\lvert\mu\rvert}}
{\mathsf{P}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
{\mathsf{Q}}_{{\lambda}}\biggl[\frac{1-qt^{n-1}/a}{1-t}\biggr]
\prod_{i,j=1}^n
\frac{(qt^{j-i-1}/a)_{{\lambda}_i-\mu_j}}{(qt^{j-i}/a)_{{\lambda}_i-\mu_j}}.\end{gathered}$$
Recalling and it follows that the right-hand side is completely factorised. Moreover, for $a=1$ the summand vanishes unless $\nu=\mu$ so that we recover the known factorisation of ${\mathsf{Q}}_{{\lambda}/\mu}[(1-q/t)/(1-t)]$, see [@Rains06 Equation (8.20)] or [@W05 Proposition 3.2]: $${\mathsf{Q}}_{{\lambda}/\mu}\biggl[\frac{1-q/t}{1-t}\biggr]
=t^{(1-n){\lvert\mu\rvert}} (qt^{n-1})_{{\lambda}} {\mathsf{P}}_{\mu}({\langle 0\rangle})
\prod_{i,j=1}^n
\frac{(qt^{j-i-1})_{{\lambda}_i-\mu_j}}{(qt^{j-i})_{{\lambda}_i-\mu_j}}.$$
In the first few steps we follow the proof of Theorem \[thmPhitrafo\] but in an asymmetric manner. That is, we take , multiply both sides by ${\mathsf{P}}_{{\lambda}}(X)$ and sum over ${\lambda}$. By the Cauchy identity followed by the $q$-binomial theorem we can perform both sums on the right to find $$\label{ass}
\sum_{{\lambda},\nu} (at^n)_{{\lambda}} (a)_{\nu} {\mathsf{Q}}_{\mu/\nu}(a{\langle {\lambda}\rangle})
{\mathsf{P}}_{{\lambda}}(X)
=(at^n)_{\mu} \prod_{i=1}^n \frac{(ax_i)_{\infty}}{(x_i)_{\infty}}
\prod_{i,j=1}^n \frac{(atx_i{\langle \mu\rangle}_j)_{\infty}}
{(ax_i{\langle \mu\rangle}_j)_{\infty}}.$$ On the left we use and (twice) to rewrite the sum over $\nu$ as $$\sum_{\nu} (a)_{\nu} {\mathsf{Q}}_{\mu/\nu}(a{\langle {\lambda}\rangle})=
{\mathsf{Q}}_{\mu}\biggl[a{\langle {\lambda}\rangle}+\frac{1-a}{1-t}\biggr]=
\sum_{\nu} {\mathsf{Q}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
{\mathsf{Q}}_{\nu}(a{\langle {\lambda}\rangle}).$$ On the right we use to trade $(at^n)_{\mu}$ for $Q_{\mu}[(1-at^n)/(1-t)]$. Also renaming the summation index ${\lambda}$ as $\omega$, thus takes the form $$\sum_{\nu,\omega} (at^n)_{\omega} {\mathsf{Q}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
{\mathsf{Q}}_{\nu}(a{\langle \omega\rangle}) {\mathsf{P}}_{\omega}(X)
={\mathsf{Q}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(x_i)_{\infty}}
\prod_{i,j=1}^n \frac{(atx_i{\langle \mu\rangle}_j)_{\infty}}
{(ax_i{\langle \mu\rangle}_j)_{\infty}}.$$ By it readily follows that we may replace all occurrences of ${\mathsf{Q}}$ in the above by ${\mathsf{P}}$. Then specialising $X=b{\langle {\lambda}\rangle}$ we find $$\begin{gathered}
\label{lhs}
\sum_{\nu,\omega} (at^n)_{\omega} {\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
{\mathsf{P}}_{\nu}(a{\langle \omega\rangle}) {\mathsf{P}}_{\omega}(b{\langle {\lambda}\rangle}) \\
={\mathsf{P}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
\prod_{i=1}^n \frac{(ab{\langle {\lambda}\rangle}_i)_{\infty}}
{(b{\langle {\lambda}\rangle}_i)_{\infty}}
\prod_{i,j=1}^n \frac{(abt{\langle {\lambda}\rangle}_i{\langle \mu\rangle}_j)_{\infty}}
{(ab{\langle {\lambda}\rangle}_i{\langle \mu\rangle}_j)_{\infty}}.\end{gathered}$$ The next few steps focus on the left-hand side of this identity. First, by homogeneity followed by an application of the evaluation symmetry , $$\text{LHS}\eqref{lhs}=
\sum_{\nu,\omega}a^{{\lvert\nu\rvert}} b^{{\lvert\omega\rvert}}(at^n)_{\omega}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
\frac{{\mathsf{P}}_{\omega}({\langle 0\rangle})}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}\,
{\mathsf{P}}_{{\lambda}}({\langle \omega\rangle}){\mathsf{P}}_{\nu}({\langle \omega\rangle}).$$ Using this can be further rewritten as $$\text{LHS}\eqref{lhs}=
\sum_{\eta,\nu,\omega} a^{{\lvert\nu\rvert}} b^{{\lvert\omega\rvert}}
(at^n)_{\omega} \, {\mathsf{f}}^{\eta}_{{\lambda}\nu}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
\frac{{\mathsf{P}}_{\omega}({\langle 0\rangle})}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}\,
{\mathsf{P}}_{\eta}({\langle \omega\rangle}).$$ By another appeal to evaluation symmetry this yields $$\text{LHS}\eqref{lhs}=
\sum_{\eta,\nu,\omega} a^{{\lvert\nu\rvert}} b^{{\lvert\omega\rvert}}
(at^n)_{\omega} \, {\mathsf{f}}^{\eta}_{{\lambda}\nu}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
\frac{{\mathsf{P}}_{\eta}({\langle 0\rangle})}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}\,
{\mathsf{P}}_{\omega}({\langle \eta\rangle}).$$ The sum over $\omega$ can now be performed by so that $$\text{LHS}\eqref{lhs}=
\sum_{\eta,\nu} a^{{\lvert\nu\rvert}} \, {\mathsf{f}}^{\eta}_{{\lambda}\nu}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
\frac{{\mathsf{P}}_{\eta}({\langle 0\rangle})}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}\,
\prod_{i=1}^n \frac{(abt^n{\langle \eta\rangle}_i)_{\infty}}
{(b{\langle \eta\rangle}_i)_{\infty}}.$$ Equating this with the right-hand side of , manipulating the (infinite) $q$-shifted factorials and finally replacing $b\mapsto bt^{1-n}$ we find $$\begin{gathered}
\sum_{\eta,\nu} a^{{\lvert\nu\rvert}}
\frac{(b)_{\eta}}{(abt^n)_{\eta}} \,
{\mathsf{f}}^{\eta}_{{\lambda}\nu}
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr]
{\mathsf{P}}_{\eta}({\langle 0\rangle}) \\
=\frac{(b)_{{\lambda}}}{(ab)_{{\lambda}}} \, {\mathsf{P}}_{{\lambda}}({\langle 0\rangle})
{\mathsf{P}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
\prod_{i,j=1}^n
\frac{(abt^{n-i-j+1})_{{\lambda}_i+\mu_j}}{(abt^{n-i-j+2})_{{\lambda}_i+\mu_j}}.\end{gathered}$$ For $a=1$ the summand vanishes unless $\nu=\mu$ and we recover [@W05 Proposition 3.1].
Next we specialise $b=q^{-N}$ and then replace ${\lambda}$ and $\eta$ by their complements with respect to the rectangular partition $(N^n)$. Denoting these complementary partitions by $\hat{{\lambda}}$ and $\hat{\eta}$, we have $\hat{{\lambda}}_i=N-{\lambda}_{n-i+1}$ for $1\leq i\leq n$ (and a similar relation between $\eta$ and $\hat{\eta}$). Using the relations [@W05 pp. 259 & 263] $${\mathsf{f}}_{\hat{{\lambda}}\nu}^{\hat{\eta}}=
(-q^N t^{1-n})^{{\lvert{\lambda}-\eta\rvert}}
q^{n(\eta')-n({\lambda}')} t^{n({\lambda})-n(\eta)}
{\mathsf{f}}_{\eta\nu}^{{\lambda}}\,
\frac{(q^{-N})_{{\lambda}}}{(q^{-N})_{\eta}}\,
\frac{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}{{\mathsf{P}}_{\eta}({\langle 0\rangle})},$$ $$\frac{(a)_{\hat{{\lambda}}}}{(b)_{\hat{{\lambda}}}}=
\Bigl(\frac{b}{a}\Bigr)^{{\lvert{\lambda}\rvert}}
\frac{(a)_{(N^n)}}{(b)_{(N^n)}}\,
\frac{(q^{1-N}t^{n-1}/b)_{{\lambda}}}{(q^{1-N}t^{n-1}/a)_{{\lambda}}},$$ and $${\mathsf{P}}_{\hat{{\lambda}}}({\langle 0\rangle})=(-1)^{{\lvert{\lambda}\rvert}}
q^{N{\lvert{\lambda}\rvert}-n({\lambda}')}
t^{2N\binom{n}{2}+n({\lambda})-2(n-1){\lvert{\lambda}\rvert}}
\frac{(q^{-N},qt^{n-1})_{{\lambda}}}{(qt^{n-1})_{(N^n)}}\,
{\mathsf{P}}_{{\lambda}}({\langle 0\rangle}),$$ as well as the fact that the summand vanishes unless ${\lvert\nu\rvert}+{\lvert\eta\rvert}={\lvert{\lambda}\rvert}$, we end up with $$\begin{gathered}
\sum_{\eta,\nu} t^{-{\lvert\nu\rvert}}
(q/at)_{\eta}
{\mathsf{f}}_{\eta\nu}^{{\lambda}}\,
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-a}{1-t}\biggr] \\
=t^{-n{\lvert\mu\rvert}} (qt^{n-1}/a)_{{\lambda}}
{\mathsf{P}}_{\mu}\biggl[\frac{1-at^n}{1-t}\biggr]
\prod_{i,j=1}^n
\frac{(qt^{j-i-1}/a)_{{\lambda}_i-\mu_j}}{(qt^{j-i}/a)_{{\lambda}_i-\mu_j}}.\end{gathered}$$ By and $$\label{bfQ}
\sum_{\nu} (b)_{\nu} {\mathsf{f}}^{{\lambda}}_{\mu\nu}=
\sum_{\nu} {\mathsf{f}}^{{\lambda}}_{\mu\nu} {\mathsf{Q}}_{\nu}\biggl[\frac{1-b}{1-t}\biggr]=
{\mathsf{Q}}_{{\lambda}/\mu}\biggl[\frac{1-b}{1-t}\biggr],$$ so that the sum over $\eta$ can be performed. By a final appeal to the proof is done.
Equipped with Theorem \[thmPQ\] it is not difficult to prove Theorem \[thmCauchy\]. To streamline the proof given below we first prepare an easy lemma.
\[Pieri\] For $X=\{x_1,\dots,x_n\}$ and $\mu$ a partition of length at most $n$, $$\begin{aligned}
\sum_{{\lambda}} {\mathsf{Q}}_{{\lambda}/\mu}\biggl[\frac{a-b}{1-t}\biggr]{\mathsf{P}}_{{\lambda}}(X)
&={\mathsf{P}}_{\mu}(X) \prod_{i=1}^n \frac{(bx_i)_{\infty}}{(ax_i)_{\infty}} \\
\intertext{and}
\sum_{{\lambda}} {\mathsf{P}}_{{\lambda}/\mu}\biggl[\frac{a-b}{1-t}\biggr]{\mathsf{Q}}_{{\lambda}}(X)
&={\mathsf{Q}}_{\mu}(X) \prod_{i=1}^n \frac{(bx_i)_{\infty}}{(ax_i)_{\infty}}.\end{aligned}$$
For $\mu=0$ this is just the $q$-binomial theorem for Macdonald polynomials.
By the two identities are in fact one and the same result and we only need to prove the first claim. To achieve this we multiply by ${\mathsf{P}}_{{\lambda}}(aX)$ and sum over ${\lambda}$. By and homogeneity this yields $${\mathsf{P}}_{\mu}(X)
\sum_{\nu} (b)_{\nu} {\mathsf{P}}_{\nu}(aX)=
\sum_{{\lambda}} {\mathsf{Q}}_{{\lambda}/\mu}\biggl[\frac{a-ab}{1-t}\biggr]{\mathsf{P}}_{{\lambda}}(X).$$ On the right we can sum over $\nu$ using the $q$-binomial theorem leading to the desired result (with $b\mapsto ab$).
Elementary manipulations show that the theorem is invariant under the simultaneous changes $n\leftrightarrow m$, $tX\leftrightarrow Y$ and $a\mapsto qt/a$. Without loss of generality we may thus assume that $m\leq n$. But $$(at^{n-1})_{{\lambda}}
\prod_{i,j=1}^n \frac{(a t^{j-i-1})_{{\lambda}_i-\mu_j}}
{(a t^{j-i})_{{\lambda}_i-\mu_j}}\bigg|_{\mu_{m+1}=\dots=\mu_n=0}
=(at^{m-1})_{{\lambda}}
\prod_{i=1}^n \prod_{j=1}^m \frac{(a t^{j-i-1})_{{\lambda}_i-\mu_j}}
{(a t^{j-i})_{{\lambda}_i-\mu_j}}$$ so that the case $m<n$ follows from the case $m=n$ by setting $y_{m+1}=\dots=y_n=0$.
In the remainder we assume that $m=n$, in which case the theorem simplifies to $$\begin{gathered}
\label{misn}
\sum_{{\lambda},\mu}
t^{{\lvert{\lambda}\rvert}-n{\lvert\mu\rvert}}
{\mathsf{P}}_{{\lambda}}(X){\mathsf{P}}_{\mu}(Y)\,
(at^{n-1})_{{\lambda}}(qt^n/a)_{\mu}
\prod_{i,j=1}^n
\frac{(a t^{j-i-1})_{{\lambda}_i-\mu_j}}
{(a t^{j-i})_{{\lambda}_i-\mu_j}} \\
=\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(tx_i)_{\infty}}
\prod_{j=1}^n\frac{(qy_j/a)_{\infty}}{(y_j)_{\infty}}
\prod_{i,j=1}^n \frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}}.\end{gathered}$$ To prove this we take Theorem \[thmPQ\], replace $a\mapsto q/a$, multiply both sides by $$t^{{\lvert{\lambda}\rvert}} {\mathsf{P}}_{{\lambda}}(X){\mathsf{Q}}_{\mu}(Y)$$ and sum over ${\lambda}$ and $\mu$. Hence $$\begin{gathered}
\sum_{{\lambda},\mu,\nu} {\mathsf{P}}_{{\lambda}}(X){\mathsf{Q}}_{\mu}(Y)
{\mathsf{P}}_{\mu/\nu}\biggl[\frac{1-q/a}{1-t}\biggr]
{\mathsf{Q}}_{{\lambda}/\nu}\biggl[\frac{t-a}{1-t}\biggr] \\
=\sum_{{\lambda},\mu} t^{{\lvert{\lambda}\rvert}-n{\lvert\mu\rvert}}
{\mathsf{P}}_{{\lambda}}(X){\mathsf{Q}}_{\mu}(Y) (at^{n-1})_{{\lambda}}\,
{\mathsf{P}}_{\mu}\biggl[\frac{1-qt^n/a}{1-t}\biggr]
\prod_{i,j=1}^n
\frac{(at^{j-i-1})_{{\lambda}_i-\mu_j}}{(at^{j-i})_{{\lambda}_i-\mu_j}}.\end{gathered}$$ On the right we apply and to rewrite $${\mathsf{Q}}_{\mu}(Y) {\mathsf{P}}_{\mu}\biggl[\frac{1-qt^n/a}{1-t}\biggr]=
(qt^n/a)_{\mu} {\mathsf{P}}_{\mu}(Y),$$ and on the left we employ Lemma \[Pieri\] to carry out the sums over ${\lambda}$ and $\mu$. Hence $$\begin{gathered}
\sum_{\nu} {\mathsf{P}}_{\nu}(X){\mathsf{Q}}_{\nu}(Y)
\prod_{i=1}^n \frac{(ax_i)_{\infty}}{(tx_i)_{\infty}}
\prod_{j=1}^n \frac{(qy_j/a)_{\infty}}{(y_j)_{\infty}} \\
=\sum_{{\lambda},\mu} t^{{\lvert{\lambda}\rvert}-n{\lvert\mu\rvert}}
{\mathsf{P}}_{{\lambda}}(X){\mathsf{P}}_{\mu}(Y) (at^{n-1})_{{\lambda}}(qt^n/a)_{\mu} \,
\prod_{i,j=1}^n
\frac{(at^{j-i-1})_{{\lambda}_i-\mu_j}}{(at^{j-i})_{{\lambda}_i-\mu_j}}.\end{gathered}$$ Performing the remaining sum on the left by results in .
We conclude this section with a remark about a generalisation of Theorem \[thmCauchy\]. Let $X=\{x_1,\dots,x_n\}$ and let ${\lambda}$ be a partition of length $n$. Then $$P_{{\lambda}}(X)=x_1\cdots x_n \, P_{\mu}(X),$$ where $\mu=({\lambda}_1-1,\dots,{\lambda}_n-1)$. Now let ${\mathcal{P}}$ denote the set of weakly decreasing integer sequences of finite length. Then we may turn things around and use the above recursion to extend $P_{{\lambda}}$ to all ${\lambda}\in{\mathcal{P}}$. It is then readily verified that $$(qt^{n-1})_{{\lambda}} {\mathsf{P}}_{{\lambda}}(X)
= t^{n({\lambda})} \frac{(qt^{n-1})_{{\lambda}}}{c'_{{\lambda}}}\,
P_{{\lambda}}(X)$$ is well-defined for ${\lambda}\in{\mathcal{P}}$ (unlike ${\mathsf{P}}_{{\lambda}}(X)$).
We now state without proof the following generalisation of Theorem \[thmCauchy\].
Let $X=\{x_1,\dots,x_n\}$ and $Y=\{y_1,\dots,y_m\}$. Then $$\begin{gathered}
\sum_{{\lambda}\in{\mathcal{P}}}\sum_{\mu} t^{{\lvert{\lambda}\rvert}-n{\lvert\mu\rvert}}
{\mathsf{P}}_{{\lambda}}(X) {\mathsf{P}}_{\mu}(Y)
\frac{(at^{m-1},qt^{n-1})_{{\lambda}}(bt^n)_{\mu}}{(abt^{n-1})_{{\lambda}}}\,
\prod_{i=1}^n \prod_{j=1}^m
\frac{(at^{j-i-1})_{{\lambda}_i-\mu_j}}{(at^{j-i})_{{\lambda}_i-\mu_j}} \\
=\prod_{i=1}^n\frac{(qt^{i-1},bt^i,ax_i,q/ax_i)_{\infty}}
{(abt^{i-1},qt^i/a,tx_i,b/x_i)_{\infty}}
\prod_{j=1}^m \frac{(by_j)_{\infty}}{(y_j)_{\infty}}
\prod_{i=1}^n \prod_{j=1}^m \frac{(tx_iy_j)_{\infty}}{(x_iy_j)_{\infty}}.\end{gathered}$$
For $m=0$ this reduces to Kaneko’s $_1\Psi_1$ sum for Macdonald polynomials [@Kaneko98] and for $n=0$ to the $_1\Phi_0$ sum . When $ab=q$ the summand on the left vanishes unless ${\lambda}$ is an actual partition and we recover Theorem \[thmCauchy\].
The ${\mathfrak{sl}_{3}}$ Selberg integral {#SecSel3}
==========================================
The integration chains $C^{k_1,k_2}_{\beta,\gamma}[0,1]$ and $C^{k_1,k_2}_{\gamma}[0,1]$ {#Secchains}
----------------------------------------------------------------------------------------
Before proving Theorem \[thmsl3\] we give two descriptions of the chain $C_{\beta,\gamma}^{k_1,k_2}[0,1]$. We also identify the special case $\beta=1$ with the chain $C_{\gamma}^{k_1,k_2}[0,1]$ defined by Tarasov and Varchenko in [@TV03].
Let $$\begin{gathered}
\label{Rk1k2}
I^{k_1,k_2}[0,1]=\{(x_1,\dots,x_{k_1},y_1,\dots,y_{k_2})\in{\mathbb R}^{k_1+k_2}:\\
0<x_1<\dots<x_{k_1}<1{\quad\text{and}\quad}0<y_1<\dots<y_{k_2}<1\},\end{gathered}$$ and fix a total ordering among the $x_i$ and $y_j$ as follows. Let $a=(a_1,\dots,a_{k_1})$ be a weakly increasing sequence of nonnegative integers not exceeding $k_2$: $$\label{aas}
0\leq a_1\leq\dots\leq a_{k_1}\leq k_2.$$ Then the domain $I_a^{k_1,k_2}[0,1]\subseteq I^{k_1,k_2}[0,1]$ is formed by imposing the additional inequalities $$x_i<y_{a_i+1}<y_{a_i+2}<\dots<y_{a_{i+1}}<x_{i+1} \quad
\text{for $\;0\leq i\leq k_1$},$$ where $x_0:=0$, $x_{k_1+1}:=1$, $a_0:=0$ and $a_{k_1+1}:=k_2$. Equivalently, $$\label{order}
\begin{cases}
0<y_1<y_2<\dots<y_{a_i}<x_i \\
x_i<y_{a_i+1}<\dots<y_{k_2-1}<y_{k_2}<1
\end{cases}
\quad \text{for $\;1\leq i\leq k_1$}.$$ Clearly, as a chain, $$\label{Ikk}
I^{k_1,k_2}[0,1]=\sum_a I_a^{k_1,k_2}[0,1],$$ where the sum is over all sequences $a=(a_1,\dots,a_{k_1})$ satisfying . To lift $I^{k_1,k_2}[0,1]$ to $C^{k_1,k_2}_{\beta,\gamma}[0,1]$ we replace the right-hand side of by a weighted sum: $$\label{Ckk}
C^{k_1,k_2}_{\beta,\gamma}[0,1]=\sum_{a}
\biggl(\:\prod_{i=1}^{k_1}
\frac{\sin\pi(\beta-(i-a_i-k_1+k_2)\gamma)}
{\sin\pi(\beta-(i-k_1+k_2)\gamma)} \biggr) I_a^{k_1,k_2}[0,1],$$ where it is assumed that $\beta,\gamma\in{\mathbb C}$ such that $$\beta+(i-k_2-1)\gamma\not\in{\mathbb Z}\quad
\text{for $1\leq i\leq \min\{k_1,k_2\}$.}$$ This is a necessary and sufficient condition for $$\prod_{i=1}^{k_1}
\frac{\sin\pi(\beta-(i-a_i-k_1+k_2)\gamma)}
{\sin\pi(\beta-(i-k_1+k_2)\gamma)}$$ to be free of poles for all admissible sequences $a$.
By viewing $(a_{k_1},\dots,a_2,a_1)$ as a partition with largest part not exceeding $k_2$ and length not exceeding $k_1$, the operations of conjugation and/or complementation yield several alternative descriptions of the chain . Below we give one such description, reflecting the ${\mathbb Z}_2$ symmetry of Theorem \[thmsl3\] with respect to the interchange of the labels $1$ and $2$ in $k_i$, $\alpha_i$ and $\beta_i$.
Assume and fix a total ordering among the $x_i$ and $y_j$ as follows. Let $b=(b_1,\dots,b_{k_2})$ be a weakly increasing sequence of nonnegative integers not exceeding $k_1$: $$\label{bb}
0\leq b_1\leq\dots\leq b_{k_2}\leq k_1.$$ Then the domain $\bar{I}_b^{k_1,k_2}[0,1]\subseteq I^{k_1,k_2}[0,1]$ is formed by assuming the further inequalities $$y_i<x_{b_i+1}<x_{b_i+2}<\dots<x_{b_{i+1}}<y_{i+1} \quad
\text{for $\;0\leq i\leq k_2$},$$ where $y_0:=0$, $y_{k_2+1}:=1$, $b_0:=0$ and $b_{k_2+1}:=k_1$. It is easily seen that if $\mu=(b_{k_2},\dots,b_1)$ and ${\lambda}=(a_{k_1},\dots,a_1)$, then $\mu'$ is the conjugate of ${\lambda}$ with respect to $(k_2^{k_1})$, i.e., $\mu'_i=k_2-{\lambda}_{k_1-i+1}=k_2-a_i$ for $1\leq i\leq k_1$. Hence, for a pair of admissible sequences $(a,b)$ related by “conjugation–complementation”, $$\bar{I}_b^{k_1,k_2}[0,1]=I_a^{k_1,k_2}[0,1]$$ and $$\prod_{i=1}^{k_2}
\frac{\sin\pi(\beta+(i-b_i+k_1-k_2-1)\gamma)}
{\sin\pi(\beta+(i-k_2-1)\gamma)}
=
\prod_{i=1}^{k_1}
\frac{\sin\pi(\beta-(i-a_i-k_1+k_2)\gamma)}
{\sin\pi(\beta-(i-k_1+k_2)\gamma)}.$$ In other words, $$\label{Ckkb}
C^{k_1,k_2}_{\beta,\gamma}[0,1]
=\sum_b \biggl(\:\prod_{i=1}^{k_2}
\frac{\sin\pi(\beta+(i-b_i+k_1-k_2-1)\gamma)}
{\sin\pi(\beta+(i-k_2-1)\gamma)} \biggr)
\bar{I}_b^{k_1,k_2}[0,1]$$ summed over all sequences $b=(b_1,\dots,b_{k_2})$ subject to . Comparing and , and using that for $\beta_1+\beta_2=\gamma+1$, $$\prod_{i=1}^{k_1}
\frac{\sin\pi(\beta_1-(i-k_1+k_2))\gamma)}
{\sin\pi(\beta_2+(i-k_1-1)\gamma)}
=
\prod_{i=0}^{k_1-1}\frac{\Gamma(\beta_1+i\gamma)}
{\Gamma(\beta_1+(i-k_2)\gamma)}
\prod_{i=0}^{k_2-1}\frac{\Gamma(\beta_2+(i-k_1)\gamma)}
{\Gamma(\beta_2+i\gamma)}$$ it readily follows that the symmetry relation holds.
To conclude this section we consider for $\beta=1$: $$C^{k_1,k_2}_{1,\gamma}[0,1]=\sum_a
\biggl(\:\prod_{i=1}^{k_1}
\frac{\sin\pi((i-a_i-k_1+k_2)\gamma)}
{\sin\pi((i-k_1+k_2)\gamma)} \biggr)
I_a^{k_1,k_2}[0,1].$$ The summand vanishes if $a_i=i-k_1+k_2$ for some $1\leq i\leq k_1$ so that we may add the additional restrictions $$a_i\neq i-k_1+k_2 \qquad\text{for $1\leq i\leq k_1$}$$ to the sum over $a$. Recalling this in fact implies that the much stronger $$a_i<i-k_1+k_2 \qquad\text{for $1\leq i\leq k_1$}.$$ Therefore, $$C^{k_1,k_2}_{1,\gamma}[0,1]=
\sum_{\substack{a \\[1pt] a_i<i-k_1+k_2}}
\biggl(\:\prod_{i=1}^{k_1}
\frac{\sin\pi((i-a_i-k_1+k_2)\gamma)}
{\sin\pi((i-k_1+k_2)\gamma)} \biggr)
I_a^{k_1,k_2}[0,1].$$ Defining $M(i)=a_i+1$, so that $$1\leq M(1)\leq M(2)\leq \dots \leq M(k_1)\leq k_2$$ and $$M(i)\leq i-k_i+k_2 \qquad\text{for $1\leq i\leq k_1$},$$ and writing $M=(M(1),\dots,M(k_1))$, we finally obtain $$\begin{aligned}
\label{chainTV}
C^{k_1,k_2}_{1,\gamma}[0,1]
&=\sum_M
\biggl(\:\prod_{i=1}^{k_1}
\frac{\sin\pi((i-M(i)-k_1+k_2+1)\gamma)}
{\sin\pi((i-k_1+k_2)\gamma)} \biggr)
I_M^{k_1,k_2}[0,1] \\[1mm]
&=:C^{k_1,k_2}_{\gamma}[0,1]. \notag\end{aligned}$$ In the above, by abuse of notation, $I_M^{k_1,k_2}[0,1]=I_a^{k_1,k_2}[0,1]$ if $M=(a_1+1,\dots,a_{k_1}+1)$. The chain $C^{k_1,k_2}_{\gamma}[0,1]$ is precisely that of Tarasov and Varchenko (up to an interchange of $k_1$ and $k_2$), see [@TV03 page 177].
Proof of Theorem \[thmsl3\]
---------------------------
We are now prepared to prove Theorem \[thmsl3\]. In fact, we will prove a more general integral, generalising Kadell’s extension of the Selberg integral [@Kadell97] to ${\mathfrak{sl}_{3}}$. To shorten some of the subsequent equations we introduce another normalised Macdonald polynomial, and for $X=\{x_1,\dots,x_n\}$ $$\tilde{{\mathsf{P}}}_{{\lambda}}(X)=\frac{{\mathsf{P}}_{{\lambda}}(X)}{{\mathsf{P}}_{{\lambda}}({\langle 0\rangle})}=
\frac{P_{{\lambda}}(X)}{P_{{\lambda}}({\langle 0\rangle})}.$$ Similarly we define a (normalised) Jack polynomial as $$\tilde{{\mathsf{P}}}_{{\lambda}}^{(\alpha)}(X)=
\lim_{q\to 1} \tilde{{\mathsf{P}}}_{{\lambda}}(X;q^{\alpha},q).$$ Hence $$\tilde{{\mathsf{P}}}_{{\lambda}}^{(\alpha)}(X)=
\frac{P_{{\lambda}}^{(\alpha)}(X)}{P_{{\lambda}}^{(\alpha)}(1^n)},$$ where $P_{{\lambda}}^{(\alpha)}(X)$ is the Jack polynomial [@Macdonald95; @Stanley89].
Set $X=\{x_1,\dots,x_{k_1}\}$, $Y=\{y_1,\dots,y_{k_2}\}$, $$\operatorname{d\!}X=\operatorname{d\!}x_1\cdots\operatorname{d\!}x_{k_1}{\quad\text{and}\quad}\operatorname{d\!}Y=\operatorname{d\!}y_1\cdots\operatorname{d\!}y_{k_1}.$$ For $\alpha_1,\alpha_2,\beta_1,\beta_2,\gamma\in{\mathbb C}$ such that ${\lvert\gamma\rvert}$ is sufficiently small, $$\min\{{\textup{Re}}(\alpha_1)+{\lambda}_{k_1},{\textup{Re}}(\alpha_2)+\mu_{k_2},
{\textup{Re}}(\beta_1),{\textup{Re}}(\beta_2)\}>0,$$
$$\beta_1+(i-k_2-1)\gamma\not\in{\mathbb Z}\quad\text{for $1\leq i\leq \min\{k_1,k_2\}$}$$ and $$\beta_1+\beta_2=\gamma+1$$ there holds $$\begin{aligned}
{\int\limits}_{C_{\beta_1,\gamma}^{k_1,k_2}[0,1]}
&\tilde{{\mathsf{P}}}^{(1/\gamma)}_{{\lambda}}(X)
\tilde{{\mathsf{P}}}^{(1/\gamma)}_{\mu}(Y)
\prod_{i=1}^{k_1} x_i^{\alpha_1-1}(1-x_i)^{\beta_1-1}
\prod_{i=1}^{k_2} y_i^{\alpha_2-1} (1-y_i)^{\beta_2-1} \\[-2mm]
&\times \prod_{1\leq i<j\leq k_1} {\lvertx_i-x_j\rvert}^{2\gamma}
\prod_{1\leq i<j\leq k_2} {\lverty_i-y_j\rvert}^{2\gamma} \:
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}\, {\lvertx_i-y_j\rvert}^{-\gamma} \;
\operatorname{d\!}X \operatorname{d\!}Y \\[2mm]
&\qquad=\prod_{i=1}^{k_1}\frac{\Gamma(\alpha_1+(k_1-i)\gamma+{\lambda}_i)
\Gamma(\beta_1+(i-k_2-1)\gamma)\Gamma(i\gamma)}
{\Gamma(\alpha_1+\beta_1+(2k_1-k_2-i-1)\gamma+{\lambda}_i)\Gamma(\gamma)} \\
&\qquad\quad\times \prod_{i=1}^{k_2}
\frac{\Gamma(\alpha_2+(k_2-i)\gamma+\mu_i)
\Gamma(\beta_2+(i-1)\gamma)\Gamma(i\gamma)}
{\Gamma(\alpha_2+\beta_2+(2k_2-k_1-i-1)\gamma+\mu_i)
\Gamma(\gamma)} \\
&\qquad\quad\times\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}
\frac{\Gamma(\alpha_1+\alpha_2+(k_1+k_2-i-j-1)\gamma+{\lambda}_i+\mu_j)}
{\Gamma(\alpha_1+\alpha_2+(k_1+k_2-i-j)\gamma+{\lambda}_i+\mu_j)}.\end{aligned}$$
Theorem \[thmsl3\] corresponds to the case special case ${\lambda}=\mu=0$, and Kadell’s integral arises by taking $k_1=0$ or $k_2=0$.
Throughout the proof $0<q<1$.
We take Theorem \[thmCauchy\] with $({\lambda},\mu)$ replaced by $(\eta,\nu)$ and $(n,m)$ replaced by $(k_1,k_2)$. If we then specialise $X=z{\langle {\lambda}\rangle}_{k_1}$ and $Y=w{\langle \mu\rangle}_{k_2}$ and use the evaluation symmetry on both Macdonald polynomials in the summand, we obtain $$\begin{gathered}
\sum_{\eta,\nu}
t^{{\lvert\eta\rvert}-k_1{\lvert\nu\rvert}}
\tilde{{\mathsf{P}}}_{{\lambda}}({\langle \eta\rangle}_{k_1})
\tilde{{\mathsf{P}}}_{\mu}({\langle \nu\rangle}_{k_2})
{\mathsf{P}}_{\eta}(z{\langle 0\rangle}_{k_1}){\mathsf{P}}_{\nu}(w{\langle 0\rangle}_{k_2}) \\[-2mm]
\times (at^{k_2-1})_{{\lambda}}(qt^{k_1}/a)_{\nu}
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}
\frac{(a t^{j-i-1})_{\eta_i-\nu_j}}{(a t^{j-i})_{\eta_i-\nu_j}} \\
=\prod_{i=1}^{k_1} \frac{(azq^{{\lambda}_i}t^{k_1-i})_{\infty}}
{(zq^{{\lambda}_i}t^{k_1-i+1})_{\infty}}
\prod_{j=1}^{k_2}\frac{(wq^{\mu_j+1}t^{k_2-j}/a)_{\infty}}
{(wq^{\mu_j}t^{k_2-j})_{\infty}}
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}
\frac{(wztq^{{\lambda}_i+\mu_j}t^{k_1+k_2-i-j})_{\infty}}
{(wzq^{{\lambda}_i+\mu_j}t^{k_1+k_2-i-j})_{\infty}}.\end{gathered}$$ Next we set $$(z,w,a,t)=(q^{\alpha_1-\gamma},q^{\alpha_2},
q^{\beta_1+(k_1-k_2)\gamma},q^{\gamma})$$ and introduce the auxiliary variable $\beta_2$ by $\beta_1+\beta_2=\gamma+1$. Equations , and imply the principal specialisation formula $${\mathsf{P}}_{{\lambda}}({\langle 0\rangle}_n)=\frac{t^{2n({\lambda})}}{(qt^{n-1})_{{\lambda}}}
\prod_{1\leq i<j\leq n}
\frac{1-q^{{\lambda}_i-{\lambda}_j}t^{j-i}}{1-t^{j-i}}\,
\frac{(t^{j-i+1})_{{\lambda}_i-{\lambda}_j}}{(qt^{j-i-1})_{{\lambda}_i-{\lambda}_j}}.$$ Using this as well as the definition of the $q$-Gamma function $$\Gamma_q(x)=\frac{(q)_{x-1}}{(1-q)^{x-1}}, \quad x\in{\mathbb C},$$ we can rewrite the above identity as $$\begin{gathered}
(1-q)^{k_1+k_2}\sum_{\eta,\nu}
\tilde{{\mathsf{P}}}_{{\lambda}}(x_1q^{(k_1-1)\gamma},x_2q^{(k_1-2)\gamma},\dots,x_{k_1}) \\
\times
\tilde{{\mathsf{P}}}_{\mu}(y_1q^{(k_2-1)\gamma},y_2q^{(k_2-2)\gamma},\dots,y_{k_1}) \\
\times
\prod_{i=1}^{k_1} x_i^{\alpha_1}
(q^{1+(k_1-i)\gamma}x_i)_{\beta_1-1}
\prod_{1\leq i<j\leq k_1} x_j^{2\gamma}
\bigl(1-q^{(j-i)\gamma}x_i/x_j\bigr)
(q^{1+(j-i-1)\gamma}x_i/x_j)_{2\gamma-1} \\
\times
\prod_{i=1}^{k_2} y_i^{\alpha_2}
(q^{1+(k_2-i)\gamma}y_i)_{\beta_2-1}
\prod_{1\leq i<j\leq k_2} y_j^{2\gamma}
\bigl(1-q^{(j-i)\gamma}y_i/y_j\bigr)
(q^{1+(j-i-1)\gamma}y_i/y_j)_{2\gamma-1} \\
\times
\prod_{i=1}^{k_1} \prod_{j=1}^{k_2} y_j^{-\gamma}
(q^{\beta_1+(k_1-k_2+j-i)\gamma}x_i/y_j)_{-\gamma} \\
=\prod_{i=1}^{k_1}\frac{\Gamma_q(\alpha_1+(k_1-i)\gamma+{\lambda}_i)
\Gamma_q(\beta_1+(i-k_2-1)\gamma)\Gamma_q(i\gamma)}
{\Gamma_q(\alpha_1+\beta_1+(2k_1-k_2-i-1)\gamma+{\lambda}_i)\Gamma_q(\gamma)} \\
\times \prod_{i=1}^{k_2}
\frac{\Gamma_q(\alpha_2+(k_2-i)\gamma+\mu_i)
\Gamma_q(\beta_2+(i-1)\gamma)\Gamma_q(i\gamma)}
{\Gamma_q(\alpha_2+\beta_2+(2k_2-k_1-i-1)\gamma+\mu_i)\Gamma_q(\gamma)} \\
\times\prod_{i=1}^{k_1} \prod_{j=1}^{k_2}
\frac{\Gamma_q(\alpha_1+\alpha_2+(k_1+k_2-i-j-1)\gamma+{\lambda}_i+\mu_j)}
{\Gamma_q(\alpha_1+\alpha_2+(k_1+k_2-i-j)\gamma+{\lambda}_i+\mu_j)}.\end{gathered}$$ Here $x_i:=q^{\eta_i}$ and $y_i:=q^{\nu_i}$, so that $$\label{xy}
0<x_1<\cdots<x_{k_1}\leq 1{\quad\text{and}\quad}0<y_1<\cdots<y_{k_1}\leq 1.$$ The above is essentially a $(k_1+k_2)$-dimensional $q$-integral (more on this in the next section) and all that remains is to let $q$ tend to $1$ from below. The resulting integrand, however, depends sensitively on the relative ordering between the $x_i$ and $y_j$. Indeed [@W09], $$\begin{gathered}
\lim_{q\to 1^{-}}
y_j^{-\gamma} (q^{\beta_1+(k_1-k_2+j-i)\gamma}x_i/y_j)_{-\gamma} \\
={\lvertx_i-y_j\rvert}^{-\gamma} \times
\begin{cases}1 & \text{if $x_i<y_j$} \\[2mm]
\displaystyle
\frac{\sin\pi (\beta_1-(i-j-k_1+k_2)\gamma)}
{\sin\pi (\beta_1-(i-j-k_1+k_2+1)\gamma)}
& \text{if $x_i>y_j$}.
\end{cases}\end{gathered}$$ Consequently, before we can take the required limit we must fix a complete ordering among the integration variables (compatible with ) and sum over all admissible orderings. This is exactly what is done at the beginning of this section and in the remainder we assume that $$(x_1,\dots,x_{k_1},y_1,\dots,y_{k_2})\in I_{a_1,\dots,a_{k_1}}^{k_1,k_2}[0,1].$$ To find how to weigh this domain we recall that according to $y_{a_i+1}>x_i>y_{a_i}>\dots>y_1$. The correct weight is thus $$\prod_{i=1}^{k_1} \prod_{j=1}^{a_i}
\frac{\sin\pi (\beta_1-(i-j-k_1+k_2)\gamma)}
{\sin\pi (\beta_1-(i-j-k_1+k_2+1)\gamma)}
=\prod_{i=1}^{k_1}
\frac{\sin\pi (\beta_1-(i-a_i-k_1+k_2)\gamma)}
{\sin\pi (\beta_1-(i-k_1+k_2)\gamma)},$$ in accordance with $C^{k_1,k_2}_{\beta_1,\gamma}[0,1]$, see .
The Askey–Habsieger–Kadell integral
===================================
For $0<q<1$ the $q$-integral on $[0,1]$ is defined as $$\label{qint}
\int_0^1 f(x){{\operatorname{d\!}\,}_q}x=(1-q)\sum_{k=0}^{\infty} f(q^k) q^k,$$ where it is assumed the series on the right converges. When $q\to 1^-$ the $q$-integral reduces, at least formally, to the Riemann integral of $f$ on the unit interval. An obvious $n$-dimensional analogue of is $${\int\limits}_{[0,1]^n} f(X){{\operatorname{d\!}\,}_q}X=(1-q)^n\sum_{k_1,\dots,k_n=0}^{\infty}
f(q^{k_1},\dots,q^{k_n}) q^{k_1+\cdots+k_n},$$ where the multiple sum on the right is assumed to be absolutely convergent and where $f(X)=f(x_1,\dots,x_n)$ and ${{\operatorname{d\!}\,}_q}X={{\operatorname{d\!}\,}_q}x_1\cdots{{\operatorname{d\!}\,}_q}x_n$.
In 1980 Askey [@Askey80] conjectured a $q$-analogue of the Selberg integral when the parameter $\gamma$ is a nonnegative integer, say $k$: $$\begin{gathered}
{\int\limits}_{[0,1]^n}\prod_{i=1}^n x_i^{\alpha-1}
(x_iq)_{\beta-1}
\prod_{1\leq i<j\leq n} x_i^{2k} (q^{1-k}x_j/x_i)_{2k}
\,{{\operatorname{d\!}\,}_q}X \\
=q^{\alpha k\binom{n}{2}+2k^2\binom{n}{3}}
\prod_{i=0}^{n-1}
\frac{\Gamma_q(\alpha+ik)\Gamma_q(\beta+ik)\Gamma_q(1+(i+1)k)}
{\Gamma_q(\alpha+\beta+(n+i-1)k)\Gamma_q(1+k)} ,\end{gathered}$$ for $\textup{Re}(\alpha)>0$ and $\beta\neq 0,-1,-2,\dots$. Askey’s conjecture was proved independently by Habsieger [@Habsieger88] and Kadell [@Kadell88].
Just as the ordinary Selberg integral, the Askey–Habsieger–Kadell integral can be generalised by the inclusion of symmetric functions in the integrand. Specifically, Kaneko [@Kaneko96] and Macdonald [@Macdonald95] proved that $$\begin{gathered}
\label{qKM}
{\int\limits}_{[0,1]^n} \tilde{{\mathsf{P}}}_{{\lambda}}(X;q,q^k)
\prod_{i=1}^n x_i^{\alpha-1}(x_iq)_{\beta-1}
\prod_{1\leq i<j\leq n} x_i^{2k}
(q^{1-k}x_j/x_i)_{2k}\, {{\operatorname{d\!}\,}_q}X \\
=q^{\alpha k\binom{n}{2}+2k^2\binom{n}{3}}
\prod_{i=1}^n
\frac{\Gamma_q(\alpha+(n-i)k+{\lambda}_i)\Gamma_q(\beta+(i-1)k)\Gamma_q(ik+1)}
{\Gamma_q(\alpha+\beta+(2n-i-1)k+{\lambda}_i)\Gamma_q(k+1)},\end{gathered}$$ for $\textup{Re}(\alpha)>-{\lambda}_n$ and $\beta\neq 0,-1,-2,\dots$.
The ${\mathfrak{sl}_{n}}$–${\mathfrak{sl}_{m}}$ transformation formula of Theorem \[thmPhitrafo\] allows the for the Askey–Habsieger–Kadell integral as well as its generalisation to be extended to a transformation between integrals of different dimensions. For ${\lambda}=({\lambda}_1,\dots,{\lambda}_n)$ and $\mu=(\mu_1,\dots,\mu_m)$ define $$\begin{gathered}
S^{(n,m)}_{{\lambda}\mu}(\alpha_1,\alpha_2,\beta;k)
={\int\limits}_{[0,1]^n}\tilde{{\mathsf{P}}}_{{\lambda}}(X;q,q^k)\prod_{i=1}^n x_i^{\alpha_1-1}
(x_iq)_{\beta-(n-1)k-1}\\
\times \prod_{i=1}^n \prod_{j=1}^m
\frac{(x_iq)_{\alpha_2+\beta+\mu_j+(m-n-j)k-1}}
{(x_iq)_{\alpha_2+\beta+\mu_j+(m-n-j+1)k-1}}
\prod_{1\leq i<j\leq n} x_i^{2k} (q^{1-k}x_j/x_i)_{2k}
\,\textup{d}_q X.\end{gathered}$$
Let ${\lambda}=({\lambda}_1,\dots,{\lambda}_n)$, $\mu=(\mu_1,\dots,\mu_m)$ be partitions, $k$ a nonnegative integer and $\alpha_1,\alpha_2,\beta\in{\mathbb C}$. Then $$\begin{aligned}
S^{(n,m)}_{{\lambda}\mu}(\alpha_1,\alpha_2,\beta;k)
&=q^{\alpha_1 k\binom{n}{2}-\alpha_2 k\binom{m}{2}+
2k^2\binom{n}{3}-2k^2\binom{m}{3}} \,
S^{(m,n)}_{\mu{\lambda}}(\alpha_2,\alpha_1,\beta;k) \\
&\quad \times \prod_{i=1}^n
\frac{\Gamma_q(\beta-(i-1)k)\Gamma_q(\alpha_1+{\lambda}_i+(n-i)k)\Gamma_q(ik+1)}
{\Gamma_q(\alpha_1+\beta+{\lambda}_i+(n-m-i)k)\Gamma_q(k+1)} \\
&\quad \times \prod_{i=1}^m
\frac{\Gamma_q(\alpha_2+\beta+\mu_i+(m-n-i)k)\Gamma_q(k+1)}
{\Gamma_q(\beta-(i-1)k)\Gamma_q(\alpha_2+\mu_i+(m-i)k)\Gamma_q(ik+1)}\end{aligned}$$ for $\textup{Re}(\alpha_1)>-{\lambda}_n$, $\textup{Re}(\alpha_2)>-\mu_m$, and generic $\beta$.
By “generic $\beta$” it is meant that $\beta$ should avoid a countable set of isolated singularities. More precisely, $\beta$ should be such that none of $\beta-(n-1)k$, $\beta-(m-1)k$, $\alpha_1+\beta+{\lambda}_j+(n-m-j)k$ and $\alpha_2+\beta+\mu_j+(m-n-j)k$ take nonpositive integer values. Since $S^{(0,n)}_{0,{\lambda}}(\alpha_2,\alpha_1,\beta;k)=1$ the $m=0$ case of the theorem corresponds to with $(\alpha,\beta)\mapsto (\alpha_1,\beta-(n-1)k)$.
The method of proof is identical to that employed in [@W05 Theorem 1.1] and we only sketch the details of what are essentially elementary manipulations.
We specialise $X\mapsto c{\langle {\lambda}\rangle}_n$ and $Y\mapsto b{\langle \mu\rangle}_m$ in Theorem \[thmPhitrafo\] and apply the evaluation symmetry to obtain $$\begin{gathered}
\sum_{\nu}
\frac{(a,abq^{\mu_1}t^{m-2},\dots,abq^{\mu_m}t^{-1})_{\nu}}
{(abq^{\mu_1}t^{m-1},\dots,abq^{\mu_m})_{\nu}}\,
{\mathsf{P}}_{\nu}(c{\langle 0\rangle}_n)\tilde{{\mathsf{P}}}_{{\lambda}}({\langle \nu\rangle}_n)\\
=\biggl(\:\prod_{i=1}^n \frac{(acq^{{\lambda}_i}t^{n-i})_{\infty}}
{(cq^{{\lambda}_i}t^{n-i})_{\infty}}
\biggr)\biggl(\:\prod_{i=1}^m \frac{(bq^{\mu_i}t^{m-i})_{\infty}}
{(abq^{\mu_i}t^{m-i})_{\infty}} \biggr) \\
\times \sum_{\nu}
\frac{(a,acq^{{\lambda}_1}t^{n-2},\dots,acq^{{\lambda}_n}t^{-1})_{\nu}}
{(acq^{{\lambda}_1}t^{n-1},\dots,acq^{{\lambda}_n})_{\nu}}\,
{\mathsf{P}}_{\nu}(b{\langle 0\rangle}_m)\tilde{{\mathsf{P}}}_{\mu}({\langle \nu\rangle}_m).\end{gathered}$$
Next we replace $t\mapsto q^k$ with $k$ a positive integer and replace $a\mapsto q^{\beta}$, $b\mapsto q^{\alpha_2}$ and $c\mapsto
q^{\alpha_1}$. Then we apply [@W05 Lemma 3.1] to write the $\nu$-sums as $n$-fold unrestricted sums, and the claim follows.
The above derivation can be repeated starting from Theorem \[thmCauchy\]. The result is an ${\mathfrak{sl}_{3}}$ variant of the $q$-integral . Problem with the theorem below is, however, that it does not converge in the $q\to 1^{-}$ limit unless $m$ or $n$ is $0$. (This can be remedied by replacing $[0,1]^{m+n}$ by appropriate multiple Pochhammer double loops).
Let ${\lambda}=({\lambda}_1,\dots,{\lambda}_n)$, $\mu=(\mu_1,\dots,\mu_m)$ be partitions, $k$ a nonnegative integer and $\alpha_1,\alpha_2,\beta_1,\beta_2\in{\mathbb C}$ such that $\beta_1+\beta_2=k+1$. Then $$\begin{aligned}
{\int\limits}_{[0,1]^{n+m}} & \tilde{{\mathsf{P}}}_{{\lambda}}(X;q,q^k)
\prod_{i=1}^n x_i^{\alpha_1} (qx_i)_{\beta_1-1}
\prod_{1\leq i<j\leq n} x_j^{2k} (q^{1-k} x_i/x_j)_{2k} \\[-2mm]
\times\, & \tilde{{\mathsf{P}}}_{\mu}(Y;q,q^k)
\prod_{i=1}^m y_i^{\alpha_2} (qy_i)_{\beta_2-1}
\prod_{1\leq i<j\leq m} y_j^{2k} (q^{1-k} y_i/y_j)_{2k} \\
&\hspace{4cm}\times\prod_{i=1}^n \prod_{j=1}^m
y_j^{-k} (q^{\beta_1}x_i/y_j)_{-k}
\: {{\operatorname{d\!}\,}_q}X \, {{\operatorname{d\!}\,}_q}Y \\
&=q^{\alpha_1 k\binom{n}{2}+\alpha_2 k\binom{m}{2}+
2k^2\binom{n}{3}+2k^2\binom{n}{3}-k^2 n\binom{m}{2}} \\
&\quad\times
\prod_{i=1}^n\frac{\Gamma_q(\alpha_1+(n-i)k+{\lambda}_i)
\Gamma_q(\beta_1+(i-m-1)k)\Gamma_q(ik+1)}
{\Gamma_q(\alpha_1+\beta_1+(2n-m-i-1)k+{\lambda}_i)\Gamma_q(k)} \\
&\quad\times
\prod_{i=1}^m \frac{\Gamma_q(\alpha_2+(m-i)k+\mu_i)
\Gamma_q(\beta_2+(i-1)k)\Gamma_q(ik+1)}
{\Gamma_q(\alpha_2+\beta_2+(2m-n-i-1)k+\mu_i)\Gamma_q(k)} \\
&\quad\times\prod_{i=1}^n \prod_{j=1}^m
\frac{\Gamma_q(\alpha_1+\alpha_2+(n+m-i-j-1)k+{\lambda}_i+\mu_j)}
{\Gamma_q(\alpha_1+\alpha_2+(n+m-i-j)k+{\lambda}_i+\mu_j)},\end{aligned}$$ for $\textup{Re}(\alpha_1)>-{\lambda}_n$, for $\textup{Re}(\alpha_2)>-\mu_m$, and $\beta_1,\beta_2\neq 0,-1,-2,\dots$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank Eric Rains for helpful discussions.
[99]{}
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[^1]: Work supported by the Australian Research Council
|
---
abstract: 'It is well known that, for fast rotating fluids with the axis of rotation being perpendicular to the boundary, the boundary layer is of Ekman-type, described by a linear ODE system. In this paper we consider fast rotating fluids, with the axis of rotation being parallel to the boundary. We show that the corresponding boundary layer is describe by a nonlinear, degenerated PDE system which is similar to the $2$ -D Prandtl system. Finally, we prove the well-posedness of the governing system of the boundary layer in the space of analytic functions with respect to tangential variable.'
address:
- 'School of Mathematics and Statistics, and Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China '
- 'Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France'
- |
Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France\
and\
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
author:
- 'Wei-Xi Li, Van-Sang Ngo and Chao-Jiang Xu'
title: Boundary layer analysis for the fast horizontal rotating fluids
---
Introduction {#section1}
============
The incompressible Navier-Stokes equation coupled with a large Coriolis term reads $$\begin{aligned}
\left\{
\begin{aligned}
&\partial_tu^{\varepsilon} - \nu\Delta u^{\varepsilon} + u^{\varepsilon}\cdot\nabla u^{\varepsilon} + \frac{\omega\times u^{\varepsilon}}{\varepsilon} + \nabla p^{\varepsilon} = 0,&&\\
&{\mbox{div }}u^{\varepsilon} = 0, &&\\
&u^{\varepsilon}|_{t=0}= u^\varepsilon_0, &&
\end{aligned}
\right. \end{aligned}$$ with Dirichlet boundary condition, where $ \frac{\omega\times u_{\varepsilon}}{{\varepsilon}}$ stands for the Coriolis force and $\omega$ is the rotation vector, ${\varepsilon}^{-1}$ the rescaled speed of rotation, $\nu$ the viscosity coefficients. The above system is sufficient to describe the rotation fluids which is a significant part of geophysics. Due to the earth’s self-rotation, we can’t neglect the Coriolis force in order to model the oceanography and meteorology dealing with large-scale magnitude. When the fluid is between a strip and the direction of rotation is not parallel to the boundary, we have the well-developed Ekman layers to match the interior flow with Dirichlet boundary condition, cf. [@CDGG3; @CDGGbook; @grenier-masmoudi; @masmoudi-2] and the references therein. The situation will be more complicated when the direction of rotation is parallel to the boundary, considering cylinder for instance and letting the fluid rotate around the vertical axis. Then we will have two types of boundaries, the horizontal boundary layer which is Ekman layers and the vertical boundary layers for which much less is known, despite various studies [@CDGGbook; @ste; @vooren]. We refer to [@CDGGbook] for detailed discussions on the problem of vertical boundary layers.
In this paper, we consider the fast rotating viscous fluids where the the axe of rotation is horizontal with respect to the boundary. We prove that the governing equation for boundary layer is nonlinear PDE system which is similar to classical $2$ -D Prandtl boundary layer system, and we also obtain the well-posedness of this vertical boundary layers in the space of analytic functions.
As a preliminary step we first consider the half space case $\mathbb R^3_+ = \mathbb{R}^2_h \times \mathbb{R}_+$. More precisely, we consider the following system $$\label{NSCeps}\tag{N-S$_\varepsilon$}
\left\{
\begin{aligned}
&\partial_tu^{\varepsilon} - \nu\Delta u^{\varepsilon} + u^{\varepsilon}\cdot\nabla u^{\varepsilon} + \frac{e_2\times u^{\varepsilon}}{\varepsilon} + \nabla p^{\varepsilon} = 0 &&\mbox{in }\; \mathbb{R}^2_h \times \mathbb{R}_+, \;\forall t \geq 0\\
&{\mbox{div }}u^{\varepsilon} = 0 &&\mbox{in }\; \mathbb{R}^2_h \times \mathbb{R}_+, \;\forall t \geq 0\\
&u^\varepsilon|_{x_3=0} = 0 &&\mbox{on }\; \mathbb{R}^2_h\\
&u^{\varepsilon}|_{t=0}= u^\varepsilon_0, &&\mbox{in }\; \mathbb{R}^2_h \times \mathbb{R}_+\, .
\end{aligned}
\right.$$ where $e_2 = (0,1,0)$ is the unit horizontal vector, $\nu>0$ the coefficient of viscosity of fluids and $\varepsilon$ the Rossby number. These equations describe the evolution of an incompressible three-dimensional viscous fluid in a rotating frame, $\frac{e_2\times u^{\varepsilon}}{\varepsilon}$ being the Coriolis force due to the rotation at high frequency $\varepsilon^{-1}$. According to the Taylor-Proudman theorem [@taylor], the fast rotation penalize the movement of the fluid in the direction of the rotation axis. As a consequence, the fluid has tendency to move in columns, parallel to the rotation axis, which are widely known as the Taylor columns. This phenomenon is well-known in oceanography and meteorology, which is observed in many large-scale atmospheric and oceanic flows. In mathematical point of view, when $\varepsilon$ goes to zero, the rotation term $\frac{e_2\times u^{\varepsilon}}{\varepsilon}$ becomes large and can only be balanced by the pressure. This means that, if $u$ is the (formal) limit of $u^\varepsilon$, as $\varepsilon \rightarrow 0$, then $e_2 \times u$ need to be a gradient term, which implies that $u$ is independent of $x_2$ (more explanations will be found in Section \[section2\]). In this paper, we will consider the case where the initial data are well prepared, *i.e.* $u^\varepsilon_0$ do not depend on $x_2$.
When there is no Coriolis force, the zero-viscosity limit for the Navier-Stokes equations for incompressible fluids in a domain with boundary, with non-slip boundary conditions, is a challenging problem due to the formation of a boundary layer which is governed by the Prandtl equations ([@oleinik]). The mathematical analysis theory of Prandtl equation is also a challenging problem, see [@awxy; @e-1; @e-2; @GV-D; @M-W] and references therein. Far from the boundary, the inviscid limit problem was treated by several authors; we can refer, for instance, to Swann [@swann] and Kato [@kato-2]. In another work, Kato [@kato-1] gives some equivalent formulations of this problem in the case of bounded domains, showing that the convergence to the Euler system is equivalent to the fact that the $L^2$ strength of the boundary layer goes to $0$. Caflisch & Sammartino [@samm] solved the problem for analytic solutions on a half space by solving the Prandtl equations via abstract Cauchy-Kowaleskaya theorem. We also refer to [@GMM; @GN; @maekawa] and the references therein for the recent progress on the inviscid limit of the Navier-Stokes equations when the initial vorticity is located away from the boundary. On the other hand, another commonly used boundary conditions are Navier-type slip boundary conditions, in which case the vanishing viscosity limit is rigorously justified; cf. [@Lions; @wangwangxin; @XiaoXin; @Yu] and references therein.
We want to say a few words to compare the system with the case where the rotation axis is vertical with respect to the boundary (the rotation axis is in the direction of $e_3 = (0,0,1)$ instead of $e_2$). If the domain considered is between two parallel plates (${\mathbb{T}}^2\times [0,1]$ or ${\mathbb{R}}^2 \times [0,1]$), it was proved in Grenier & Masmoudi [@grenier-masmoudi], Masmoudi [@masmoudi-2; @masmoudi-3] and Chemin *et al.* [@CDGG3] that for the rotating fluids with anisotropic viscosity $-\nu \Delta_h-{\varepsilon}\partial_{x_3}^2$, all the weak solutions of Navier-Stokes equation converge to the solution of the 2D Euler or 2D Navier-Stokes system (with damping term - effect of the Ekman pumping). The vertical rotation and the specific form of the domain (between two parallel plates) permit to explicitly construct the boundary layer velocity term from the interior velocity term (which satisfies a 2D damped Euler system), without using the Prandtl equations. We also want to mention the work of Dalibard and Gérard-Varet [@DaGV] in the case of fast rotating fluids on a rough domain with non-slip boundary conditions. The boundary layer is also proved to be of size $\varepsilon$ (contrary to the case of Prandtl equations where the boundary layer is of size $\sqrt{\varepsilon}$). We also refer to a series of work for the rotating fluids with anisotopic viscosity (see for exemple [@CDGG], [@CDGG2], [@IG1], [@IGLSR], [@gongguowang],[@Iftimie3], [@VSN], [@Paicu1]).
We want to emphasize that the formation of the boundary layers in the case of vertical rotation axis is due to the incompatibility of the Dirichlet boundary conditions with the columnar movement of the limit fluid (as $\varepsilon \rightarrow 0$). Indeed, as the rotation axis is $e_3$, the limiting velocity of the fluid is independent of $x_3$, and so, the Dirichlet boundary conditions imply that the limit velocity should be zero. This incompatibility leads to the fact that a thin layer (Ekman’s layer) is formed near the boundary, and the fluid’s evolution is violent in this small scale zone, in a way that stops the fluid on the boundary.
In the case of horizontal rotation axis (in the direction of $e_2$), the incompatibility of boundary conditions will be more complicated, because of the fact that the limit velocity is independent of $x_2$ instead of $x_3$. In Section \[section2\], we prove that the limit system is a 2D Euler-like system. This means that we are no longer in the case considered by Ekman. The techniques of [@grenier-masmoudi] and [@CDGG3] do not work and we can not explicitly calculate the boundary layer. The fast rotation only penalizes the fluid motion in the $x_2$ direction, and leads to a problem very close to the inviscid limit of two-dimensional Navier-Stokes system. It is then relevant to look for a boundary layer of size $\sqrt{\varepsilon}$ and we will show in Section \[section2\] that in this boundary layer of size $\sqrt{\varepsilon}$, the fluid velocity actually satisfies a two-dimensional Prandtl-like system. Finally, we remark that in this paper, we only consider the case where $\nu=\varepsilon$. Indeed, as explained in [@grenier-masmoudi] and also in [@CDGGbook], if the ratio $\nu/\varepsilon$ goes to infinity, the fluid rapidly stops after a few evolutions. It is then more interesting to consider the case where $\nu \lesssim \varepsilon$, which moreover better fits physical observations.
In this work, we study the formation of the boundary layer when $\nu=\varepsilon \to 0$. We suppose the existence of a boundary layer of size $\sqrt{\varepsilon}$ near the boundary ${\left\{x_3=0\right\}}$ of ${\mathbb{R}}^3_+$. We will derive the limit equation and the boundary layer equation by using a formal asymptotic expansion in the Section \[section2\]. We refer to the book of Pedlovsky [@pedlovsky] for more detail about this formal expansion. To this end, we suppose that the solution of accepts the following asymtotic expansion
[align]{} \[eq:AnsatzU\] u\^(t,x\_1,x\_2,x\_3) &= \_[j=0]{}\^1 \^[2]{} + ,\
\[eq:AnsatzP\] p\^(t,x\_1,x\_2,x\_3) &= \_[j=-2]{}\^1 \^[2]{} p\^[I,j]{}(t,x\_1,x\_2,x\_3) + \_[j=-2]{}\^0 \^[2]{} p\^[B,j]{}[(t,x\_1,x\_2,)]{} +,
where $u^{B,j}(t,x_1,x_2,y)$ and $p^{B,j}(t,x_1,x_2,y)$ exponentially go to zero as $y\stackrel{\rm def}{=}\frac{x_3}{\sqrt{\varepsilon}} \to +\infty$. The remaining terms is supposed to be very small (at least of order 3).
Throughout this paper, we will always use ${\partial}_t$, ${\partial}_i$ (or ${\partial}_{x_i}$), $i=1,2,3$, and ${\partial}_y$ to respectively denote the derivatives with respect to the time variable $t$, the space variables $x_i$, $i=1,2,3$, and the boundary layer variable $y = \frac{x_3}{\sqrt{\varepsilon}}$. Using the above asymptotic expansion, we first deduce that the behavior of the fluid near the boundary is governed by the following 2D Prandtl-like equation
[equation]{} \[eq:PrRotE2\] {
&\_t \^[p,0]{}\_1 - \_y\^2 \^[p,0]{}\_1 + \^[p,0]{}\_1 \_1 \^[p,0]{}\_1 + \^[p,1]{}\_3 \_y \^[p,0]{}\_1 + \_1 p\^[B,0]{} + = 0,\
&\_t \^[p,1]{}\_3 - \_y\^2 \^[p,1]{}\_3 + \^[p,0]{}\_1 \_1 \^[p,1]{}\_3 + \^[p,1]{}\_3 \_y \^[p,1]{}\_3 + + y = 0,\
&\_1 \^[p,0]{}\_1 + \_y \^[p,1]{}\_3 = 0,\
&\^[p, 0]{}\_1|\_[y=0]{} =0,\_[y+]{} \^[p,0]{}\_1(t,x\_1,y)=,\
&\^[p, 1]{}\_3|\_[y=0]{} =0,\_y \^[p,1]{}\_3 |\_[y=0]{}= 0,\
& ( \^[p,0]{}\_1, \^[p,1]{}\_3)|\_[t=0]{} = ( \^[p,0]{}\_[1, 0]{}, \^[p,1]{}\_[3, 0]{}),
.
with the unknown functions ${\left({\mathcal{U}}^{p,0}_1, {\mathcal{U}}^{p,1}_3, p^{B,0}\right)},$ and the horizontal second component satisfies a parabolic type equation
[equation]{} \[eq:PrRotE2-b\] {
&\_t \^[p,0]{}\_2 - \_y\^2 \^[p,0]{}\_2 + \^[p,0]{}\_1 \_1 \^[p,0]{}\_2 + \^[p,1]{}\_3 \_y \^[p,0]{}\_2 = 0\
&\^[p, 0]{}\_2|\_[y=0]{} =0,\_[y+]{} \^[p,0]{}\_2(t,x\_1,y)) =\
& \^[p, 0]{}\_2 |\_[t=0]{} = \^[p, 0]{}\_[2, 0]{}.
.
Here $$\partial_2 {\mathcal{U}}^{p,0}_1 = \partial_2 {\mathcal{U}}^{p,0}_2 = \partial_2 {\mathcal{U}}^{p,1}_3 = 0.$$ Here, we emphasize the “Prandtl-like” property of our system by using the new unknown functions $$\begin{aligned}
&{\mathcal{U}}^{p,0}_j= u^{B,0}_j + \overline{u^{I,0}_j}, \qquad j=1, 2\\
&{\mathcal{U}}^{p,1}_3 = u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3 u^{I,0}_3}\end{aligned}$$ where $\overline{u^{I,j}}, \overline{p^{I,j}}, j=1, 2 $ are the values on the boundary of the tangential velocity and pressure of the outflow satisfying the Bernoulli-type law $$ \left\{
\begin{aligned}
&\partial_t \overline{u^{I,0}_1}+\overline{u^{I,0}_1}\partial_1\overline{u^{I,0}_1}+ \overline{\partial_1p^{I,0}}=0\\
&\partial_t \overline{u^{I,0}_2}+\overline{u^{I,0}_1}\partial_1\overline{u^{I,0}_2}+ \overline{\partial_2p^{I,0}}=0\\
&\partial_t \overline{u^{I,1}_3}+\overline{u^{I,0}_1}\partial_1\overline{u^{I,1}_3}+\overline{u^{I,1}_3}\partial_3\overline{u^{I,0}_3}+ \overline{\partial_3p^{I,1}}=0
\end{aligned}
\right.$$which is the restriction of the Euler system and linearized Euler system on the boundary $x_3=0$, so that they depend only on the variavles $(t, x_1)$. More precise description will be found in Section \[section2\].
Note that the boundary layer equation look very close to that of classical 2D Prandtl equation, but the fast rotating produces the boundary layer pressure for the first components, so that the boundary layer equation is now really a system of 3 equations with both the velocity $ ( {\mathcal{U}}^{p,0}_1, {\mathcal{U}}^{p,1}_3)$ and the boundary pressure $p^{B, 0}$ to determined. We remark that on one side, the first equation in admits the similar structure of Prandtl equation, i.e., the degeneracy in $x_1$ coupled with the nonlocal property arising from the term ${\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_1$, so that the system is quite similar to Prandtl equation. Therefore we can only expect the local well-posedness for analytic initial data if no additional assumptions are imposed. On the other hand, there is a crucial difference between Prandtl equation and the first equation in , due to the unknown pressure $p^{B,0}.$ Recall the pressure term in Prandtl equation is from outflow and can be defined by the Bernoulli law, so that the pressure therein is a given function and therefore Prandtl equation is a kind of degenerate parabolic equation. But here the situation is quite complicated since we have the unknown pressure $p^{B,0}$ in , which arises because of the fast rotation parallel to the boundary, and can’t be defined by the Bernoulli law anymore. So the classical theory for Prandtl equation is not applicable directly to our case and moreover we can’t follow the same strategy as in Prandtl equation to treat the the first equation in . To overcome the difficulty due to the unknown pressure term in the first equation of , we will firstly solve the second equation for ${\mathcal{U}}^{p,1}_3$, and then use the divergence-free property to find ${\mathcal{U}}^{p,0}_1$ (see Section \[section3\] for detail). Finally we mention that the mathematical justification of the inviscid limit for solutions to , is also complicated as classical Prandtl boundary layer theory. We only concentrate in this work on the well-posedness of boundary layer and will investigate this inviscid limit problem in the future work.
On the other hand, we will prove in Section \[section2\] that the limiting velocity of the outer flow satisfies a classical 2D Euler-type equation, which is, $$\label{eq:Ordre0Ibis}
\left\{
\begin{aligned}
&\partial_t u^{I,0}_1 + u^{I,0}_1 \partial_1 u^{I,0}_1 + u^{I,0}_3 \partial_3 u^{I,0}_1 + \partial_1 p^{I,0} = 0\\
&\partial_t u^{I,0}_2 + u^{I,0}_1 \partial_1 u^{I,0}_2 + u^{I,0}_3 \partial_3 u^{I,0}_2 = 0\\
&\partial_t u^{I,0}_3 + u^{I,0}_1 \partial_1 u^{I,0}_3 + u^{I,0}_3 \partial_3 u^{I,0}_3 + \partial_3 p^{I,0} = 0\\
&\partial_2 u^{I,0}_1 = \partial_2 u^{I,0}_2 = \partial_2 u^{I,0}_3 = {\partial}_2 p^{I,0} = 0\\
&\partial_1 u^{I,0}_1 + \partial_3 u^{I,0}_3 = 0\\
&u^{I,0}_3|_{x_3=0} = 0\\
&u^{I,0} |_{t=0}= u^{I,0}_0(x_1,x_3).
\end{aligned}
\right.$$
In the system [(\[eq:Ordre0Ibis\])]{}, the components $(u^{I, 0}_1, u^{I, 0}_3, p^{I, 0})$ satisfy exactly a 2-D incompressible Euler equation on the half-plane, so that the existence and regularity in Gevery class of local in time solution is well know, (see Vicol [@KV] and references therein), but in the study of boundary layer equation, we need some weighted on the tangential variables, we cite in particular the results of [@cheng-li].
Let $\frac{1}{2}<\ell\leq 1$ be given. We denote by $\mathcal A_\tau$ the space of analytic functions with analytic radius $\tau>0$, which is consist of all functions $f\in L^2({\mathbb{R}}^2_+)$ such that $${\big\Vertf\big\Vert}_{\mathcal A_\tau}\stackrel{\rm def}{=} \sup_{{\left\vert\alpha\right\vert}\geq 0} \frac{\tau^{{\left\vert\alpha\right\vert}}}{{\left\vert\alpha\right\vert}!}{\big\Vert{\left<z\right>}^\ell \partial_{z}^\alpha f\big\Vert}_{L^2(\mathbb R_+^2)} < +\infty.$$
\[th:E2D\] Suppose that the initial data $u^{I,0}_0=(u^{I,0}_{1,0},u^{I,0}_{2,0}, u^{I,0}_{3,0})$ in [(\[eq:Ordre0Ibis\])]{} satisfies $$\begin{aligned}
u^{I,0}_{1,0},\; u^{I,0}_{2,0},\; u^{I,0}_{3,0} \in \mathcal A_{\tau_0} \end{aligned}$$ for some $\tau_0>0$, the divergence-free condition and the compatibility condition. Then Euler-type system [(\[eq:Ordre0Ibis\])]{} admits a unique solution $(u^{I,0}_1, u^{I,0}_2, u^{I,0}_3)\in L^\infty{\left([0,T];~\mathcal A_\tau\right)}$ for some $T>0$ and $\tau>0.$
The construction of the components $(u^{I, 0}_1, u^{I, 0}_3, p^{I, 0})$ is given in [@cheng-li]. The construction of $u^{I,0}_2$ is standard, using the classical theory of transport equation.
Now we list several estimates, which are just immediate consequences of the definition of ${\big\Vert\cdot\big\Vert}_{\mathcal A_{\tau}}$ and Sobolev inequalities. For $u^{I,0}_1\in L^\infty{\left([0,T];~\mathcal A_\tau\right)},$ we have, for all $p,q\geq 0,$ $$\begin{aligned}
\label{eseuler}
{\big\Vert{\left<x_1\right>}^\ell\partial_1^{p}\partial_3^q u^{I,0}_3(x_1,x_3)\big\Vert}_{L^\infty{\left(\mathbb R_+;~L^2(\mathbb R_{x_1})\right)}}\leq C {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau} \frac{(p+q+3)!}{ \tau^{p+q+3}}.\end{aligned}$$ Using the equation $$\begin{aligned}
\partial_t u^{I,0}_1 + u^{I,0}_1 \partial_1 u^{I,0}_1 + u^{I,0}_3 \partial_3 u^{I,0}_1 + \partial_1 p^{I,0}=0, \end{aligned}$$ we can calculate, by virtue of Leibniz formula, $$\begin{aligned}
\label{dt}
&&{\big\Vert{\left<x_1\right>}^\ell \partial_t\partial_1^{p}\partial_3^q u^{I,0}_3\big\Vert}_{L^\infty{\left(\mathbb R_+;~L^2(\mathbb R_{x_1})\right)}}\qquad\qquad\\
&&\leq C_\tau {\left( {\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau}^2+{\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau}{\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau}+{\big\Vertp^{I,0}\big\Vert}_{\mathcal A_\tau}\right)} \frac{2^{p+q} (p+q)!}{\tau^{p+q}}\,.\nonumber
\end{aligned}$$
In order to completely give the solutions of the systems and , we also need the following linearized Euler system, which describes the evolution of the fluids in the interior part of the domain, far from the boundary, at the order $\sqrt{\varepsilon}$.
[equation]{} \[eq:Ordre0.5I\] {
&\_t u\^[I,1]{}\_1 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_1 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_1 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_1 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_1 + \_1 p\^[I,1]{} = 0\
&\_t u\^[I,1]{}\_2 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_2 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_2 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_2 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_2 = 0\
&\_t u\^[I,1]{}\_3 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_3 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_3 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_3 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_3 + \_3 p\^[I,1]{} = 0\
&\_2 u\^[I,1]{}\_1 = \_2 u\^[I,1]{}\_2 = \_2 u\^[I,1]{}\_3 = \_2 p\^[I,1]{} = 0\
&\_1 u\^[I,1]{}\_1 + \_3 u\^[I,1]{}\_3 = 0\
&u\^[I,1]{}\_3|\_[x\_3=0]{}= -u\^[B,1]{}\_3(t,x\_1,0)\
&u\^[I,1]{}|\_[t=0]{} = u\^[I,1]{}\_0(x\_1,x\_3).
.
For this linearized Euler system [(\[eq:Ordre0.5I\])]{}, we have
\[th:E2Dlin\] Let $\ell>1/2$, $\tau_0>0$ and $u^{B,1}_3(t,x_1,0)$ a given function such that $$\sum_{m\leq 2 } {\big\Vert{\left<x_1\right>}^\ell \partial_1^m u^{B,1}_3(t,x_1,0)\big\Vert}_{L^2(\mathbb R_{x_1})}^2 + \sum_{ m\geq 3 } {\left[\frac{\tau_0^{m-1}}{ (m-3)!}\right]}^2{\big\Vert{\left<x_1\right>}^\ell \partial_1^m u^{B,1}_3(t,x_1,0)\big\Vert}_{L^2(\mathbb R_{x_1})}^2 <+\infty.$$ Suppose that the initial data $u^{I,1}_0=(u^{I,1}_{1,0},u^{I,1}_{2,0}, u^{I,1}_{3,0})$ in [(\[eq:Ordre0.5I\])]{} satisfies the divergence-free condition, the compatibility condition and $$\begin{aligned}
u^{I,1}_{1,0},\;u^{I,1}_{2,0},\; u^{I,1}_{3,0}\in \mathcal A_{\tau_0}.
\end{aligned}$$ Then the linearized Euler system [(\[eq:Ordre0.5I\])]{} admits a unique solution $(u^{I,1}_1, u^{I,1}_2, u^{I,1}_3)\in L^\infty{\left([0,T];~\mathcal A_\tau\right)}$ for some $T>0$ and $\tau>0.$
We remark that, the compatibility condition ask $$u^{I,1}_{3,0} (x_1, 0)=-u^{B,1}_{3,0}(x_1, 0).$$ It is exactly the non-slip condition of at order $1$. Because of its linearity, treating the system is still much easier than treating the system , even with the presence of the given boundary function $u^{B,1}_3(t,x_1,0)$. So, to prove Theorem \[th:E2Dlin\], we can simply follow the lines of the proof of Theorem \[th:E2D\] as in [@cheng-li].
Before giving the well-posedness results on and , we need the following weighted analytic function spaces in tangential variable. We also remark that there is no coupling between $({\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,1}_3)$ and ${\mathcal{U}}^{p,0}_2$. Then, the strategy consists in separately solving the systems and .
\[xrho\] Let $1/2<\ell\leq 1$ be given throughout the paper. With each pair $(\rho, a) $ with $\rho>0$ and $a>0$ we associate a space $X_{\rho,a} $ of all functions $u(x_1,y)\in H^\infty(\mathbb R_{x_1};~H^2(\mathbb R_+))$ such that $$\begin{aligned}
\sum_{m\leq 2 } {\left(\sum_{0\leq j\leq 1} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}^2\right)}+\sum_{ m\geq 3 } {\left(\sum_{0\leq j\leq 1} {\left[\frac{\rho^{m-1}}{ (m-3)!}\right]}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}^2 \right)}<+\infty,\end{aligned}$$ where we use the convention $0!=1$. We endow $X_{\rho,a}$ with the norm $$\begin{aligned}
{\left\vertu\right\vert}_{X_{\rho, a}}^2=\sum_{m\leq 2 } {\left(\sum_{0\leq j\leq 1} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}^2\right)}+\sum_{ m\geq 3 } {\left(\sum_{0\leq j\leq 1} {\left[\frac{\rho^{m-1}}{ (m-3)!}\right]}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}^2 \right)}.\end{aligned}$$
The well-posedness of the system can be stated as follows.
\[th:Prandtl\] Suppose that the initial data $$\begin{aligned}
{\mathcal{U}}^{p,1}_{3,0} = u^{B,1}_{3,0} + \overline{u^{I,1}_{3,0}} + y \overline{\partial_3 u^{I,0}_{3,0}}\end{aligned}$$ in [(\[eq:PrRotE2\])]{} satisfies that $$\begin{aligned}
u^{B,1}_{3,0}\in X_{\rho_0, a_0}, \quad u^{I,1}_{3,0}, \,\, u^{I,0}_{3,0}\in \mathcal A_{\tau_0}\end{aligned}$$ for some $a_0>0$, $\rho_0 >0$ and $\tau_0>0$ and $${\mathcal{U}}^{p,0}_{1, 0}(x_1, y) =-\int_{-\infty}^{x_1} \partial_yu^{B,1}_{3, 0}(z, y)dz+
\overline{u^{I,0}_{1, 0}}(x_1).$$ Then there exist $T>0$, $\tau>0$ and a pair ${\left(\rho, a\right)}$ with $\rho, a>0$, such that the system admits a unique solution $({\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,1}_3, \partial_1p^{B, 0})$, and moreover $$\begin{aligned}
&&{\mathcal{U}}^{p,1}_3 = u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3 u^{I,0}_3}\\
&&{\mathcal{U}}^{p,0}_1(t,x_1,y) =-\int_{-\infty}^{x_1} \partial_yu^{B,1}_3(t,z,y)dz+ \overline{u^{I,0}_1}(t,x_1),\end{aligned}$$ with $u^{B,1}_3\in L^\infty{\left([0,T]; ~X_{\rho,a}\right)}$ and $u^{I,0}_1, u^{I,0}_3, u^{I,1}_3 \in L^\infty{\left([0,T]; ~\mathcal A_\tau\right)}.$
(i) Here we consider the well prepared initial data, that is the initial data are independent of $x_2.$
(ii) We want to remark that once we find ${\mathcal{U}}^{p,1}_3$, we can obtain ${\mathcal{U}}^{p,0}_1$ using the divergence-free property in the third equation of the system .
Let ${\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,1}_3$ be the solutions to the system given by the theorem above. Then we see is a linear parabolic equation, and we have the following theorem concerned with its well-posedness.
\[th:SystemP2\] Let $\rho_0 > 0$, $a_0 > 0$, $\tau_0 > 0$ be given. For any initial data $${\mathcal{U}}^{p,0}_{2,0} = u^{B,0}_{2,0} + \overline{u^{I,0}_{2,0}}$$ where $u^{B,0}_{2,0} \in X_{\rho_0, a_0}$ and $u^{I,0}_{2,0}\in \mathcal A_{\tau_0}$, there exist $T>0$, $0<\tau<\tau_0$ and $0<a<a_0$, such that the equation admits a unique solution ${\mathcal{U}}^{p,0}_2$ satisfying ${\mathcal{U}}^{p,0}_2 = u^{B,0}_2 + \overline{u^{I,0}_2}$ with $$u^{B,0}_2 \in L^{\infty}{\left([0,T], X_{\rho_0, a}\right)},\quad u^{I,0}_2 \in L^{\infty}{\left([0,T], \mathcal A_\tau\right)}.$$
By the two theorems above we obtain the well-posedness for the boundary layer equation of the system in the frame of analytic space in tangential variable. The paper is organized as follows. In section \[section2\], we formally derive the governing equations of the outer flow inside the domain and the systems and which describe the fluid motion inside the boundary layer. The sections \[section3\]-\[section4\] are devoted to proving the well-posedness of the system . Finally, we give some brief ideas of the proof of Theorem \[th:SystemP2\] for the well-posedness of equation in the section \[section5\].
Formal asymtotic expansion {#section2}
==========================
First of all, we want to give a few words to explain our special choice of the order of the expansions of the velocity and the pressure. Indeed, we remark that as for the formulation of Prandtl boundary layer equations, we are only interested in the leading orders which are necessary to allow us to formally obtain the governing equations of the evolution of the boundary layer. By using the asymptotic expansions and , we have the following asymptotic identities for the leading terms up to order $\varepsilon^{1/2}$ and all the remaining terms are of higher order in ${\varepsilon}$.
[equation]{} \[eqs:AnsatzSyst\] {
\_t u\^&=\_[j=0]{}\^1 \^[2]{} [(\_t u\^[I,j]{} + \_t u\^[B,j]{})]{} +\
-u\^&=-\_y\^2 u\^[B,0]{} - \^[2]{}\_y\^2 u\^[B,1]{} - \_[j=0]{}\^1 \^[1+2]{} [(u\^[I,j]{} + \_h u\^[B,j]{})]{} +\
u\^u\^&=\_[j=0]{}\^1 \^[2]{} + \_[j=0]{}\^1 \^[2]{}\
&+ \_[j=0]{}\^1 \^[2]{} +\
&=\_[j=0]{}\^1 \^[2-1]{} +\
p\^&=\^[-2]{}
0\
0\
\_y p\^[B,-1]{}
+ \_[j=-2]{}\^[-1]{} \^j
\_1 p\^[B,j]{}\
\_2 p\^[B,j]{}\
\_y p\^[B,j+1]{}
+
\_1 p\^[B,0]{}\
\_2 p\^[B,0]{}\
0
+ \_[j=-2]{}\^1 \^[2]{} p\^[I,j]{} + .
.
Formal derivation of the fluid behavior far from the boundary
-------------------------------------------------------------
We put all the asymptotic identities into the system and we deduce that
[equation]{} \[eq:Interior\] \_[j=0]{}\^1 \^[2]{} \_t u\^[I,j]{} - \_[j=0]{}\^1 \^[1+2]{} u\^[I,j]{} + \_[j=0]{}\^1 \^[2]{} \_[k=0]{}\^j u\^[I,k]{} u\^[I,j-k]{} + \_[j=0]{}\^1 \^[2-1]{}
u\^[I,j]{}\_3\
0\
-u\^[I,j]{}\_1
+ \_[j=-2]{}\^1 \^[2]{} p\^[I,j]{} = 0().
Taking the limit $y = \frac{x_3}{\sqrt{{\varepsilon}}} \to +\infty$ (${\varepsilon}\to 0$), the divergence-free property writes
[equation]{} \[eq:divI\] u\^[I,j]{} = 0, j0.
**At the leading term of $\varepsilon^{-1}$** in , we simply have
[equation]{} \[eq:minus1I\]
u\^[I,0]{}\_3\
0\
-u\^[I,0]{}\_1
+
\_1 p\^[I,-2]{}\
\_2 p\^[I,-2]{}\
\_3 p\^[I,-2]{}
= 0.
Then, classical calculations (see Grenier-Masmoudi [@grenier-masmoudi] or Chemin [*et al.*]{} [@CDGGbook]) give
[equation]{} \[eq:minus1IuA\] \_2 p\^[I,-2]{} = \_2 u\^[I,0]{}\_1 = \_2 u\^[I,0]{}\_2 = \_2 u\^[I,0]{}\_3 = 0.
**At the order $\varepsilon^{-1/2}$** in , we have
[equation]{} \[eq:minus0.5I\]
u\^[I,1]{}\_3\
0\
-u\^[I,1]{}\_1
+
\_1 p\^[I,-1]{}\
\_2 p\^[I,-1]{}\
\_3 p\^[I,-1]{}
= 0,
which imply
[equation]{} \[eq:minus0.5Ip\] \_2 p\^[I,-1]{} = \_2 u\^[I,1]{}\_1 = \_2 u\^[I,1]{}\_2 = \_2 u\^[I,1]{}\_3 = 0.
Identities and mean that the limit behaviour of the outer flow is two-dimensional, as predicts the Taylor-Proudman theorem.
**At the order $\varepsilon^{0}$** in , taking into account and the divergence-free condition , we obtain
[equation]{} \[eq:Ordre0Iter\] {
&\_t u\^[I,0]{}\_1 + u\^[I,0]{}\_1 \_1 u\^[I,0]{}\_1 + u\^[I,0]{}\_3 \_3 u\^[I,0]{}\_1 + \_1 p\^[I,0]{} = 0\
&\_t u\^[I,0]{}\_2 + u\^[I,0]{}\_1 \_1 u\^[I,0]{}\_2 + u\^[I,0]{}\_3 \_3 u\^[I,0]{}\_2 + \_2 p\^[I,0]{}= 0\
&\_t u\^[I,0]{}\_3 + u\^[I,0]{}\_1 \_1 u\^[I,0]{}\_3 + u\^[I,0]{}\_3 \_3 u\^[I,0]{}\_3 + \_3 p\^[I,0]{} = 0\
&\_2 u\^[I,0]{}\_1 = \_2 u\^[I,0]{}\_2 = \_2 u\^[I,0]{}\_3 = 0\
&\_1 u\^[I,0]{}\_1 + \_3 u\^[I,0]{}\_3 = 0
.
Now, by applying ${\partial}_2$ to the second equation of the system , we obtain $${\partial}_2^2 p^{I,0} = 0,$$ which means that there exist $g_1(x_1,x_3)$ and $g_2(x_1,x_3)$ such that $$p^{I,0} = x_2 g_1 + g_2.$$ Now, differentiating the first and third equations of with respect to $x_2$, we obtain $${\partial}_1 g_1 = {\partial}_3 g_1 = 0.$$ By taking ${\left\vertx\right\vert} \to +\infty$ in the second equation of , we conclude that $g_1 \equiv 0$. Thus, the system becomes the following 2D Euler-type system with three components in the half-plane, which is the formal limiting system of far from the boundary as ${\varepsilon}\to 0$ $$ \left\{
\begin{aligned}
&\partial_t u^{I,0}_1 + u^{I,0}_1 \partial_1 u^{I,0}_1 + u^{I,0}_3 \partial_3 u^{I,0}_1 + \partial_1 p^{I,0} = 0\\
&\partial_t u^{I,0}_2 + u^{I,0}_1 \partial_1 u^{I,0}_2 + u^{I,0}_3 \partial_3 u^{I,0}_2 = 0\\
&\partial_t u^{I,0}_3 + u^{I,0}_1 \partial_1 u^{I,0}_3 + u^{I,0}_3 \partial_3 u^{I,0}_3 + \partial_3 p^{I,0} = 0\\
&\partial_2 u^{I,0}_1 = \partial_2 u^{I,0}_2 = \partial_2 u^{I,0}_3 = {\partial}_2 p^{I,0} = 0\\
&\partial_1 u^{I,0}_1 + \partial_3 u^{I,0}_3 = 0\\
&u^{I,0}_3|_{x_3=0} = 0.
\end{aligned}
\right.$$Since this system is independent of $x_2$, for the compatibility, we need to impose the well prepared initial data, which means that $$u^{I,0}(0,x_1,x_3) = u^{I,0}_0(x_1,x_3).$$ The boundary condition will be discussed in .
The system will be completed with a boundary condition for the second component $u^{I,0}_2$. In fact, the trace function $\overline{u^{I,0}_2}(t,x_1)$ on the boundary ${\left\{x_3 = 0\right\}}$ satisfies the following system $$ \left\{
\begin{aligned}
&\partial_t\overline{u^{I,0}_2} + \overline{u^{I,0}_1} \partial_1 \overline{u^{I,0}_2} = 0\\
&\overline{u^{I,0}_2}(0,x_1) = u^{I,0}_{0,2}(x_1,0).
\end{aligned}
\right.$$ **At the order $\varepsilon^{1/2}$** in , using and the divergence-free condition , we obtain the system
[equation\*]{} {
&\_t u\^[I,1]{}\_1 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_1 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_1 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_1 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_1 + \_1 p\^[I,1]{} = 0\
&\_t u\^[I,1]{}\_2 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_2 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_2 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_2 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_2 + \_2 p\^[I,1]{} = 0\
&\_t u\^[I,1]{}\_3 + u\^[I,0]{}\_1 \_1 u\^[I,1]{}\_3 + u\^[I,0]{}\_3 \_3 u\^[I,1]{}\_3 + u\^[I,1]{}\_1 \_1 u\^[I,0]{}\_3 + u\^[I,1]{}\_3 \_3 u\^[I,0]{}\_3 + \_3 p\^[I,1]{} = 0\
&\_2 u\^[I,1]{}\_1 = \_2 u\^[I,1]{}\_2 = \_2 u\^[I,1]{}\_3 = \_2 p\^[I,1]{} = 0\
&\_1 u\^[I,1]{}\_1 + \_3 u\^[I,1]{}\_3 = 0
.
We also remark that we can not obtain any determined boundary condition for $u^{I,1}$, but only a condition depending on the boundary condition of $u^{B,1}$. Indeed, on the boundary, we recall the value of $u^{I,j}_i$ is related to the value of $u^{B,j}_i$ by the equation $$u^{I,j}_i(t,x_1,0) + u^{B,j}_i(t,x_1,0) = 0\qquad j=0, 1; \quad i=1, 2, 3.$$
Using the same argument, we can prove that ${\partial}_2 p^{I,1} = 0$, and we obtain the following 2D linearized Euler-type system with three components in the half-plane $$ \left\{
\begin{aligned}
&\partial_t u^{I,1}_1 + u^{I,0}_1 \partial_1 u^{I,1}_1 + u^{I,0}_3 \partial_3 u^{I,1}_1 + u^{I,1}_1 \partial_1 u^{I,0}_1 + u^{I,1}_3 \partial_3 u^{I,0}_1 + {\partial}_1 p^{I,1} = 0\\
&\partial_t u^{I,1}_2 + u^{I,0}_1 \partial_1 u^{I,1}_2 + u^{I,0}_3 \partial_3 u^{I,1}_2 + u^{I,1}_1 \partial_1 u^{I,0}_2 + u^{I,1}_3 \partial_3 u^{I,0}_2 = 0\\
&\partial_t u^{I,1}_3 + u^{I,0}_1 \partial_1 u^{I,1}_3 + u^{I,0}_3 \partial_3 u^{I,1}_3 + u^{I,1}_1 \partial_1 u^{I,0}_3 + u^{I,1}_3 \partial_3 u^{I,0}_3 + {\partial}_3 p^{I,1} = 0\\
&\partial_2 u^{I,1}_1 = \partial_2 u^{I,1}_2 = \partial_2 u^{I,1}_3 = {\partial}_2 p^{I,1} = 0\\
&\partial_1 u^{I,1}_1 + \partial_3 u^{I,1}_3 = 0\\
&u^{I,1}_3(t,x_1,0) = -u^{B,1}_3(t,x_1,0)\\
&u^{I,1}(0,x_1,x_3) = u^{I,1}_0(x_1,x_3).
\end{aligned}
\right.$$Here, we also suppose that the initial data are well prepared, *i.e.* independent of $x_2$.
Formal asymptotic expansions inside the boundary layer {#subsection2.2}
------------------------------------------------------
Inside the boundary layer (in the domain $0<x_3\le \sqrt{\varepsilon}$), we consider the Taylor expansions
[align\*]{} u\^[I,j]{}\_i(t,x\_h,x\_3) &= u\^[I,j]{}\_i(t,x\_h,0) + x\_3 \_3 u\^[I,j]{}\_i(t,x\_h,0) + 2 \_3\^2 u\^[I,j]{}\_i(t,x\_h,0) + …\
p\^[I,j]{}(t,x\_h,x\_3) &= p\^[I,j]{}(t,x\_h,0) + x\_3 \_3 p\^[I,j]{}(t,x\_h,0) + 2 \_3\^2 p\^[I,j]{}(t,x\_h,0) + …
Performing the change of variable $y = \frac{x_3}{\sqrt{\varepsilon}}$, we have
[equation]{} \[eq:traceUI\] {
u\^[I,j]{}\_i(t,x\_h,x\_3) &= + \^[2]{} y + +\
p\^[I,j]{}(t,x\_h,x\_3) &= + \^[2]{} y + +
.
where $\overline{f} = f(t,x_1,x_2,0)$ is the trace of $f$ on ${\left\{x_3 = 0\right\}}$. Now, we will rewrite the identities , taking into account the expansion . First, we have
[align]{} \[eqs:AnsatzBound00\] u\^&= [(u\^[B,0]{} + )]{} + \^[2]{} [(u\^[B,1]{} + + y)]{} + \_[k=2]{}\^3 \^[2]{} [( + )]{} +\
&= [\^[p,0]{} + \^[2]{} \^[p,1]{} + \_[k=2]{}\^3 \^[2]{} [( + )]{} +.]{}
where we note $$\label{eqs:AnsatzBounddt}
{\mathcal{U}}^{p,0}=u^{B,0} + \overline{u^{I,0}},\qquad {\mathcal{U}}^{p,1}=u^{B,1} + \overline{u^{I,1}} + y\overline{\partial_3 u^{I,0}}.$$ The derivatives of $u^\varepsilon$ with respect to tangential variables write $$ \partial^m_{t, 1, 2} u^\varepsilon = \partial^m_{t, 1, 2} {\mathcal{U}}^{p,0} + \varepsilon^{\frac{1}2} \partial^m_{t, 1, 2} {\mathcal{U}}^{p,1} + \sum_{k=2}^3 \varepsilon^{\frac{k}2} \partial^m_{t, 1, 2} {\left(\frac{y^{k-1}}{(k-1)!} \overline{\partial_3^{k-1} u^{I,1}} + \frac{y^k}{k!} \overline{\partial_3^k u^{I,0}}\right)} + \cdots.$$where $m=1, 2$. For the normal variable, we have $$\partial_3 u^\varepsilon
= \varepsilon^{-\frac{1}2} \partial_y u^{B,0} + {\left(\partial_y u^{B,1} + \overline{\partial_3 u^{I,0}}\right)} + \sum_{k=1}^3 \varepsilon^{\frac{k}{2}} {\left(\frac{y^{k-1}}{(k-1)!} \overline{\partial_3^{k} u^{I,1}} + \frac{y^k}{k!} \overline{\partial_3^{k+1} u^{I,0}}\right)} + \cdots$$ and $$ \partial_3^2 u^\varepsilon = \varepsilon^{-1} \partial_y^2 u^{B,0} + \varepsilon^{-\frac{1}{2}} \partial_y^2 u^{B,1} + \overline{\partial_3^2 u^{I,0}} + \sum_{k=1}^3 \varepsilon^{\frac{k}{2}} {\left(\frac{y^{k-1}}{(k-1)!} \overline{\partial_3^{k+1} u^{I,1}} + \frac{y^k}{k!} \overline{\partial_3^{k+2} u^{I,0}}\right)} + \cdots.$$Thus, $$\begin{aligned}
-\varepsilon \Delta u^\varepsilon &= -\varepsilon \Delta_h {\mathcal{U}}^{p,0} - \varepsilon^{\frac{3}2} \Delta_h {\mathcal{U}}^{p,1}-\partial_y^2 {\mathcal{U}}^{p,0} - \varepsilon^{\frac{1}2} \partial_y^2 {\mathcal{U}}^{p,1} - \varepsilon \overline{\partial_3^2 u^{I,0}}\\
&\qquad\qquad - \varepsilon^{\frac{3}2} {\left(\frac{y^{k-1}}{(k-1)!} \overline{\partial_3^{k+1} u^{I,1}} + \frac{y^{k}}{k!} \overline{\partial_3^{k+2} u^{I,0}}\right)} + \cdots .\notag\end{aligned}$$
For the non-linear term, we only give the explicit calculations for the first orders of its expansion. We write $$u^\varepsilon\cdot\nabla u^\varepsilon = u^\varepsilon_h\cdot \nabla_h u^\varepsilon + u^\varepsilon_3 \partial_3 u^\varepsilon.$$ Then, we have $$\begin{aligned}
u^\varepsilon_h\cdot \nabla_h u^\varepsilon&={\mathcal{U}}^{p,0}_h \cdot \nabla_h {\mathcal{U}}^{p,0}_h + \varepsilon^{\frac{1}2} {\mathcal{U}}^{p,0}_h \cdot \nabla_h {\mathcal{U}}^{p,1} + \varepsilon^{\frac{1}2} {\mathcal{U}}^{p,1}_h \cdot \nabla_h {\mathcal{U}}^{p,0}_h + \cdots\\ u^\varepsilon_3 \partial_3 u^\varepsilon
&={\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_h + \varepsilon^{\frac{1}{2}} {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,1} + \varepsilon^{\frac{1}2} {\left(y\overline{\partial_3 u^{I,1}_3} + \frac{y^2}{2} \overline{\partial_3^2 u^{I,0}_3}\right)} \partial_y {\mathcal{U}}^{p,0}_h + \cdots .\end{aligned}$$
For the Coriolis forcing term (the rotation term), we have $$\begin{aligned}
&\frac{e_2\times u^\varepsilon}{\varepsilon}\\
&=\varepsilon^{-1} \begin{pmatrix} 0\\0\\-{\mathcal{U}}^{p,0}_1 \end{pmatrix} + \varepsilon^{-\frac{1}2} \begin{pmatrix} {\mathcal{U}}^{p,1}_3\\0\\-{\mathcal{U}}^{p,1}_1 \end{pmatrix} + \sum_{k=2}^3 \varepsilon^{\frac{k}2-1} {\left[\frac{y^{k-1}}{(k-1)!} \begin{pmatrix} \overline{\partial_3^{k-1} u^{I,1}_3} \\0\\-\overline{\partial_3^{k-1} u^{I,1}_1} \end{pmatrix} + \frac{y^k}{k!} \begin{pmatrix} \overline{\partial_3^k u^{I,0}_3} \\0\\-\overline{\partial_3^k u^{I,0}_1} \end{pmatrix} \right]} + \cdots. \notag\end{aligned}$$ Finally, the pressure term is
[align]{} \[eqs:AnsatzBound05\]
\_[x\_1]{} p\^\
\_[x\_2]{} p\^\
\_[x\_3]{} p\^
&= \^[-2]{}
0\
0\
\_y \^[p,-2]{}
+ \_[j=-2]{}\^[-1]{} \^[2]{}
\_1 \^[p,j]{}\
\_2 \^[p,j]{}\
\_y \^[p,j+1]{}
+\
&+ \^[2]{} \_[j=-2]{}\^1
\
\
+ .
where
[align]{} \[eq:Pprandtl2\] &\^[p,-2]{} = p\^[B,-2]{} + ,\^[p,-1]{} = p\^[B,-1]{} + + y\
\[eq:Pprandtl0\] &\^[p,0]{} = p\^[B,0]{} + + y + 2.
Incompressibility and boundary conditions
-----------------------------------------
The divergence-free property of the velocity field is rewritten as follows
[align\*]{} 0 &= u\^= \^[-2]{} \_y u\^[B,0]{}\_3[(t,x\_h,)]{} +\
& + \^[2]{} + .
Inside the boundary layer, using the expansion and , we deduce the following divergence-free condition
[equation\*]{} \^[-2]{} \_y u\^[B,0]{}\_3 + [(\_1 u\^[B,0]{}\_1 + \_2 u\^[B,0]{}\_2 + \_y u\^[B,1]{}\_3)]{} + \^[2]{} [(\_1 u\^[B,1]{}\_1 + \_2 u\^[B,1]{}\_2)]{}= 0.
Thus, we obtain the incompressibility of the boundary layer
[align]{} \[eq:divV0b\] &\_1 u\^[B,0]{}\_1 + \_2 u\^[B,0]{}\_2 + \_y u\^[B,1]{}\_3 = 0,\
&\_1 u\^[B,1]{}\_1 + \_2 u\^[B,1]{}\_2 = 0.
Moreover, we have $$\partial_y u^{B,0}_3 = 0,$$ which, by taking $y\to +\infty$, gives $$\label{eq:ub03} u^{B,0}_3 = 0.$$ For the boundary condition in on ${\left\{x_3 = 0\right\}}$, we have $$\label{eq:boundarycond} \sum_{j=0}^1 \varepsilon^{\frac{j}2} {\left[u^{I,j} (t,x_h,0) + u^{B,j} (t,x_h,0)\right]} = 0,$$ which implies that $$\overline{u^{I,0}(t)} + u^{B,0} (t,x_h,0) = 0,$$
[align]{} \[eq:bound1\] + u\^[B,1]{} (t,x\_h,0) = 0.
In particular, $u^{B,0}_3 = 0$ imply $$\label{eq:ZerothBC}
u^{I,0}_3|_{x_3=0}= \overline{u^{I,0}_3} = 0,$$ which is the boundary condition for Euler equation in , and the third components in gives the boundary condtion of linearized Euler equation in .
Formal derivation of the governing equations of the fluid in the boundary layer
-------------------------------------------------------------------------------
Now, we consider the system near ${\left\{x_3 = 0\right\}}$, using the asymptotic formal - .
**At the order $\varepsilon^{-\frac{3}{2}}$**, we have
[equation\*]{} \_y p\^[B,-2]{} = 0,
which implies that $p^{B,-2} = 0$ because $p^{B,-2}$ goes to zero as $y\to +\infty$. Using the new notation of the pressure defined in , we get $$\partial_y {\mathcal{P}}^{p,-2} = 0.$$
**At the order $\varepsilon^{-1}$**, using the fact that $\overline{u^{I,0}_3} = 0$, $u^{B,0}_3 = 0$ and $p^{B,-2} = 0$, we get
[equation\*]{} \[eq:minus1\]
0\
0\
-u\^[B,0]{}\_1
+
0\
0\
-
+
0\
0\
\_y p\^[B,-1]{}
+ = 0,
which implies that $\overline{\partial_1 p^{I,-1}} = \overline{\partial_2 p^{I,-1}} = 0$ and
[equation]{} \[eq:minus1B\] - u\^[B,0]{}\_1 - + \_y p\^[B,-1]{} + = 0.
Using the new velocity and pressure defined in and taking into account the fact that ${\mathcal{U}}^{p,0}_3 = 0$, we can also write
[equation]{} \[eq:Pminus1B\]
0\
0\
-\^[p,0]{}\_1
+
\_1\^[p,-2]{}\
\_2 \^[p,-2]{}\
\_y \^[p,-1]{}
= 0.
**At the order $\varepsilon^{-1/2}$**, we have
[equation\*]{}
u\^[B,1]{}\_3\
0\
-u\^[B,1]{}\_1
+
\
0\
-
+ y
\
0\
-
+
\_1 p\^[B,-1]{}\
\_2 p\^[B,-1]{}\
\_y p\^[B,0]{}
+
\
\
+ y
\
\
= 0,
or in a equivalent way, using the new velocity and pressure defined in ,
[equation]{} \[eq:Pminus0.5B\]
\^[p,1]{}\_3\
0\
-\^[p,1]{}\_1
+
\_1 \^[p,-1]{}\
\_2 \^[p,-1]{}\
\_y \^[p,0]{}
= 0.
then $$\partial_2 {\mathcal{P}}^{p,-1} = 0.$$ and
[align\*]{} &\_2 \^[p,0]{}\_1 = \_2 \_y \^[p,-1]{} = \_y \_2 \^[p,-1]{} = 0\
&\_2 \^[p,1]{}\_3 = -\_2 \_1 \^[p,-1]{} = -\_1 \_2 \^[p,-1]{} = 0.
Using the divergence-free properties and , we also have
[equation\*]{} \_2 \^[p,0]{}\_2 = - \_1 \^[p,0]{}\_1 - \_y \^[p,1]{}\_3 = - \_1 \_y \^[p,-1]{} - [(-\_y \_1 \^[p,-1]{})]{} = 0.
We deduce that $({\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,0}_2,{\mathcal{U}}^{p,1}_3)$ is a divergence-free vector field which is independent on $x_2$. The fact that ${\partial}_2 u^{I,0} = {\partial}_2 u^{I,1} = 0$ implies that $$\label{c2}
{\partial}_2 u^{B,0}_1 = {\partial}_2 u^{B,0}_2 = {\partial}_2 u^{B,1}_3 = 0.$$
The leading order of the velocity of the fluid inside the boundary layer also obeys the Taylor-Proudman theorem.
**At the order $\varepsilon^{0}$**, recalling that $u^{B,0}_3 = \overline{u^{I,0}_3} = 0$, we get the following equation
[multline\*]{} \_t [(u\^[B,0]{}\_h + )]{} - \_y\^2 u\^[B,0]{}\_h + [(u\^[B,0]{}\_h + )]{}\_h [(u\^[B,0]{}\_h + )]{} + [(u\^[B,1]{}\_3 + + y)]{} \_y u\^[B,0]{}\_h\
+ y
\
0\
-
+ 2
\
0\
-
+
\_1 p\^[B,0]{}\
\_2 p\^[B,0]{}\
0
+
\
\
+ y
\
\
+
\
\
= 0.
From and , we deduce that $$-y \overline{\partial_3 u^{I,1}_1} - \frac{y^2}2 \overline{\partial_3^2 u^{I,0}_1} + y \overline{\partial_3^2 p^{I,-1}} + \frac{y^2}2 \overline{\partial_3^3 p^{I,-2}} = 0.$$ We also remark that the boundary condition applying to the third equation of the Euler system implies that $$\overline{\partial_3 p^{I,0}} = 0.$$ Then, using the new velocity and pressure defined in and , we get
[multline\*]{} \_t \^[p,0]{}\_h - \_y\^2 \^[p,0]{}\_h + \^[p,0]{}\_h\_h \^[p,0]{}\_h + \^[p,1]{}\_3 \_y \^[p,0]{}\_h +
\_1 p\^[B,0]{} +\
\_2 \^[p,0]{}
= 0.
Taking into account the divergence-free condition , the identities and , and $({\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,0}_2,{\mathcal{U}}^{p,1}_3)$ is independs on $x_2$, we deduce that $({\mathcal{U}}^{p,0}_1,{\mathcal{U}}^{p,0}_2,{\mathcal{U}}^{p,1}_3)$ satisfies the following system $$\left\{
\begin{aligned}
&\partial_t {\mathcal{U}}^{p,0}_1 - \partial_y^2 {\mathcal{U}}^{p,0}_1 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_1 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_1 + \partial_1 p^{B,0} + \overline{\partial_1 p^{I,0}} = 0\\
&\partial_t {\mathcal{U}}^{p,0}_2 - \partial_y^2 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_2 + \partial_2 {\mathcal{P}}^{p,0} = 0\\
&\partial_1 {\mathcal{U}}^{p,0}_1 + \partial_y {\mathcal{U}}^{p,1}_3 = 0,\\
&\partial_2 {\mathcal{U}}^{p,0}_1 = \partial_2 {\mathcal{U}}^{p,0}_2 = \partial_2 {\mathcal{U}}^{p,1}_3 = 0.
\end{aligned}
\right.$$ We remark that the above system is not complete, since we need another equation for the component ${\mathcal{U}}^{p,1}_3$.
**At the order $\varepsilon^{1/2}$**, we have
[multline\*]{} \_t \^[p,1]{} - \_y\^2 \^[p,1]{} + \^[p,0]{}\_h \_h \^[p,1]{} + \^[p,1]{}\_h \_h \^[p,0]{}\_h + \^[p,1]{}\_3 \_y \^[p,1]{} + [(y + )]{} \_y \^[p,0]{}\_h\
+ + \_[k=0]{}\^3
\
\
= 0.
Here, we are only interested in the component ${\mathcal{U}}^{p,1}_3$. Using the fact that $\partial_2 {\mathcal{U}}^{p,1}_3 = 0$, we obtain $$\partial_t {\mathcal{U}}^{p,1}_3 - \partial_y^2 {\mathcal{U}}^{p,1}_3 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,1}_3 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,1}_3 + \overline{\partial_3 p^{I,1}} + y\overline{\partial_3^2 p^{I,0}} = 0.$$
Collect all the above formal calculations, we deduce the following governing equations of the boundary layer
[equation]{} \[eq:PrRotE2P1\] {
&\_t \^[p,0]{}\_1 - \_y\^2 \^[p,0]{}\_1 + \^[p,0]{}\_1 \_1 \^[p,0]{}\_1 + \^[p,1]{}\_3 \_y \^[p,0]{}\_1 + \_1 p\^[B,0]{} + = 0\
&\_t \^[p,1]{}\_3 - \_y\^2 \^[p,1]{}\_3 + \^[p,0]{}\_1 \_1 \^[p,1]{}\_3 + \^[p,1]{}\_3 \_y \^[p,1]{}\_3 + + y = 0\
&\_1 \^[p,0]{}\_1 + \_y \^[p,1]{}\_3 = 0\
&\_2 \^[p,0]{}\_1 = \_2 \^[p,1]{}\_3 = 0\
&\^[p,0]{}\_1(t,x\_1,0) = 0, \_[y+]{} \^[p,0]{}\_1(t,x\_1,y) = (x\_1)\
&\^[p,1]{}\_3(t,x\_1,0) = 0, \_y \^[p,1]{}\_3(t,x\_1,0) = 0\
&\^[p,0]{}\_1(0,x\_1,y) = u\^[B,0]{}\_[0,1]{} (x\_1,y) + (x\_1)\
&\^[p,1]{}\_3(0,x\_1,y) = u\^[B,1]{}\_[0,3]{} (x\_1,y) + (x\_1) + y(x\_1).
.
and
[equation]{} \[eq:P2\] {
&\_t \^[p,0]{}\_2 - \_y\^2 \^[p,0]{}\_2 + \^[p,0]{}\_1 \_1 \^[p,0]{}\_2 + \^[p,1]{}\_3 \_y \^[p,0]{}\_2 + \_2 \^[p,0]{} = 0\
&\^[p,0]{}\_2(0,x\_1,y) = u\^[B,0]{}\_[0,2]{} (x\_1,y) + (x\_1)\
&\^[p,0]{}\_2(t,x\_1,0) = 0,\_[y+]{} \^[p,0]{}\_2(t,x\_1,y) = (x\_1)\
&\^[p,0]{}\_2(0,x\_1,y) = u\^[B,0]{}\_[0,2]{} (x\_1,y) + (x\_1).
.
**Claim:** [*The pressure term of the satisfies $\partial_2{\mathcal{P}}^{p,0} = 0$.*]{}
Indeed, applying $\partial_2$ to the first equation of the systems and , and using the fact that $$\partial_2{\mathcal{U}}^{p,0}_1 = \partial_2{\mathcal{U}}^{p,0}_2 = \partial_2{\mathcal{U}}^{p,1}_3 = 0,$$ we deduce that $$\partial_1\partial_2{\mathcal{P}}^{p,0} = \partial_2^2{\mathcal{P}}^{p,0} = 0.$$ This means that, modulo a contant, we have $${\mathcal{P}}^{p,0} = x_2 G_1(t,y) + \int_{-\infty}^{x_1} \tilde{f}(t,x,y)dx,$$ where $$G_1 = - {\left(\partial_t {\mathcal{U}}^{p,0}_2 - \partial_y^2 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_2\right)}$$ is to be determined and $$\tilde{f} = \partial_1 {\mathcal{P}}^{p,0} = -\partial_t {\mathcal{U}}^{p,0}_1 + \partial_y^2 {\mathcal{U}}^{p,0}_1 - {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_1 - {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_1 - {\left(\frac{y^2}{2} \overline{\partial_3^2 u^{I,0}_3} + y\overline{\partial_3 u^{I,1}_3}\right)}.$$
We recall that, from , we have $$\partial_y {\mathcal{P}}^{p,0} = {\mathcal{U}}^{p,1}_1,$$ where ${\mathcal{U}}^{p,1}_1$ is the solution of the system $$\left\{
\begin{aligned}
&\partial_t {\mathcal{U}}^{p,1}_1 - \partial_y^2 {\mathcal{U}}^{p,1}_1 + {\mathcal{U}}^{p,0}_1 \cdot {\partial}_1 {\mathcal{U}}^{p,1}_1 + {\mathcal{U}}^{p,0}_2 \cdot {\partial}_2 {\mathcal{U}}^{p,1}_1 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,1}_1 + {\mathcal{U}}^{p,1}_1 \partial_1 {\mathcal{U}}^{p,0}_1\\
&\qquad\qquad + {\left(y\overline{\partial_3 u^{I,1}_3} + \frac{y^2}{2} \overline{\partial_3^2 u^{I,0}_3}\right)} \partial_y {\mathcal{U}}^{p,0}_1 + {\left[\frac{y^2}{2} \overline{\partial_3^2 u^{I,1}_3} + \frac{y^3}{6} \overline{\partial_3^3 u^{I,0}_3} \right]} + \sum_{k=0}^3 \frac{y^k}{k!} \overline{\partial_1 \partial_3^k p^{I,1-k}} = 0\\
&{\mathcal{U}}^{p,1}_1(0,x_1,x_2,y) = u^{B,1}_{0,1}(x_1,y) + \overline{u^{I,1}_{0,1}}(x_1) + y\overline{\partial_3u^{I,0}_{0,1}}(x_1) + \alpha_1(y)x_2\\
&{\mathcal{U}}^{p,1}_1(t,x_1,0) = 0.
\end{aligned}
\right.$$ We remark that ${\partial}_yG_1(t,y) = {\partial}_2 {\mathcal{U}}^{p,1}_1$ and we recall that ${\partial}_1{\partial}_2 {\mathcal{U}}^{p,1}_1 = {\partial}_2^2 {\mathcal{U}}^{p,1}_1$. So, in fact, we will find ${\partial}_y G_1$ by solving the following system $$\label{eq:G1}
\left\{
\begin{aligned}
&\partial_t(\partial_y G_1) - \partial_y^2 (\partial_y G_1) + (\partial_1 {\mathcal{U}}^{p,0}_1)(\partial_y G_1) + {\mathcal{U}}^{p,1}_3 {\partial}_y ({\partial}_y G_1) = 0\\
&{\partial}_yG_1(0,y) = \alpha_1(y)\\
&{\partial}_yG_1(t,0) = 0.
\end{aligned}
\right.$$ where $\alpha_1$ is a given function, with $\alpha_1(0) = 0$. For the case of well prepared data, we consider the initial data to be independent of $x_2$, so $\alpha_1 \equiv 0$ and it is easy to see that the system admits $0$ as a trivial solution. Then, the uniqueness of this solution implies ${\partial}_yG_1(t,.) \equiv 0$. Replacing $y=0$ in , we obtain $G_1(t,0) = 0$, and so $G_1(t,.) \equiv 0$, for any $t \in {\mathbb{R}}_+$.
Well-posedness of the boundary layer system {#section3}
===========================================
In this section we will prove the well-posedness for system . Since the pressure term in the first equation of is unknown, we begin with handling the second one to prove the existence of $ {\mathcal{U}}^{p,1}_3$ and then use the divergence-free property to find ${\mathcal{U}}^{p,0}_1$ . To do so we insert the representations $$\begin{aligned}
{\mathcal{U}}^{p,0}_1= u^{B,0}_1+ \overline{u^{I,0}_1}, \quad
{\mathcal{U}}^{p,1}_3 = u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3 u^{I,0}_3}
\end{aligned}$$ into the second equation of , and then make use of the equations and of $u^{I,0}_3$ and $u^{I,1}_3$. It then follows that the unknowns $u^{B,1}_3, u^{B,0}_1$ and $u^{I,1}_3$ satisfy the equation $$\begin{aligned}
&&{\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} u^{B,1}_3 + {\left( u^{B,0}_1+ \overline{u^{I,0}_1}\right)} \partial_1 u^{B,1}_3\\
&&\qquad \qquad + {\left(u^{B,1}_3 +\overline{u^{I,1}_3} \right)} \partial_y u^{B,1}_3+ \overline{\partial_3 u^{I,0}_3} u^{B,1}_3+ {\left( -\partial_1\overline{u^{I,1}_3} + y\overline{\partial_1\partial_3 u^{I,0}_3}\right)} u^{B,0}_1= 0,
\end{aligned}$$ and the divergence-free properties and yield $$\begin{aligned}
u^{B,0}_1= - \int_{-\infty}^{x_1}\partial_y u^{B,1}_3(t, z,y)dz.
\end{aligned}$$ Thus the above is just a equation for $u^{B,1}_3$. To solve the system , we consider the following nonlinear initial-boundary problem, $$\label{linsys}
\left\{
\begin{aligned}
&{\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} u + {\left( v+ \overline{u^{I,0}_1}\right)} \partial_1 u\\
&\qquad \qquad + {\left( u - u(t,x_1,0)\right)} \partial_y u+ \overline{\partial_3 u^{I,0}_3} u+ {\left(\partial_1 u(t,x_1,0) + y\overline{\partial_1\partial_3 u^{I,0}_3}\right)} v= 0,\\
&\partial_y u|_{y=0} =- \overline{\partial_3u^{I, 0}_3}(t,x_1),\quad\lim_{y\to +\infty} u(t,x_1,y) = 0,\\
&u|_{t=0}= u_{0}(x_1,y),
\end{aligned}
\right.$$ where the unknown functions $ u$ and $ v$ are linked by the relation $$\begin{aligned}
\label{relauv}
v(t,x_1,y)= - \int_{-\infty}^{x_1}\partial_y u(t, z,y)dz.\end{aligned}$$ Recall the functions $u^{I,0}_1, u^{I,0}_3$ are the solutions to the Euler-type system . By Theorem \[th:E2D\], we see $u^{I,0}_1, u^{I,0}_3\in \mathcal A_\tau$ for some $\tau>0.$
The main result of this section can be stated as follows.
\[th41\] Suppose the initial data $u_{0}\in X_{\rho_0, a_0}$ for some $\rho_0>0$ and $a_0>0$ and satisfies the compatibility conditions. Then the system admits a unique solution $$\begin{aligned}
u\in L^\infty{\left([0,T_*]; X_{\rho_*, a}\right)} \end{aligned}$$ for some $\rho_*>0$, $a>0$ and $T_*>0. $
We now proceed the proof of the theorem \[th41\] through the following parabolic approximations.
[**The approximate solutions.**]{} Consider the following regularized system, for ${\varepsilon}>0$, $$\label{linsys+}
\left\{
\begin{aligned}
&{\left(\partial_t - {\varepsilon}\partial_1^2-\partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y \right)} u^{\varepsilon}+ {\left( v^{\varepsilon}+ \overline{u^{I,0}_1}\right)} \partial_1 u^{\varepsilon}\\
&\qquad \qquad + {\left( u^{\varepsilon}- u^{\varepsilon}(t,x_1,0)\right)} \partial_y u^{\varepsilon}+ \overline{\partial_3 u^{I,0}_3} u^{\varepsilon}+ {\left(\partial_1 u(t,x_1,0) + y\overline{\partial_1\partial_3 u^{I,0}_3}\right)} v= 0,\\
&\partial_y u^{\varepsilon}(t, x_1,0) = \overline{\partial_1u^{I,0}_1}(t,x_1),\quad
\lim_{y\to +\infty} u^{\varepsilon}(t,x_1,y) = 0, \\
&u^{\varepsilon}|_{t=0}= u_{0}(x_1,y).
\end{aligned}
\right.$$
The above is a nonlinear parabolic equation, and from classical theory we can deduce the following local well-posedness result.
\[th42\] Suppose the initial data $u_{0}\in X_{2\rho_0, a_0}$ for some $\rho_0>0$, $a_0>0$ and satisfies the compatibility conditions. Then the system admits a unique solution $$\begin{aligned}
u^{\varepsilon}\in L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_0, a}\right)} \end{aligned}$$ for some $0<a<a_0$ independent of ${\varepsilon}$ and $ T_{\varepsilon}>0$ depends on ${\varepsilon}$ .
[**Uniform estimates for the approximate solutions.**]{} We will perform the uniform estimate with respect to ${\varepsilon}$ for the approximate solutions $u^{\varepsilon}$ given in the previous Theorem. The main result here can be stated as follows.
\[propapri\] Suppose $u^{\varepsilon}\in L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_0, a}\right)}$ is a solution to the initial-boundary problem . Then there exists $0<\rho_*\le \rho_0$, depending only on ${\left\vertu_0\right\vert}_{X_{\rho_0,a_0}}$, such that $u^{\varepsilon}\in L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_*, a}\right)}$ for all ${\varepsilon}>0.$ Moreover $$\begin{aligned}
\label{uniest}
{\big\Vertu^{\varepsilon}\big\Vert}_{L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_*,a} \right)}} \leq C {\left\vertu_0\right\vert}_{X_{\rho_0,a_0} },\end{aligned}$$ where $C$ is a constant depending only on $a_0, \rho_0, \tau, {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau}$ and ${\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} $, but independent of ${\varepsilon}$.
To prove the above proposition, we need another two [*auxiliary norms*]{} $ {\left\vert\cdot\right\vert}_{Y_{\rho, a}}$ and $ {\left\vert\cdot\right\vert}_{Z_{\rho, a}}$ which are defined by $$\label{Yrho}
\begin{split}
&{\left\vertu\right\vert}_{Y_{\rho, a}}^2=\sum_{ m\leq 2 } {\left(\sum_{0\leq j\leq 1} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}\right)}^2\\
&\qquad\qquad\qquad\qquad+\sum_{m\geq 3} {\left(\sum_{0\leq j\leq 1} (m-1)^{1/2} \rho^{-1/2} \frac{\rho^{m-1}}{ (m-3)!}{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)}\right)}^2,
\end{split}$$ and $$\begin{split}
{\left\vertu\right\vert}_{Z_{\rho, a}}^2&=\sum_{ m\leq 2 } {\left(\sum_{1\leq j\leq 2} {\big\Vert {\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)} \right)}^2\\
&\qquad+\sum_{ m\geq 3 } {\left(\sum_{1\leq j\leq 2} \frac{\rho^{m-1}}{ (m-3)!}{\big\Vert {\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)} \right)}^2.
\end{split}$$
The following energy estimate is a key part to prove Proposition \[propapri\].
\[prenes\] Let $u^{\varepsilon}\in L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_0, a}\right)}$ be a solution to the initial-boundary problem and $0<\rho(t)\le \min{\left\{\rho_0/2, \tau/3\right\}}$ a smooth function. Then for any $ t\in[0, T_{\varepsilon}], $ $$\label{53}
\begin{split}
&{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho(t),a }}^2
+\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho(t), a }}^2dt -\int_0^T \rho'(t) {\left\vertu^{\varepsilon}(t)\right\vert}_{Y_{\rho(t), a }}^2 dt\\
\leq &{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+ C \int_0^{T_{\varepsilon}} {\left({\left\vert\rho'(t)\right\vert}\rho(t)^{-2}{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho(t), a} }+{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho(t), a} }^2+ {\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho(t), a}}^4\right)}dt\\
&+C \int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho(t), a}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Y_{\rho(t), a} }^2dt\, .
\end{split}$$
The proof of the proposition above is postponed to the next section, and we now use it to prove Proposition \[propapri\].
To simplify the notations we will use $C$ in the following discussion to denote different suitable constants, which depend only on $a_0, \rho_0, \tau, {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau}$ and ${\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} $, but independent of ${\varepsilon}$.
Let $\rho_{\varepsilon}$ be the solution to the differential equation: $$\label{eq+}
\left\{
\begin{split}
\rho_{\varepsilon}'(t)=-{\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon}(t), a}}, \\
\rho|_{t=0}=\min{\left\{\rho_0/2, \tau/3\right\}},
\end{split}
\right.$$ or equivalently $$\begin{aligned}
\label{des++}
\rho_{\varepsilon}(t)=\min{\left\{\rho_0/2, \tau/3\right\}}-\int_0^t {\left\vertu^{\varepsilon}(s)\right\vert}_{Z_{\rho_{\varepsilon}(s), a}} ds.\end{aligned}$$ Observe, for any $0<\rho, \,\tilde\rho \leq \rho_0/2,$ we have $$\begin{aligned}
{\left\vert {\left\vertu^{\varepsilon}\right\vert}_{Z_{\rho, a}} - {\left\vertu^{\varepsilon}\right\vert}_{Z_{\tilde\rho, a}} \right\vert}\leq C {\left\vertu^{\varepsilon}\right\vert}_{Z_{\rho_0, a}} {\left\vert\rho-\tilde\rho\right\vert},\end{aligned}$$ which along with Cauchy-Lipschitz Theorem gives the existence of $\rho_{\varepsilon}$ to equation . Now choosing $\rho(t)=\rho_{\varepsilon}(t)$ in and observing , we can rewrite as $$\begin{aligned}
&&{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho_{\varepsilon},a} }^2+\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon},a} }^2 dt \\
&\leq&{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+ C \int_0^{T_{\varepsilon}} {\left({\left\vert\rho_{\varepsilon}'(t)\right\vert}\rho_{\varepsilon}^{-2}{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho,a} }+{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a} }^2+ {\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a}}^4\right)}dt.\end{aligned}$$ Thus, using , $$\label{Eq+++}
\begin{split}
&{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho,a} }^2+\frac{1}{2}\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon},a} }^2 dt \\
\leq& {\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+ C \int_0^{T_{\varepsilon}} {\left( \rho_{\varepsilon}^{-4}{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a} }^2+{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a} }^2 +{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a}}^4\right)}dt.
\end{split}$$ In view of for $T_{\varepsilon}$ be small sufficiently we have $$\begin{aligned}
\forall~t\in[0,T_{\varepsilon}], \quad \rho_{\varepsilon}(t)\geq \frac{1}{8}\min{\left\{\rho_0,\tau/3\right\}}, \end{aligned}$$ and thus it follows from that, for any $t\in[0,T_{\varepsilon}], $ $$\begin{aligned}
{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho,a} }^2+\frac{1}{2}\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon},a} }^2 dt \leq {\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+ C \int_0^{T_{\varepsilon}} {\left( {\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a} }^2 +{\left\vertu^{\varepsilon}\right\vert}_{X_{\rho_{\varepsilon},a}}^4\right)}dt,\end{aligned}$$ with $C$ depending only on $a_0, \rho_0, \tau, {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau}$ and ${\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} $, but independent of ${\varepsilon}$. Thus by general Gronwall inequality, we conclude $$\begin{aligned}
\label{ueps}
{\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho_{\varepsilon},a} }^2\leq C {\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2,\end{aligned}$$ and $$\begin{aligned}
\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon},a} }^2 dt \leq 3{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^4.\end{aligned}$$ As a result, in view of we see $$\begin{aligned}
\rho_{\varepsilon}(t)&=&\min{\left\{\rho_0/2, \tau/3\right\}}-\int_0^t {\left\vertu^{\varepsilon}(s)\right\vert}_{Z_{\rho_{\varepsilon}(s), a}} ds\\
&\geq& \min{\left\{\rho_0/2, \tau/3\right\}}-t^{1/2}{\left(\int_0^{T_{\varepsilon}} {\left\vertu^{\varepsilon}(t)\right\vert}_{Z_{\rho_{\varepsilon},a} }^2 dt\right)}^{1/2}\\
&\geq& \min{\left\{\rho_0/2, \tau/3\right\}}-t^{1/2}{\left(2{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^4\right)}^{1/2}.\end{aligned}$$ So if we choose $T_*$ such that $$\begin{aligned}
\label{tstar}
T_*= 4^{-1}{\left(3{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^4\right)} ^{-1}\Big(\min{\left\{\rho_0/2, \tau/3\right\}}\Big)^2\end{aligned}$$ Then $$\begin{aligned}
\forall~t\in [0,T_{\varepsilon}] \subset [0,T_*], \quad \rho_{\varepsilon}(t) \geq \rho_*\stackrel{\rm def}{=} \frac{1}{4}\min{\left\{\rho_0, \tau/3\right\}}.\end{aligned}$$ By , it follows that $$\begin{aligned}
\forall~t\in [0,T_{\varepsilon}] \subset [0,T_*] ,\quad {\left\vertu^{\varepsilon}(t)\right\vert}_{X_{\rho_*,a} }^2\leq C{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2.\end{aligned}$$ This completes the proof of Proposition \[propapri\].
Due to the uniform estimate , we can extend the lifespan $T_{\varepsilon}$ to $T_*$ with $T_*$ defined in , following the standard bootstrap arguments. Thus we see for any ${\varepsilon}>0$ the system admits a unique solution $u^{\varepsilon}\in L^\infty{\left([0,T_*]; X_{\rho_*, a}\right)} $ such that $$\begin{aligned}
{\big\Vertu^{\varepsilon}\big\Vert}_{L^\infty{\left([0,T_{\varepsilon}]; X_{\rho_*,a} \right)}} \leq C {\left\vertu_0\right\vert}_{X_{\rho_0,a_0} },\end{aligned}$$ with $T_*, \rho_*, a, C$ independent of ${\varepsilon}.$ Thus letting ${\varepsilon}\rightarrow 0,$ the compactness arguments show that the limit $u\in L^\infty{\left([0,T_*]; X_{\rho_*, a}\right)}$ solves the system , proving Theorem \[th41\].
[**Proof of Theorem \[th:Prandtl\] .** ]{} Taking $$u=u^{B,1}_3,\quad v= - \int_{-\infty}^{x_1}\partial_y u^{B,1}_3(t, z,y)dz,$$ the system implies that the function $${\mathcal{U}}^{p,1}_3 = u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3 u^{I,0}_3}$$ satisfies $$\left\{
\begin{aligned}
&\partial_t {\mathcal{U}}^{p,1}_3 - \partial_y^2 {\mathcal{U}}^{p,1}_3 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,1}_3 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,1}_3 + \overline{\partial_3 p^{I,1}} + y\overline{\partial_3^2 p^{I,0}} = 0,\\
&\partial_y {\mathcal{U}}^{p,1}_3(t,x_1,0) = 0,\\
&{\mathcal{U}}^{p,1}_3(0,x_1,y) = u^{B,1}_{0,3} (x_1,y) + \overline{u^{I,1}_{0,3}}(x_1) + y\overline{{\partial}_3u^{I,0}_{0,3}}(x_1)\, ,
\end{aligned}
\right.$$ with $${\mathcal{U}}^{p,0}_1= - \int_{-\infty}^{x_1}\partial_y u^{B,1}_3(t, z,y)dz+ \overline{u^{I,0}_1}\,.$$ So we need to check the boundary condition $$\label{bounadry3}
{\mathcal{U}}^{p,1}_3|_{y=0}= u^{B,1}_3(t, x_1, 0)+\overline{u^{I,1}_3}(t, x_1)=0.\,\,$$ For this purpose, we first use Theorem \[th41\] to determine $u^{B,1}_3$, then use Theorem \[th:E2Dlin\] to solve the linearized Euler system with the boundary condition $${u^{I ,1}_3}|_{x_3=0}=-u^{B,1}_3(t, x_1, 0)\,.$$ For the component $ {\mathcal{U}}^{p,0}_1$, using the divergence-free properties of $u^{I, 0}$, we have firstly $$\begin{aligned}
{\mathcal{U}}^{p, 0}_1|_{y=0}&=&- \int_{-\infty}^{x_1}\partial_y u^{B,1}_3(t, z, 0)dz+
\overline{u^{I,0}_1}(t, x_1)\\
&=& \int_{-\infty}^{x_1}\overline{\partial_3u^{I, 0}_3}(t, z)dz+ \overline{u^{I,0}_1}(t, x_1)\\
&=&- \int_{-\infty}^{x_1}\overline{\partial_1u^{I, 0}_1}(t, z)dz+ \overline{u^{I,0}_1}(t, x_1)\\
&=&0.\end{aligned}$$ On the other hand, since $ u^{B,1}_3\in L^\infty{\left([0,T_*]; X_{\rho_*, a}\right)}$, we have the limit $$\begin{aligned}
\lim_{y\to +\infty}{\mathcal{U}}^{p, 0}_1(t, x_1, y)&=&- \lim_{y\to +\infty}\int_{-\infty}^{x_1}\partial_y u^{B,1}_3(t, z, y)dz+
\overline{u^{I,0}_1}(t, x_1)\\
&=&\overline{u^{I,0}_1}(t, x_1).\end{aligned}$$ So the boundary conditions for $ {\mathcal{U}}^{p,0}_1$ are satisfied. Finally, for the pressure term of the first equation in , once we obtain ${\mathcal{U}}^{p, 1}_3$, ${\mathcal{U}}^{p, 0}_1$ and $\overline{\partial_1 p^{I,0}}$, it is enough to put $$\partial_1 p^{B,0}=
-\partial_t {\mathcal{U}}^{p,0}_1 + \partial_y^2 {\mathcal{U}}^{p,0}_1 - {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_1 - {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_1 - \overline{\partial_1 p^{I,0}} .$$ We then complete the proof of Theorem \[th:Prandtl\].
Uniform energy estimates {#section4}
========================
In this section we proceed through the following lemmas to prove Proposition \[prenes\]. To simplify the notations in the following proof we will write $u$ instead of $u^{\varepsilon},$ omitting the superscript ${\varepsilon}$, and use $C$ in the following discussion to denote different suitable constants, which depend only on $a_0, \rho_0, \tau, {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau}$ and ${\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} $.
In view of the definition of ${\left\vert\cdot\right\vert}_{X_{\rho,a}}$ it suffices to estimate terms $$\begin{aligned}
\label{est1or}
\sum_{ m\leq 2 } {\left({\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m u\big\Vert}_{L^2(\mathbb R_+^2)} \right)}+\sum_{ m\geq 3 } {\left(\frac{\rho^{m-1}}{ (m-3)!}{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m u\big\Vert}_{L^2(\mathbb R_+^2)}\right)}\end{aligned}$$ and $$\begin{aligned}
\label{est2or}
\sum_{ m\leq 2 } {\left({\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y u\big\Vert}_{L^2(\mathbb R_+^2)} \right)}+\sum_{ m\geq 3 } {\left(\frac{\rho^{m-1}}{ (m-3)!}{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y u\big\Vert}_{L^2(\mathbb R_+^2)} \right)}\end{aligned}$$ Here we first treat the terms in , and the ones in can be deduced similarly with simpler arguments. To do so, we use the notation $\omega=\partial_y u$. Then it follows from that $$\label{linsysomega}
\left\{
\begin{aligned}
&{\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} \omega + {\left( v+ \overline{u^{I,0}_1}\right)} \partial_1 \omega\\
&\qquad \qquad + {\left( u - u(t,x_1,0)\right)} \partial_y \omega+ 2\overline{\partial_3 u^{I,0}_3} \omega+ {\left(\partial_1 u(t,x_1,0) + y\overline{\partial_1\partial_3 u^{I,0}_3}\right)} \partial_yv\\
&\qquad \qquad +{\left(\partial_y v\right)} \partial_1u+ \omega^2 +\overline{\partial_1\partial_3 u^{I,0}_3}v= 0,\\
&\omega|_{y=0} = \overline{\partial_1u^{I,0}_1}(t,x_1),\\
&\omega|_{t=0}= \partial_y u^{B,1}_{3, 0}.
\end{aligned}
\right.$$ Thus the function, defined by $$\begin{aligned}
\label{defver}
\varphi_m ={\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\omega (t)={\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y u (t),
\end{aligned}$$ solves the equation $$\begin{cases}
{\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} \varphi_m -a'(t)y^2 \varphi_m + {\left(v + \overline{u^{I,0}_1}\right)} \partial_1 \varphi_m + {\left(u - u (t,x_1,0)\right)} \partial_y \varphi_m =\mathcal R^{m}(t),\\[10pt]
\varphi_m \big|_{y=0}={\left<x_1\right>}^\ell\overline{\partial_1^{m+1} u^{I,0}_1}(t,x_1),\\[10pt]
\varphi_m \big|_{t=0}={\left<x_1\right>}^\ell e^{a y^2}\partial_1^m \partial_y u^{B,1}_{3, 0},
\end{cases}$$ where $$\begin{aligned}
\mathcal R^{m}(t)=\sum_{j=1}^{11}\mathcal R_j^{m}(t)\end{aligned}$$ with $$\begin{aligned}
\mathcal R_1^{m}&=&-4a y\partial_y\varphi_m+4a^2 y^2\varphi_m -2a\varphi_m, \\
\mathcal R_2^{m}&=&2ay^2 \overline{\partial_3u^{I,0}_3} \varphi_m+2ay{\left(u-u(t,x_1,0)\right)}\varphi_m, \\
\mathcal R_3^{m}&=&{\left(\partial_1 {\left<x_1\right>}^\ell \right)}e^{a y^2}{\left( v+\overline{u^{I,0}_1}\right)} \partial_1^{m} \omega, \\
\mathcal R_4^{m}&=&
- \sum_{k=1}^m {m\choose k} {\left<x_1\right>}^\ell e^{a y^2}{\left(\partial_1^k \overline{\partial_3u^{I,0}_3}\right)} y\partial_y\partial_1^{m-k}\omega, \\
\mathcal R_5^{m}&=& - \sum_{k=1}^m {m\choose k} {\left<x_1\right>}^\ell e^{a y^2}{\left(\partial_1^k v+\partial_1^k \overline{u^{I,0}_1}\right)} \partial_1^{m-k+1}\omega, \\
\mathcal R_6^{m} &=& - \sum_{k=1}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2}{\left(\partial_1^k u -\partial_1^k u (t,x_1,0)\right)} \partial_1^{m-k} \partial_y\omega,\\
\mathcal R_7^{m} &=& - \sum_{k=0}^m {m\choose k} {\left<x_1\right>}^\ell e^{a y^2}{\left(\partial_1^k \overline{\partial_3u^{I,0}_3}\right)} \partial_1^{m-k}\omega,\\
\mathcal R_8^{m}&=& - \sum_{k=0}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2} {\left(\partial_1^{k+1} u(t,x_1,0)+y \overline{\partial_1^{k+1} \partial_3u^{I,0}_3}\right)} \partial_1^{m-k}\partial_yv,\\
\mathcal R_9^{m}&=& - \sum_{k=0}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2} {\left(\partial_1^{k+1} u\right)} \partial_1^{m-k}\partial_yv,\\
\mathcal R_{10}^{m}&=& - \sum_{k=0}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2} {\left({\left(\partial_1^{k} \omega\right)} \partial_1^{m-k}\omega+ \overline{\partial_1^{k+1}\partial_3 u^{I,0}_3}\partial_1^{m-k}v\right)}.\end{aligned}$$ From the first equation in , it follows that $$\begin{aligned}
\label{maiequ}
& {\left({\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} \varphi_m , ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)} - {\left(a'(t)y^2 \varphi_m ,~\varphi_m \right)}_{L^2(\mathbb R_+^2)}\nonumber \\
&\qquad\qquad+{\left( {\left(v+ \overline{u^{I,0}_1}\right)} \partial_1\varphi_m + {\left(u - u (t,x_1,0)\right)} \partial_y \varphi_m (t), ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)} \\
=& {\left( \mathcal R^{m}(t), ~\varphi_m (t)\right)}_{L^2(\mathbb R_+^2)},\notag\end{aligned}$$ with $\mathcal R^{m}$ given above.
In the following lemmas, let $ 0<a(t)<a_0$ to be determined later, and let $0<\rho=\rho(t)\leq \min{\left\{\rho_0/2,\tau/3\right\}}$ be an arbitrary smooth function of $t.$
\[R1+2\] A constants $C$ exists such that for any $N\geq 3$, $$\begin{aligned}
\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_1^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}
\leq C \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2,
\end{aligned}$$ and $$\begin{aligned}
\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_2^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}
\leq C \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+ C {\left\vertu\right\vert}_{X_{\rho,a}}^4.
\end{aligned}$$
We have, integrating by parts, $$\begin{aligned}
{\left( \mathcal R_1^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}=
{\left( 4a^2y^2\varphi_m, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}= 4a^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2
\end{aligned}$$ Direct verification shows $$\begin{aligned}
{\left( \mathcal R_2^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}\leq
{\left(2a {\big\Vert\overline{\partial_3u^{I,0}_3}\big\Vert}_{L^\infty}+ a^2\right)} {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2 + 4{\big\Vertu\big\Vert}_{L^\infty(\mathbb R_+^2)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2.
\end{aligned}$$ Observe $$\begin{aligned}
{\big\Vert\overline{\partial_3u^{I,0}_3}\big\Vert}_{L^\infty}\leq C {\big\Vertu^{I,0}_3\big\Vert}_{G_\tau}
\end{aligned}$$ and $$\begin{aligned}
\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\leq \sum_{m=3}^\infty {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\leq {\left\vertu\right\vert}_{X_{\rho,a}},
\end{aligned}$$ and thus the desired results follow, completing the proof.
\[R8\] There exists a constant $C$ such that for any $ \rho$ with $0<\rho\leq \tau/3$, we have $$\begin{aligned}
\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_8^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}
\leq {1\over 8} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 +C \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2. \end{aligned}$$
Recall $\mathcal R_8^{m,j}$ can be written as, for any $\tilde{\varepsilon}>0,$ $$\begin{aligned}
{\left( \mathcal R_8^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}
&= & {\left(-\sum_{k=0}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2} {\left(\partial_1^{k+1} u(x_1,0)\right)} \partial_1^{m-k}\partial_y v, ~ \varphi_m\right)}_{L^2(\mathbb R_+^2)} \\
&&+ {\left(- \sum_{k=0}^m{m\choose k} {\left<x_1\right>}^\ell e^{a y^2} {\left( y \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \right)}\partial_1^{m-k}\partial_y v, ~ \varphi_m\right)}_{L^2(\mathbb R_+^2)}\\
&\leq &\sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \\
&&+\tilde{\varepsilon}\bigg[\sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \big\Vert}_{ L_{x_1}^2(\mathbb R)}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}\bigg]^2 \\
&&+C_{\tilde{\varepsilon}} {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2.\end{aligned}$$ Then it suffices to show that $$\label{cru1}
\begin{split}
& \sum_{m=3}^N{\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \\
\leq & C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2\, ,
\end{split}$$ and $$\label{cru2}
\begin{split}
& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left[ \sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \big\Vert}_{ L_{x_1}^2(\mathbb R)}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}\right]}^2\\
\leq & C {\left\vertu\right\vert}_{Z_{\rho,a}} ^2\, .
\end{split}$$ We will proceed to prove the above estimate through the following steps.
[**Step (a)**]{} We begin with several estimates to be used later in the proof. Firstly in view of the definition of ${\left\vert\cdot\right\vert}_{Y_{\rho,a}}$ given in , we may write $$\begin{aligned}
{\left\vertu\right\vert}_{Y_{\rho,a}}^2=\sum_{m=0}^{+\infty}{\left\vertu\right\vert}_{Y_{\rho,a,m}}^2\end{aligned}$$ where ${\left\vertu\right\vert}_{Y_{\rho,a,m}} $ is defined by $$\begin{aligned}
{\left\vertu\right\vert}_{Y_{\rho,a,m}} =
\begin{cases}
\sum_{0\leq j\leq 1} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)},\quad 0\leq m\leq 2\\[8pt]
\sum_{0\leq j\leq 1} (m-1)^{1/2} \rho^{-/2} \frac{\rho^{m-1}}{ (m-3)!} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)},\quad m\geq 3.
\end{cases}\end{aligned}$$ Thus $$\begin{aligned}
\label{ta}
{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}\leq
\begin{cases}
{\left\vertu\right\vert}_{Y_{\rho,a}},\quad 0\leq m\leq 2,\\[8pt]
{\left\vertu\right\vert}_{Y_{\rho,a,m}} m^{-1/2}\rho^{1/2}\frac{(m-3)!}{\rho^{m-1}},\quad m\geq 3,
\end{cases}\end{aligned}$$
Next from the relations it follows that $$\begin{aligned}
{\big\Vert e^{a y^2} \partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} \leq C {\big\Vert{\left<x_1\right>}^\ell e^{a y^2} \partial_y^2 u\big\Vert}_{L^2{\left(\mathbb R_+^2\right)}} \leq C{\left\vertu\right\vert}_{Z_{\rho,a}},\end{aligned}$$ and that for $j\geq 1,$ $$\begin{aligned}
{\big\Vert e^{a y^2}\partial_1^j \partial_yv\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}&=&{\big\Vert e^{a y^2} \partial_1^{j-1}\partial_y^2 u \big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}\\
&\leq& C {\big\Vert{\left<x_1\right>}^\ell e^{a y^2} \partial_1^{j}\partial_y^2 u\big\Vert}_{L^2{\left(\mathbb R_+^2\right)}} \\
&\leq &\left\{
\begin{array}{lll}
C {\left\vertu\right\vert}_{Z_{\rho,a}},\quad && 1\leq j\leq 2,\\[6pt]
C {\left\vertu\right\vert}_{Z_{\rho,a,j}} \frac{(j-3)!}{\rho^{j-1}},\quad && j\geq 3,
\end{array}
\right.\end{aligned}$$ where ${\left\vertu\right\vert}_{Z_{\rho,a, k}} $ is defined by the relation ${\left\vertu\right\vert}_{Z_{\rho,a}}=\sum_{k\geq 0}{\left\vertu\right\vert}_{Z_{\rho,a, k}}^2$, so that $$\begin{aligned}
{\left\vertu\right\vert}_{Z_{\rho,a,k}} =
\begin{cases}
\sum_{1\leq j\leq 2} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^k\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)},\quad 0\leq k\leq 2\\[8pt]
\sum_{1\leq j\leq 2} \frac{\rho^{k-1}}{ (k-3)!} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^k\partial_y^j u\big\Vert}_{L^2(\mathbb R_+^2)},\quad k\geq 3.
\end{cases}\end{aligned}$$ Thus we conclude $$\begin{aligned}
\label{tc+}
{\big\Vert e^{a y^2}\partial_1^j \partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} \leq \left\{
\begin{array}{lll}
C {\left\vertu\right\vert}_{Z_{\rho,a}},\quad && 0\leq j \leq 2,\\[6pt]
C {\left\vertu\right\vert}_{Z_{\rho,a,j}} \frac{(j-3)!}{\rho^{j-1}},\quad && j\geq 3.
\end{array}
\right.\end{aligned}$$ Using the Sobolev inequality $$\begin{aligned}
{\big\Vert{\left<x_1\right>}^\ell \partial_1^{j} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}} \leq C {\big\Vert{\left<x_1\right>}^\ell \partial_1^{j} u\big\Vert}_{L^2(\mathbb R_+)}+C {\big\Vert{\left<x_1\right>}^\ell \partial_1^{j} \partial_y u\big\Vert}_{L^2(\mathbb R_+)}, \end{aligned}$$ gives $$\begin{aligned}
\label{tb}
{\big\Vert{\left<x_1\right>}^\ell \partial_1^{j} u \big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}} \leq \left \{ \begin{array}{lll}
C{\left\vertu\right\vert}_{Y_{\rho,a}},\qquad{\rm if}~0\leq j\leq 2,\\[6pt]
C{\left\vertu\right\vert}_{Y_{\rho,a,j} } j^{-1/2}\rho^{1/2}\frac{(j-3)!}{\rho^{j-1}},\qquad{\rm if}~j\geq 3.
\end{array}\right.\end{aligned}$$ Finally, $$\begin{aligned}
\label{eseu}
\forall~k\geq 0,\quad {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \big\Vert}_{ L_{x_1}^2(\mathbb R)}\leq C {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau} \frac{(k+3)!}{\tau^{k+3}}\end{aligned}$$ due to .
[**Step (b)**]{}. We now prove . For this purpose we use and to calculate $$\begin{aligned}
&& \sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \big\Vert}_{ L_{x_1}^2(\mathbb R)}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}\\
&\leq & C {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau} \sum_{k=0}^{m-3} \frac{m!}{k!(m-k)! }\frac{(k+3)!}{\tau^{k+3}}\frac{(m-k-3)!}{\rho^{m-k-1}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\\
&&+ C {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau} \sum_{k=m-2}^{m} \frac{m!}{k!(m-k)! }\frac{(k+3)!}{\tau^{k+3}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}} \\
&\leq& C \frac{
(m-3)!}{\rho^{m-1}} \sum_{k=0}^{m-3} \frac{m^3}{k^3(m-k-2)^3}\frac{2^k\rho^k}{\tau^{k+3}}{\left\vertu\right\vert}_{Z_{\rho,a, m-k}}+ C \frac{
(m-3)!}{\rho^{m-1}}\sum_{k=m-2}^{m} \frac{2^k\rho^{m-1}}{\tau^{k+3}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\\
&\leq& C \tau^{-3}\frac{
(m-3)!}{\rho^{m-1}} \sum_{k=0}^{m-3} \frac{2^k\rho^k}{\tau^{k}}{\left\vertu\right\vert}_{Z_{\rho,a, m-k}}+ C\tau^{-3} \frac{
(m-3)!}{\rho^{m-1}}\sum_{k=m-2}^{m} \frac{2^k\rho^{m-1}}{\tau^{k}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}},\end{aligned}$$ which yields $$\begin{aligned}
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left[ \sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{k+1}\partial_3u^{I,0}_3} \big\Vert}_{ L_{x_1}^2(\mathbb R)}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}}\right]}^2\\
&\leq& C\sum_{m=3}^N {\left( \sum_{k=0}^{m-3} \frac{2^k\rho^k}{\tau^{k}}{\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\right)}^2+ C\sum_{m=3}^N {\left(\sum_{k=m-2}^{m} \frac{2^k\rho^{m-1}}{\tau^{k}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\right)}^2 \end{aligned}$$ On the other hand, by virtue of Young’s inequality for discrete convolution (cf. [@HR Theorem 20.18] ) we have $$\begin{aligned}
\sum_{m=3}^N {\left( \sum_{k=0}^{m-3} \frac{2^k\rho^k}{\tau^{k}}{\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\right)}^2
\leq C {\left(\sum_{k=0}^{N} \frac{2^k\rho^k}{\tau^{k}} \right)}^2 \sum_{k=0}^{N}{\left\vertu\right\vert}_{Z_{\rho,a, k}}^2 \leq C {\left\vertu\right\vert}_{Z_{\rho,a}}^2,\end{aligned}$$ since $\rho\leq \tau/3.$ And direct computation yields $$\begin{aligned}
\sum_{m=3}^N {\left(\sum_{k=m-2}^{m} \frac{2^k\rho^{m-1}}{\tau^{k}} {\left\vertu\right\vert}_{Z_{\rho,a, m-k}}\right)}\leq C {\left\vertu\right\vert}_{Z_{\rho,a}}^2.
\end{aligned}$$ Then we obtain , combining the above inequalities.
[**Step (c)**]{}. Now we check and write $$\begin{aligned}
&& \sum_{k=0}^m{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \\
&\leq & S_1+S_2+S_3\end{aligned}$$ with $$\begin{aligned}
S_1&=& \sum_{k=0}^2{m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)},\\
S_2 &=& \sum_{k=3}^{m-3} {m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}\end{aligned}$$ and $$\begin{aligned}
S_3 &=& \sum_{k=m-2}^{m} {m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}.\end{aligned}$$ For the term $S_{2,m}$, we use [(\[ta\])]{}, [(\[tc+\])]{} and [(\[tb\])]{} to compute $$\begin{aligned}
S_{2,m} &=&{\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2\,\, \sum_{k=3}^{m-3} {m\choose k} {\big\Vert{\left<x_1\right>}^\ell \partial_1^{k+1} u\big\Vert}_{L^\infty_y{\left(\mathbb R_+;~L_{x_1}^2(\mathbb R)\right)}}{\big\Vert e^{a y^2} \partial_1^{m-k}\partial_y v\big\Vert}_{L^2_y{\left(\mathbb R_+;~L_{x_1}^\infty(\mathbb R)\right)}} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}\\
&\leq &C {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2\,\,\sum_{k=3}^{m-3}\frac{m!}{k!(m-k)! }\left[ k^{-1/2}\rho^{1/2}\frac{(k-2)!}{\rho^{k}} {\left\vertu\right\vert}_{Y_{\rho,a,k+1} } \right] \frac{(m-k-3)!}{\rho^{m-k-1}}{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\\
&&\qquad\qquad\qquad \times m^{-1/2}\rho^{1/2} \frac{(m-3)!}{\rho^{m-1}}{\left\vertu\right\vert}_{Y_{\rho,a,m} } \\
&\leq & C \rho {\left\vertu\right\vert}_{Y_{\rho,a,m} } \sum_{k=3}^{m-3}\frac{m^3}{k^2(m-k-2)^3 }k^{-1/2}m^{-1/2} {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\\
&\leq & C \rho {\left\vertu\right\vert}_{Y_{\rho,a,m} } {\left(\sum_{k=3}^{m-3}\frac{m^3}{k^2(m-k-2)^3 }k^{-1/2}m^{-1/2} {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }^2\right)}^{1/2} \\
&&\qquad\qquad\qquad \times {\left(\sum_{k=3}^{m-3}\frac{m^3}{k^2(m-k-2)^3 }k^{-1/2}m^{-1/2}{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}^2\right)}^{1/2}\\
&\leq & C \rho {\left\vertu\right\vert}_{Z_{\rho,a,m}} {\left\vertu\right\vert}_{Y_{\rho,a,m} }^2,\end{aligned}$$ and thus $$\begin{aligned}
\sum_{m=3}^N S_{2,m}& \leq& C\rho {\left(\sum_{m=3}^N {\left\vertu\right\vert}_{Y_{\rho,a,m} }^2\right)}^{1/2}{\left(\sum_{m=3}^N {\left[\sum_{k=3}^{m-3}\frac{m^3}{k^2(m-k-2)^3 }k^{-1/2}m^{-1/2} {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\right]}^2 \right)}^{1/2}\\
& \leq& C\rho {\left\vertu\right\vert}_{Y_{\rho,a}} {\left(\sum_{m=3}^N {\left[\sum_{k=3}^{m-3}\frac{1}{k^2 } {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\right]}^2 \right)}^{1/2}\\
&&\qquad\qquad \qquad + C\rho {\left\vertu\right\vert}_{Y_{\rho,a}} {\left(\sum_{m=3}^N {\left[\sum_{k=3}^{m-3}\frac{1}{(m-k-2)^3 } {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\right]}^2 \right)}^{1/2} \end{aligned}$$ the last inequality following from the fact that $$\begin{aligned}
\forall~ 3\leq k\leq m-3,\quad \frac{m^3}{k^2(m-k-2)^3 }k^{-1/2}m^{-1/2}\leq C {\left(\frac{1}{k^2}+ \frac{1}{(m-k-2)^3}\right)}. \end{aligned}$$ Moreover, by virtue of Young’s inequality for discrete convolution (cf. [@HR Theorem 20.18] ) we obtain $$\begin{aligned}
{\left(\sum_{m=3}^N {\left[\sum_{k=3}^{m-3}\frac{1}{k^2 } {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\right]}^2 \right)}^{1/2}&\leq &C{\left(\sum_{m=3}^{+\infty} {\left\vertu\right\vert}_{Z_{\rho,a,m}}^2 \right)}^{1/2}\sum_{k=3}^{+\infty} \frac{1}{k^2} {\left\vertu\right\vert}_{Y_{\rho,a,k}}\\
&\leq &C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left( \sum_{k=1}^{+\infty} {\left\vertu\right\vert}_{Y_{\rho,a,k}}^2\right)}^{1/2}{\left( \sum_{k=1}^{+\infty} \frac{1}{k^4}\right)}^{1/2}\\
&\leq &C {\left\vertu\right\vert}_{Z_{\rho,a}}{\left\vertu\right\vert}_{Y_{\rho,a}}.\end{aligned}$$ Similarly $$\begin{aligned}
{\left(\sum_{m=3}^N {\left[\sum_{k=3}^{m-3}\frac{1}{(m-k-2)^3 } {\left\vertu\right\vert}_{Y_{\rho,a,k+1} }{\left\vertu\right\vert}_{Z_{\rho,a,m-k}}\right]}^2 \right)}^{1/2}
\leq C {\left\vertu\right\vert}_{Z_{\rho,a}}{\left\vertu\right\vert}_{Y_{\rho,a}}.\end{aligned}$$ Combining these inequality we conclude $$\begin{aligned}
\sum_{m=3}^N S_{2,m}\leq C \rho {\left\vertu\right\vert}_{Z_{\rho,a}}{\left\vertu\right\vert}_{Y_{\rho,a}}^2\leq C {\left\vertu\right\vert}_{Z_{\rho,a}}{\left\vertu\right\vert}_{Y_{\rho,a}}^2.\end{aligned}$$ The estimates on the rest two terms $S_1$ and $S_3$ can be deduced similarly and directly, and we have $$\begin{aligned}
\sum_{m=3}^N {\left(S_{1,m}+S_{3,m}\right)}
\leq C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2,\end{aligned}$$ proving . The proof of Lemma \[R8\] is complete.
\[Rothers\] A constant $C$ exists such that $$\begin{aligned}
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_3^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left({\left\vertu\right\vert}_{X_{\rho,a,m}}^3 + {\left\vertu\right\vert}_{X_{\rho,a,m} }^2\right)},\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_4^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq {1\over8} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 +C\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2,\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_5^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left({\left\vertu\right\vert}_{X_{\rho,a,m}}^3 + {\left\vertu\right\vert}_{X_{\rho,a,m} }^2\right)},\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_6^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2,\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_7^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left\vertu\right\vert}_{X_{\rho,a} }^2,\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_9^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2,\\
&& \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left( \mathcal R_{10}^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)} \leq C {\left({\left\vertu\right\vert}_{X_{\rho,a}}^3 + {\left\vertu\right\vert}_{X_{\rho,a} }^2\right)}.
\end{aligned}$$
The treatment of $\mathcal R_6, \mathcal R_9$ is exactly the same as in the proof of . The other terms can be deduced similarly by following the proof in Lemma \[R8\] with slightly changes, and the arguments here will be simpler since there is no the highest derivative $\partial_1^{m+1}$ involved. This means we can perform the estimates with the norm $Y_{\rho,a}$ in Lemma \[R8\] replaced by $X_{\rho,a}$ here. So we omit the proof for brevity.
Combining the estimates in Lemma \[R1+2\]-Lemma \[Rothers\], we have
\[cor0222\] There are two constants $C, C_0$ such that $$\begin{aligned}
&&\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left(\sum_{k=1}^{10}{\left( \mathcal R_k^{m}, ~\varphi_m\right)}_{L^2(\mathbb R_+^2)}\right)}\\
& \leq& {1\over 4} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 +C_0 \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2+C {\left({\left\vertu\right\vert}_{X_{\rho,a}}^2 + {\left\vertu\right\vert}_{X_{\rho,a} }^4\right)}.
\end{aligned}$$
\[lowleft\] We have $$\begin{aligned}
&&{\left({\left(\partial_t - \partial_y^2 + \overline{\partial_3 u^{I,0}_3} y\partial_y\right)} \varphi_m , ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)} - {\left(a'(t)y^2 \varphi_m ,~\varphi_m \right)}_{L^2(\mathbb R_+^2)}\nonumber \\
&&\qquad\qquad+{\left( {\left(v+ \overline{u^{I,0}_1}\right)} \partial_1\varphi_m + {\left(u - u (t,x_1,0)\right)} \partial_y \varphi_m (t), ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)} \\
&\geq& \frac{1}{2}\frac{d}{dt} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+{\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2-a'(t) {\big\Verty \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2
\\
&&+\frac{d}{dt}\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1-C {\left( \frac{(m-3)!}{\rho^{m-1}}\right)}^2{\left\vertu\right\vert}_{X_{\rho,a,m} }^2.\end{aligned}$$
Firstly we calculate, integrating by parts and using the relation , $$\begin{aligned}
\label{seesf}
&& {\left\vert{\left( \overline{\partial_3 u^{I,0}_3} y\partial_y\varphi_m , ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)} \right\vert}+{\left\vert{\left( {\left(v+ \overline{u^{I,0}_1}\right)} \partial_1\varphi_m + {\left(u - u (t,x_1,0)\right)} \partial_y \varphi_m, ~ \varphi_m\right)}_{L^2(\mathbb R_+^2)}\right\vert}\nonumber \\
&\leq &\frac{1}{2} {\left({\big\Vert\overline{\partial_3 u^{I,0}_3} \big\Vert}_{L^\infty}+{\big\Vert\overline{\partial_3 u^{I,0}_1} \big\Vert}_{L^\infty}\right)}{\big\Vert \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2.\end{aligned}$$ Integrating by parts and using the boundary condition in , we have $$\begin{aligned}
\label{fiest}
{\left({\left(\partial_t - \partial_y^2 \right)} \varphi_m , ~ \varphi_m \right)}_{L^2(\mathbb R_+^2)}
&=&\frac{1}{2}\frac{d}{dt} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+{\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2\nonumber\\
&&+\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left(\partial_y\varphi_m\right)}(t, x_1, 0) \,dx_1.\end{aligned}$$ Now we check the boundary value of $\partial_y \varphi$. In view of we see $$\begin{aligned}
\partial_y\varphi_m\big |_{y=0}={\left<x_1\right>}^\ell \partial_1^m\partial_y^2 u \big|_{y=0}.\end{aligned}$$ And moreover, using the relation $$\begin{aligned}
{\left<x_1\right>}^\ell\partial_y^2 u\big |_{y=0}=\partial_t u(x_1, 0)-\overline{\partial_3 u^{I,0}_3} u(x_1, 0)+\overline{u^{I,0}_1}(\partial_1u)(x_1, 0)\end{aligned}$$ which follows from , we conclude $$\begin{aligned}
\partial_y\varphi_m\big |_{y=0}=\partial_t {\left<x_1\right>}^\ell\partial_1^mu(t, x_1, 0)-{\left<x_1\right>}^\ell\partial_1^m {\left(\overline{\partial_3 u^{I,0}_3} u(t,x_1, 0)\right)}+{\left<x_1\right>}^\ell\partial_1^m{\left(\overline{u^{I,0}_1}(\partial_1u)(t,x_1, 0)\right)}.\end{aligned}$$ As a result, $$\begin{aligned}
&&\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1){\left<x_1\right>}^\ell {\left(\partial_y\varphi_m\right)}(t, x_1, 0) \,dx_1\\
&=&\frac{d}{dt}\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&-\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_t\partial_1^{m+1}u^{I,0}_1}(t,x_1){\left<x_1\right>}^\ell \partial_1^mu(t,x_1, 0)\,dx_1\\
&&-\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) \partial_1^m {\left(\overline{\partial_3 u^{I,0}_3} u(x_1, 0)\right)}\,dx_1\\
&&+\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) \partial_1^m{\left(\overline{u^{I,0}_1}(\partial_1u)(t,x_1, 0)\right)}\,dx_1.\end{aligned}$$ Moreover, In view of , we can repeat the arguments in Lemma \[R8\] and Lemma \[Rothers\], to obtain, observing $\rho<\tau/4,$ $$\begin{aligned}
{\left\vert\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_t\partial_1^{m+1}u^{I,0}_1}(t,x_1){\left<x_1\right>}^\ell \partial_1^mu(t,x_1, 0)\,dx_1\right\vert}\leq C {\left( \frac{(m-3)!}{\rho^{m-1}}\right)}^2 {\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} {\left\vertu\right\vert}_{X_{\rho,a,m} },\end{aligned}$$ $$\begin{aligned}
{\left\vert\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) \partial_1^m {\left(\overline{\partial_3 u^{I,0}_3} u(x_1, 0)\right)}\,dx_1\right\vert}
\leq C {\left( \frac{(m-3)!}{\rho^{m-1}}\right)}^2 {\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} {\big\Vertu^{I,0}_3\big\Vert}_{\mathcal A_\tau} {\left\vertu\right\vert}_{X_{\rho,a,m} }\end{aligned}$$ and $$\begin{aligned}
{\left\vert\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) \partial_1^m{\left(\overline{u^{I,0}_1}(\partial_1u)(t,x_1, 0)\right)}\,dx_1\right\vert}
\leq C {\left( \frac{(m-3)!}{\rho^{m-1}}\right)}^2 {\big\Vertu^{I,0}_1\big\Vert}_{\mathcal A_\tau} ^2 {\left\vertu\right\vert}_{X_{\rho,a,m} }.\end{aligned}$$ Combing these inequalities above, we conclude $$\begin{aligned}
&&\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1){\left<x_1\right>}^\ell {\left(\partial_y\varphi_m\right)}(t, x_1, 0) \,dx_1\\
&\geq &\frac{d}{dt}\int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1-C {\left( \frac{(m-3)!}{\rho^{m-1}}\right)}^2{\left\vertu\right\vert}_{X_{\rho,a,m} },
\end{aligned}$$ which, along with and , yields the conclusion, completing the proof.
\[lem0222\] Let $ a(t)=a_0-{\left(2a_0^2+C_0\right)}t$ with $C_0$ the constants given in Corollary \[cor0222\]. Then for any $N$, $$\begin{aligned}
&& \frac{1}{2}\frac{d}{dt} \sum_{m=3}^N{\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+\frac{1}{2}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^2 u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2 \\
&&\qquad\qquad\qquad\qquad-\rho'(t)\sum_{m=3}^N {\left((m-1)^{1/2} \rho^{-1/2} \frac{\rho^{m-1}}{(m-3)!}{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \right)}^2\\
&\leq &{1\over 4} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 -\frac{d}{dt}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&+C {\left( {\left\vert\rho'\right\vert}\rho^{-2} {\left\vertu\right\vert}_{X_{\rho,a}} +{\left\vertu\right\vert}_{X_{\rho,a}}^2 + {\left\vertu\right\vert}_{X_{\rho,a} }^4\right)}.
\end{aligned}$$
Using the equality and Lemma \[lowleft\], we obtain $$\begin{aligned}
&&\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left[\frac{1}{2}\frac{d}{dt} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+ {\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2-a'(t){\big\Verty \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\right]}\\
&\leq& -\frac{d}{dt}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&+\sum_{m=3}^N(2m-2)\frac{\rho'(t) \rho^{2m-3}}{\left[(m-3)!\right]^2} \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1+{\left\vertu\right\vert}_{\rho,a}\\
&&+\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left(\sum_{k=1}^{10}{\left( \mathcal R_k^m(t), ~\varphi_m (t)\right)}_{L^2(\mathbb R_+^2)}\right)}\\
&\leq& -\frac{d}{dt}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&+\sum_{m=3}^N(2m-2)\frac{\rho'(t) \rho^{2m-3}}{\left[(m-3)!\right]^2} \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1+{\left\vertu\right\vert}_{\rho,a}\\
&&+ {1\over 4} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 +C_0 \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2+C {\left({\left\vertu\right\vert}_{X_{\rho,a}}^2 + {\left\vertu\right\vert}_{X_{\rho,a} }^4\right)},
\end{aligned}$$ the last inequality following from Corollary \[cor0222\]. On the other hand, $$\begin{aligned}
&&\sum_{m=3}^N(2m-2)\frac{\rho'(t) \rho^{2m-3}}{\left[(m-3)!\right]^2} \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1 \\
&\leq& C \sum_{m=3}^N \frac{2^m \rho'(t) \rho^{2m-3}}{\left[(m-3)!\right]^2} {\big\Vert{\left<x_1\right>}^\ell \overline{\partial_1^{m+1}u^{I,0}_1}\big\Vert}_{L^2{\left(\mathbb R_{x_1}\right)}} {\left({\big\Vert{\left<x_1\right>}^\ell \partial_1^mu\big\Vert}_{L^2}+{\big\Vert{\left<x_1\right>}^\ell \partial_1^m\partial_y u\big\Vert}_{L^2}\right)}\\
&\leq& C \sum_{m=3}^N \frac{2^m \rho'(t) \rho^{m-2}}{\tau^{m+3}} {\big\Vertu^{I,0}_1\big\Vert}_{G_\tau} \left[\frac{ \rho^{m-1}}{(m-3)!} {\left({\big\Vert{\left<x_1\right>}^\ell \partial_1^mu\big\Vert}_{L^2}+{\big\Vert{\left<x_1\right>}^\ell \partial_1^m\partial_y u\big\Vert}_{L^2}\right)}\right]
\\
&\leq& C {\left\vert\rho'\right\vert}\rho^{-2} {\left\vertu\right\vert}_{X_{\rho,a}}
\end{aligned}$$ the last inequality using the fact that $\rho<\tau/3$. As a result, combining the equalities above yields $$\begin{aligned}
&&\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left[\frac{1}{2}\frac{d}{dt} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+ {\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2-a'(t){\big\Verty \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\right]} \\
&\leq& {1\over 4} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 -\frac{d}{dt}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&+C_0 \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2+C {\left( {\left\vert\rho'\right\vert}\rho^{-2} {\left\vertu\right\vert}_{X_{\rho,a}} +{\left\vertu\right\vert}_{X_{\rho,a}}^2 + {\left\vertu\right\vert}_{X_{\rho,a} }^4\right)}.
\end{aligned}$$ Moreover from the relations $$\begin{aligned}
{\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2 \geq \frac{1}{2} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^2 u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2-2a_0^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2\end{aligned}$$ and $$\begin{aligned}
&& \frac{1}{2}\frac{d}{dt} {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+ {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2-a'(t) {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\\
&&\qquad\qquad\qquad\qquad-\rho'(t){\left((m-1)^{1/2} \rho^{-1/2} \frac{\rho^{m-1}}{(m-3)!}{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \right)}^2\\
&=& {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\left[\frac{1}{2}\frac{d}{dt} {\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+ {\big\Vert\partial_y\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2-a'(t){\big\Verty \varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2\right]},
\end{aligned}$$ it follows that $$\begin{aligned}
&& \frac{1}{2}\frac{d}{dt} \sum_{m=3}^N{\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2+\frac{1}{2}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^2 u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2 \\
&&\qquad\qquad\qquad -{\left(a'(t)+2a_0^2\right)}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2 \\
&&\qquad\qquad\qquad\qquad-\rho'(t)\sum_{m=3}^N {\left((m-1)^{1/2} \rho^{-1/2} \frac{\rho^{m-1}}{(m-3)!}{\big\Vert\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)} \right)}^2\\
&\leq &{1\over 4} {\left\vertu\right\vert}_{Z_{\rho,a}}^2 -\frac{d}{dt}\sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 \int_{\mathbb R_{x_1}} {\left<x_1\right>}^\ell\overline{\partial_1^{m+1}u^{I,0}_1}(t,x_1) {\left<x_1\right>}^\ell\partial_1^mu(t,x_1, 0)\,dx_1\\
&&+C_0 \sum_{m=3}^N {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Verty\varphi_m\big\Vert}_{L^2(\mathbb R_+^2)}^2+C {\left\vertu\right\vert}_{Z_{\rho,a}} {\left\vertu\right\vert}_{Y_{\rho,a} }^2+C {\left( {\left\vert\rho'\right\vert}\rho^{-2} {\left\vertu\right\vert}_{X_{\rho,a}} +{\left\vertu\right\vert}_{X_{\rho,a}}^2 + {\left\vertu\right\vert}_{X_{\rho,a} }^4\right)}.
\end{aligned}$$ Now observing $
a(t)=a_0-{\left(2a_0^2+C_0\right)}t,
$ we complete the proof.
By Lemma \[lem0222\], we integrate both sides over $[0, t]\subset[0,T]$ and then let $N\rightarrow +\infty$, to obtain that for any $t\in [0,T],$ $$\begin{aligned}
&& \sum_{m=3}^{+\infty} {\left(\frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Vert\varphi_m(t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2+\int_0^T \sum_{m=3}^{+\infty} {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^2 u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2dt\\
&&\qquad-\int_0^T \rho'(t) \sum_{m=3}^{+\infty} \frac{m-1}{\rho}{\left(\frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Vert \varphi_m(t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2 dt\\
&\leq &{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+\frac{1}{2}\int_0^T{\left\vertu(t)\right\vert}_{Z_{\rho,a}}^2dt+C \int_0^T {\left({\left\vert\rho'(t)\right\vert}\rho^{-2}{\left\vertu(t)\right\vert}_{X_{\rho,a} }+{\left\vertu(t)\right\vert}_{X_{\rho,a} }^2+ {\left\vertu(t)\right\vert}_{X_{\rho,a}}^4\right)}dt\\
&&+C \int_0^T {\left\vertu(t)\right\vert}_{Z_{\rho,a}} {\left\vertu(t)\right\vert}_{Y_{\rho,a} }^2dt.
\end{aligned}$$ Direct computation also gives $$\begin{aligned}
&& \sum_{m\leq 2} {\big\Vert\varphi_m(t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2+\int_0^T \sum_{m\leq 2} {\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y^2 u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2dt-\int_0^T \rho'(t) \sum_{m\leq 2}^{+\infty} {\big\Vert \varphi_m(t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2 dt\\
&\leq &{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+\frac{1}{2}\int_0^T{\left\vertu(t)\right\vert}_{Z_{\rho,a}}^2dt+C \int_0^T {\left({\left\vert\rho'(t)\right\vert}\rho^{-2}{\left\vertu(t)\right\vert}_{X_{\rho,a} }+{\left\vertu(t)\right\vert}_{X_{\rho,a} }^2+ {\left\vertu(t)\right\vert}_{X_{\rho,a}}^4\right)}dt\\
&&+C \int_0^T {\left\vertu(t)\right\vert}_{Z_{\rho,a}} {\left\vertu(t)\right\vert}_{Y_{\rho,a} }^2dt.
\end{aligned}$$ Similarly, using the notation $$\begin{aligned}
\psi_m ={\left<x_1\right>}^\ell e^{a y^2}\partial_1^m u (t),
\end{aligned}$$ we can deduce, following the proof of the above two inequalities with slight modification and simpler arguments, $$\begin{aligned}
&&\sum_{m\leq 2}{\big\Vert\psi_m(t)\big\Vert}_{L^2}^2+\sum_{m=3}^{+\infty} {\left(\frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Vert\psi_m(t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2\\
&&\qquad+\int_0^T{\left( \sum_{m\leq 2}{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2+\sum_{m=3}^{+\infty} {\left( \frac{\rho^{m-1}}{(m-3)!}\right)}^2{\big\Vert{\left<x_1\right>}^\ell e^{a y^2}\partial_1^m\partial_y u (t)\big\Vert}_{L^2(\mathbb R_+^2)} ^2\right)} dt\\
&&\qquad-\int_0^T \rho'(t) {\left( \sum_{m\leq 2}{\big\Vert \psi_m\big\Vert}_{L^2}^2+\sum_{m=3}^{+\infty} \frac{m-1}{\rho}{\left(\frac{\rho^{m-1}}{(m-3)!}\right)}^2 {\big\Vert \psi_m\big\Vert}_{L^2(\mathbb R_+^2)} ^2\right)} dt\\
&\leq &{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+\frac{1}{2}\int_0^T{\left\vertu(t)\right\vert}_{Z_{\rho,a}}^2dt+C \int_0^T {\left({\left\vert\rho'(t)\right\vert}\rho^{-2}{\left\vertu(t)\right\vert}_{X_{\rho,a} }+{\left\vertu(t)\right\vert}_{X_{\rho,a} }^2+ {\left\vertu(t)\right\vert}_{X_{\rho,a}}^4\right)}dt\\
&&+C \int_0^T {\left\vertu(t)\right\vert}_{Z_{\rho,a}} {\left\vertu(t)\right\vert}_{Y_{\rho,a} }^2dt
\end{aligned}$$ Combining these inequalities we conclude, observing the definition of ${\left\vert\cdot\right\vert}_{X_{\rho,a }}, {\left\vert\cdot\right\vert}_{Y_{\rho,a }}$ and ${\left\vert\cdot\right\vert}_{Z_{\rho,a }}$ and any $\rho\leq \min{\left\{\rho_0, \tau/3\right\}}$, $$\begin{aligned}
&&{\left\vertu(t)\right\vert}_{X_{\rho,a }}^2
+\int_0^T {\left\vertu(t)\right\vert}_{Z_{\rho,a }}^2dt -\int_0^T \rho'(t) {\left\vertu(t)\right\vert}_{Y_{\rho,a }}^2 dt\\
&\leq &{\left\vertu_0\right\vert}_{X_{\rho_0,a_0} }^2+\frac{1}{2}\int_0^T{\left\vertu(t)\right\vert}_{Z_{\rho,a}}^2dt+C \int_0^T {\left({\left\vert\rho'(t)\right\vert}\rho^{-2}{\left\vertu(t)\right\vert}_{X_{\rho,a} }+{\left\vertu(t)\right\vert}_{X_{\rho,a} }^2+ {\left\vertu(t)\right\vert}_{X_{\rho,a}}^4\right)}dt\\
&&+C \int_0^T {\left\vertu(t)\right\vert}_{Z_{\rho,a}} {\left\vertu(t)\right\vert}_{Y_{\rho,a} }^2dt
\end{aligned}$$ Thus Claim follows and the proof is complete.
Existence of solution for second component {#section5}
==========================================
In this section, we determine the second component ${\mathcal{U}}^{p,0}_2$ by solving the parabolic-type equation $$\left\{
\begin{aligned}
&\partial_t {\mathcal{U}}^{p,0}_2 - \partial_y^2 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,0}_1 \partial_1 {\mathcal{U}}^{p,0}_2 + {\mathcal{U}}^{p,1}_3 \partial_y {\mathcal{U}}^{p,0}_2 = 0,\\ &{\mathcal{U}}^{p,0}_2(t,x_1,0) = 0,\quad\lim_{y\to +\infty} {\mathcal{U}}^{p,0}_2(t,x_1,y) = \overline{u^{I,0}_2}(x_1),\\
&{\mathcal{U}}^{p,0}_2(0,x_1,y) = u^{B,0}_{0,2} (x_1,y) + \overline{u^{I,0}_{0,2}}(x_1)\, .
\end{aligned}
\right.$$ We recall that $$\label{eq:traceUI02bis} \partial_t\overline{u^{I,0}_2} + \overline{u^{I,0}_1} \partial_1 \overline{u^{I,0}_2} = 0\, .$$ Then, the system becomes $$\label{eq:P2B}\tag{P2bis}
\left\{
\begin{aligned}
&\partial_t u^{B,0}_2 - \partial_y^2 u^{B,0}_2 + {\left(u^{B,0}_1 + \overline{u^{I,0}_1}\right)} \partial_1 u^{B,0}_2 + {\left(u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3u^{I,0}_3}\right)} \partial_y u^{B,0}_2 + \overline{\partial_1 u^{I,0}_2} u^{B,0}_1 = 0,\\
&\partial_2 u^{B,0}_2 = 0,\\
&u^{B,0}_2(t,x_1,0) = -\overline{u^{I,0}_2},\quad
\lim_{y\to +\infty} u^{B,0}_2(t,x_1,y) = 0,\\
&u^{B,0}_2(0,x_1,y) = u^{B,0}_{0,2} (x_1,y).
\end{aligned}
\right.$$ We have the following results
\[th:P2sys\] Let $\rho_0 > 0$, $a_0 > 0$. For any initial data $u^{B,0}_{2,0} \in X_{\rho_0, a_0}$, there exists $T>0$, $\tau>0$ and $0<a<a_0$, such that the system admits a unique solution $u^{B,0}_2 \in L^{\infty}{\left([0,T], X_{\rho_0, a}\right)}$.
**Proof.** In order to prove Theorem \[th:P2sys\], the idea is to define an auxiliary function $$v = u^{B,0}_2 + e^{-2a_0y^2} \overline{u^{I,0}_2},$$ which satisfies the following boundary conditions $$v(t,x_1,0) = \lim_{y\to +\infty} v(t,x_1,y) = 0.$$ Then, the first equation of the system becomes $$\begin{gathered}
\partial_t {\left(v - e^{-2a_0y^2} \overline{u^{I,0}_2}\right)} - \partial_y^2 {\left(v - e^{-2a_0y^2} \overline{u^{I,0}_2}\right)} + {\left(u^{B,0}_1 + \overline{u^{I,0}_1}\right)} \partial_1 {\left(v - e^{-2a_0y^2} \overline{u^{I,0}_2}\right)}\\
+ {\left(u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3u^{I,0}_3}\right)} \partial_y {\left(v - e^{-2a_0y^2} \overline{u^{I,0}_2}\right)} + \overline{\partial_1 u^{I,0}_2} u^{B,0}_1 = 0.\end{gathered}$$ Using , we can rewrite the system as $$\label{eq:P2Bv}\tag{$P2_v$}
\left\{
\begin{aligned}
&\partial_t v - {\partial}_y^2 v + {\left(u^{B,0}_1 + \overline{u^{I,0}_1}\right)} \partial_1 v + {\left(u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3u^{I,0}_3}\right)} \partial_y v + R = 0,\\
&\partial_2 v = 0,\\
&v(t,x_1,0) = 0,\quad\lim_{y\to +\infty} u^{B,0}_2(t,x_1,y) = 0,\\
&v(0,x_1,y) = u^{B,0}_{0,2} (x_1,y) + e^{-2a_0y^2} \overline{u^{I,0}_{0,2}}(x_1),
\end{aligned}
\right.$$ where $$R = {\left(16 a_0^2 y^2 - 4a_0\right)} e^{-2a_0y^2} \overline{u^{I,0}_2} + 4a_0 {\left(u^{B,1}_3 + \overline{u^{I,1}_3} + y \overline{\partial_3u^{I,0}_3}\right)} y e^{-2a_0y^2} \overline{u^{I,0}_2} + {\left(1 - e^{-2a_0y^2}\right)} \overline{\partial_1 u^{I,0}_2} u^{B,0}_1.$$ We remark that the system is in the same form as the system with Dirichlet boundary conditions. Thus, we can prove Theorem \[th:P2sys\] in the same way (with a lot of simplifications) as we did to prove Theorem \[th41\].
[**Acknowledgements.**]{} The research of the first author was supported by NSF of China(11422106) and Fok Ying Tung Education Foundation (151001), and he would like to thank the invitation of “ laboratoire de mathématiques Raphaël Salem” of the Université de Rouen. The second author would like to express his sincere thanks to School of mathematics and statistics of Wuhan University for the invitations. The research of the last author is supported partially by “The Fundamental Research Funds for Central Universities of China".
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---
abstract: 'We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi–direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi–direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group.'
address: 'Sobolev Institute of Mathematics, Novosibirsk 630090, Russia'
author:
- 'Valerij G. Bardakov'
title: THE VIRTUAL AND UNIVERSAL BRAIDS
---
[^1]
Recently some generalizations of classical knots and links were defined and studied: singular links [@Vas; @Bir93], virtual links [@Ka; @GPV] and welded links [@FRR].
One of the ways to study classical links is to study the braid group. Singular braids [@Baez; @Bir93], virtual braids [@Ka; @Ver01], welded braids [@FRR] were defined similar to the classical braid group. Theorem of A. A. Markov [@Bir Ch. 2.2] reduces the problem of classification of links to some algebraic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generalizations of Markov theorem for singular links [@Ge], virtual links, and welded links [@Kam].
There are some different ways to solve the word problem for the singular braid monoid and singular braid group [@Dash; @Corran; @Ver]. The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group [@FRR]. A normal form of words in the welded braid group was constructed in [@Kr].
In this paper we study the structure of the virtual braid group $VB_n$. Similar to the classical braid group $B_n$ and welded braid group $WB_n$, the group $VB_n$ contains the normal subgroup $VP_n$ which is called [*virtual pure braid group*]{}. The quotient group $VB_n/VP_n$ is isomorphic to the symmetric group $S_n$. In the article we find generators and defining relations of $VP_n$. Since $VB_n$ is a semi–direct product of $VP_n$ and $S_n$, we should study the structure of $VP_n$. It will be proved that $VP_n$ is representable as the following semi–direct product $$VP_n = V_{n-1}^* \leftthreetimes VP_{n-1} = V_{n-1}^* \leftthreetimes
(V^*_{n-2} \leftthreetimes (\ldots \leftthreetimes
(V_2^* \leftthreetimes V_1^*))\ldots),$$ where $V_i^*$ is some (in general infinitely generated for $i > 1$) free subgroup of $VP_n$. From this result it follows that there exists a normal form of words in $VB_n$.
In the last section we define the universal braid group $UB_n$ which contains the braid group $B_n$ and has as its quotient groups the singular braid group $SG_n$, virtual braid group $VB_n$, welded braid group $WB_n$, and braid group $B_n$. It is known [@FRR] that $VB_n$ has as its quotient the group $WB_n$. It will be proved that the quotient homomorphism maps $VP_n$ into the welded pure braid group $WP_n$. This homomorphism agrees with the decomposition of this group into the semi-direct product given by Theorem \[theorem2\] and by [@Bar; @BarP].
By Artin theorem, $B_n$ is embedded into the automorphism group $\mbox{Aut}(F_n)$ of the free group $F_n$. In [@FRR] it was proved that $WB_n$ is also embedded into $\mbox{Aut}(F_n)$. It is not known if it is true that $SG_n$ and $VB_n$ are embedded into $\mbox{Aut}(F_n)$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I am very grateful to Joanna Kania-Bartoszynska, Jozef Przytycki, Pawel Traczyk, and Bronislaw Wajnryb for organizing of the Mini–semester on Knot Theory (Poland, July, 2003) and for the invitation to participate in this very interesting and very well–organized Mini–semester. I would also like to thank Vladimir Vershinin and Andrei Vesnin for their interest to this work. Special thanks goes to the participants of the seminar “Evariste Galois” at Novosibirsk State University for their kind attention to my work.
different classes of braids and their properties
================================================
In this section we remind (see references from the introduction) some known facts about braid groups, singular braid monoids, virtual braid groups and welded braid groups.
The braid group and the group of conjugating automorphisms
----------------------------------------------------------
The braid group $B_n$, $n\geq 2$, on $n$ strings can be defined as a group generated by $\sigma_1,\sigma_2,\ldots,\sigma_{n-1}$ (see Fig. \[f:1\])
with the defining relations $$\sigma_i \, \sigma_{i+1} \, \sigma_i = \sigma_{i+1} \, \sigma_i \, \sigma_{i+1},~~~ i=1,2,\ldots,n-2, \label{eq1}$$ $$\sigma_i \, \sigma_j = \sigma_j \, \sigma_i,~~~|i-j|\geq 2. \label{eq2}$$
There exists a homomorphism of $B_n$ onto the symmetric group $S_n$ on $n$ letters. This homomorphism maps $\sigma_i$ to the transposition $(i,i+1)$, $i=1,2,\ldots,n-1$. The kernel of this homomorphism is called [*pure braid group*]{} and denoted by $P_n$. The group $P_n$ is generated by $a_{ij}$, $1\leq i < j\leq n$ (see Fig. \[f:3\]).
These generators can be expressed by the generators of $B_n$ as follows $$a_{i,i+1}=\sigma_i^2,$$ $$a_{ij} = \sigma_{j-1} \, \sigma_{j-2} \ldots \sigma_{i+1} \, \sigma_i^2 \, \sigma_{i+1}^{-1} \ldots
\sigma_{j-2}^{-1} \, \sigma_{j-1}^{-1},~~~i+1< j \leq n.$$
The group $P_n$ is the semi–direct product of the normal subgroup $U_n$ which is a free group with free generators $a_{1n}, a_{2n},\ldots,a_{n-1,n},$ and $P_{n-1}$. Similarly, $P_{n-1}$ is the semi–direct product of the free group $U_{n-1}$ with free generators $a_{1,n-1}, a_{2,n-1},\ldots,a_{n-2,n-1}$ and $P_{n-2},$ and so on. Therefore, $P_n$ is decomposable (see [@Mar]) into the following semi–direct product $$P_n=U_n\leftthreetimes (U_{n-1}\leftthreetimes (\ldots \leftthreetimes
(U_3\leftthreetimes U_2))\ldots),~~~U_i\simeq F_{i-1}, ~~~i=2,3,\ldots,n.$$
The group $B_n$ has a faithful representation as a group of automorphisms of ${\rm
Aut}(F_n)$ of the free group $F_n = \langle x_1, x_2, \ldots, x_n \rangle.$ In this case the generator $\sigma_i$, $i=1,2,\ldots,n-1$, defines the automorphism $$\sigma_{i} : \left\{
\begin{array}{ll}
x_{i} \longmapsto x_{i} \, x_{i+1} \, x_i^{-1}, & \\ x_{i+1} \longmapsto
x_{i}, & \\ x_{l} \longmapsto x_{l}, & l\neq i,i+1.
\end{array} \right.$$
By theorem of Artin [@Bir Theorem 1.9], an automorphism $\beta $ from ${\rm
Aut}(F_n)$ lies in $B_n$ if and only if $\beta $ satisfies to the following conditions: $$~~~~~1)~~ \beta(x_i) = a_i^{-1} \, x_{\pi(i)} \, a_i,~~~1\leq i\leq n,$$ $$2)~~ \beta(x_1x_2 \ldots x_n)=x_1x_2 \ldots x_n,$$ where $\pi $ is a permutation from $S_n$ and $a_i\in F_n$.
An automorphism is called a [*conjugating automorphism*]{} (or a permutation–conjugating automorphism according to the terminology from [@FRR]) if it satisfies to condition 1). The group of conjugating automorphisms $C_n$ is generated by $\sigma_i$ and automorphisms $\alpha_i$, $i=1, 2, \ldots,$ $n-1$, where $$\alpha_{i} : \left\{
\begin{array}{ll}
x_{i} \longmapsto x_{i+1}, & \\
x_{i+1} \longmapsto x_{i}, &\\
x_{l} \longmapsto x_{l}, & l\neq i, i+1.
\end{array} \right.$$ It is not hard to see that the automorphisms $\alpha_i$ generate the symmetric group $S_n$ and, hence, satisfy the following relations $$\alpha_i \, \alpha_{i+1} \, \alpha_i = \alpha_{i+1} \, \alpha_i \, \alpha_{i+1},~~~ i=1,2,\ldots,n-2, \label{eq3}$$ $$\alpha_i \, \alpha_j = \alpha_j \, \alpha_i,~~~ |i-j|\geq 2, \label{eq4}$$ $$\alpha_i^2 = 1,~~~ i=1,2,\ldots,n-1. \label{eq5}$$ The group $C_n$ is defined by relations (\[eq1\])–(\[eq2\]) of $B_n$, relations (\[eq3\])–(\[eq5\]) of $S_n$, and the mixed relations (see [@FRR; @Sav]) $$\alpha_i \, \sigma_j = \sigma_j \, \alpha_i,~~~ |i-j|\geq 2, \label{eq6}$$ $$\sigma_i \, \alpha_{i+1} \, \alpha_i = \alpha_{i+1} \, \alpha_i \, \sigma_{i+1},~~~ i=1,2,\ldots,n-2, \label{eq7}$$ $$\sigma_{i+1} \, \sigma_{i} \, \alpha_{i+1} = \alpha_{i} \, \sigma_{i+1} \, \sigma_{i},~~~ i=1,2,\ldots,n-2. \label{eq8}$$
If we consider the group generated by automorphisms $\varepsilon_{ij}$, $1\leq i\neq j \leq n$, where $$\varepsilon_{ij} : \left\{
\begin{array}{ll}
x_{i} \longmapsto x_{j}^{-1} \, x_i \, x_j, & i\neq j, \\
x_{l} \longmapsto x_{l}, & l\neq i,
\end{array} \right.$$ then we get the group of [*basis–conjugating automorphisms*]{} $Cb_n$. The elements of $Cb_n$ satisfy condition 1) for the identical permutation $\pi $, i. e., map each generator $x_i$ to the conjugating element. J. McCool [@Mac] proved that $Cb_n$ is defined by the relations (from here different letters stand for different indices) $$\varepsilon_{ij} \, \varepsilon_{kl} = \varepsilon_{kl} \, \varepsilon_{ij},
\label{eq9}$$ $$\varepsilon_{ij} \, \varepsilon_{kj} = \varepsilon_{kj} \, \varepsilon_{ij},
\label{eq10}$$ $$(\varepsilon_{ij} \, \varepsilon_{kj}) \, \varepsilon_{ik} = \varepsilon_{ik} \,
(\varepsilon_{ij} \,
\varepsilon_{kj}).
\label{eq11}$$
The group $C_n$ is representable as the semi–direct product: $C_n = Cb_n \leftthreetimes S_n$, where $S_n$ is generated by the automorphisms $\alpha_1, \alpha_2, \ldots, \alpha_{n-1}$. The following equalities are true (see [@Sav]): $$\varepsilon_{i,i+1} = \alpha_i \, \sigma^{-1}_i,~~~
\varepsilon_{i+1,i} = \sigma^{-1}_i \, \alpha_i,$$ $$\varepsilon_{ij} = \alpha_{j-1} \, \alpha_{j-2} \ldots \alpha_{i+1} \, \varepsilon_{i,
i+1} \,
\alpha_{i+1} \ldots \alpha_{j-2} \, \alpha_{j-1}~~~i <j,$$ $$\varepsilon_{ji} = \alpha_{j-1} \, \alpha_{j-2} \ldots \alpha_{i+1} \,
\alpha_i \,
\varepsilon_{i, i+1} \, \alpha_i \,
\alpha_{i+1} \ldots \alpha_{j-2} \, \alpha_{j-1}~~~i <j.$$
The structure of $Cb_n$ was studied in [@Bar; @BarP]. There it was proved that $Cb_n$, $n\geq 2$, is decomposable into the semi–direct product $$Cb_n = D_{n-1}\leftthreetimes (D_{n-2}\leftthreetimes (\ldots
\leftthreetimes (D_2\leftthreetimes D_1))\ldots ),$$ of subgroups $D_i,$ $i=1,2,\ldots,n-1,$ generated by $\varepsilon_{i+1,1},$ $\varepsilon_{i+1,2},$ $\ldots,\varepsilon_{i+1,i},$ $\varepsilon_{1,i+1},$ $\varepsilon_{2,i+1},$ $\ldots,\varepsilon_{i,i+1}$. The elements $\varepsilon_{i+1,1},$ $\varepsilon_{i+1,2},$ $
\ldots,\varepsilon_{i+1,i}$ generate a free group of rank $i$, elements $\varepsilon_{1,i+1},$ $\varepsilon_{2,i+1},$ $\ldots,\varepsilon_{i,i+1}$ generate a free abelian group of rank $i$.
The pure braid group $P_n$ is contained in $Cb_n$ and the generators of $P_n$ can be written in the form $$a_{i,i+1}=\varepsilon_{i,i+1}^{-1} \, \varepsilon_{i+1,i}^{-1},~~~~~i=1,2,\ldots,n-1,$$ $$a_{ij} = \varepsilon_{j-1,i} \, \varepsilon_{j-2,i} \ldots
\varepsilon_{i+1,i} \,
(\varepsilon_{ij}^{-1} \, \varepsilon_{ji}^{-1}) \, \varepsilon_{i+1,i}^{-1} \ldots
\varepsilon_{j-2,i}^{-1} \, \varepsilon_{j-1,i}^{-1} =$$ $$= \varepsilon_{j-1,j}^{-1} \, \varepsilon_{j-2,j}^{-1} \ldots
\varepsilon_{i+1,j}^{-1} \,
(\varepsilon_{ij}^{-1} \, \varepsilon_{ji}^{-1}) \, \varepsilon_{i+1,j} \ldots
\varepsilon_{j-2,j} \, \varepsilon_{j-1,j},~~~~~2 \leq i+1 < j \leq n.$$
The singular braid monoid.
--------------------------
[*The Baez–Birman monoid*]{} [@Baez; @Bir93] or [*the singular braid monoid*]{} $SB_n$ is generated (as monoid) by elements $\sigma_i,$ $\sigma_i^{-1}$, $\tau_i$, $i = 1, 2, \ldots, n-1$. The elements $\sigma_i,$ $\sigma_i^{-1}$ generate the braid group $B_n$. The generators $\tau_i$ satisfy the defining relations $$\tau_i \, \tau_j = \tau_j \, \tau_i, ~~~|i - j| \geq 1, \label{eq12}$$ other relations are mixed: $$\tau_{i} \, \sigma_{j} = \sigma_{j} \, \tau_{i}, ~~~|i - j| \geq 1, \label{eq13}$$ $$\tau_{i} \, \sigma_{i} = \sigma_{i} \, \tau_{i},~~~ i=1,2,\ldots,n-1, \label{eq14}$$ $$\sigma_{i} \, \sigma_{i+1} \, \tau_i = \tau_{i+1} \, \sigma_{i} \, \sigma_{i+1},~~~ i=1,2,\ldots,n-2,
\label{eq15}$$ $$\sigma_{i+1} \, \sigma_{i} \, \tau_{i+1} = \tau_{i} \,
\sigma_{i+1} \, \sigma_{i}, ~~~ i=1,2,\ldots,n-2.
\label{eq16}$$
In the work [@FKR] it was proved that the singular braid monoid $SB_n$ is embedded into the group $SG_n$ which is called the [*singular braid group*]{} and has the same defining relations as $SB_n$.
The virtual braid group and welded braid group
----------------------------------------------
The virtual braid group $VB_n$ was introduced in [@Ka]. In [@Ver01] it was found more short system of defining relations (see below). The group $VB_n$ is generated by $\sigma_i$, $\rho_i$, $i = 1, 2, \ldots, n-1$ (see Fig. \[f:2\]).
The elements $\sigma_i$ generate the braid group $B_n$ with defining relations (\[eq1\])–(\[eq2\]) and the elements $\rho_i$ generate the symmetric group $S_n$ which is defined by the relations $$\rho_{i} \, \rho_{i+1} \, \rho_{i} = \rho_{i+1} \, \rho_{i} \, \rho_{i+1},~~~ i=1,2,\ldots,n-2,
\label{eq17}$$ $$\rho_{i} \, \rho_{j} = \rho_{j} \, \rho_{i},~~~|i-j| \geq 1,
\label{eq18}$$ $$\rho_{i}^2 = 1~~~ i=1,2,\ldots,n-1.
\label{eq19}$$ Other relations are mixed: $$\sigma_{i} \, \rho_{j} = \rho_{j} \, \sigma_{i},~~~|i-j| \geq 1,
\label{eq20}$$ $$\rho_{i} \, \rho_{i+1} \, \sigma_{i} = \sigma_{i+1} \, \rho_{i} \, \rho_{i+1},~~~
i=1,2,\ldots,n-2.
\label{eq21}$$
Note that the last relation is equivalent to the following relation: $$\rho_{i+1} \, \rho_{i} \, \sigma_{i+1} = \sigma_{i} \, \rho_{i+1} \, \rho_{i}.$$
In the work [@GPV] it was proved that the relations $$\rho_{i} \, \sigma_{i+1} \, \sigma_{i} = \sigma_{i+1} \, \sigma_{i}
\, \rho_{i+1},~~~~~
\rho_{i+1} \, \sigma_{i} \, \sigma_{i+1} = \rho_{i} \, \sigma_{i+1}
\, \sigma_{i}.$$ are not fulfilled in $VB_n$.
In the work [@FRR] it was introduced the welded braid group $WB_n$. This group is generated by $\sigma_i$, $\alpha_i$, $i=1, 2, \ldots, n-1$. The elements $\sigma_i$ generate the braid group $B_n$. The elements $\alpha_i$ generate the symmetric group $S_n$ and the following mixed relations hold $$\alpha_{i} \, \sigma_{j} = \sigma_{j} \, \alpha_{i}, ~~~|i - j| \geq 1, \label{eq22}$$ $$\sigma_{i} \, \alpha_{i+1} \, \alpha_i = \alpha_{i+1} \, \alpha_{i} \, \sigma_{i+1},
~~~ i=1,2,\ldots,n-2, \label{eq23}$$ $$\sigma_{i+1} \, \sigma_{i} \, \alpha_{i+1} = \alpha_{i} \, \sigma_{i+1} \, \sigma_{i},~~~ i=1,2,\ldots,n-2.
\label{eq24}$$
In the work [@FRR] it was proved that $WB_n$ is isomorphic to the group of conjugating automorphisms $C_n$.
Comparing the defining relations of $VB_n$ with the defining relations of $WB_n$, we see that $WB_n$ can be obtained from $VB_n$ by adding some new relation. Therefore, there exists a homomorphism $$\varphi_{VW} : VB_n \longrightarrow WB_n,$$ taking $\sigma_i$ to $\sigma_i$ and $\rho_i$ to $\alpha_i$ for all $i$. Hence, $WB_n$ is the homomorphic image of $VB_n$.
In [@FRR] it was proved that the following relation (symmetric to (\[eq23\])) $$\sigma_{i+1} \, \alpha_{i} \, \alpha_{i+1} = \alpha_{i} \, \alpha_{i+1} \, \sigma_{i},$$ is true in $WB_n$. But the following relation is not fulfilled $$\alpha_{i+1} \, \sigma_{i} \, \sigma_{i+1} = \sigma_{i} \, \sigma_{i+1}
\, \alpha_{i}.$$
In [@Ver01] it was constructed the linear representations of $VB_n$ and $WB_n$ by matrices from $\mbox{GL}_n(\mathbb{Z}[t, t^{-1}])$ which continue the well known Burau representation. The linear representation of $C_n \simeq
WB_n$ it was constructed in [@BarP]. This representation continue (with some conditions on parameters) the known Lawrence–Krammer representation.
Generators and defining relations of the virtual pure braid group
=================================================================
In this section we introduce a virtual pure braid group and find its generators and defining relations.
Define the map $$\nu : VB_n \longrightarrow S_n$$ of $VB_n$ onto the symmetric group $S_n$ by actions on generators $$\nu(\sigma_i) = \nu(\rho_i) = \rho_i, ~~~ i = 1, 2, \ldots, n-1,$$ where $S_n$ is the group generated by $\rho_i$. The kernel $\mbox{ker}(\nu)$ of this map is called the [*virtual pure braid group*]{} and denoted by $VP_n$. It is clear that $VP_n$ is a normal subgroup of index $n!$ of $VB_n$. Moreover, since $VP_n \bigcap S_n = e$ and $VB_n = VP_n \cdot S_n$, then $VB_n = VP_n \leftthreetimes S_n$, i. e., the virtual pure braid group is the semi–direct product of $VP_n$ and $S_n$.
Define the following elements $$\lambda_{i,i+1} = \rho_i \, \sigma_i^{-1},~~~
\lambda_{i+1,i} = \rho_i \, \lambda_{i,i+1} \, \rho_i = \sigma_i^{-1} \, \rho_i,
~~~i=1, 2, \ldots, n-1,$$ $$\lambda_{ij} = \rho_{j-1} \, \rho_{j-2} \ldots \rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1}
\ldots \rho_{j-2} \, \rho_{j-1},$$ $$\lambda_{ji} = \rho_{j-1} \, \rho_{j-2} \ldots \rho_{i+1} \, \lambda_{i+1,i} \, \rho_{i+1}
\ldots \rho_{j-2} \, \rho_{j-1}, ~~~1 \leq i < j-1 \leq n-1.$$ Obviously, all these elements belong to $VP_n$ and have the following geometric interpretation (Fig. \[f:5\], \[f:6\])
The next lemma hold
\[lemma1\] Let $1 \leq i < j \leq n$. The following conjugating rules are fulfilled in $VB_n$:\
1) for $k < i-1$ or $i < k < j-1$ or $k > j$ $$\rho_k \, \lambda_{ij} \, \rho_k = \lambda_{ij},~~~~~\rho_k \, \lambda_{ji} \, \rho_k =
\lambda_{ji};$$\
2) $
\rho_{i-1} \, \lambda_{ij} \, \rho_{i-1} = \lambda_{i-1,j},
~~~\rho_{i-1} \, \lambda_{ji} \, \rho_{i-1} =
\lambda_{j,i-1};\\
$ 3) for $i < j-1$ $$\begin{array}{ll}
\rho_i \, \lambda_{i,i+1} \, \rho_i = \lambda_{i+1,i}, & \rho_i \, \lambda_{ij} \, \rho_i =
\lambda_{i+1,j}, \\
\rho_i \, \lambda_{i+1,i} \, \rho_i = \lambda_{i,i+1}, & \rho_i \, \lambda_{ji} \, \rho_i =
\lambda_{j,i+1}; \\
\end{array}$$ 4) for $i+1 < j$ $$\rho_{j-1} \, \lambda_{ij} \, \rho_{j-1} = \lambda_{i,j-1},
~~~~~\rho_{j-1} \, \lambda_{ji} \, \rho_{j-1} =
\lambda_{j-1,i};$$ 5) $
\rho_{j} \, \lambda_{ij} \, \rho_{j} = \lambda_{i,j+1},~~~\rho_{j} \, \lambda_{ji} \, \rho_{j} =
\lambda_{j+1,i}.
$
We consider only the rules containing $\lambda_{ij}$ for $i < j$ (the remaining rules can be considered analogously). Recall that $$\lambda_{ij} = \rho_{j-1} \, \rho_{j-2} \ldots \rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1}
\ldots \rho_{j-2} \, \rho_{j-1}.$$ If $k < i - 1$ or $k > j$ then $\rho_k$ is permutable with $\rho_i, \rho_{i+1}, \ldots, \rho_{j-1}$ in view of relation (\[eq18\]) and with $\sigma_i$ in view of relation (\[eq20\]). Hence, $\rho_k$ is permutable with $\lambda_{ij}$.
Let $i < k < j-1$. Then $$\rho_k \, \lambda_{ij} \, \rho_{k} = \rho_{k} \, (\rho_{j-1} \ldots \rho_{k+2} \, \rho_{k+1}
\, \rho_{k} \ldots
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots \rho_{k} \, \rho_{k+1} \, \rho_{k+2}
\ldots \rho_{j-1}) \, \rho_k.$$ Permuting $\rho_k$ to $\lambda_{i,i+1}$ while it is possible, we get $$\rho_{j-1} \, \ldots \rho_{k+2} \, (\rho_{k} \, \rho_{k+1} \, \rho_{k}) \ldots
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots (\rho_{k} \, \rho_{k+1} \, \rho_{k})
\, \rho_{k+2}
\ldots \rho_{j-1}.$$ Using the relation $\rho_{k} \, \rho_{k+1} \, \rho_{k} =
\rho_{k+1} \, \rho_{k} \, \rho_{k+1}$, rewrite the last formula as follows: $$\rho_{j-1} \, \ldots \rho_{k+2} \, \rho_{k+1} \, \rho_{k} \, (\rho_{k+1} \, \rho_{k-1} \ldots
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots \rho_{k-1} \, \rho_{k+1}) \times$$ $$\times \rho_{k}
\, \rho_{k+1} \, \rho_{k+2}
\ldots \rho_{j-1}
= \rho_{j-1} \ldots \rho_{k} \, (\rho_{k+1} \, \lambda_{i,k} \, \rho_{k+1}) \, \rho_{k}
\ldots \rho_{j-1}.$$ In view of the case considered earlier, we have $$\rho_{k+1} \, \lambda_{ik} \, \rho_{k+1} = \lambda_{ik}$$ and, hence, $$\rho_{j-1} \ldots \rho_{k} \, (\rho_{k+1} \, \lambda_{ik} \, \rho_{k+1}) \, \rho_{k}
\ldots \rho_{j-1} =
\lambda_{ij}.$$ Thus, the first rule from 1) is proven.
2\) Consider $$\rho_{i-1} \, \lambda_{ij} \, \rho_{i-1} = \rho_{i-1} \, (\rho_{j-1} \, \rho_{j-2} \ldots
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots \rho_{j-2}
\, \rho_{j-1}) \, \rho_{i-1}.$$ Using relation (\[eq18\]), let as permute $\rho_{i-1}$ to $\lambda_{i,i+1}$ as long as it is possible. We get $$\rho_{i-1} \, \lambda_{ij} \, \rho_{i-1} = \rho_{j-1} \ldots \rho_{i+2} \, \rho_{i+1}
(\rho_{i-1} \, \lambda_{i,i+1} \, \rho_{i-1}) \, \rho_{i+1}
\rho_{i+2} \ldots \rho_{j-2}.
\label{eq27}$$ The expression in the brackets can be rewritten in the following form $$\rho_{i-1} \, \lambda_{i,i+1} \, \rho_{i-1} = \rho_{i-1} \, \rho_{i} \, \sigma_i^{-1}
\, \rho_{i-1}
= \rho_{i-1} \, \rho_{i} \, \sigma_i^{-1} \, \rho_{i-1}
\rho_{i} \, \rho_{i}.$$ Using the relation $\sigma_i^{-1} \, \rho_{i-1} \, \rho_{i} = \rho_{i-1} \, \rho_i \, \sigma_{i-1}^{-1}$ (it follows from (\[eq21\])) and (\[eq18\]), (\[eq19\]), we obtain $$\rho_{i-1} \, \rho_{i} \, (\sigma_{i}^{-1} \, \rho_{i-1} \, \rho_{i}) \, \rho_{i} =
\rho_{i-1} \, ( \rho_i \, \rho_{i-1} \, \rho_i) \, \sigma_{i-1}^{-1} \, \rho_{i} =$$ $$= (\rho_{i-1} \, \rho_{i-1})
\, \rho_i \, \rho_{i-1} \,
\sigma_{i-1}^{-1} \, \rho_{i} = \rho_{i} \, \lambda_{i-1,i} \, \rho_{i}.$$ Then from (\[eq27\]) we obtain $$\rho_{i-1} \, \lambda_{ij} \, \rho_{i-1} = \lambda_{i-1,j}.$$ Therefore, the desired relations are proven.
3\) The first formula follows from the definitions of $ \lambda_{i,i+1}$ and $ \lambda_{i+1,i}$. Let us consider $$\rho_{i} \, \lambda_{ij} \, \rho_{i} = \rho_{i} \, (\rho_{j-1} \, \rho_{j-2} \ldots
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \ldots \rho_{j-2} \,
\rho_{j-1}) \, \rho_{i}.$$ Permuting $\rho_i$ to $\lambda_{i,i+1}$ while it is possible, we obtain $$\rho_{i} \, \lambda_{ij} \, \rho_{i} = \rho_{j-1} \ldots \rho_{i+2} \, (\rho_{i} \,
\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \, \rho_{i}) \,
\rho_{i+2} \, \ldots \rho_{j-1}.$$ Rewrite the expression in the brackets as follows $$\rho_{i} \, \rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \, \rho_i = \rho_{i}
\, \rho_{i+1} \, \rho_{i} \,
(\sigma_i^{-1} \, \rho_{i+1} \, \rho_{i}) = \rho_{i} \, \rho_{i+1} \, (\rho_{i} \,
\rho_{i+1} \, \rho_{i}) \, \sigma_{i+1}^{-1} =$$ $$= \rho_{i} \, \rho_{i+1} \, \rho_{i+1} \,
\rho_{i} \, \rho_{i+1} \, \sigma_{i+1}^{-1} = \rho_{i+1} \, \sigma_{i+1}^{-1}.$$ Hence, $$\rho_{i} \, \lambda_{ij} \, \rho_{i} = \rho_{j-1} \ldots \rho_{i+2} \,
(\rho_{i+1} \,
\sigma_{i+1}^{-1}) \, \rho_{i+2} \ldots \rho_{j-1} =
\lambda_{i+1,j}.$$ Therefore, the desired relations are proven.
4\) follows from the relation $\rho_{j-1}^2 = e$ and the definition of $\lambda_{ij}$.
5\) is an immediate consequence of the definition of $\lambda_{ij}$.
\[cor1\] The group $S_n$ acts by conjugation on the set $\{ \lambda_{kl} ~ \vert 1 ~ \leq k \neq l \leq n \}.$ This action is transitive.
In view of Lemma \[lemma1\], the subgroup $\langle \lambda_{kl} ~ \vert ~ 1 \leq k \neq l \leq n \rangle$ of $VP_n$ is normal in $VB_n$. Let us prove that this group coincides with $VP_n$ and let us find its generators and defining relations. For this purpose we use the Reidemeister–Schreier method (see, for example, [@KMS Ch. 2.2]).
Let $m_{kl} = \rho_{k-1} \, \rho_{k-2} \ldots \rho_l$ for $l < k$ and $m_{kl} = 1$ in other cases. Then the set $$\Lambda_n = \left\{ \prod\limits_{k=2}^n m_{k,j_k} \vert 1 \leq j_k
\leq k \right\}$$ is a Schreier set of coset representatives of $VP_n$ in $VB_n$.
\[theorem1\] The group $VP_n$ admits a presentation with the generators $\lambda_{kl},$ $1 \leq k \neq l \leq
n$, and the defining relations: $$\lambda_{ij} \, \lambda_{kl} = \lambda_{kl} \, \lambda_{ij},
\label{eq29}$$ $$\lambda_{ki} \, (\lambda_{kj} \, \lambda_{ij}) = (\lambda_{ij} \, \lambda_{kj}) \, \lambda_{ki},
\label{eq28}$$ where distinct letters stand for distinct indices.
Define the map $^- : VB_n \longrightarrow \Lambda_n$ which takes an element $w \in VB_n$ into the representative $\overline{w}$ from $\Lambda_n$. In this case the element $w \overline{w}^{-1}$ belong to $VP_n$. By Theorem 2.7 from [@KMS] the group $VP_n$ is generated by $$s_{\lambda, a} = \lambda a \cdot (\overline{\lambda a})^{-1},$$ where $\lambda$ run the set $\Lambda_n$ and $a$ run the set of generators of $VB_n$.
It is easy to establish that $s_{\lambda, \rho_i} = e$ for all representatives $\lambda $ and generators $\rho_i$. Consider the generators $$s_{\lambda, \sigma_i} = \lambda \sigma_i \cdot (\overline{\lambda
\rho_i})^{-1}.$$ For $\lambda =e$ we get $s_{e,\sigma_i} = \sigma_i \rho_i = \lambda_{i,i+1}^{-1}$. Note that $\lambda \rho_i$ is equal to $\overline{\lambda \rho_i}$ in $S_n$. Therefore, $$s_{\lambda, \sigma_i} = \lambda (\sigma_i \rho_i) \lambda^{-1}.$$ From Lemma \[lemma1\] it follows that each generator $s_{\lambda, \sigma_i}$ is equal to some $\lambda_{kl}$, $1 \leq k \neq l \leq n$. By Corollary \[cor1\], the inverse statement is also true, i. e., each element $\lambda_{kl}$ is equal to some generator $s_{\lambda, \sigma_i}$. The first part of the theorem is proven.
To find defining relations of $VP_n$ we define a rewriting process $\tau $. It allows us to rewrite a word which is written in the generators of $VB_n$ and presents an element in $VP_n$ as a word in the generators of $VP_n$. Let us associate to the reduced word $$u = a_1^{\varepsilon_1} \, a_2^{\varepsilon_2} \ldots
a_{\nu}^{\varepsilon_{\nu}},~~~\varepsilon_l = \pm 1,~~~a_l \in
\{\sigma_1, \sigma_2, \ldots, \sigma_{n-1}, \rho_1, \rho_2, \ldots, \rho_{n-1}
\},$$ the word $$\tau(u) = s_{k_1,a_1}^{\varepsilon_1} \, s_{k_2,a_2}^{\varepsilon_2}
\ldots s_{k_{\nu},a_{\nu}}^{\varepsilon_{\nu}}$$ in the generators of $VP_n$, where $k_j$ is a representative of the ($j-1$)th initial segment of the word $u$ if $\varepsilon_j = 1$ and $k_j$ is a representative of the $j$th initial segment of the word $u$ if $\varepsilon_j = -1$.
By [@KMS Theorem 2.9], the group $VP_n$ is defined by relations $$r_{\mu,\lambda} = \tau (\lambda \, r_{\mu} \, \lambda^{-1}),~~~\lambda \in
\Lambda_n,$$ where $r_{\mu}$ is the defining relation of $VB_n$.
Denote by $$r_1 = \sigma_i \, \sigma_{i+1} \, \sigma_i \, \sigma_{i+1}^{-1} \, \sigma_i^{-1}
\, \sigma_{i+1}^{-1}$$ the first relation of $VB_n$. Then $$r_{1,e} = \tau(r_1) = s_{e,\sigma_i} \, s_{\overline{\sigma_i},\sigma_{i+1}} \,
s_{\overline{\sigma_i \sigma_{i+1}} ,\sigma_{i}} \,
s_{\overline{\sigma_i \sigma_{i+1} \sigma_i \sigma_{i+1}^{-1}} ,\sigma_{i+1}}^{-1} \,
s_{\overline{\sigma_i \sigma_{i+1} \sigma_i \sigma_{i+1}^{-1} \sigma_i^{-1}} ,\sigma_{i}}^{-1} \,
s_{\overline{r_1} ,\sigma_{i+1}}^{-1} =$$ $$= \lambda_{i,i+1}^{-1} \, (\rho_i \, \lambda_{i+1,i+2}^{-1} \, \rho_i) \,
(\rho_i \, \rho_{i+1} \, \lambda_{i,i+1}^{-1} \, \rho_{i+1} \, \rho_i) \times$$ $$\times (\rho_{i+1} \, \rho_{i} \, \lambda_{i+1,i+2} \, \rho_{i} \, \rho_{i+1}) \,
(\rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1}) \,
\lambda_{i+1,i+2}.$$ Using the conjugating rules from Lemma \[lemma1\], we get $$r_{1,e} = \lambda_{i,i+1}^{-1} \, \lambda_{i,i+2}^{-1} \, \lambda_{i+1,i+2}^{-1} \,
\lambda_{i,i+1} \, \lambda_{i,i+2} \, \lambda_{i+1,i+2}.$$ Therefore, the following relation $$\lambda_{i,i+1} \, (\lambda_{i,i+2} \, \lambda_{i+1,i+2}) = (\lambda_{i+1,i+2} \,
\lambda_{i,i+2}) \,
\lambda_{i,i+1}$$ is fulfilled in $VP_n$. The Remaining relations $r_{1,\lambda}$, $\lambda \in \Lambda_n$, can be obtained from this relation using conjugation by $\lambda^{-1}$. Bu the formulas from Lemma \[lemma1\], we obtain relations (\[eq28\]).
Let us consider the next relation of $VB_n$: $$r_2 = \sigma_i \, \sigma_j \, \sigma_i^{-1} \, \sigma_j^{-1},~~~|i - j| > 1.$$ For it we have $$r_{2,e} = \tau(r_2) = s_{e,\sigma_i} \, s_{\overline{\sigma_i},\sigma_{j}} \,
s_{\overline{\sigma_i \sigma_{j} \sigma_i^{-1}} ,\sigma_{i}}^{-1} \,
s_{\overline{r_2} ,\sigma_{j}}^{-1} =$$ $$= \lambda_{i,i+1}^{-1} \, \lambda_{j,j+1}^{-1} \, \lambda_{i,i+1} \,
\lambda_{j,j+1}.$$ Hence, the relation $$\lambda_{i,i+1} \, \lambda_{j,j+1} = \lambda_{j,j+1} \, \lambda_{i,i+1},
~~~|i - j| > 1$$ holds in $VP_n$. Conjugating this relation by all representatives from $\Lambda_n$, we obtain relations (\[eq29\]).
Let us prove that only trivial relations follow from all other relations of $VB_n$. It is evident for relations (\[eq17\])–(\[eq19\]) defining the group $S_n$ because $s_{\lambda,\rho_i} = e$ for all $\lambda \in \Lambda_n$ and $\rho_i$.
Consider the mixed relation (\[eq21\]) (relation (\[eq20\]) can be considered similarly): $$r_3 = \sigma_{i+1} \, \rho_i \, \rho_{i+1} \, \sigma_i^{-1} \, \rho_{i+1} \,
\rho_i.$$ Using the rewriting process, we get $$r_{3,e} = \tau(r_3) = s_{e,\sigma_{i+1}} \,
s_{\overline{\sigma_{i+1} \rho_i \rho_{i+1} \sigma_i^{-1}}
,\sigma_{i}}^{-1} =$$ $$= \lambda_{i+1,i+2}^{-1} \, (\rho_i \, \rho_{i+1} \, \lambda_{i,i+1} \, \rho_{i+1} \, \rho_i) =
\lambda_{i+1,i+2}^{-1} \, \lambda_{i+1,i+2} = e.$$ Therefore, $VP_n$ is defined by relations (\[eq29\]) –(\[eq28\]).
The structure of the virtual braid group
========================================
From the definition of $VP_n$ and Lemma \[lemma1\] it follows that $VB_n = VP_n \leftthreetimes S_n$, i. e., $VB_n$ is the splittable extension of the group $VP_n$ by $S_n$. Consequently, we have to study the structure of the virtual pure braid group $VP_n$. Let us define the subgroups $$V_i = \langle \lambda_{1,i+1}, \lambda_{2,i+1}, \ldots, \lambda_{i,i+1};
\lambda_{i+1,1}, \lambda_{i+1,2}, \ldots, \lambda_{i+1,i} \rangle ,~~~i=1,
2, \ldots, n-1,$$ of $VP_n$. Each $V_i$ is a subgroup of $VP_{i+1}$. Let $V_i^*$ be the normal closure of $V_i$ in $VP_{i+1}$. The following theorem is the main result of this section.
\[theorem2\] The group $VP_n$, $n \geq 2$, is representable as the semi–direct product $$VP_n = V_{n-1}^* \leftthreetimes VP_{n-1} = V_{n-1}^* \leftthreetimes
(V^*_{n-2} \leftthreetimes (\ldots \leftthreetimes
(V_2^* \leftthreetimes V_1^*))\ldots),$$ where $V_1^*$ is a free group of rank $2$ and $V_i^*$, $i=2,
3, \ldots, n-1,$ are free infinitely generated subgroups.
Let us prove the theorem by induction on $n$. For $n = 2$, we have $$VP_2 = V_1 = V_1^*$$ and, by Theorem \[theorem1\], the group $V_1$ is free generated by $\lambda_{12}$ and $\lambda_{21}$.
To make the general case more clear consider the case $n=3$.
The structure of $VP_3$.
------------------------
By Theorem \[theorem1\], the group $VP_3$ is generated by subgroups $V_1$, $V_2$ and defined by the relations $$\lambda_{12} \, (\lambda_{13} \, \lambda_{23}) = (\lambda_{23} \,
\lambda_{13}) \, \lambda_{12},~~~~~
\lambda_{21} \, (\lambda_{23} \, \lambda_{13}) = (\lambda_{13} \,
\lambda_{23}) \, \lambda_{21},$$ $$\lambda_{13} \, (\lambda_{12} \, \lambda_{32}) = (\lambda_{32} \,
\lambda_{12}) \, \lambda_{13},~~~~~
\lambda_{31} \, (\lambda_{32} \, \lambda_{12}) = (\lambda_{12} \,
\lambda_{32}) \, \lambda_{31},$$ $$\lambda_{23} \, (\lambda_{21} \, \lambda_{31}) = (\lambda_{31} \,
\lambda_{21}) \, \lambda_{23},~~~~~
\lambda_{32} \, (\lambda_{31} \, \lambda_{21}) = (\lambda_{21} \,
\lambda_{31}) \, \lambda_{32}.$$
From these relations we obtain the next lemma.
\[lemma2\] In $VP_3$ the following equalities hold:\
1) $$\begin{array}{lll}
\lambda_{13}^{\lambda_{12}} = \lambda_{32}^{\lambda_{12}} \, \lambda_{13} \,
\lambda_{32}^{-1},&
\lambda_{31}^{\lambda_{12}} = \lambda_{32} \, \lambda_{31} \,
\lambda_{32}^{-\lambda_{12}},&
\lambda_{23}^{\lambda_{12}} = \lambda_{13} \, \lambda_{23} \,
\lambda_{32} \, \lambda_{13}^{-1} \, \lambda_{32}^{-\lambda_{12}},\\
& & \\
\lambda_{13}^{\lambda_{12}^{-1}} = \lambda_{32}^{-1} \, \lambda_{13} \,
\lambda_{32}^{\lambda_{12}^{-1}},&
\lambda_{31}^{\lambda_{12}^{-1}} =
\lambda_{32}^{-\lambda_{12}^{-1}} \,
\lambda_{31} \lambda_{32},&
\lambda_{23}^{\lambda_{12}^{-1}} =
\lambda_{32}^{-\lambda_{12}^{-1}} \,
\lambda_{13}^{-1} \, \lambda_{32} \, \lambda_{23} \, \lambda_{13},\
\end{array}$$ 2) $$\begin{array}{lll}
\lambda_{23}^{\lambda_{21}} = \lambda_{31}^{\lambda_{21}} \, \lambda_{23} \,
\lambda_{31}^{-1},&
\lambda_{32}^{\lambda_{21}} = \lambda_{31} \, \lambda_{32} \,
\lambda_{31}^{-\lambda_{21}},&
\lambda_{13}^{\lambda_{21}} = \lambda_{23} \, \lambda_{13} \,
\lambda_{31} \, \lambda_{23}^{-1} \, \lambda_{31}^{-\lambda_{21}},\\
& & \\
\lambda_{23}^{\lambda_{21}^{-1}} = \lambda_{31}^{-1} \, \lambda_{23} \,
\lambda_{31}^{\lambda_{21}^{-1}},&
\lambda_{32}^{\lambda_{21}^{-1}} =
\lambda_{31}^{-\lambda_{21}^{-1}} \,
\lambda_{32} \, \lambda_{31},&
\lambda_{13}^{\lambda_{21}^{-1}} =
\lambda_{31}^{-\lambda_{21}^{-1}} \,
\lambda_{23}^{-1} \, \lambda_{31} \, \lambda_{13} \, \lambda_{23},\\
\end{array}$$ where $a^b$ stand for $b^{-1} a b$.
The first and second relations from 1) immediately follow from the third and forth relations of $VP_3$ (see the relations before the lemma). Similarly, the first and second relations from 2) immediately follow from the fifth and sixth relations of $VP_3$.
Further, from the first and second relations of $VP_3$ we obtain $$\lambda_{23}^{\lambda_{1,2}} = \lambda_{13} \, \lambda_{23} \,
\lambda_{13}^{-\lambda_{12}},~~~
\lambda_{13}^{\lambda_{21}} = \lambda_{23} \, \lambda_{13} \,
\lambda_{23}^{-\lambda_{21}}.$$ Using the proved formulas for $\lambda_{13}^{\lambda_{12}}$ and $\lambda_{23}^{\lambda_{21}}$, we get the third formulas from 1) and 2) respectively.
The formulas for conjugation by $\lambda_{12}^{-1}$ and $\lambda_{21}^{-1}$ can be obtained analogously.
Note that there exists an epimorphism $$\varphi_3 : VP_3 \longrightarrow VP_2,$$ which takes the generators of $V_2 = \langle \lambda_{13}, \lambda_{23}, \lambda_{31},
\lambda_{32} \rangle $ into the unit and fixes the generators of $V_1 = \langle \lambda_{12}, \lambda_{21} \rangle $. The kernel of this epimorphism is the normal closure of $V_2$ in $VP_3$, i. e., $\mbox{ker}(\varphi_3) = V_2^*$.
Let $u$ be the empty word or a reduced word beginning with non-zero power of $\lambda_{12}$ and representing an element from $V_1$. Let $\lambda_{32}(u) = \lambda_{32}^u$ $= u^{-1} \, \lambda_{32} \, u$. We call this element [*the reduced power of the generator*]{} $\lambda_{32}$ with the power $u$. Analogously, if $v$ is the empty word or a reduced word beginning with non-zero power of $\lambda_{21}$ and representing an element from $V_1$, then we put $\lambda_{31}(v) = \lambda_{13}^v$ and call this element [*the reduced power of generator*]{} $\lambda_{31}$ with the power $v$.
\[lemma3\] The group $V_2^*$ is a free group with generators $\lambda_{13}$, $\lambda_{23}$ and all reduced powers of $\lambda_{31}$ and $\lambda_{32}$.
To prove the lemma we can use the Reidemeister–Shreier method, but it is simpler to use the definitions of normal closure and semi-direct product. Evidently, the group $V_2^*$ is generated by the elements $$\lambda_{13}^w,~~~ \lambda_{23}^w,~~~ \lambda_{31}^w,~~~
\lambda_{32}^w,~~~w \in V_1.$$ In view of Lemma \[lemma2\], it is sufficient to take from these elements only $\lambda_{13}$, $\lambda_{23}$ and all reduced powers of the generators $\lambda_{31}$ and $\lambda_{32}$.
The freedom of $V_2^*$ follows from the representation of $VP_3$ as the semi–direct product. Indeed, since $V_1 \bigcap V_2^* = e$, $V_1 V_2^* = VP_3$, then $VP_3 = V_2^* \leftthreetimes V_1$. In this case the defining relations of $VP_3$ are equivalent to the conjugating rules from Lemma \[lemma2\]. Therefore, all relations define the action of the group $V_1$ on the group $V_2^*$. Since there are not other relations, this means that $V_1$ and $V_2^*$ are free groups.
As a consequence of this Lemma, we obtain the normal form of words in $VP_3$. Any element $w$ from $VP_3$ can be written in the form $w = w_1 w_2$, where $w_1$ is a reduced word over the alphabet $\{ \lambda_{12}^{\pm 1}, \lambda_{21}^{\pm 1} \}$ and $w_2$ is a reduced word over the alphabet $\{ \lambda_{13}^{\pm 1}, \lambda_{23}^{\pm 1}, \lambda_{31}(u)^{\pm 1},
\lambda_{32}(v)^{\pm 1} \}$, where $\lambda_{31}(u)$, $\lambda_{32}(v)$ are reduced powers of the generators $\lambda_{31}$ and $\lambda_{32}$ respectively.
The proof of Theorem \[theorem2\]
---------------------------------
Now, we introduce the following notation. By $\lambda_{ij}^*$ denote any $\lambda_{ij}$ or $\lambda_{ji}$ from $VP_n$.
\[lemma4\] For every $n \geq 2$ there exists a homomorphism $$\varphi : VP_n \longrightarrow VP_{n-1}$$ which takes the generators $\lambda_{ij}^*$, $i = 1, 2, \ldots , n-1,$ to the unit and fixes other generators.
It is sufficient to prove that all defining relations turn to the defining relations by such defined map. For the defining relations of $VP_{n-1}$ it is evident. If the relation of commutativity (see relation (\[eq29\])) contains some generator of $V_{n-1}$ then by acting with $\varphi_n$ it turns to the trivial relation. Let us consider the left hand side of relation (\[eq28\]). We see that it contains every index two times. Hence, if this part includes some generator of $V_{n-1}$ (i. e., one of the indices is equal to $n$) then some other generator contains the index $n$. Therefore, there are two generators of $V_{n-1}$ in the left hand side of the relation. Since the right hand side contains all generators from the left hand side, then by acting with $\varphi_n$ this relation turns to the trivial relation.
\[lemma5\] The following formulas are fulfilled in the group $VP_n$:\
1) $ \lambda_{kl}^{\lambda_{ij}^{\varepsilon}} = \lambda_{kl},
~~~\mbox{max}\{i, j\} < \mbox{max}\{k, l\},~~~\varepsilon = \pm 1;
$\
2) $\lambda_{ik}^{\lambda_{ij}} = \lambda_{kj}^{\lambda_{ij}} \lambda_{ik}
\lambda_{kj}^{-1},~~~
\lambda_{ik}^{\lambda_{ij}^{-1}} = \lambda_{kj}^{-1} \lambda_{ik}
\lambda_{kj}^{\lambda_{ij}^{-1}},~~~i < j < k~~ \mbox{or}~~ j < i < k;
$\
3) $ \lambda_{ki}^{\lambda_{ij}} = \lambda_{kj} \lambda_{ki}
\lambda_{kj}^{-\lambda_{ij}},~~~
\lambda_{ki}^{\lambda_{ij}^{-1}} = \lambda_{kj}^{-\lambda_{ij}^{-1}}
\lambda_{ki} \lambda_{kj},~~~i < j < k~~ \mbox{or}~~ j < i < k;
$\
4) $ \lambda_{jk}^{\lambda_{ij}} = \lambda_{ik} \lambda_{jk}
\lambda_{kj} \lambda_{ik}^{-1} \lambda_{kj}^{-\lambda_{ij}},~~~
\lambda_{jk}^{\lambda_{ij}^{-1}} = \lambda_{jk}^{-\lambda_{ik}^{-1}}
\lambda_{ij}^{-1} \lambda_{jk} \lambda_{kj} \lambda_{ij},~~~i < j < k ~~ \mbox{or}~~ j < i <
k,
$\
where, as usual, different letters stand for different indices.
The formula 1) immediately follows from the first relation of Theorem \[theorem1\].
Consider relation (\[eq28\]) from Theorem \[theorem1\]: $$\lambda_{ki} \, (\lambda_{kj} \, \lambda_{ij}) = (\lambda_{ij} \, \lambda_{kj}) \, \lambda_{ki}.$$ Note that the indices of generators are connected by one of the inequalities: $$a)~~ k < j < i,~~~b)~~ j < k < i,~~~c)~~ i < j < k,$$ $$d)~~ j < i < k,~~~e)~~ k < i < j,~~~f)~~i < k < j.$$ If the indices are connected by inequality $a)$ or $b)$ then from (\[eq28\]) we obtain $$\lambda_{ki}^{\lambda_{kj}} = \lambda_{ij}^{\lambda_{kj}} \,
\lambda_{ki} \,
\lambda_{ij}^{-1},$$ and it is the first formula from 2).
If the indices in relation (\[eq28\]) are connected by inequality c) or d) we obtain $$\lambda_{ki}^{\lambda_{ij}} = \lambda_{kj} \, \lambda_{ki} \,
\lambda_{kj}^{-\lambda_{ij}},$$ and it is the first formula from 3).
If indices in relation (\[eq28\]) are connected by inequality e) or f) then $$\lambda_{ij}^{\lambda_{ki}} = \lambda_{kj} \, \lambda_{ij} \,
\lambda_{kj}^{-\lambda_{ki}}.$$ Using the formula from 2), we obtain $$\lambda_{ij}^{\lambda_{ki}} = \lambda_{kj} \, \lambda_{ij} \, \lambda_{ji} \,
\lambda_{kj}^{-1} \,
\lambda_{ji}^{-\lambda_{ki}},$$ and it is the first formula from 4).
The formulas of conjugations by elements $\lambda_{ij}^{-1}$ can be established similarly.
Assume that the theorem is proven for the group $VP_{n-1}$. Hence, any element $w \in VP_{n-1}$ can be written in the form $$w \, = \, w_1 \, w_2 \ldots w_{n-2},~~~w_i \in V_i^*,$$ where each word $w_i$ is a reduced word over the alphabet consisting if generators $\lambda_{ki}^{\pm 1}$, $1 \leq k \leq i-1,$ and reduced powers of generators $\lambda_{ki}$, $1 \leq k \leq i-1,$ and their inverse. Let us define reduced powers of generators in the group $V_{n-1}^*$. We say that the element $\lambda_{nk}(w) = \lambda_{nk}^w$ is [*the reduced power of the generator*]{} $\lambda_{nk}$ if $w$ is the empty word or a word written in the normal form and begin with a reduced power of some generator $\lambda_{lk}$ or its inverse.
The statement about decomposition in to the semi–direct product $VP_n = V_n^* \leftthreetimes VP_{n-1}$ is quite evident. It remains to find generators of $V_n^*$ and prove its freedom.
\[lemma6\] The group $V_{n-1}^*$ is a free group. It is generated by $\lambda_{1n}, \lambda_{2n},$ $\ldots, $ $\lambda_{n-1,n}$ and all reduced powers of the generators $\lambda_{n1}, \lambda_{n2}, \ldots, \lambda_{n,n-1}$.
The proof is similar to that of Lemma \[lemma3\]. From Lemma \[lemma5\] it follows that this set is the set of generators of $V_{n-1}^*$. Further, since the set of defining relations of $VP_n$ is equivalent to the set of conjugating formulas defining the action of $VP_{n-1}$ on $V_{n-1}^*$, only trivial relations are fulfilled in $V_{n-1}^*$.
Theorem \[theorem2\] follows from these results.
As a consequence of this theorem we obtain the normal form of words in $VB_n$.
\[cor2\] Every element from $VB_n$ can be written uniquely in the form $$w = w_1 \, w_2 \ldots w_{n-1} \, \lambda,~~~\lambda \in
\Lambda_n,~~~w_i \in V_i^*,$$ where $w_i$ is a reduced word in generators, reduced powers of generators and their inverse.
The defined above homomorphism of the virtual braid group onto the welded braid group agrees with the decomposition from Theorem \[theorem2\] and with the decomposition of $C_n \simeq WB_n$ described in the first section.
\[cor3\] The homomorphism $\varphi_{VW} : VB_n \longrightarrow WB_n$ agrees with the decomposition of these groups, i. e., it maps the group $VP_n$ onto $Cb_n \simeq WP_n$ and the factors $V_i^*$ onto the factors $D_i$, $i = 1, 2, \ldots, n-1$.
The universal braid group
=========================
Let us define [*the universal braid group*]{} $UB_n$ as the group with generators $\sigma_1, \sigma_2, \ldots, \sigma_{n-1},$ $c_1, c_2, \ldots, c_{n-1},$ defining relations (\[eq1\])–(\[eq2\]), the relations: $$c_i \, c_j = c_j \, c_i,~~~ |i-j| \geq 2,$$ and the mixed relations: $$c_i \, \sigma_j = \sigma_j \, c_i~~~ |i-j| \geq 2.$$
Recall (see [@BS]) that Artin’s group of the type $I$ is called the group $A_I$ with generators $a_i$, $i \in I$, and the defining relations $$a_i \, a_j \, a_i \ldots = a_j \, a_i \, a_j \ldots ,~~~ i, j \in I,$$ where words from the left and right hand sides consist of $m_{ij}$ alternating letters $a_i$ and $a_j$.
\[proposition1\] 1) The group $UB_n$ has as a subgroup the braid group $B_n$.
2\) There exist homomorphisms $$\varphi_{US} : UB_n \longrightarrow SG_n,~~~\varphi_{UV} : UB_n \longrightarrow
VB_n,~~~\varphi_{UB} : UB_n \longrightarrow B_n.$$
3\) The group $UB_n$ is Artin’s group.
1\) Evidently, there exists a homomorphism $B_n \longrightarrow UB_n$. On the other hand, assuming $\psi(\sigma_i) = \sigma_i$, $\psi(c_i) = e$, $i = 1, 2, \ldots, n-1,$ we obtain the retraction $\psi$ of $UB_n$ onto $B_n$. Therefore, the subgroup $\langle \sigma_1, \sigma_2, \ldots, \sigma_{n-1} \rangle $ of $UB_n$ is isomorphic to the braid group $B_n$.
2\) Let us define the map $\varphi_{US}$ as follows $$\varphi_{US}(\sigma_i) = \sigma_i,~~~ \varphi_{US}(c_i) = \tau_i,~~~i = 1, 2, \ldots,
n-1.$$ Comparing the defining relations of $UB_n$ and $SG_n$, we see that this map is a homomorphism. Analogously, we can show that the map $$\sigma_i \longmapsto \sigma_i,~~~ c_i \longmapsto \rho_i,$$ is extendable to the homomorphism $\varphi_{UV}$ and the map $$\sigma_i \longmapsto \sigma_i,~~~ c_i \longmapsto e,$$ is extendable to the homomorphism $\varphi_{UB}$.
3\) immediately follows from the defining relations of $UB_n$ and the definition of Artin’s group.
It should be noted that none of the groups $SG_n$, $VB_n$, $WB_n$ (in the natural presentations) is not Artin’s group.
The following questions naturally arise in the context of the results obtained above.
[**Problems.**]{} 1) Solve the word and conjugacy problems in $UB_n$, $n > 2$.\
2) Is it possible to give some geometric interpretation for elements of $UB_n$ similar to the geometric interpretation for elements of the braid groups $B_n$, $SG_n$, $VB_n$, $UB_n$?
[99]{}
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[^1]: Authors were supported in part by the Russian Foundation for Basic Research (grant 02–01–01118).
|
---
abstract: 'We study the form factors appearing in the inclusive decay $b\rightarrow s g^*$, in the framework of the noncommutative standard model. Here $g^*$ denotes the virtual gluon. We get additional structures and the corresponding form factors in the noncommutative geometry. We analyse the dependencies of the form factors to the parameter $p\,\Theta\, k$ where $p$ ($k$) are the four momenta of incoming (outgoing) $b$ quark (virtual gluon $g^*$), $\Theta$ is a parameter which measures the noncommutativity of the geometry. We see that the form factors are weakly sensitive to this parameter.'
author:
- |
\
[**E. O. Iltan**]{} [^1]\
Physics Department, Middle East Technical University\
Ankara, Turkey\
title: ' [**The form factors existing in the $b\rightarrow s g^*$ decay and the possible CP violating effects in the noncommutative standard model**]{}'
---
Introduction
============
The quantum field theory over noncommutative spaces [@Connes] has been reached a great interest in recent years . The string theory arguments are the reason for the re-motivation of the physics on the noncommutative spaces [@Connes2; @Witten]. Noncommutative field theories (NCFT’s) are difficult to handle since they have non-local structure. Besides this, it has been argued that they were sensible field theories and they have been studied extensively in the literature. The renormalizability of NCFT’s in general have been studied in [@Gonzales]. The unitarity in noncommutative theories has been discussed in [@Gomis]. In [@Hewett], the unitarity properties of spontaneously broken noncommutative gauge theories have been examined. The noncommutative quantum electrodynamics (NCQED) and anomalous magnetic moments have been studied in [@Riad] and a detailed calculation for the noncommutative electron-photon vertex has been presented. The noncommutative Yang-Mills theory has been studied in [@Krajewski]. In [@Hinchliffe] the noncommutative CP violating effects has been examined at low energies and it was ephasized that CP violation due to noncommutative geometry was comparable to the one due to the standard model (SM) only, for a noncommutative scale $\Lambda \leq 2\,
TeV$. Noncommutative SM (NCSM) building has been studied in [@Chaichian] and recently, the determination of triple neutral gauge boson couplings has been done in [@Deshpande].
In noncommutative geometry the space-time coordinates $x_{\mu}$ are replaced by the Hermitian operators $\hat{x}_{\mu}$ where they do not commute $$\begin{aligned}
[\hat{x}_{\mu},\hat{x}_{\nu}]=i\,\Theta_{\mu\nu} \,\, .
\label{com1}\end{aligned}$$ Here $\Theta_{\mu\nu}$ is real and antisymmetric tensor. On the other hand noncommutative field theory is equivalent to the ordinary one except that the usual product is replaced by the $*$ product $$\begin{aligned}
(f*g)(x)=e^{i\,\Theta_{\mu\nu} \,\partial^y_{\mu}\,\partial^z_{\nu}} f(y)\,
g(z)|_{y=z=x}\,.
\label{product}\end{aligned}$$ The commutation of the Hermitian operators $\hat{x}_{\mu}$ (see eq. (\[com1\])) holds with this new product, namely, $$\begin{aligned}
[\hat{x}_{\mu},\hat{x}_{\nu}]_*=i\,\Theta_{\mu\nu} \,\, .
\label{com2}\end{aligned}$$ For constructing the effective low energy theory it is convenient to choose the energy scale as $\Lambda=\frac{1}{\sqrt{\Theta}}$ [@Hinchliffe]. Here the parameter $\Theta$ is taken the average magnitude of the tensor $\Theta_{\mu\nu}$. NCSM can be constructed at least up to $O(\Theta)$ by replacing the ordinary products by $*$ product. This replacement modifies the Feynman rules considerably (see the appendix of [@Hinchliffe]).
In our work, we study the form factors appearing in the inclusive decay $b\rightarrow s g^*$, in the framework of the NCSM. When the noncommutative effects are switched on, the form factors due to the SM are modified and new structures with the corresponding form factors arise. The noncommutative effects are at least at the order of $p\,\Theta\, k$ where $p$ ($k$) are the four momenta of incoming (outgoing) $b$ quark (virtual gluon). Here, we take the noncommutative scale $\Lambda=\frac{1}{\sqrt{\Theta}}\leq 1\, Tev$ and at these low energies, the problems of unitarity and causality are supressed. The combination $p\,\Theta\, k$, which appears in the expressions are at the order of the magnitude of $10^{-6}-10^{-4}$ for our process. This is a small number which creates weak noncommutative effects in the calculation of form factors. However, these noncommutative effects may be stronger for the decays including heavy flavors. On the other hand, this parameter is a new source for the CP violation which exists with the help of the complex Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in the ordinary SM.
The paper is organized as follows: In Section 2, we present the structures and the form factors appearing in the $b\rightarrow s g^*$ decay in the SM, including the non-commutative effects. Section 3 is devoted to the analysis of these form factors and our discussions.
The form factors existing in the $b\rightarrow s g^*$ decay, in the SM including the non-commutative effects
============================================================================================================
In this section, we calculate the form factors of the decay $b\rightarrow s g^*$ in the framework of the SM, including the noncommutative effects. As it is well known, $b\rightarrow s g^*$ decay is created by flavor changing neutral currents at loop level in the SM. The possible interactions at one loop level are self energy and vertex type (see Fig.\[fig1\]). At this stage, we use the on-shell renormalization scheme to get rid of the divergences appearing in the ordinary SM and obtain a gauge invariant vertex function. Notice that, in this scheme, the self energy diagrams do not contribute and only the vertex part survives. When the non-commutative effects are switched on, there appears new structures and the corresponding form factors, that contain new UV and IR divergences. Use of on-shell renormalization scheme helps one to get a gauge invariant vertex function however, the form factors appearing due to the noncommutative effects still need renormalization. Now, we will present the calculations [@Riad] to get the form factors for the decay underconsideration.
The starting point of the calculation is to use the exponential representation for the propagators, i.e. the Schwinger parametrization $$\begin{aligned}
\frac{i}{p^2-m^2+i\epsilon}=\int_0^{\infty}\, d x_1
e^{i\,x_1 \,(p^2-m^2+i\epsilon)}
\label{Schwinger}\end{aligned}$$ and to obtain the denominator of the momentum integral for the vertex function, which we call the core integral, $$\begin{aligned}
I=-i\, \int \frac{d^d\,q}{(2\pi)^d}\, e^{i\,(q\,z-\frac{1}{2} (p-q)\,
(\widetilde{p'-q})}\, \frac{1}{(q^2-m_W^2)\,\Big( (p-q)^2-m_i^2 \Big)
\Big( (p'-q)^2-m_i^2 \Big)}\,\, .
\label{int1}\end{aligned}$$ where $\widetilde{v_{\mu}}=\Theta_{\mu\nu}\, v^{\nu}$, $p$ ($p'$) is four momentum of $b$ ($s$) quark, $p=p'-k$ and $k$ is virtual gluon four momentum. Here, the new factor $e^{-i\,\frac{1}{2} (p-q)\,\widetilde{p'-q}}$ is due to the non-commutative geometry and it can be rewritten as $e^{-\frac{i}{2}\,p\,\widetilde{p'}}\,\,e^{-\frac{i}{2}\,q\,\widetilde{k}}$. The factor $e^{i\,q\,z}$ is introduced to obtain the expressions appearing in the numerator of the momentum integral, by differentiation [@Zuber]. Using the parametrization in eq. (\[Schwinger\]) and making the momentum integration, the integral I in eq. (\[int1\]) can be written as $$\begin{aligned}
I=\int_{0}^{\infty}\, dx_1\,dx_2\,dx_3\,
\frac{e^{\frac{-i\,(\widetilde{k}+4\,x_2\,p'+4\,x_3\,p-2\,z)^2+
16\,(x_1+x_2+x_3)\,m_W^2\,(x_1+x_i\,(x_2+x_3)-x_b\,x_3)}
{16\,(x_1+x_2+x_3)}}}{(4\pi)^2\,(x_1+x_2+x_3)^2}
\,\, ,
\label{int2}\end{aligned}$$ where $x_i=\frac{m_i^2}{m_W^2}$, $x_b=\frac{m_b^2}{m_W^2}$. Notice that we take $m_s=0$ in the expressions.
Here we will summarize the procedure used in the following:
- Calculating the the numerator of the integral by differentiating the momentum integrated core integral with respect to the auxilary variable $z$ and set $z$ to zero at the end.
- Applying the Wick rotation $x_i\rightarrow \frac{x_i}{i}$ and using the identity $\int_{0}^{\infty}\,d\rho\, \delta(\rho-(x_1+x_2+x_3) )=1$
- Making the rescaling $x_i\rightarrow \rho \, x_i$
- Redefining the core integral by introducing the UV regulator $e^{\frac{i}{(x_1+x_2+x_3)\,\Lambda^2}}$ as $$\begin{aligned}
I=-\frac{1}{16\,\pi^2}\, e^{-\frac{i}{2}p \widetilde{p'}}\,\int_{0}^{\infty}
d\rho \int_{0}^{1} dx \int_{0}^{1-x} dy\,
exp\, [\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p\, (1-x)]
\,\, ,
\label{int3}\end{aligned}$$ with the functions $e_1$ and $e_2$ $$\begin{aligned}
e_1&=&-\frac{1}{\Lambda_{eff}^2} \,\, , \nonumber \\
e_2&=&-m_W^2\,\Big( x+x_i\,(1-x)-x_b\,(1-x-y)\,(x+s\,y) \Big ) \,\, ,
\label{e12}\end{aligned}$$ and $\Lambda_{eff}$ $$\begin{aligned}
\Lambda_{eff}^2=\frac{1}{\frac{1}{\Lambda^2}-\frac{\widetilde{k}^2}{16}}
\,\, ,
\label{Lambda}\end{aligned}$$
where $s=\frac{k^2}{m_b^2}$. With this procedure one obtains all the structures and the corresponding raw coefficients due to the decay under consideration. Now, we use the on-shell renormalization scheme and extract the nonvanishing structures to get a gauge invariant result. In this scheme, only the vertex diagram (Fig. \[fig1\]) contributes and the self energy diagrams vanish. Using the raw bare vertex function, $\Gamma^{0\, a}_{\mu}$, introducing the counterms $\Gamma^{C\, a}_{\mu}$ to satisfy the expression $$\begin{aligned}
k^{\mu} \Gamma^{(Raw)\,Ren,\, a}_{\mu}=0 \,\, ,
\label{renorm1}\end{aligned}$$ where $\Gamma^{(Raw)\,Ren,\, a}_{\mu}$ is $\Gamma^{(Raw)\,Ren,\, a}_{\mu}=\Gamma^{0\,a}_{\mu}+\Gamma^{C\,a}_{\mu}$ and neglecting the $s$-quark mass we get $$\begin{aligned}
\Gamma^{Ren\, a}_{\mu}&=&
\frac{-i\, g^2\,g_s}{32\,m_W^2\,\pi^2}\,\lambda^a \Big( F_1^{Raw} (k^2)\,
(k_{\mu} \slash \!\!\!{k}-k^2 \gamma_{\mu} ) L + i\,F_2^{Raw} (k^2)\, m_b\,
\sigma_{\mu\nu} \, k^{\nu} R + i\,F_3^{Raw} (k^2) m_b\, \widetilde{k}_{\mu}
R \nonumber \\ &+&
i\, F_4^{Raw} (k^2) (\gamma_{\mu} \slash \!\!\!{k} \slash \!\!\!
{\widetilde{k}}-k_{\mu} \slash \!\!\!{\widetilde{k}}) L +
F_5^{Raw} (k^2) \widetilde{k}_{\mu} \slash \!\!\!{\widetilde{k}} L
\label{vertexop}\end{aligned}$$ where $k_{\mu}$ is the gluon momentum 4-vector and $k^2$ dependent functions $F_{1}^{Raw}(k^2)$ and $F_{2}^{Raw}(k^2)$ are proportional to the charge radius and dipole form factors. $F_{3}^{Raw}(k^2)$, $F_{4}^{Raw}(k^2)$ and $F_{5}^{Raw}(k^2)$ are the new form factors appearing when the noncommutative effects are switched on and they read as $$\begin{aligned}
F_1^{raw} (k^2)&=&\frac{m_W^2}{2}\,\sum_{i=,u,c,t} V_{ib} V^*_{is}
\int_{0}^{\infty} d\rho \int_{0}^{1} dx \int_{0}^{1-x} dy
\nonumber \\ & & \!\!\!\!\!\!\!
e^{-\frac{i}{2} p \widetilde{p'}}\,
e^{\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p (x-1)}
\, \Big( x_i\, (x^2-(2-3\,y)\,x+1+2\,y^2-4\,y)+2\,
(x^2+2\,y\,(y-1) \nonumber \\&+&
x\, (3\,y-2) \Big )
\nonumber \,\, ,\\
F_2^{raw} (k^2)&=&\frac{m_W^2}{2}\,\sum_{i=,u,c,t} V_{ib} V^*_{is}
\int_{0}^{\infty} d\rho \int_{0}^{1}\!\! dx \int_{0}^{1-x}\!\!\!\! dy
\nonumber \\ & & \!\!\!\!\!\!\!
e^{-\frac{i}{2} p \widetilde{p'}}\,
e^{\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p (x-1)}
\, \Big( x_i+ x^2\, (2+x_i)+ x\, (x_i\, (y-2)+ 2\,y ) \Big)
\nonumber \,\, ,\\
F_3^{raw} (k^2)&=&\frac{m_W^2}{4}\, \sum_{i=,u,c,t} V_{ib}
V^*_{is} \int_{0}^{\infty} d\rho
\int_{0}^{1}\!\! dx \int_{0}^{1-x}\!\!\!\! dy\,
e^{-\frac{i}{2} p \widetilde{p'}}\,
\frac{e^{\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p (x-1)}}{\rho}
\,(2+x_i)\,(1-x-y)
\, \, , \nonumber \\
F_4^{raw} (k^2)&=&\frac{m_W^2}{8}\,\sum_{i=,u,c,t}
V_{ib} V^*_{is} \int_{0}^{\infty} d\rho \int_{0}^{1}\!\! dx
\int_{0}^{1-x}\!\!\!\! dy\, \frac{e^{-\frac{i}{2} p \widetilde{p'}}\,
e^{\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p (x-1)}}{\rho}
\,(2-\,x_i+x\,(2+x_i))
\, \, , \nonumber \\
F_5^{raw} (k^2)&=&-\frac{m_W^2}{16}\,\sum_{i=,u,c,t}
V_{ib} V^*_{is} \int_{0}^{\infty} d\rho \int_{0}^{1}\!\! dx
\int_{0}^{1-x}\!\!\!\! dy\, e^{-\frac{i}{2} p \widetilde{p'}}\,
\frac{e^{\frac{e_1}{\rho}+e_2 \rho-\frac{i}{2} \widetilde{k}.p
(x-1)}}{\rho^2}\,(2+x_i) \,\, ,
\label{Fcofro}\end{aligned}$$ In the calculation of the coefficients, at first, the $\rho$ integrations are taken. These integrations bring the modified Bessel functions of first and second type. For the high energy limit, namely $\Lambda^2 \rightarrow
\infty$, or the low energy limit $k\rightarrow 0$ simultaneously, the integration of $F_1^{raw}$ and $F_2^{raw}$ over $\rho$ does not bring any divergence. However, the $\rho$ integrations of $F_3^{raw}$ and $F_4^{raw}$ result in the modified Bessel functions of first type, where the logarithmic divercences appear. Here we assume that these divercences can be overcome by adding the necessary counter terms, i.e. the NCSM model is renormalizable at least at one loop level, similar to NCQED [@Hayakawa]. The result of the integration of the form factor $F_5^{raw}$ over $\rho$ is proportional to the cut-off factor $\Lambda_{eff}^2$. Fortunately, this term is irrelevant because of the following reason (see [@Riad]). If we consider the UV limit, namely, $\frac{1}{\Lambda^2}<< \widetilde{k}^2$ or $\Lambda_{eff}^2 \sim \frac{1}{\widetilde{k}^2}$, this term is finite. However as $\Lambda^2$ tends to zero, we have $\frac{1}{\Lambda^2}>> \widetilde{k}^2$ and therefore $\Lambda^2 \widetilde{k}^2<<1$. Since the structure due to $F_5^{raw}$ contains the term proportional to $\widetilde{k}^2$, this term is irrelevant in the IR limit. Finally, we end up with the form factors $F_1(s)$, $F_2(s)$, $F_3(s)$ and $F_4(s)$: $$\begin{aligned}
F_1 (s)&=&e^{-\frac{i}{2} p.\widetilde{p'}}\,\sum_{i=,u,c,t} V_{ib} V^*_{is}
\int_{0}^{1} dx \int_{0}^{1-x} dy
\nonumber \\ & & e^{-\frac{i}{2}
p.\widetilde{k} (1-x)}\,\frac{x_i\, (x^2-(2-3\,y)\,x+1+2\,y^2-4\,y)+2\,
(x^2+2\,y\,(y-1) + x\, (3\,y-2)}
{2\,(x_i+ x_b\, y\, (y-1)\,s-x\,(-1+x_i+ x_b\,y\,(1-s) )}
\nonumber \,\, ,\\
F_2 (s)&=& e^{-\frac{i}{2} p.\widetilde{p'}}\,\sum_{i=,u,c,t}
V_{ib} V^*_{is} \int_{0}^{1} dx \int_{0}^{1-x} dy
\nonumber \\ & & e^{-\frac{i}{2} p.\widetilde{k}(1-x)}\,
-\frac{x_i+ x^2\, (2+x_i)+ x\, (x_i\, (y-2)+ 2\,y )}
{2\,(x_i+ x_b\, y\, (y-1)\,(s-1)-x\,(-1+x_i+ x_b\,y\,(1-s) )}
\nonumber \,\, ,\\
F_3 (s)&=&e^{-\frac{i}{2} p.\widetilde{p'}}\,\frac{\mbox{EulerGamma}
\, m_b\, m_W^2}{2} \sum_{i=,u,c,t} V_{ib}
V^*_{is} \int_{0}^{1} dx \int_{0}^{1-x} dy e^{-\frac{i}{2}
p.\widetilde{k}(1-x)}\,(2+x_i)\,(1-x-y) \, \, , \nonumber \\
F_4 (s)&=&\!\!\!\!-\,e^{-\frac{i}{2} p.\widetilde{p'}}\,\frac{EulerGamma\,
m_W^2}{4} \sum_{i=,u,c,t} V_{ib} V^*_{is}
\int_{0}^{1} dx \int_{0}^{1-x} \!\!\! dy e^{-\frac{i}{2}
p.\widetilde{k} (1-x)}\,(2-\,x_i+x\,(2+x_i)) \, \, .
\label{Fcof}\end{aligned}$$ $F_3 (s)$ and $F_4 (s)$, the integration over the parameters $x$ and $y$ can be performed easily and we get $$\begin{aligned}
F_3 (s)&=& \sum_{i=,u,c,t} V_{ib} V^*_{is}\,\,\mbox{EulerGamma}\,\,m_W^2\,
\frac{e^{-i \,p.\widetilde{p'}}\, \Bigg( 4\, p.\widetilde{p'}+
i\,\Big( (p. \widetilde{p'})^2+8\,(e^{\frac{i \,p.\widetilde{p'}}{2}}-1)
\Big)\, (2+x_i) \Bigg) }{2 (p.\widetilde{p'})^3}
\nonumber \,\, ,\\
F_4 (s)&=&\sum_{i=,u,c,t} V_{ib} V^*_{is}\,\,\mbox{EulerGamma}\,\,m_W^2\,
\nonumber \\ & &
\frac{e^{-i \,p.\widetilde{p'}}\,\Bigg( 4\, p.\widetilde{p'}\,
(2\,e^{\frac{i \,p.\widetilde{p'}}{2}}
+x_i)+i\, \Big( (p.\widetilde{p'})^2\,
(x_i-2)+ 8\,(e^{\frac{i \,p.\widetilde{p'}}{2}}-1)\,(2+x_i) \Big ) \Bigg )}
{2\, (p. \widetilde{p'})^3} \, \, .
\label{Fcof2}\end{aligned}$$ For $s > \frac{4\,m_i^2}{m_b^2}$, $i=u,c$, the internal u and c quarks are on mass-shell and an absorptive part appears in the coefficients related with the light quark part. Notice that when the non-commutative effects are switched off the form factors $F_3(s)$ and $F_4(s)$ disappears and we obtain the form factors in the ordinary SM.
Discussion
==========
This section is devoted to the analysis of the form factors appearing in the $b\rightarrow s g^*$ process in the framework of the SM including noncommutative effects. The new parameter existing in this geometry is $p\widetilde{k}$ and it is at the order of the magnitude of $10^{-6}-10^{-4}$ for the process underconsideration. This factor is also a new source for the CP violating effects in addition to the complex CKM matrix elements in the SM, $V_{ub}$ in our case. It enters into expressions as an exponential and its odd powers in the expansion of the exponential factor bring new CP violating effects, even for real CKM matrix elements. In our work, we study $p\widetilde{k}$ and the $sin\delta$ dependence of the real and imaginary parts of the form factors. Here, we use the parametrization $V_{ub}=e^{i\delta}\, |V_{ub}|$ and $sin\delta$ is proportional to the imaginary part of $V_{ub}$, which is the only source of the CP violating effects in the commutative SM. Note that we take the numerical value $|V_{ub}\,V_{us}|=6\times 10^{-3}$.
In Fig. \[ReF34\] we present the $sin\delta$ dependences of the real parts of the form factors $F_3$ and $F_4$, for different values of the parameter $p\widetilde{k}$, namely $10^{-6}$, $10^{-5}$ and $10^{-4}$. It is seen that $Re[F_3]$ and $Re[F_4]$ are not sensitive to $p\widetilde{k}$. However, there is a weak sensitivity to $sin\delta$. $Re[F_3]$ ($Re[F_4]$) decreases (increases) with the increasing values of $sin\delta$. Both form factors are almost at the order of the magnitude of $58.5\pm 0.05\, GeV^2$.
Fig. \[ImF3\] (\[ImF4\]) is devoted to the $sin\delta$ dependence of the imaginary part of the form factor $F_3$ ($F_4$), for three different values of the parameter $p\widetilde{k}$, $10^{-6}$, $10^{-5}$ and $10^{-4}$. It is seen that $Im[F_3]$ ($Im[F_4]$) is sensitive to $p\widetilde{k}$ and it increases (decreases) as a magnitude when $p\widetilde{k}$ decreases. $Im[F_3]$ can have both signs for large values of $p\widetilde{k}$, depending on the parameter $sin\delta$. The sensitivity of $Im[F_3]$ ($Im[F_4]$) to the parameter $sin\delta$ is strong. $Im[F_3]$ ($Im[F_4]$) can reach to the values $1.7\, GeV^2$ ($-0.9\, GeV^2$)
Fig. \[ReF1\] (\[ReF2\]) shows the $sin\delta$ dependence of the real part of the form factor $F_1$ ($F_2$), for three different values of the parameter $p\widetilde{k}$, $10^{-6}$, $10^{-5}$ and $10^{-4}$. $F_1$ ($F_2$) is not sensitive to $p\widetilde{k}$ and weakly sensitive to $sin\delta$. It is predicted that its magnitude is $0.0845\pm 0.0005$ ($0.0038\pm 0.0001$).
Finally Fig. \[ImF12\] is devoted to the $sin\delta$ dependence of the imaginary part of the form factor $F_1$ and $F_2$. $Im[F_1]$ and $Im[F_2]$ are not sensitive to $p\widetilde{k}$, however the sensitivity to $sin\delta$ not small. Their magnitudes are at the order of $0.001\pm 0.001$.
As a summary, there are additional structures and corresponding form factors in the non-commutative geometry. The factor $p\widetilde{k}$ is the new source for the CP violating effects and its magnitude depends on the process studied. In our case $p\widetilde{k}$ is small, namely $p\widetilde{k}\sim 10^{-5}$ and the form factors are not so much sensitive to this parameter. In the calculation of the CP violation of any decay which is based on the process $b\rightarrow s g^*$, the non-commutative effects probably weak. However, we belive that these effects would be stronger when the processes with heavy flavors have been considered, even in the framework of the SM.
Acknowledgement
===============
This work was supported by Turkish Academy of Sciences (TUBA/GEBIP).
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[^1]: E-mail address: eiltan@heraklit.physics.metu.edu.tr
|
---
address: |
Institut für Kernphysik, TU Darmstadt, Schlossgartenstr. 9\
D-64289 Darmstadt\
Germany
author:
- 'B.-J. SCHAEFER, O. BOHR AND J. WAMBACH'
title: FLOW EQUATIONS AND THE CHIRAL PHASE TRANSITION
---
=cmr8
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\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
The $N$-component scalar field theory with an $O(N)$-symmetry often serves as a prototype for symmetry restoration investigations at finite temperature. The manifestation of universality of critical phenomena enables the applicability of the $O(N)$-symmetric model to a wide class of very different physical systems around the critical temperature.
Wilczek and Pisarski have argued that the two flavor chiral transition is either of second order or of first order depending on the $U_A (1)$ anomaly. In case that the transition is of second order Wilczek and Rajagopal have shown that two flavor QCD belongs to the same universality class as the three dimensional four component Heisenberg magnet [@wilc]. Thus for $N=4$ this theory is used as an effective model for the chiral phase transition in two flavor QCD enabling deeper insights into the chiral phase transition in QCD provided that the chiral order parameter remains close to thermal equilibrium through the transition. On the lattice it is still not clear whether QCD for two flavors and the $O(4)$-model lie in the same universality class [@laer].
In a second order transition, the system is at an infrared fixed point of the renormalization group. This means that the physics is scale invariant and the flow equations show a scaling solution. Due to a diverging correlation length in the chiral limit the order parameter fluctuates on all length scales and the correlation functions exhibit a power law behavior with critical exponents.
The article is organized as follows: In Sect. \[floweq\] we summarize the concept of the renormalization group (RG) method. In Sect. \[finiteT\] we present some results for the $O(4)$-model at finite temperature. Finally Sect. \[criticalbeh\] is dedicated to the critical behavior of the system at exactly the critical temperature.
Concept of the Renormalization Group Method {#floweq}
===========================================
RG flow equations predict the behavior of a given theory in different momentum regimes. The main ingredient of the RG approach is the integration of irrelevant short-distance modes in order to derive a low-energy effective theory in the infrared in analogy to a discrete block-spin transformation on the lattice. For this purpose one introduces an infrared scale $k$ that separates the fast-fluctuating short-distance modes from the slowly-varying modes. This idea – pioneered by Wilson and Kadanoff in the early seventies [@kada] – can be systematically incorporated e.g. in a momentum regularization in a transparent way. It results in an effective action parameterized by the averaged blocked field at the scale $k$. Thus the full one-particle irreducible (1PI) Feynman graphs or vertex functions are generated by the renormalized effective action in the limit $k \to 0$ and the $k$-dependent effective action provides a smooth interpolation between the bare action defined at the UV scale $\Lambda$, where no fluctuations are considered, and the renormalized effective action in the IR. The flow pattern of the given theory is obtained by studying an infinitesimal change of the IR scale $k$ in the effective $k$-dependent action and is governed by differential flow equations.
Of course, the integration step cannot be performed in an exact way and one has to resort to some approximation such as the loop expansion. However a sharp momentum cutoff is in conflict with important underlying symmetries of the considered theory. The task is then to implement both the UV and the IR cutoff scales in a symmetry-conserving way, which can be accomplished by an operator cutoff regularization in Schwinger’s proper-time representation.
The one-loop contribution to the effective action in general yields a non-local UV diverging logarithm, which can be rewritten in a proper-time representation and leads to \[eq\] = - d\^4 x \_[1/\^2]{}\^ e\^[-(q\^2 + V”)]{} where the primes on the potential $V$ denote differentiation w.r.t. $\phi$. Here, the UV (IR) divergences appear for $\tau \to 0$ ($\infty$). We modify the above expression by introducing a regulating $a$ $priori$ unknown smearing function $f_k (\tau)$ in the proper-time integrand. Differentiating the resulting $k$-dependent effective action w.r.t. the scale $k$ yields the flow equations. It can be shown that all universal results in the infrared are independent of the choice of this regulating smearing function [@papp].
Finite Temperature {#finiteT}
==================
It is straightforward to generalize the above approach to finite temperature. Within the imaginary time (Matsubara) approach the four-dimensional theory at $T=0$ is conveniently matched to the effectively three-dimensional behavior at the critical temperature by a replacement of the four-dimensional momentum integration in Eq. (\[eq\]) with a three-dimensional integration and a Matsubara summation over the corresponding bosonic frequencies. Thus at finite temperature the scale $k$ serves as a generalized IR cutoff for a combination of the three-dimensional momenta and Matsubara frequencies.
To solve the flow equations numerically we start the evolution in the broken phase deep in the UV region $k = \Lambda$ and use a tree-level parameterization of the potential with two initial values. These initial values are fixed at $T=0$ and are constant in a relatively large temperature region.
In the $k\to 0$ limit the temperature dependence of the potential $V(\phi )$ (left panel of Fig. \[figcoupling\]) and the order parameter $\phi_{k\to 0} (T)$ (right panel) signal a second-order phase transition. Our calculations coincide with chiral perturbation theory up to $T \sim 45$ MeV. Around $T_c$ we obtain a scaling behavior of the order parameter with a critical exponent $\beta
\propto 0.4$ (cf. right panel).
Critical Behavior {#criticalbeh}
=================
At the critical temperature only properly rescaled quantities asymptotically exhibit scaling and the evolution of the system is purely three-dimensional [@wett]. This is the well-known dimensional reduction phenomenon. The rescaled dimensionless minimum $\kappa (t)$ of the potential approaches a constant (the fixed point $\kappa^*$) as the dimensionless flow “time” $t = \ln (k/\Lambda) $ tends to $-\infty$ in the infrared (see left panel of Fig. \[figfixpoint\]). For a starting value near the critical value $\kappa_{cr}$ at the ultraviolet scale $t=0$ the evolution moves towards either the spontaneously broken $\kappa \neq 0$ or the symmetric $\kappa = 0$ phase in the infrared. Due to the rescaling, which in the infrared tends to a constant value or zero during the $k$-evolution, the dimensionless minimum $\kappa = \rho_0/k^2$ diverges for $k \to 0$ in the broken phase. The time the system spends on this scaling solution can be rendered arbitrarily long by appropriate fine tuning of the initial values at $t=0$.
Besides of the trivial high-temperature and low-temperature fixed points the $O(N)$-symmetric model exhibits a nontrivial mixed fixed point inherent in the flow equations. The region around this fixed point is displayed in right panel of Fig. \[figfixpoint\] where the rescaled dimensionless quantities $\kappa $ and $\lambda $ are plotted for $N = 1$ and for different initial values during the evolution toward the infrared.
In general there are two relevant physical parameters for the $O(N)$-model which must be adjusted to bring the system to the critical fixed point according to universality class arguments. Since we here work in the chiral limit (no external sources or masses) only one relevant eigenvalue from the linearized renormalization group equations is left (the temperature). This can also be seen in Fig. \[figfixpoint\]. The quantity $\kappa$ is the relevant variable because repeated renormalization group iterations (which are the discrete analog to the continuous evolution with respect to $t \to -\infty$) drive the variable away from the fixed point value while $\lambda$ is the irrelevant variable being iterated toward the fixed point if the initial values are chosen sufficiently close to it. Thus there is a one-dimensional curve of points attracted to the fixed point which is the so-called critical surface for this model.
One nicely sees that this critical line separates both phases. When we choose initial values near the critical line the system spends a long time near the critical point. E.g. starting with $\kappa >
\kappa_{cr}$ and $\lambda$ finite the evolution ultimately tends to the zero-temperature fixed point corresponding to very large values of $\kappa$. On the other hand starting below the critical line the evolution of $\kappa$ tends towards the high-temperature fixed point corresponding to small $\kappa$’s. Only for exactly $\kappa =
\kappa_{cr}$ at $t=0$ the renormalization group trajectory flows into the critical mixed fixed point, which means that the long distance behavior at the critical point is the same as that of the fixed point.
Here the predictive power of this nonperturbative approach becomes visible: During the evolution near the scaling solution the system loses memory of the initial starting value in the UV and the effective three-dimensional dynamics near the transition is completely determined by the fixed point being independent of the details of the considered microscopic interaction at short distances. Exactly at a second-order phase transition the potential should be described by a $k$-independent solution. The equation of state drives the potential away from the critical temperature. As a result, after the potential has evolved away from the scaling solution, its shape is independent of the choice of the initial values and exhibits universal behavior.
In the vicinity of the critical point we obtain a scaling behavior of the system governed by critical exponents which parameterize the singular behavior of the free energy around the phase transition. Out of the six critical exponents for the $O(N)$-model only two are independent due to four scaling relations which we have verified by explicit calculations of $\alpha$, $\beta$, $\gamma$ and $\nu$ in three dimensions for various $N$ (see tab. \[tabcritical\]).
The critical exponent $\beta$ parameterizes the behavior of the spontaneous magnetization or the order parameter for the broken phase $( \kappa_{\Lambda} > \kappa_{cr} )$ in the vicinity of $T_c$. The difference $\kappa_{\Lambda} - \kappa_{cr}$ is a measure of the distance from the phase transition irrespective of any given value of $\lambda_\Lambda$. If $\kappa_{\Lambda}$ is interpreted as a function of temperature this difference is $\propto (T - T_c)$, i.e. $\kappa_{cr}$ defines the critical temperature in three dimensions.
All calculated critical exponents nicely converge to the large–N values. In addition for $N=1$ we have found $\eta = 0.0439$ and $\delta = 4.748$.
$ \hspace{4mm}N$ $\nu$ $\beta$ $\alpha$ $\gamma$
------------------ ------- --------- ---------- ----------
N=1 0.643 0.335 0.071 1.258
N=2 0.695 0.3475 -0.085 1.39
N=3 0.75 0.375 -0.25 1.50
N=4 0.79 0.395 -0.37 1.58
N=10 0.911 0.4555 -0.733 1.822
N=100 0.993 0.4965 -0.979 1.986
1.0 0.5 -1.0 2.0
: \[tabcritical\] Critical exponents for different $N$ compared with the large–N result.
The structure of the regularized RG-improved flow equations depends on the used $a$ $priori$ unknown cutoff functions $f_k$. Employing the same class of the cutoff function as in ref. [@papp] we have calculated critical exponents for several cutoff functions with more and more monomials on a grid for the full potential. A small systematic decrease in the values for the critical exponents with the number of included monomials of the smearing function is observed. Restricting to less monomials in the cutoff function $f_k$ seems to accelerate the IR convergence at the critical point. This is visible in the evolution of the minimum of the potential which stays longer on the critical trajectory in the vicinity of the scaling solution. For more monomials the evolution of the minimum escapes faster from the critical trajectory to the symmetric as well as the broken phase.\
To conclude we have presented a powerful nonperturbative method based on the RG approach applied to the $O(N)$-model at finite temperature, which, in fact, is consistent with pertinent lattice simulations. It remains to be seen in how far the critical $O(4)$-behavior is realized in full two flavor QCD simulations.
Acknowledgments {#acknowledgments .unnumbered}
===============
One of the authors (BJS) would like to express his gratitude to the organizer of the XVII. Autumn School for financial support and for providing a most stimulating environment. This work was supported by NFS-grant NFS-PHY98-00978 and by GSI Darmstadt.
References {#references .unnumbered}
==========
[9999]{}
R. Pisarski and F. Wilczek, ; K. Rajagopal and F. Wilczek, .
E. Laermann, these proceedings.
L. P. Kadanoff, [*Physica*]{} [**2**]{} (1966) 263; K. G. Wilson, .
G. Papp, B.-J. Schaefer, H.J. Pirner and J. Wambach, [hep-ph/9909246]{}.
J. Berges, D.-U. Jungnickel and C. Wetterich, .
|
---
author:
- 'M. Ravasio'
- 'G. Tagliaferri'
- 'G. Ghisellini'
- 'F. Tavecchio'
- 'M. Böttcher'
- 'M. Sikora'
date: 'Received ...; accepted ...'
title: '[*Beppo*]{}SAX and multiwavelength observations of BL Lacertae in 2000'
---
Introduction
============
Blazars are radio–loud Active Galactic Nuclei producing variable non–thermal radiation in relativistic jets oriented close to the line of sight: the emission is therefore beamed and Doppler boosted (Blandford & Rees, 1978). They are characterized by a Spectral Energy Distribution (SED) displaying two broad features: the first, extending from radio to UV/X–ray is usually ascribed to synchrotron emission; the second, ranging from X–ray to $\gamma$–ray, sometimes up to TeV energies, is attributed to inverse Compton scattering of seed photons by the same population of synchrotron emitting electrons. The seed radiation field could be constituted by the synchrotron photons themselves (SSC model, Maraschi, Ghisellini & Celotti, 1992) or by external photons, produced by the accretion disk (Dermer & Schlickeiser, 1993), by the Broad Line Region (Sikora, Begelmann & Rees, 1994; Ghisellini & Madau, 1996) or by hot dust (Blazejowski et al., 2000; Arbeiter et al., 2002). Different contributions of these fields can explain the observed blazar spectra. The BL Lac subclass is characterized by the absence or weakness of broad emission features and is divided in HBL and LBL (High and Low energy peaked BL Lacs) according to the radio to X–ray flux ratio (Padovani & Giommi, 1995).\
BL Lac itself (1ES 2200+420) has been classified as a LBL on the basis of its radio–to–X broad band spectral index $\alpha_{rx} = 0.85$ (Sambruna et al., 1996). It was first identified as the optical counterpart of the radio source VRO 42.22.01 by Schmitt (1968); the presence of weak narrow emission lines in its spectrum allowed an accurate measurement of the redshift $z=0.069$ of the host elliptical galaxy (Miller & Hawley, 1977; Miller et al., 1978). It shows superluminal motions on m.a.s. scale ($\beta_{app}\sim 3-4 ~h^{-1}$, Mutel et al., 1990; $\beta_{app} \sim 2.2-5.0 ~h^{-1}$, Denn et al., 2000). In spite of the definition of BL Lac objects as having featureless continua, in 1995, a survey performed by Vermeulen et al. (1995) revealed the presence of an H$\alpha$ emission line with equivalent width of 7 Å, confirmed by subsequent observations (Corbett et al., 1996). During the same year EGRET, aboard the Compton Gamma Ray Observatory, detected a 4.4 $\sigma$ excess above 100 MeV from its direction (Catanese et al., 1997). In the summer of 1997, BL Lac entered an exceptional flaring state with the highest X–ray flux ever recorded (Sambruna et al., 1999; Madejski et al., 1999) and a $\gamma$–ray flux 4 times higher than in 1995 (Bloom et al., 1998) .\
In the X-ray band, BL Lac has been detected for the first time in 1980 by the IPC (0.1-4 keV) and the MPC (2-10 keV) aboard the Einstein Observatory (Bregman et al., 1990). Since then the source has been observed many times by different satellites such as EXOSAT (Bregman et al., 1990), GINGA (Kawai et al., 1991), ROSAT (Urry et al., 1996; Madejski et al., 1999), ASCA (Sambruna et al., 1999; Madejski et al., 1999), RXTE (Madejski et al., 1999) and finally [*BeppoSAX*]{}, which observed it in 1997 (Padovani et al., 2001) and twice in 1999 (Ravasio et al., 2002).\
During the second half of 2000, from July to December, the source has been the object of an intensive multiwavelength campaign (Böttcher et al., in prep.) which included two X-ray observations performed by [*Beppo*]{}SAX and was supplemented by a continuous long–term monitoring program by the Rossi X–ray Timing Explorer (RXTE), with 3 short pointings per week (Marscher et al., in prep.).\
During this campaign BL Lac has been observed in the radio band by the telescopes of the University of Michigan and of the Metsähovi Radio Observatory, while in the optical band it has been observed almost continuously by 24 telescopes in the context of an extensive WEBT campaign (Villata et al., 2002). Finally, HEGRA set an upper limit of 25% of the Crab flux above 0.7 TeV, after having accumulated a total of 10.5 h of on-source time during the autumn of 2000 (Mang et al., 2001). In this paper we will analyze in detail the [*Beppo*]{}SAX data of this campaign, comparing them with RXTE simultaneous ones and discussing the spectral and temporal behaviour of BL Lac in the X–ray band and in the whole radio–to–TeV energy range.
[*Beppo*]{}SAX observations
===========================
Thanks to its uniquely wide energy range (0.1-200 keV), the italian–dutch satellite [*Beppo*]{}SAX represents an ideal experiment for looking at blazars and expecially objects such as BL Lac, since it can detect the transition between the synchrotron and inverse Compton components of the SED (Tagliaferri et al., 2000; Ravasio et al., 2002). Therefore it allows to compare the simultaneous behaviour of extremely different parts of the emitting electron distribution. Boella et al. (1997 and references therein) report an extensive summary of the mission.\
[*Beppo*]{}SAX observed BL Lac (1ES 2200+420) twice during 2000, since the July 26–27 measurements were soon interrupted (on–source time $=1.69 \times 10^4$ s). Therefore we were given a second chance and a new observation started in October 31 lasting until November 2, with a duration of $2.49 \times 10^4$ s. In Table \[tab1\] we report the exposures and the mean count rates for each [*BeppoSAX*]{} instrument.\
----------------- ---------- ------------------------------- ---------- ------------------------------- ------------------ -------------------------------
Date exposure count rate$^a$ exposure count rate$^b$ exposure count rate$^c$
(s) $\times 10^{-2}$ cts s$^{-1}$ (s) $\times 10^{-2}$ cts s$^{-1}$ (s) $\times 10^{-2}$ cts s$^{-1}$
26-27 July 2000 16917 $4.99 \pm 0.20$ 23309 $8.23\pm0.21$ $1.04\times10^4$ $-1.20\pm5.73$
31 October-
2 November 2000 24899 $28.5\pm0.38$ 33661 $33.5\pm0.33$ $1.88\times10^4$ $12.92\pm4.27$
----------------- ---------- ------------------------------- ---------- ------------------------------- ------------------ -------------------------------
We performed our analysis on linearized and cleaned event files available at the [*Beppo*]{}SAX Science Data Center (SDC) online archive (Giommi & Fiore, 1998) using the software contained in the FTOOLS Package (XIMAGE 2.63c, XSELECT 1.4b, XSPEC 10.00) and XRONOS 4.02. Data from MECS2 & MECS3 were merged by the SDC team in a single event file. Using XSELECT we extracted spectra and light curves from circular regions around the source of 8 and 4 arcmin radius for LECS and MECS, respectively. We extracted event files also from off–source circular regions, in order to monitor the background behaviour during our measurements. Since the LECS and MECS backgrounds are not uniformly distributed across the detectors, after having checked the constancy of the extracted background light curves, we choose to use the background files obtained from long blank field exposures, available at the SDC public ftp site (Fiore et al., 1999; Parmar et al., 1999).
Spectral analysis
-----------------
The spectral analysis was performed with XSPEC 10.0, using the updated response matrices and blank-sky background files (01/2000) available at the SDC public ftp site. The LECS/MECS normalization factor was fixed at 0.72, while the PDS/MECS was fixed at 0.9 to be consistent with our previous works (Ravasio et al., 2002): these values are within the acceptable range indicated by SDC (LECS: 0.67-1, PDS: 0.77-0.93, Fiore et al., 1999).\
### July 26-27
During this observation, the source was not detected by PDS because of the short on–source time and because of the intrinsic weakness of the source itself. For the same reasons and because of the high galactic absorption, the detection was uncertain also at LECS low energies: therefore we proceeded to the analysis only in the \[0.6-10\] keV range, fitting the extracted spectrum with a single power law model.\
We repeated the procedure three times, letting the interstellar absorption parameter either free to vary, or fixed at the value N$_{\rm H}=3.6\times10^{21}$ cm$^{-2}$. The latter is obtained adding the galactic value from 21 cm measurements N$_{\rm H} = (2.0 \pm 1.0) \times 10^{21}$ cm$^{-2}$ (Elvis et al., 1989) and the absorption due to the molecular clouds observed along the line of sight (Lucas & Liszt, 1993). Finally, we used the fixed value N$_{\rm H}=2.5\times10^{21}$ cm$^{-2}$, to be consistent with previous works (Ravasio et al., 2002; Sambruna et al., 1999; Madejski et al., 1999).\
Letting the absorption parameter N$_{\rm H}$ free, the 0.6-10 keV spectrum is well fitted by a single power law with spectral index $\alpha = 0.75 \pm 0.15$ ($\chi_r^2/d.o.f.=0.96/50$). We obtained N$_{\rm H} = 2.0 \pm 1\times 10^{21}$ cm$^{-2}$, the large uncertainties are due to the low count rate in the soft X-ray band.\
A single power law model with N$_{\rm H}$ fixed at $3.6\times 10^{21}$ cm$^{-2}$ leaves large positive residuals below $\sim 1$ keV. This suggests that we are using an exceedingly high absorption column or that near the low boundary of our energy range we are detecting the transition between a soft and a hard spectral component.\
These residuals become less relevant adopting the intermediate value N$_{\rm H} = 2.5 \times10^{21}$ cm$^{-2}$ (fig. \[pow-j2000\]): in this case a single power law model with $\alpha_X=0.8\pm 0.1$ fits the data well ($\chi^2_r/d.o.f. = 0.96/51$).\
According to an F-test, a broken power law model does not improve significantly the quality of the fit. The best fit parameters for each model are shown in table \[tab2\], which reports also the flux at 1 keV and the integrated flux in the 2-10 keV energy range.
[lcccclc]{}\
N$_{\rm H}$ & $\alpha_1$ & E$_b$ & $\alpha_2$ & F$_{1 keV}$ & F$_{2-10 keV}$ & $\chi^2_r/d.o.f.$\
$\times 10^{21}$ cm$^{-2}$ & & keV & & $\mu$Jy & erg cm$^{-2}$ s$^{-1}$ &\
free & & & & & &\
$2.0 \pm 1$ & $0.75 \pm 0.15 $ & & & 1.1 & $ 5.8 \times10^{-12}$ & 0.96/50\
fixed & & & & & &\
3.6 & $0.9\pm0.1$ & & & 1.3 & $5.8\times10^{-12}$ & 1.07/51\
fixed & & & & & &\
2.5 & $0.8\pm0.1$ & & & 1.1 & $5.8 \times10^{-12}$ & 0.96/51\
& & & & & &\
\
N$_{\rm H}$ & $\alpha_1$ & E$_b$ & $\alpha_2$ & F$_{1 keV}$ & F$_{2-10 keV}$ & $\chi^2_r/d.o.f.$\
$\times 10^{21}$ cm$^{-2}$ & & keV & & $\mu$Jy & erg cm$^{-2}$ s$^{-1}$ &\
\
free & & & & & &\
$3.0\pm0.3$ & $1.63 \pm 0.06 $ & & & 13.0 & $2.1\times10^{-11}$ & 0.69/76\
fixed & & & & & &\
3.6 & $ 1.71 \pm 0.03$ & & & 14.6 & $2.1\times10^{-11}$ & 0.96/77\
fixed & & & & &\
2.5 & $1.56\pm0.03$ & & & 11.9 & $2.1\times10^{-11}$ & 0.87/77\
\
fixed & & & & & &\
2.5 & $1.45^{+0.1}_{-0.5}$ & $2.2^{+0.7}_{-1.3}$ & $1.65\pm0.08$ & 11.4 & $2.1\times10^{-11}$ & 0.68/75\
### October 30- November 2
The October observation is more interesting because of the longer duration and the higher state of the source. This allows us to analyze a wider spectral range, from 0.3 keV up to 50 keV, thanks to the detection by the PDS.\
As before, we performed the fits letting N$_{\rm H}$ free, then fixing it to its maximum value and finally to N$_{\rm H}=2.5\times 10^{21}$ cm$^{-2}$. In the first case, the LECS + MECS spectrum was well fitted by a soft single power law model, with energy index $\alpha_X = 1.63 \pm 0.06$ (N$_{\rm H} = 3.0\pm 0.3 \times 10^{21}$ cm$^{-2}$). A similar result is obtained also keeping N$_{\rm H}$ fixed to $2.5 \times10^{21}$ cm$^{-2}$, while the model with the highest absorption value leaves positive residuals increasing towards low energies.\
We repeated the procedure using a broken power law model and fixing N$_{\rm H} =2.5\times 10^{21}$ cm$^{-2}$, for consistence with previous works (e.g. Ravasio et al., 2002). The LECS+MECS spectrum in the latter case is well fitted by a convex curve steepening beyond $\sim 2$ keV. The best–fit parameters of each LECS + MECS spectral model are listed in Table \[tab2\]. An F-test suggests that the addition of two parameters gives a $ 99.9 \%$ probability of improving the quality of the fit.\
Subsequently we included also PDS data in the analysis. A single power law model leaves large positive residuals towards high energies (fig. \[pow+pds-oct\]): the \[0.1–50\] keV spectrum seems to be concave. PDS data lie largely above the extrapolation of the LECS and MECS spectrum: this can be explained as the rising of a second, hard spectral component. However, the error bars are too large and we cannot constrain a second component in the model. Thus, in order to have an idea of this hard component spectral shape, we fitted the MECS + PDS spectra with a broken power law model, keeping the low energy spectral index fixed to the value obtained from the fit of the LECS + MECS spectrum ($\sim 1.65$, independently from the absorption parameter chosen). The best fit of the two extra parameters are E$_b \sim 9.9$ keV and $\alpha_{\rm {PDS}} \sim 0.56$, but we are not able to estimate the uncertainties.
### Simultaneous RXTE observations
In addition to [*Beppo*]{}SAX, the 2000 campaign was covered in the X–ray band also by Rossi X–ray Timing Explorer (Bradt, Rotschild & Swank, 1993), which provided three short exposures a week, covering the whole duration (Marscher et al, in prep.). We analyzed the two RXTE pointings temporally closest to the [*Beppo*]{}SAX ones. In July the two observations were exactly simultaneous while in November RXTE was lagging [*Beppo*]{}SAX only by 1 hour. Therefore we are given the opportunity to test our results with a totally independent set of data.\
We restrict our analysis to PCA data (Jahoda et al., 1996), an instrument composed by 5 passively collimated independent X–ray detectors sensitive to the 2-60 keV range.\
We choose to compare the \[3-15\] keV RXTE spectrum to the \[1-10\] keV MECS using a power law model with fixed absorption parameter (N$_{\rm H} = 2.5 \times10^{21}$ cm$^{-2}$). The model parameters are reported in Table \[xte-mecs\], together with the log of the observations.\
Both RXTE spectra are well fitted by power law models: during the observation of July, RXTE detected a hard spectrum ($\alpha= 0.9$), while in October, it detected a soft component ($\alpha=1.45$). The slope and the normalization of the best-fit models of RXTE spectra are consistent with those of the MECS, confirming the results obtained from [*Beppo*]{}SAX spectral analysis.
[cccccccc]{}\
Instrument & Time start & Time end & Duration &$\alpha$ & F$_{1 keV}$ & F$^{*}_{2-10 keV}$ & $\chi^2_r$/d.o.f.\
& & & (sec) & & ($\mu$Jy) & ($\times10^{-12}$) &\
PCA & 26/7/00 & 26/7/00 & 2208 & $0.9^{+0.7}_{-0.6}$& 1.4 & 6.3 & 0.42/25\
3-15 keV & 18:23:44 & 19:00:32 & & & & &\
MECS & 26/7/2000 & 27/7/2000 & 23309 & $0.8\pm0.1$ & 1.18 & 5.8 & 0.86/43\
1-10 keV & 10:12:39 & 06:43:33 & & & & &\
\
PCA & 2/11/2000 & 2/11/2000 & 2000 & $1.45^{+0.3}_{-0.25}$ & 10.3 & 19.7 & 0.45/25\
3-15 keV & 10:56:16 & 11:29:36 & & & & &\
MECS & 31/10/2000 & 2/11/2000 & 33661 & $1.6\pm0.05$ & 12.7 & 19.7 & 0.61/58\
1-10 keV & 20:46:55 & 09:59:28 & & & & &\
Temporal analysis
-----------------
We performed this analysis with the XRONOS 5.16 package, grouping data in different time bins to reduce uncertainties. During the observation of July 2000 the source was in a faint state of activity so we choose bins of 3600 s. During the second observation, the source emission was stronger so we could rebin our data in smaller bins of 600 s. In our analysis we excluded the bins with less than $20\%$ of effective exposure time, to reduce weighting errors.
### 26-27 July
In order to have information on the behaviour of the source at different X-ray wavelenghts we extracted a soft (\[0.7–2\] keV, LECS) and a hard (\[2–10\] keV, MECS) X–ray light curve. The source displays large variability in the low energy band: the 0.7-2 keV flux increases by a factor $>2$ on time scales of $\sim 4$ hours, then fades to previous values in $\sim 5-6$ hours (Fig. \[2curve-JULY\]). In the harder energy band, variability events are less evident but the source is still active, with flux variations of $25-30\%$ ($\chi^2$-test constancy probability $\sim 10\%$). The two light curves are only marginally correlated: the null correlation probability is quite high ($\sim 25\%$).\
This behaviour was noticed also in the June 1999 [*BeppoSAX*]{} observation of BL Lac (Ravasio et al., 2002): in the high energy MECS band (4-10 keV) the source was not variable, while the \[0.3-2\] keV LECS and the \[2-4\] keV MECS light curves were displaying a flare of a factor $\sim 4$.
### 31 October-2 November
During the October [*Beppo*]{}SAX observation, BL Lac displayed very fast and remarkable flux variations in the whole energy range covered by the LECS and the MECS (Fig. \[2curveratio-oct\]). The light curves are characterized by a similar, erratic behaviour, with flux variations of factors $\sim 3-4$, even on timescales of $\sim 1$ hour. The three light curves are highly correlated (P$>99.9 \%$) even with a short temporal binning of 10 min.\
We analyzed also the PDS light curve, which turned out to be constant: a test gives a $\sim 96\% $ constancy probability. Because of the large errorbars, however, we would not be able to detect variations smaller than a factor of 3. For BL Lac, this X–ray behaviour is not unprecedented: as already mentioned, during the June 1999 [*Beppo*]{}SAX observation, the \[0.3–2\] keV flux doubled in 20 min and faded to previous values in a similar time. In less than 2 hours a complete flare with a flux variation of a factor $\sim 4$ was observed. The amplitude of this event was highly frequency–dependent: while the flare was extremely prominent in the \[0.3–2\] keV and in the \[2–4\] keV curves, at higher frequencies the flux remained constant.\
The best way to characterize the temporal behaviour of the source would be the calculation of the power density spectrum: however this tool is not appropriate for unevenly sampled light curves like [*Beppo*]{}SAX ones. The large observational gaps, the short duration of the run and the limited statistics caused by the faintness of the source also makes the use of alternative techniques very difficult, such as the discrete correlation function (Edelson & Krolik, 1988) or the structure function calculation (Simonetti, Cordes & Heeschen, 1985).\
We can still characterize the variability degree of the source using two common estimators: the [*normalized excess variance*]{} parameter $\sigma^2_{\rm rms}$ and the [*minimum doubling timescale*]{} T$_{short}$ (Zhang et al., 1999; Fossati et al., 2000). The first parameter is defined as the normalized difference between the variance of the light curve and the variance due to measurement errors: it quantifies the mean variability of the source. The second parameter, instead, represent a measure of the fastest significant timescale of the source (Edelson, 1992). Assuming that each point of the light curve is described as ($t_i$,$f_i$) we define the “doubling time” as $$T_{ij}=\Big|\frac{f_{ij}\Delta T_{ij}}{\Delta f_{ij}}\Big|$$ where $\Delta T_{ij}=t_i-t_j$, $\Delta f_{ij}=f_i-f_j$ and $f_{ij}=(f_i+f_j)/2$.\
For each of the discussed energy ranges (\[0.5–2\] keV–LECS; \[2–4\] keV–MECS; \[4–10\] keV–MECS) we produced 4 different light curves with time bin of 500, 1000, 1500 and 2000 sec, respectively. For each curve we computed the $\sigma^2_{\rm rms}$ and the average of the 5 smallest $T_{ij}$ (calculated for each data pair (i,j) of the light curve) with fractional error lower than 25%. We define $T_{short}$ as their weighted average. In Table \[var\] we summarize our results.\
The highest frequency light curve is the least variable, as can be noticed also by looking at Fig. \[2curveratio-oct\]: probably this can be explained partially as an effect of the larger measurement error caused by poorer statistics and partially as a lower variability at higher energies.\
Within the relatively large errors, no significant differences are found between the $T_{short}$ values of the three light curves: the source seems to have a characteristic minimum variability timescale of $\sim 1.5-2$ hrs.
[lcccc]{}\
Energy band & $<$Count rate$>^a$ & $\sigma^2_{rms}$$^a$ & $< \sigma^2_{rms}>$$^b$ & T$_{short}^b$\
(keV) & ($10^{-2}$ cts/s) & ($10^{-2}$) & ($10^{-2}$) & (ks)\
LECS & 17.6 & $15.2$ & $12.8$ & 6.3\
0.5-2 keV & $\pm7.2$ & $\pm3.3$ & $\pm 1.6$ & $\pm 0.6$\
MECS & 14.7 &11.4 & 10.1 & 5.5\
2-4 keV & $\pm 5.3$ & $\pm 2.5$ & $\pm1.5$ & $\pm 0.5$\
MECS & 7.4 & 7.9 & 7.5 & 7.1\
4-10 keV & $ \pm2.5$ & $\pm 2.1$ & $\pm 1.1$ & $\pm 1.3$\
Spectral Energy Distributions
=============================
After having performed the spectral analysis we are now able to build the SEDs of BL Lac relative to the [*Beppo*]{}SAX 2000 observations and to compare them with other historical multiwavelength SEDs.\
As we have discussed above, during 2000 [*Beppo*]{}SAX detected the source in two completely different states of activity: in July the source was detected only in the \[0.6–10\] keV range, displaying a faint (F$_{2-10}=5.8\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$) hard spectrum. In October–November, instead, BL Lac was displaying a very intense (F$_{2-10}=2.1\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$) soft spectrum up to $\sim 10$ keV, while at higher energies a hard component, detected by the PDS up to 45 keV, was dominant.\
Also in the optical band the source was in different states: the increase of the optical flux, in fact, was the reason for prolonging the multiwavelength campaign beyond August 2000. The optical fluxes measured simultaneously to the [*Beppo*]{}SAX observations differed by 40% between the two pointings: in the core of the campaign (26-27 July) the source average R–band magnitude was m$_R=14.08$, while during the second X–ray run it was m$_R=13.74$ (Villata et al., 2002). After having dereddened the data using A$_{\rm B}=1.42$ (Schlegel et al., 1998), we calculated the optical spectral indices using weight–averaged UBVRI fluxes (see values in fig. \[sed-2000\] caption): when fainter, the spectrum is softer ($\alpha_{opt}\sim1.84\pm 0.01$), while during the autumn it is harder ($\alpha_{opt}\sim 1.58\pm0.04$).\
In Fig. \[sed-2000\] we plot the two simultaneous multiwavelength SEDs of BL Lac. We report also the upper limit by HEGRA, $7.7\times10^{-12}$ photons cm$^{-2}$ s$^{-1}$ above 0.7 TeV (Mang et al., 2001), which is $25 \%$ of the Crab level: this means F$_{0.7 \rm TeV} = 5.13\times10^{-15}\times(\alpha_{\rm TeV})$ Jy, where $\alpha_{\rm TeV}$ is the energy index of the TeV spectrum (assumed to be $=1.5$ in the plot).
Historical SEDs
---------------
As mentioned in the introduction, blazar SEDs are characterized by two broad humps extending, the first, from radio to UV/X–rays and the second from X–rays to $\gamma$-ray or even TeV energies. The first component is usually interpreted as synchrotron emission from high–energy electrons in relativistic motion, while the second is explained as inverse Compton scattering of seed photons of still debated origin.\
----------------- ---------------------- ------------ --------------------- ----------------- -------------
Date $\alpha_{1keV}$ F$_{1keV}$ F$_{2-10 keV}$ $\alpha_{opt}$ F$_{\rm V}$
($\mu$Jy) ($10^{-12}$ c.g.s.) (mJy)
June 1980 $2.2\pm1.0$ 1.6 7.1$^{*}$ $2.2\pm0.2$ 5.75
December 1983 (2.2) 0.8 $2.1\pm0.4$ 9.55
June 1988 $0.71\pm0.07$ 1.27 7.7$^{*}$ $2.2\pm0.1$ 4.2
July 1988 $1.16\pm0.24$ 1.35 4.1$^{*}$ $3.21\pm0.02$ 6.46
December 1992 $0.94\pm0.46$ 0.88 3.7$^{*}$
November 1995 $0.96\pm0.04$ 2.22 8.9 $0.8\pm0.2$ 5.7
July 1997 $0.59\pm0.03$ 3.55 26 $1.85\pm0.4$ 27.9
November 1997 $0.89\pm0.13$ 2.4 11.2
June 1999 $1.57\pm0.25$ 0.77 6.5 $1.2\pm0.2$ 26.4
December 1999 $0.63\pm0.06$ 0.96 12.3 $1.8\pm0.3$ 18.6
July 2000 $0.8\pm0.1$ 1.1 5.8 $1.84\pm0.01^a$ 12.08$^a$
$0.8\pm0.1$ 1.1 5.8 $0.94\pm0.01^b$ 25.60$^b$
October $1.45^{+0.1}_{-0.5}$ 11.4 21 $1.58\pm0.04^a$ 17.56$^a$
November 2000$$ $1.45^{+0.1}_{-0.5}$ 11.4 21 $0.64\pm0.02^b$ 37.23$^b$
----------------- ---------------------- ------------ --------------------- ----------------- -------------
The multiwavelength history of BL Lac follows that of its X–ray observations which can be traced back to 1980, the year of the first X–ray detection, carried out by Einstein (Bregman et al., 1990). BL Lac was then observed also by EXOSAT (Bregman et al., 1990), GINGA (Kawai et al., 1991), ROSAT (Urry et al., 1996; Madejski et al., 1999), ASCA (Sambruna et al., 1999; Madejski et al., 1999), RXTE (Madejski et al., 1999) and by [*Beppo*]{}SAX (Padovani et al., 2001; Ravasio et al., 2002).\
In table \[history\] we report the published spectral parameters for each X–ray observation, together with simultaneous optical ones.
The X–ray spectra of BL Lac exhibit a variety of different shapes: sometimes they are concave, hardening towards high energies, and sometimes they are hard in the whole X–ray band. In 1980, IPC (\[0.1-4\] keV) and MPC (\[2-10\] keV) aboard Einstein detected quite different spectral indices ($\alpha_1=2.2$; $\alpha_2=0.8$; Bregman et al., 1990) and a similar shape was detected also by ASCA in November 1995 (Sambruna et al., 1999) and by [*Beppo*]{}SAX in June 1999 (Ravasio et al., 2002), as well as during this campaign. In other epochs the X–ray spectrum of BL Lac was displaying only a hard component. In June and July 1988 GINGA detected faint X–ray spectra which can be attributed to inverse Compton emission in a leptonic jet model (Kawai et al., 1991), similarly to the observation of ROSAT (1982), to those performed by [*Beppo*]{}SAX in November 1997 (Padovani et al., 2001), December 1999 (Ravasio et al., 2002) and to our observation of July 2000. All the hard X–ray spectra are similar to each other except for one case: during July 1997 BL Lac was in a high state, consistent with Compton emission, with a 2-10 keV flux $\sim 3-4$ times higher than that observed in 1995 by ASCA or in June 1999 by [*Beppo*]{}SAX; the high state was clearly detected also in the $\gamma$–ray by EGRET (Bloom et al., 1997). The complexity of the behaviour of the source, suggested by the data reported in table \[history\], is further highlighted by the comparison of the multiwavelength SEDs: in fig. \[SED\] we plot the results of all the multiwavelength campaigns performed on BL Lac.\
As evidenced in fig. \[sed-2000\] & \[SED\] and in table \[history\], while during July 2000 we observed the source in a normal state, in the end of October BL Lac was very active. During this run, in fact, [*Beppo*]{}SAX detected the highest soft X–ray flux and an integrated \[2–10\] keV flux which is only slightly smaller than that of the flare of July 1997, when BL Lac was displaying a hard X–ray spectrum.
Discussion
==========
X–ray observations are crucial for blazar study because at these energies we can often observe the transition between the two emission mechanisms. The two X–ray observations that we presented here clearly show this:
- [**26–27 July**]{}: during this observation [*Beppo*]{}SAX detected a hard spectrum in the LECS–MECS range, with large positive residuals below 1 keV. The hard component can be attributed to inverse Compton emission while the residuals can be explained as the very soft tail of the synchrotron component (the data however does not allow us to set limits on the spectral index of this component neither to asses its consistency with the optical data). This picture is supported by the temporal behaviour of the source: the \[2-10\] keV light curve represents the inverse Compton spectrum produced by the less energetic particles and is therefore almost constant on the timescale of a single observation. The \[0.7-2\] keV curve, instead, is influenced also by the soft synchrotron emission produced by highly energetic electrons which can account for the observed variability.
- [**31 October – 2 November**]{}: also in this observation [*Beppo*]{}SAX detected the transition between two emission mechanisms. The LECS+MECS spectrum is soft, while the PDS observed a hard component. In the usual scenario, the hard component is attributed to inverse Compton emission, while the soft spectrum is explained as synchrotron emission by very energetic electrons ($\gamma \sim 2.5 \times 10^4-10^5$ (B/1G)$^{-1/2} (\delta/10)^{-1/2}$) which have a very short cooling time (t$_{syn} \sim 3\times 10^3-10^4$(B/1G)$^{-3/2} (\delta/10)^{1/2}$ s). This could also explain the very fast variability detected in the whole LECS–MECS energy range.\
This scenario, however, could be inadequate. Looking at fig. \[sed-2000\], one can notice immediately a strange feature in the SED of October–November: the X–ray data lie above the extrapolation of the optical spectrum, while both of them should be produced by the same emission mechanism, the synchrotron. Even correcting for the host galaxy contribution (which is almost negligible for the level of activity of BL Lac during our observations) we are not able to reconcile X–ray and optical data. However, we can conceive at least four scenarios which under certain conditions can explain the observed spectral “glitch”. They are: I) a variable local absorbing column along the line of sight; II) bulk Compton radiation; III) two different synchrotron emitting regions; IV) Klein-Nishina effect on the synchrotron spectrum .\
Differential absorption
-----------------------
From the [*Beppo*]{}SAX X–ray analysis we had N$_{\rm H}=2.5\times 10^{21}$ cm$^{-2}$, consistent with previous observations: using the dust–to–gas ratio suggested by Ryter (1996) $$\rm{A}_V=4.5\times10^{-22}~ \rm{N}_{\rm H} ~\rm{cm}^{-2}$$ and using E(B-V)=0.329 as indicated by the NED, the corresponding optical absorption is A$_{\rm B}=1.45$, very similar to the value reported by Schlegel et al. (1998), A$_{\rm B}=1.42$. However, using these values, the X–ray spectrum lies largely above the optical spectrum extrapolation, as illustrated in Fig. \[abs\].
Even correcting the optical data for the host galaxy contribution the two sections do not connect continuously. In Fig. \[abs\] we plotted the R and B band BL Lac fluxes corrected using R$_{host}=15.55$ and B$_{host}=17.15$ (Villata et al., 2002): the misalignement is even worsened.\
If we try to account for this discrepancy using different absorption values, e.g. A$_{\rm B}=2.5$, and consequently N$_{\rm H}=4.8\times 10^{21}$ cm$^{-2}$, we still obtain two spectral sections which can not be smoothly connected (Fig. \[abs\]).\
Therefore we fixed the X–ray absorption to N$_{\rm H}=2.5\times 10^{21}$ cm$^{-2}$ and varied the optical absorption.\
Using A$_{\rm B} \sim 4.2$, U–filter data lie on the X–ray extrapolation, but the optical spectrum would be very hard and the connection discontinuous. Using an intermediate value, A$_{\rm B}=2.5$, the optical and X–ray spectra connect continuously on a parabolic curve, evidenced in Fig. \[abs\]. With this absorption value, we obtain a hard optical spectrum ($\alpha_{Jul}=0.94\pm0.01$ and $\alpha_{Oct}=0.64\pm0.02$, for the summer and the autumn observations respectively) and consequently higher synchrotron peak frequencies. A dust–to–gas ratio higher than that reported by Ryter (1996) could therefore account for the observed optical to X–ray misalignement.\
We can compare this behaviour with other multiwavelength campaigns: in Fig. \[SED\] we plot all the published SEDs of BL Lacertae. Only in the SED of 1980 and in that of June 1999 we observe synchrotron emission both in the optical and in the X–ray band. In all the other cases, only the Compton component was detected, thus one can not see any misalignement between the optical and X–ray synchrotron components. While the [*Beppo*]{}SAX synchrotron spectrum of June 1999 connect continuously with the simultaneous optical data (using the galactic dust–to–gas ratio; Ravasio et al., 2002), in the 1980 SED the optical–UV data seem to lie below the extrapolation of the Einstein IPC spectrum (Bregman et al., 1990), recreating the misalignement observed in October–November 2000.\
During the 1980 IPC observation, in fact, BL Lac was displaying a very steep synchrotron spectrum in the \[0.1–4\] keV range ($\alpha=2.2\pm1.0$, using however a very high absorption parameter N$_{\rm H} = 1.25\times10^{22}$ cm$^{-2}$; Bregman et al., 1990), hardly connectable with the very soft simultaneous optical–UV spectrum. The Einstein data, however, are affected by very large uncertainties, which make difficult to determine the exact shape of the SED: this is evidenced by the results published by Worrall & Wilkes (1990), which reported an IPC spectral index $\alpha=1.34^{+2.5}_{-1.3}$. The uncertainties on the IPC spectral index are such that no firm conclusion about the reality of the optical–to–X–ray misalignement in the 1980 SED can be drawn.\
The strange optical/X–ray misalignement observed in 2000 (and maybe in 1980) is not detected in the only other multiwavelength campaign that shows synchrotron emission in the optical and X–ray bands (June 1999, [*Beppo*]{}SAX). Nevertheless, we can rely on the goodness of our data: the [*Beppo*]{}SAX spectrum is confirmed by simultaneous RXTE data, as described in the previous sections, while the optical data are confirmed by different observatories (Villata et al., 2002). The optical/X–ray misalignement is therefore real. The absorption along the line of sight can not account for it, unless we assume a sudden, large increase of the dust–to–gas ratio in the interstellar material.\
This hypothesis, however, is not unlikely, since the line of sight towards BL Lac is partially covered by a low surface brightness interstellar nebulosity which is very variable (Penston & Penston, 1973; Sillanpää, priv. comm.). If these clouds are dusty, they could account for the dust–to–gas ratio excess needed to reconcile the optical and X–ray data in the SED of October–November 2000 (and possibly also in the 1980 SED; Sillanpää et al., 1993). Their proper motion could indeed explain the misalignement in the SED of autumn 2000 and its absence in that of June 1999.
Bulk Compton radiation
----------------------
Alternatively, the observed misalignement could be claimed as the first detection of the so called “bulk Compton” emission.\
This possibility was predicted by Sikora et al. (1997): he postulated the existence of a population of “cold” electrons in addition to the relativistic ones producing synchrotron and inverse Compton emission. These cold electrons, however, have bulk relativistic motion with respect to the radiation fields produced by the accretion disk and by the broad line region and can inverse Compton scatter these photons up to frequencies $$\nu_{\rm BC} \simeq \Gamma^2 \nu_0
\label{nubc}$$ where $\nu_0$ is a characteristic frequency of the external radiation field and $\Gamma$ is the bulk Lorentz factor of the jet. If the seed radiation peaks in the UV band, as the accretion disk and the broad line region emission do, the bulk Compton component will peak in the soft X–ray band accounting for the misalignement we observe in the 2000 BL Lac SED.\
Assuming a conical jet with opening angle $\theta_j\sim 1/\Gamma$ and the conservation of the flux of electrons along the jet ($n_e\propto1/r^2$), Sikora et al.(1997) estimated the observed amount of bulk Compton radiation $L_{\rm BC}$: $$L_{\rm BC} \sim 2\Gamma^2 ~\frac{n_e(r_{\rm min}) r_{\rm min} \sigma_T ~\zeta L_{\rm UV}}{4}
% \sim 2\Gamma^2_j \frac{\tau(r_{\rm min}) \zeta L_{\rm UV}}{4 ~\theta_j}$$ where $r_{\rm min}$ is the distance from the apex of the cone at which the jet is fully developed, $L_{\rm UV}$ is the external radiation field and $\zeta=\zeta(r)$ is the fraction of the UV radiation contributing to the isotropic field at a distance $r$ from the jet apex (Sikora et al., 1997).\
In October–November the X–ray luminosity was $L_{\rm X}=10^{45}$ erg s$^{-1}$. Assuming that the bulk Compton emission peaks at $\nu = 10^{17}$ Hz and $\nu_0 = 10^{15}$ Hz, from Eq. \[nubc\] we have $\Gamma_j \sim 10$.\
Since Vermeulen et al. (1995) observed broad H$\alpha$ and H$\beta$ emission lines (confirmed by Corbett et al., 2000), from their data we can evaluate $L_{\rm UV}$ in an indirect way. Postulating a fixed line ratio and following the method described in Celotti, Padovani & Ghisellini (1997) we obtain an average value L$_{\rm BLR}\sim5\times10^{42}$ erg s$^{-1}$. If the Broad Line Region covering factor is $\sim 10\%$, the disk luminosity is L$_{\rm UV}\sim 5\times 10^{43}$ erg s$^{-1}$.\
Using these values and assuming $L_{\rm BC}=L_{\rm X}$ we obtain the particle number density in the observer frame $$n_e(r_{\rm min}) \simeq \frac{6\times10^9}{\Gamma^2_{10} \zeta_{0.1} ~r_{\rm min,15}} ~{\rm cm}^{-3}$$ where $\Gamma_{10} = \Gamma$/10, $\zeta_{0.1}=\zeta/0.1$ and $r_{\rm min,15}=r_{\rm min}/10^{15} {\rm cm}$.\
This number particle density, necessary to produce the observed X–ray spectrum, puts constraints on the jet composition. In fact, if we suppose a one–to–one proton–electron plasma, we can easily evaluate the total kinetic power of the jet (Celotti & Fabian, 1993): $$\begin{aligned}
L_{\rm KIN} \sim \pi (r_{\rm min} \theta_j)^2 n'_p m_p c^2 \Gamma^2 c \sim
\nonumber\\
\sim 10^{47} ~\frac{r_{\rm min,15} ~\theta^2_{j,0.1}}{\Gamma_{10} ~\zeta_{0.1}}
~{\rm erg ~s}^{-1}\end{aligned}$$ where $n'_p = n_p/\Gamma$ is the proton number density in the jet comoving frame. This value exceeds the $L_{\rm KIN}$ estimated by Celotti, Padovani & Ghisellini (1997) by a factor $\sim 30$. To produce the observed X–ray spectrum via bulk Compton mechanism and to be consistent with the result of Celotti, Padovani & Ghisellini (1997) the proton–electron ratio must be smaller: the jet must be pair loaded.\
This is not forbidden since the optical thickness for Thomson scattering ($\tau(r) = n_e \sigma_T \theta_j r$) at $r_{\rm min}$ is $$\tau(r_{\rm min})=\frac{2}{5} \frac{\theta_{j,0.1}}{\Gamma^2_{10} ~\zeta_{0.1}}$$ Since from our calculations $\tau(r_{\rm min}) < 1$ (and decreasing further out) and the expansion timescale at $r_{\rm min}$ ($t'_{\rm exp}\sim 8\times10^4 r_{\rm min, 15} \Gamma_{10}$ s) is smaller than the annihilation timescale ($t_{ann} \sim 2\times 10^5 ~\Gamma^3_{10} ~\zeta_{0.1} r_{\rm min,15}$ s) the pairs will survive along the jet.\
After having ruled out the hypothesis of a proton–electron jet and having requested the presence of pairs in the jet, it is now fundamental to understand if the particle number necessary to produce the bulk Compton emission is sufficient to produce the observed SEDs. The particle number conservation we have just demonstrated implies that $n_e \propto 1/r^2$. Therefore, at the distance where the radiation is emitted, $r_{\rm SED} \sim 10^{16} - 10^{17}$ cm, the particle number density would be $$\begin{aligned}
n(r_{\rm SED}) = n(r_{\rm min}) \Big( \frac{r_{\rm min}}{r_{\rm SED}} \Big)^2 \sim
\nonumber\\
\sim 6\times 10^7 \frac{1}{\Gamma^2_{10} ~\zeta_{0.1}}\frac{r_{\rm min,15}}{r^2_{\rm SED,17}} ~{\rm cm}^{-3}\end{aligned}$$ This value is greater than what we needed to model old BL Lac seds: $n_e \sim 10^{7}/\gamma_{\rm min}$ cm$^{-3}$. Only a fraction of the electrons carried by the jet need to be accelerated in order to produce the observed radio to $\gamma$–ray emission.\
What we have proved to this point is that the bulk Compton can account for the X–ray data as long as the jet is pair rich. Further constraints about this emission mechanism come from the investigation of the pair loading processes occurring in the protojet, e.g. photon–photon interaction (Svensson, 1987) or the interaction with the X–ray corona field (Sikora & Madejski, 2002), but this is beyond the goal of this paper.\
However a simple viability test of the bulk Compton scenario can be performed which is based on the X–ray spectral shape. If this mechanism is working, in the X–ray band we should see the exponential tail of the blueshifted multi–temperature blackbody emission of the accretion disk superposed to the usual hard inverse Compton emission. Therefore we tried to fit [*Beppo*]{}SAX data with a new model described by: $$A(\nu) = e^{-\rm N_{H} \sigma(\nu)}(k_1\nu^{-\Gamma_1} e^{-[(\nu_0-\nu)/\nu_1]}+k_2\nu^{-\Gamma_2})
\label{newmod}$$ where the second power law parameters are the best fit values obtained from modelling PDS data alone with a power law model ($\Gamma_2 = 1.76$; $k_2 =3.19\times 10^{-3}$ photons/keV/cm$^2$/s); N$_{\rm H}$ was fixed to $2.5\times 10^{21}$ cm$^{-2}$. This model is not able to reproduce the data unless $\Gamma_1 >1$ (see e.g. in fig. \[bbody\] the residuals below 1 keV). This implies that a blueshifted Shakura–Sunyaev disk spectrum cannot reproduce the observed soft X–ray spectrum unless being characterized by a soft power law slope in the UV band.\
Two synchrotron components
--------------------------
We considered then the hypothesis that the optical and the X–ray spectra we observed are created by two populations of emitting electrons possibly located in two regions at different distances from the nucleus. One produces a synchrotron component peaking in the IR band and is responsible for the WEBT section while the second accounts for the [*Beppo*]{}SAX data. In order not to overproduce the optical flux, the X–ray component must peak above the UV band. This implies that the emitting particle population must have a break ($\gamma_{\rm break}$). If $N(\gamma) \propto \gamma^{-2}$ below $\gamma_{\rm break}$, we require $\gamma_{\rm break} > 1.6 \times 10^{4} ~\delta^{-1/2}_{10} B^{-1/2}$ to not overproduce the optical flux.\
The observed fast X–ray variability implies that the presence of a break in the particle distribution could not be an effect of the cooling. Since $t_{\rm var}\sim 6\times10^3$ s (see table \[var\]) the X–ray emitting region must be located at a distance from the nucleus $R\lesssim \delta ~c ~t_{\rm var}/ \psi \sim 2\times 10^{16} ~\Gamma^2_{10}$ cm, where $\psi \sim 1/\Gamma$ is the jet opening angle and $\delta = 1/(\Gamma (1-\beta \cos{\theta_{obs}}))$ is the Doppler factor. The dimension of the BLR of BL Lac is $R_{\rm BLR} \sim 8.3 \times 10^{16} ~{\rm M}_{BH,8}$ cm (Böttcher & Bloom, 2000), where ${\rm M}_{BH,8}$ is the black hole mass in units of $10^8$ solar masses. Since M$_{BH,8} \sim 1.7$ (Woo & Urry, 2002), $R_{\rm BLR} \sim 1.4 \times 10^{17}$ cm: the X–ray emitting region should be inside the Broad Line Region, where the cooling rate is high.\
If we assume that a continuous power law distribution of particles N$(\gamma) \propto \gamma^{-n}$ is injected for a time $t_{\rm var}$, then an electron above $\gamma_c$ can cool in a time $t_{\rm var}$. After this time, above $\gamma_c$ the particle population steepens to N$(\gamma) \propto \gamma^{-(n+1)}$, while below $\gamma_c$ it remains unchanged. The value of $\gamma_c$ is: $$\begin{aligned}
\lefteqn{\gamma_{\rm c} \sim \frac{3 m_e c^2}{4 \sigma_T c ~U'_{\rm BLR}(1+U'_B/U'_{\rm BLR}+U'_{\rm s}/U'_{\rm BLR})~\delta t_{\rm var}} \sim}
\nonumber\\
& & \sim 7 \times 10^3 \frac{1}{\Gamma^2_{10} ~\delta_{10}}\end{aligned}$$ where we have used $U'_{BLR}= 7\times10^{-2} ~\Gamma^2_{10}$ erg cm$^{-3}$. We can see that $\gamma_c$ is hardly consistent with the required $\gamma_{\rm break}$, but, if during our observation the luminosity of the lines was fainter than the average L$_{\rm BLR}$ we have considered, then it is possible that $\gamma_c \approx \gamma_{\rm break}$. In this case the required break in the distribution can be accounted by the radiative cooling. Otherwise, not to overproduce the optical flux, the X–ray emitting particle distribution should be injected with an intrinsic break.
Klein-Nishina effect on the synchrotron spectrum
------------------------------------------------
Since the Compton cooling rate of electrons with energies $$\gamma > \gamma_1 = {m_e c^2 \over \Gamma h\nu_{BEL}} \simeq
{10^4 \over \Gamma_{10} \nu_{BEL,15}}$$ is reduced because of the decline of the Compton scattering cross section in the Klein–Nishina regime, for a given electron injection rate, the energy distribution of electrons hardens beyond $\gamma_1$. The effect is particularly strong for $f \equiv U_{BLR}'/U_B' \gg 1$ and causes flattening of the synchrotron spectrum at $$\begin{aligned}
\lefteqn{ \nu > \nu_1 \simeq 3.6 \times 10^6 ~{\rm Hz} \, B ~\gamma_1^2 ~\delta
\simeq } \nonumber\\
& & \simeq 5 \times 10^{15} { (U_{BLR}/0.005~{\rm erg\,cm}^{-3})^{1/2}
(\delta/\Gamma)
\over (f/10)^{1/2} \nu_{BEL,15}^2} ~{\rm Hz}\end{aligned}$$ The hardened part of the electron energy distribution extends up to $\gamma_2 \sim f^{1/2} \gamma_1$. At $\gamma > \gamma_2$ the cooling of electrons becomes to be dominated by synchrotron radiation and the slope of the electron energy distribution becomes the same as at $\gamma \ll \gamma_1$. Consequently, at $\nu > \nu_2 \sim f \nu_1$, the synchrotron spectrum steepens, regaining the slope from the optical band but vertically shifted up by a factor $f$ above the extrapolation of the optical spectral portion. Such a scenario can reproduce the observed spectral “glitch”, provided $f \sim 10$. Details of the model will be presented elsewhere; here we would like to mention only that a similar scenario has been proposed by Dermer and Atoyan (2002) to explain spectral glitches between the optical and X-ray spectra of kiloparsec scale jets.
Conclusions
===========
BL Lac has been the target of a multiwavelength campaign during 2000, extending from June to November. [*Beppo*]{}SAX observed the source twice, in July and at the end of October, while in different optical state of activity. As evidenced also from previous observations, BL Lac displays a very complex behaviour. We summarize the [*Beppo*]{}SAX results as follows:
- In July, while optically weak, the source displayed a faint, hard X–ray spectrum with positive residuals towards low energies. The soft X–ray flux varied in a timescale of a few hours, while the hard X–ray flux was almost constant.
- In October, while BL Lac was bright in the optical band, we observed in the X–ray the transition from an extremely strong, soft component to a hard component dominating above 10–20 keV. The soft spectrum displayed an erratic temporal behaviour with large and fast variations on timescales down to $6\times 10^3$ s, while the hard component remained almost constant.
- [*Beppo*]{}SAX spectral results are confirmed by simultaneous RXTE data: PCA and MECS spectra are consistent within the uncertainties.
The frequency dependent temporal variability is consistent with the spectral analysis results: we observed fast and large flux variations when the soft component, interpreted as synchrotron emission, was dominating, while we observed constant light curves in the hard section of the spectra which can be reproduced as inverse Compton radiation in a leptonic jet model. This can be explained since synchrotron X–ray emitting electrons are more energetic than those that are producing inverse Compton emission at X–ray energies, so they cool faster. This behaviour was already observed in BL Lac (Ravasio et al., 2002) as well as in other similar objects, such as ON 231 (Tagliaferri et al., 2000) or S5 0716+714 (Tagliaferri et al., 2003).\
The analysis of the multiwavelength Spectral Energy Distributions and the comparison with other historical SEDs, evidences the exceptionality of the X–ray spectrum of October 2000: during this observation BL Lac was displaying the highest soft X–ray flux ever recorded and an integrated \[2-10\] keV flux which was only sligthly smaller than that detected in July 1997, while BL Lac was in an exceptional flaring state and was displaying a hard X–ray spectrum (Madejski et al., 1999).\
Moreover, the SED of October 2000 displayed another very interesting feature: the soft X–ray data laid above the extrapolation of the optical spectrum, while they should be both produced via the same synchrotron emission. To account for this inconsistency we have investigated 4 possibilities, among which, however, we cannot at present discriminate:
1. The dust–to–gas ratio towards BL Lac is higher than the interstellar one. However, since this misalignement is not seen in the other multiwavelength SED of BL Lac with a well defined synchrotron component (except that of 1980, where data have large uncertainties), a sudden large increase of the dust–to–gas ratio is needed. This hypothesis is not unlikely since dusty galactic nebulosities are indeed observed towards the source.
2. This is the first detection of the so called bulk Compton emission. In the hypothesis that the jet is pair loaded, we can reproduce the observed soft X–ray spectrum via the bulk Compton mechanism while the optical and the hard X–ray spectra can be explained via the usual synchrotron and inverse Compton models. However, a blueshifted Shakura–Sunyaev disk spectrum cannot model [*Beppo*]{}SAX data. Nevertheless, we cannot exclude this scenario since the effective spectral shape of accretion disks in the far UV is still uncertain and could have a power law rather than an exponential slope (see e.g. Laor et al., 1996).
3. Two synchrotron components present at different distances from the nucleus account for the optical and the X–ray spectra, respectively. In order to not overproduce the optical flux and to account for the X–ray fast variability, the X–ray emitting particle population must have a break at energies greater than $\gamma \sim 1.6 \times 10^4 \delta^{-1/2}_{10} B^{-1/2}$. The break can be accounted by the particle cooling or must be intrinsic if the line luminosity level is lower or higher than the calculated average value, respectively.
4. Through the synchrotron emitting population, we observe the transition from inverse Compton cooling dominated to synchrotron cooling dominated particles. Between these two conditions, there is a range in which the electrons are dominated by an inefficient inverse Compton cooling in the Klein–Nishina regime: this produces a hardening in the particle population and therefore in the synchrotron spectrum which accounts for the observed optical to X–ray misalignement.
In the SED of October 2000 of BL Lac, above the synchrotron peak, we have detected a well determined misalignement between two sections of the synchrotron spectrum. To explain this feature, we suggested four scenarios presenting different favorable and critical points: at present, however, the theoretical models and the data do not allow us to discriminate between them, although the hypothesis of a variable dust–to–gas ratio is the most plausible one.\
We are grateful to Dr. M. Villata for sending us informations about the optical data published in Villata et al. (2002). This research was financially supported by the Italian Ministry for University and Research. M.S. acknowledges partial support from Polish KBN grants: 5P03D00221 and 2P03C006 19p1,2.
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abstract: 'Future 5G wireless networks will rely on agile and automated network management, where the usage of diverse resources must be jointly optimized with surgical accuracy. A number of key wireless network functionalities (e.g., traffic steering, power control) give rise to hard optimization problems. What is more, high spatio-temporal traffic variability coupled with the need to satisfy strict per slice/service SLAs in modern networks, suggest that these problems must be constantly (re-)solved, to maintain close-to-optimal performance. To this end, we propose the framework of *Online Network Optimization* (ONO), which seeks to maintain both agile *and* efficient control over time, using an arsenal of data-driven, online learning, and AI-based techniques. Since the mathematical tools and the studied regimes vary widely among these methodologies, a theoretical comparison is often out of reach. Therefore, the important question “*what is the right ONO technique?*” remains open to date. In this paper, we discuss the pros and cons of each technique and present a direct quantitative comparison for a specific use case, using real data. Our results suggest that carefully combining the insights of problem modeling with state-of-the-art AI techniques provides significant advantages at reasonable complexity.'
author:
- |
Spyridon Vassilaras, Luigi Vigneri, Nikolaos Liakopoulos, Georgios S. Paschos,\
Apostolos Destounis, Thrasyvoulos Spyropoulos, and Mérouane Debbah
bibliography:
- 'ONO\_AI\_references.bib'
title: 'Problem-Adapted Artificial Intelligence for Online Network Optimization'
---
Introduction
============
Online Network Optimization (ONO) {#sec:ono}
=================================
ONO Techniques
==============
Performance Evaluation
======================
Conclusions
===========
In this paper, we have stressed the need for automating online network optimization, and have examined several approaches such as online learning, data-driven optimization, and model-based AI techniques. We have performed a direct comparison of these techniques for the challenging problem of traffic steering and load balancing in dense, heterogeneous networks. Our study demonstrated how the inherent variability of network traffic can be successfully addressed by self-optimizing methodologies. The most prominent of them, is based on artificial intelligence, but also makes use of deep modeling insights of the problem. We have therefore demonstrated that the efficiency, robustness and scalability of online AI techniques can be substantially improved by applying the knowledge of the problem specifics.
Biographies {#biographies .unnumbered}
===========
**Spyridon Vassilaras** is a principal researcher at the Huawei France Research Center since Dec. 2014. From 2003 till 2014 he was a researcher and professor at Athens Information Technology (AIT). He received the Dipl. Eng. degree from the National Technical University of Athens (1995) and the M.S. and Ph.D. degrees from Boston University (1997, 2001). His main research interests include machine learning and network optimization. **Luigi Vigneri** is a research scientist at IOTA Foundation. Previously, he was postdoctoral researcher at Huawei Technologies, France. He received his M.Sc. in Computer Engineering from Politecnico di Torino and Télécom ParisTech (2014), and his Ph.D. from EURECOM (2017). His main research interests include network optimization and distributed systems.
**Nikolaos Liakopoulos** received his B.S. in Physics and M.S. in Control and Computing from National Kapodistrian University of Athens, in 2012 and 2015. Since 2016 he has joined Huawei FRC, working towards his PhD in collaboration with Eurecom and UPMC. His research focuses in distributed and centralized control for wireless networks.
**Georgios S. Paschos** is a principal researcher at Huawei Technologies, France, since Nov. 2014. Previously, he held research positions at LIDS, MIT (USA), CERTH-ITI and University of Thessaly, Greece, and VTT, Finland. He received his ECE diploma from Aristotle University of Thessaloniki 2002, and his PhD degree from University of Patras, 2006. He served as an associate editor for IEEE/ACM Trans. on Networking, IEEE Networking Letters, and as a TPC member of INFOCOM, WiOPT, and Netsoft. **Apostolos Destounis** is with Huawei since 2014, where he is now a Senior Research Engineer. During 2011-2014 he was with Alcatel-Lucent Bell Labs France. He received his Ph.D. degree from Supelec in 2014, the M.Sc. from Imperial College London in 2010 and the Dipl. Eng. Degree from National Technical University of Athens in 2009. His research interests include optimization and machine learning applied to communication systems.
**Thrasyvoulos Spyropoulos** is an Assoc. Professor at EURECOM, France, since Oct. 2010. He has also been a postdoctoral researcher at INRIA, Sophia-Antipolis (2006-2007) and Senior Researcher and Lecturer at ETH, Zurich (2007-2010). He received the Dipl. Eng. degree from the National Technical University of Athens (2000), and the Ph.D. degree from the University of Southern California (2006).
**Mérouane Debbah** entered the Ecole Normale Supérieure de Cachan (France) in 1996 where he received his M.Sc. and Ph.D. degrees. Since 2014, he is Vice-President of the Huawei France R&D center and director of the Mathematical and Algorithmic Sciences Lab. He is an IEEE Fellow and a WWRF Fellow and has received more than 16 best paper awards. He has managed 8 EU projects, 24 national and international projects and received more than 16 best paper awards.
|
---
abstract: 'The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for finite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined.'
address: ' Department of Mathematics, University of Genova, Italy'
author:
- 'C. Fassino, E. Riccomagno, M-P Rogantin'
bibliography:
- 'FassinoRiccomagnoRogantin.bib'
title: Cubature rules and expected value of some complex functions
---
Introduction
============
Evaluations of integrals is a recurrent task in statistics and probability for example when computing marginal distributions, in the analysis of contingency tables, when estimating the moments of some known distribution or when evaluating the marginal likelihood integrals in Bayesian inference, spectral analysis of time series [@Brillinger2001] and probability Mora than in statistics, complex valued random vectors and their integration find application in many other fields such as electromagnetism and quantum mechanics, and largely in digital communication [@Lapidoth2009] and signal processing (e.g. [@ZhuBlumLin2016] and for a setting similar to ours [@SaylorSmolarski2001]). Interestingly the usefulness of complex random vectors has also been argued in actuarial science [@Halliwell2016] besides time series analysis. An introduction to the statistical analysis based on complex Gaussian distributions is given in [@Goodman1963] and a recent paper on second order estimation with complex-valued data focused on digital signal processing can be found in [@LangHuemer2017].
In this paper we address the problem of computing *exactly* the expected value with respect to a generic probability measure $\lambda$, of a complex valued function $g: \mathbb C^k \rightarrow \mathbb C $ of the $k$-variate complex random vector $Z$. We do so by using the indicator functions from the algebraic statistics theory of design of experiments. The measure $\lambda$ could be discrete or continuous. General results are presented for the discrete case and specific results are given for the multivariate complex Gaussian distributions.
The expected value of $g$ is approximated by using interpolatory cubature rules of the form $$\label{eq1} \int_{\mathbb C^k} g \, d\lambda = \sum_{d \in \mathcal D} w_d g(d) +R(g)$$ where $ \mathcal D$ is a finite set giving the cubature nodes. For us the coordinates of the cubature nodes are in suitable subsets of the $m$-th roots of the unity. The weights $\{w_d\}_{d \in \mathcal D}$ are obtained from a vectorial basis of the quotient space $\mathbb C[z_1,\dots,z_k] / I(\mathcal D)$, where $I(\mathcal D)$ is the polynomial ideal of $\mathcal D$ [@PRW2001]. Finally $R(g)$ is the error committed when approximating the integral with the finite sum in (\[eq1\]). Given a set of nodes and weights, it is of interest to determine classes of functions $g$ for which the error is zero. This set is called the precision space of the cubature rule. Quadrature rules (i.e. bi-dimensional cubature rules) with complex valued nodes have been studied e.g. in [@Panja2015Mandal]. Here we work in a multi-dimensional setting.
Our work unveils a connection between cubature rules and design of experiments which, to our knowledge, has been unnoticed so far in the literature. We find this connection somewhat natural because both in cubature rule theory and design of experiment theory a key point is to determine a suitable finite set of points $\mathcal D$ and their weights $\{w_d\}_{d \in \mathcal D}$ for achieving some specific task, although this can be different between the two theories and also within them. Another common task to the two theories is, given $\mathcal D$ and $\{w_d\}_{d \in \mathcal D}$, find their range of applicability, e.g. power of estimation, precision space.
This paper deals with this second task and it does so by the synergic use of tools and techniques from commutative algebra, numerical analysis and algebraic statistics. In particular, some results in [@FaPR2014; @FaR2015] are generalised to the complex case. The link between the above cubature problem and the algebraic statistics theory of fractional factorial design of experiments is made through the representation of a fractional factorial experiment as a polynomial indicator function [@PR2008]. This is similar to [@FRR2012] which instead unhearthed the connection between Markov bases for contingency tables and design of experiments.
In Section \[Sc:Eq\_weights\] we focus our attention on the special case with equal weights and we obtain some specific results for the Gaussian density in Section \[Sc:Gauss\]. While in Section \[Sc:Interp\_Rule\] we provide necessary and sufficient conditions for obtaining such cubature rules and we analyse their precision space, that is the vector space of polynomials $p$ whose expected value is equal to $\sum_{d \in \mathcal D} w_dfp(d)$, namely with zero error $R(p) $. The weights are found in Section \[Sc:Comp\_weights\].
Interpolatory rules {#Sc:Interp_Rule}
===================
Let $\lambda$ be a measure on $\mathbb C^k$ with finite moments (at least up to a certain degree) and $g$ be a complex integrable function, $g: \mathbb C^k \to \mathbb C$. Let $\mathcal D\subset \mathbb C^k$ be a set with $n$ elements and let $w \in \mathbb C^k$ be the vector $[w_d]_{d\in \mathcal D}$.
A *cubature rule* $(\mathcal D,w)$ is a formula of the type $$\int_{\mathbb C^k} g \ d\lambda= \sum_{d \in \mathcal D} w_d \ g(d) +R_{\mathcal D,w}(g)$$ where the sum provides an approximation to the integral and $R_{\mathcal D,w}$ is the respective error. The $w_d$’s are called the *weights* and the elements of $\mathcal D$ the *nodes* of the cubature rule.
Let $\mathbb C[z_1,\dots,z_k]$ be the ring of polynomials with complex coefficients in the indeterminates $z_1,\dots,z_k$ and let $\mathcal P$ be a set of polynomials contained in $\mathbb C[z_1,\dots,z_k]$. A cubature rule $(\mathcal D,w)$ is *exact* for $\mathcal P$ if for all elements $p$ of $\mathcal P$ $$\int_{\mathbb C^k} p \ d\lambda=\sum_{d \in \mathcal D} w_d \ p(d)$$ or, equivalently, if $R_{\mathcal D,w}(p)=0$.
A cubature rules $(\mathcal D,w)$ is called *interpolatory* if it is exact for a set $\mathcal P$ of interpolatory polynomials over $\mathcal D$. This definition generalises the definition of univariate interpolatory quadrature rules, where $\mathcal P$ is the set of univariate polynomials with degree strictly lower than the cardinality of $\mathcal D$, that is the set of the interpolatory polynomials over $\mathcal D$ [@Gautschi2004].
In this paper, given a set of nodes $\mathcal D$, we only consider sets of polynomials $\mathcal P$ such that, for any function $g: \mathbb C^k \to \mathbb C$, there exists a *unique* interpolatory polynomial $p_{g,\mathcal{D}}\in \mathcal P$ with $g(d)=p_{g,\mathcal{D}}(d)$ for all $d\in \mathcal D$. The pair $(\mathcal D,\mathcal P)$ is called *correct*. For instance, the pair $(\mathcal D,\mathcal P)$ is correct if $\mathcal D=\{d_1,\dots,d_n\}\subset \mathbb C$ and $\mathcal P=\operatorname{Span}_{\mathbb C}\left( x^\alpha \ | \ \alpha=0,\dots,n-1\right)$.
Let $\alpha \in \mathbb Z_{\ge 0}^{k}$, let $z^\alpha = z_1^{\alpha_1} \dots z_k^{\alpha_k}$ be a monomial in the indeterminates $z_1,\dots,z_k$ and let $\mathbb T = \left\{z^\alpha \ | \ \alpha \in \mathbb Z_{\ge 0}^k \right\}$ be the set of all monomials. Let $S \subset \mathbb T$ be a set of monomials such that each $p \in \mathcal P$ can be expressed as $p=\sum_{s\in S} c_s s$, $c_s \in \mathbb C$, that is $\mathcal P$ is a vectorial space over $\mathbb C$ with basis $S$. We denote $\mathcal P= \operatorname{Span}_{\mathbb C}(S)$.
An interpolatory polynomial $p_g \in \operatorname{Span}_{\mathbb C}(S)$ of a function $g$ over $\mathcal D$ is such that $p_g(d)= g(d)$, that is $\sum_{s\in S} c_s s(d)=g(d) $, for each $d \in \mathcal D$. Denoting by $X_{\mathcal D, S} =[s(d)]_{d\in \mathcal D, s\in S}$ the evaluation matrix of the elements of $S$ at $\mathcal D$ and by $[g(d)]_{d\in \mathcal D}$ the evaluation vector of $g$ at $\mathcal D$, the coefficient vector $c=[c_s]_{s\in S}$ of the polynomial $p_g$ satisfies the linear system $X_{\mathcal D,S} c =[g(d)]_{d\in \mathcal D}$. If the pair $(\mathcal D, \mathcal P)$ is correct, for each function $g$, that is for each vector $[g(d)]_{d\in \mathcal D}$, there exists a unique coefficient vector $c$ solution of this linear system, and so the pair is correct if and only if $X_{\mathcal D, S}$ is a square non singular matrix, that is if and only if $\# S = \# \mathcal D$ and the evaluation vectors $[s(d)]_{d \in \mathcal D}$ are linear independent vectors.
In the following we consider cubature rules $(\mathcal D, w) $ which are interpolatory with respect to a polynomial set $\mathcal P$ such that the pair $(\mathcal D,\mathcal P)$ is correct. A correct pair $(\mathcal D,\mathcal P)$ can be obtained considering $\mathbb C$-vector spaces $\mathcal P$ which are isomorphic to the quotient space $\mathbb C[z_1,\dots , z_k]/I(\mathcal D)$, that is considering monomial sets $S$ isomorphic to a basis of the quotient space $\mathbb C[z_1,\dots , z_k]/I(\mathcal D)$. [There exist algebraic algorithms to compute monomial bases of such a vector space, for instance the Buchberger-Möller algorithm [@BuchbergerMoller1982]]{}.
By definition, an interpolatory cubature rule $ (\mathcal D,w)$ is exact for each polynomial in $\mathcal P$ but, in general, there exist polynomials $p \notin \mathcal P$ such that $R_{\mathcal D,w}(p)=0$. In order to study the set of these polynomials, we introduce the notions of precision basis and precision space.
A finite monomial set $\mathcal B_{\mathcal D,w} \subset \mathbb T$ is a *precision basis* for $(\mathcal D,w)$ if $$\int_{\mathbb C^k} z^\alpha \ d\lambda =\sum_{d \in \mathcal D} w_d z^\alpha(d) \ \text{ for all } \; z^\alpha \in \mathcal B_{\mathcal D,w} \ \textrm{and} \
\int_{\mathbb C^k} z^\alpha \ d\lambda \neq \sum_{d \in \mathcal D} w_d z^\alpha(d) \text{ for all } z^\alpha \notin \mathcal B_{\mathcal D,w} \ .$$ The precision basis is the largest set of monomials for which $(\mathcal D,w)$ is exact. The *precision space* of $(\mathcal D,w)$ is the $\mathbb C$-vector space $\operatorname{Span}(\mathcal B_{\mathcal D,w})$ generated by $\mathcal B_{\mathcal D,w}$. In the univariate case, if $\mathcal D$ is a subset of $ \mathbb R$ of cardinality $n$ and if $\mathcal P$ is the interpolation space is generated $\{1,x,x^2,\dots,x^{n-1}\}$, the precision space of Gaussian quadrature rule is generated by $\mathcal B_{\mathcal D,w}=\{1,x,x^2,\dots,x^{2n-1}\}$ [@Gautschi2004].
In the univariate case, the precision degree of a quadrature rule is the maximal degree of the elements of $\mathcal P$ on which the quadrature rule is exact. Generalising this notion, we define the *precision degree* of $(\mathcal D,w)$ as the $\max_{z^\alpha \in \mathcal B_{\mathcal D,w}} \left\{\sum_{i=1}^k \alpha_i \right\}$.
Weights for points in $\mathbb C$-vector space with basis $S$ {#Sc:Comp_weights}
=============================================================
Let $\mathcal D$ be a set of $n$ nodes in $\mathbb C^k$, let $S \subset \mathbb T$ be a set of monomials in $\mathbb C[z_1,\dots,z_k]$ and let $\mathcal P=\operatorname{Span}_{\mathbb C}(S)$ be the $\mathbb C$-vector space of polynomials in $\mathbb C[z_1,\dots,z_k]$ generated by $S$ such that $(\mathcal P, \mathcal D)$ is correct.
The following proposition gives the vector of weights $w_S$ that makes $(\mathcal D,w)$ exact on $\mathcal P$, [that is the weights $w_S$ of the interpolatory cubature rule on $\mathcal P$]{}.
\[pr:int-cub-r\] Let $\mathcal P$ be a $\mathbb C$-vector space with basis $S$ with $\#S = \# \mathcal D$, and let $(\mathcal D,\mathcal P)$ be correct. Let $X_{\mathcal D,S}=[s(d)]_{d \in \mathcal D,s \in S}$ be the evaluation matrix of the elements of $S$ over $\mathcal D$. The cubature rule $(\mathcal D,w_S)$ is exact on $\mathcal P$ if and only if $$w_S=\left(X_{\mathcal D,S}^t\right)^{-1} \left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S} \ .$$ Furthermore, the weights $w_S$ are unique.
Each $p \in \mathcal P$ can be written uniquely as $$p=\sum_{s\in S} c_s s \ , \textrm{ for } c_s \in \mathbb C \ ,$$ and so $$[p(d)]_{d \in \mathcal D}=X_{\mathcal D,S} [c_s]_{s \in S} \ ,$$ that is $$%\label{eq:ws}
[c_s]_{s \in S}= X_{\mathcal D,S}^{-1} [p(d)]_{d \in \mathcal D} \ .$$ It follows that, for each $p \in \mathcal P$, $$\begin{aligned}
\int_{\mathbb C^k} p\ d\lambda&= \int_{\mathbb C^k} \sum_{s\in S} c_s\ s \ d\lambda= \sum_{s\in S} c_s \int_{\mathbb C^k} s \ d\lambda=\left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S}^t \left[c_s\right]_{s \in S} \\ & =
\left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S}^t \ X_{\mathcal D,S}^{-1} \ \left[p(d)\right]_{d \in \mathcal D}
\end{aligned}$$
The cubature rule $(\mathcal D,w_S)$ is exact on $\mathcal P$ if and only if $\int_{\mathbb C^k} p\ d\lambda=w_S^t \left[p(d)\right]_{d \in \mathcal D}$ for each $p \in \mathcal P$, that is if and only if $$\left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S}^t \ X_{\mathcal D,S}^{-1} \ \left[p(d)\right]_{d \in \mathcal D} =
w_S^t \left[p(d)\right]_{d \in \mathcal D}$$ or, equivalently, $$w_S= \left(X_{\mathcal D,S}^t \right)^{-1} \ \left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S} +\rho$$ where $\rho$ is orthogonal to each evaluation vector $\left[p(d)\right]_{d \in \mathcal D} $. In particular, $\rho$ is orthogonal to $\left[s(d)\right]_{d \in \mathcal D} $, $ s\in S$, that is to the columns of $X_{\mathcal D,S}$. Since $X_{\mathcal D,S}$ is a square non singular matrix, $\rho $ is the null vector and so the [vector $w_S$ ]{} of the weights of the cubature rule $(\mathcal D, \operatorname{Span}(S))$ is unique.
The weights $w_S$ do not change, if a different basis $T$ for the $\mathbb C$-vector space [$\mathcal P=\operatorname{Span}(S)$]{} is chosen. Each monomial $t \in T$ can be expressed as $t=\sum_{s \in S} m_{t,s} s$, and so, denoting by $M=[m_{t,s}]_{t \in T, s \in S}$, we have, by linearity, $$\left[\int_{\mathbb C^k} t \ d\lambda \right]_{t \in T}=M\left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S}\ .$$ Moreover, the evaluation matrix $X_{\mathcal D,T}$ of the elements of $T$ over $\mathcal D$ can be written as $X_{\mathcal D,T}=X_{\mathcal D,S}M^t$. Let $w_T$ be the weights computed using $\mathcal D$ and $T$. From Proposition \[pr:int-cub-r\] we have $$\begin{aligned}
w_T & = \left(X_{\mathcal D,T}^t\right)^{-1} \ \left[\int_{\mathbb C^k} t \ d\lambda \right]_{t\in T}
= \left( M X_{\mathcal D,S}^t \right)^{-1} M \left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S} \\
& = \left(X_{\mathcal D,S}^t \right)^{-1} \left[\int_{\mathbb C^k} s \ d\lambda \right]_{s\in S} = w_S\ .
\end{aligned}$$
$\mathbb C$-Fractional factorial designs
========================================
In this section we consider interpolatory cubature rules with set of nodes $\mathcal D$, whose elements are $k$-uple of $m$-th roots of the unit, and with interpolatory space $\mathcal P$, the $\mathbb C$-vector space generated by a monomial set $S \subset \mathbb T$, such that the pair $(\mathcal D ,\mathcal P)$ is correct.
We briefly recall some topics about of the roots of the unity. Let $m \in \mathbb Z_{\ge0}$ and $\Omega_m=\{\omega_0,\dots, \omega_{m-1}\}$ be the set of the $m$-th roots of the unity, $\omega_j=\exp (-\mathbf{i} (m/2\pi)j)$, where $\mathbf{i}=\sqrt{-1}$ is the imaginary unity.
Denoting, for $j \in \mathbb Z$, by $[j]_m$ the residue of $j \mod m$ and by $\overline j$ the class $[m-j]_m$ we have that, given $c \in \mathbb Z$ and $\omega_j, \omega_i \in \Omega$, it holds $\omega_j^c = \omega_{[c j]_m}$, $\omega_i\omega_j=\omega_{[i+j]_m}$, and the complex conjugate of $\omega_j$ is $\overline{\omega_j}=\omega_{\overline j}$. Furthermore, we denote by $ \mathbb Z_m$ the set of all congruence classes of the integers for a modulus $m$, and by $\mathbb Z_m^k$ its cartesian product.
We consider a set of $n$ nodes $\mathcal D$ contained in $\Omega^k_m \subset \mathbb C^k$. Let $f$ be the indicator function of $\mathcal D$ over $\Omega_m^k$, defined as $f(d)=1$, for $d \in \mathcal D$, and $f(d)=0$, for $d \in \Omega_m^k\setminus \mathcal D$.
Let $S_m=\{z^\alpha: \ \alpha \in \mathbb Z_m^k\}$ be the monomial basis of [the]{} $\mathbb C$-vector space which is isomorphic to the quotient space $\mathbb C[z_1,\dots , z_k]/I(\Omega_m^k)$. As presented in [@PR2008], by interpolating the values $[f(d)]_{d \in \Omega_m^k}$ with polynomials in $\operatorname{Span}(S_m)$, we obtain a representation of the indicator function $f$ as follows $$\label{ind}
f = \sum_{\alpha \in \mathbb Z_m^k} b_\alpha z^\alpha \qquad \textrm{where} \quad b_\alpha = \frac 1 {m^k} \sum_{d \in {\mathcal D}} z^{\overline \alpha}(d)$$ and, since $\alpha= [\alpha_1, \dots, \alpha_k]\in \mathbb Z_m^k$, $\overline \alpha= [m-\alpha_1, \dots, m-\alpha_k] $.
Let $(\mathcal D, w_S)$ be the cubature rule with nodes $\mathcal D\subset \Omega_m^k$, $S\subset S_m$ and weights $w_S = (X_{\mathcal D,S}^t)^{-1} \left[ \int_{\mathbb C^k} s \ d\lambda \right]_{s\in S}$.
In the next we consider non-negative and normalized weights, i.e. such that $w_S^t1_n =1$, where $1_n$ is a $n$-vector with elements equal to $1$, and the weights are all equal, that is $w_S= \frac1 n 1_n$.
Equal weights {#Sc:Eq_weights}
-------------
If the weights are non-negative, normalized and all equal, then $w = \frac 1 n {1_n}$. The following theorem characterises a cubature rule with equal weights using the indictor function of $\mathcal D$.
\[teo\_pesi\_uguali\] Let $\mathcal D \subset \Omega_m^k$ be a set of $n$ nodes and let $f$ its indicator function, as in Eq. . Let $S \subset S_m$ be a monomial set such that the pair $(\mathcal D, \operatorname{Span}(S))$ is correct and let $w_S$ as in Proposition \[pr:int-cub-r\].
Let $A$ be the set: $$A=\left \{ z^\alpha \in\mathbb T, \ \alpha \in \mathbb Z^k_{\ge 0} \ \left| \ \int_{\mathbb C^k} z^\alpha \ d\lambda = \frac {m^k} n \ b_{\overline{\alpha}}\right. \right\} \ .$$
Then $w_S= \frac 1 n 1_n$ if and only if $ S \subseteq A$.
From Proposition \[pr:int-cub-r\] and Eq. (\[ind\]) (restricted to the $\alpha$ such that $z^\alpha \in S$) it holds: $$w_S= \left(X_{\mathcal D,S}^t\right)^{-1} \left[ \int z^\alpha \ d\lambda\right]_{z^\alpha\in S} \quad \textrm{and} \quad \left[ b_{\overline \alpha} \right]_{z^\alpha\in S}=\frac 1{m^k} X_{\mathcal D,S}^t\ 1_n$$ If $ w_S =\frac 1 n 1_n$ then $$\left [\int z^\alpha \ d \lambda \right ]_{z^\alpha \in S} = \frac 1 n X_{\mathcal D,S}^t\ 1_n=\frac{m^k}n \left[ b_{\overline \alpha} \right]_{z^\alpha\in S}$$ so that $S \subseteq A$.
Vice-versa, if $S \subseteq A$, for $z^\alpha \in S$, it holds: $$\left[ b_{\overline \alpha} \right]_{z^\alpha\in S} = \frac n {m^k} \left[ \int z^\alpha \ d\lambda \right]_{z^\alpha\in S}=
\frac {n} {m^k} X_{\mathcal D,S}^t w_S$$ and, from $\left[ b_{\overline \alpha} \right]_{z^\alpha\in S}=\frac 1{m^k} X_{\mathcal D,S}^t\ 1_n$, it follows $w_S=\frac 1 n 1_n$, being $X_{\mathcal D,S}$ invertible.
Given a set $\mathcal D \subseteq \Omega_m^k$ with $n$ points, Theorem \[teo\_pesi\_uguali\] suggests an algorithm for finding, if there exists, a cubature rule with nodes $\mathcal D$ and equal weights. Fix $X^{(0)}=1_n$ and $S=\{1\}$. At the $r$-th step an element $z^\alpha$ of $A \cap \mathbb Z_m^k$ is considered. If the vector $v=[z^\alpha(d)]_{d \in \mathcal D}$ is such that the matrix $\left[X^{(r-1)},v \right]$ has full rank, then $v$ is added to the matrix $X^{(r-1)}$ for obtaining the new matrix $X^{(r)}=\left[X^{(r-1)},v\right]$ and $z^\alpha$ is added to $S$. Otherwise a different element of $A \cap \mathbb Z_m^k$ is considered.
The algorithm stops when a square non singular matrix $X^{(n-1)}$ is computed or if all the elements of $A \cap \mathbb Z_m^k$ are analysed. In the former case the basis $S$ is such that the cubature rule $(\mathcal D, S)$ has equal weights. In the latter case there not exists any basis such that the associated cubature rule with node $\mathcal D$ has equal weights. Notice that the algorithm stops because the elements of $A \cap \mathbb Z_m^k$ are finite.
The following theorem characterises the precision basis of cubature rules with equal weights.
\[prec\_pesi\_uguali\] Let $\mathcal D \subset \Omega_m^k$ be a set of $n$ nodes and let $f$ its indicator function, as in Eq. . Let $S \subset S_m$ be a monomial set such that the pair $(\mathcal D, \operatorname{Span}(S))$ is correct and let $w_S$ as in Proposition \[pr:int-cub-r\].
If $w_S=\displaystyle \frac 1 n 1_n$ then the set $A$ defined in Theorem \[teo\_pesi\_uguali\] is the precision basis $\mathcal B$ for $(\mathcal D,w_S)$.
From Eq. (\[ind\]), for each $\alpha \in[0,\dots, m-1\
]^k$, $b_{\overline \alpha} = \displaystyle \frac 1 {m^k} \sum_{d \in {\mathcal D}} z^{\alpha}(d)$, that is $\sum_{d \in {\mathcal D}} z^{\alpha}(d)= m^k b_{\overline \alpha} $.
For each $\alpha \in \mathbb Z^k_{\ge 0}$ it holds $z^\alpha(d)=z^{[\alpha]_m}(d)$. Then: $$\sum_{d\in \mathcal D} z^\alpha(d)=\sum_{d\in \mathcal D} z^{[\alpha]_m}(d)= m^k \ b_{\overline \alpha}$$ Let $w = \frac 1 n 1_n$ be the weights. Then the precision space for $(\mathcal D, \frac 1 n 1_n)$ is the largest set of monomials for which $$\int z^\alpha \ d\lambda = \frac 1 n\sum_{d \in \mathcal D} \ z^\alpha(d) \quad \textrm{i.e.} \quad \int z^\alpha \ d\lambda = \frac{m^k} n b_{\overline\alpha}$$ that is $A$.
The following result [describes the behaviour of a]{} the cubature rule with equal weights, when it is applied to [the integral of]{} monomials of the form $z^\alpha\overline z^\beta$. The connection to evaluation of moments of a distribution is evident.
\[monomio\_coniugato\] Let $(\mathcal D, w_s)$ be a cubature rule with $w_s =\frac 1 n 1_n$. The cubature rule $(\mathcal D, w_s)$ is exact for $z^\alpha$
1. if and only if $(\mathcal D, w_s)$ is exact for $ \overline z^\alpha$.
2. if and only if $(\mathcal D, w_s)$ is exact for $z^{\alpha +\gamma}\, \overline z^{\gamma} $ for each $\gamma \in \mathbb Z^k$ such that $\int_{\mathbb C^k} z^{\alpha +\gamma} \, \overline z^{\gamma} \, d\lambda = \int_{\mathbb C^k} z^\alpha \, d \lambda$.
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1. If $(\mathcal D, w_s)$ is exact for $z^\alpha$, then $$\int_{\mathbb C^k} z^\alpha d\lambda = \frac 1 n \sum_{d \in \mathcal D} z^\alpha (d) \ ,$$ and so $$\begin{aligned}
\int_{\mathbb C^k} \overline z^\alpha \ d\lambda
= \int_{\mathbb C^k} \overline {z^\alpha} \ d\lambda
= \overline{\int_{\mathbb C^k} z^\alpha \ d\lambda}
= \frac 1 n \overline{\sum_{d \in \mathcal D} z^\alpha (d) }
=\frac 1 n \sum_{d \in \mathcal D} \overline {z ^\alpha(d)} = \frac 1 n \sum_{d \in \mathcal D} \overline z ^\alpha(d)\end{aligned}$$ and we conclude that $(\mathcal D, w_s)$ is exact for $\overline z^\alpha$. The vice-versa is analogous.
2. Let $(\mathcal D, w_s)$ be exact for $z^\alpha$. By assumptions $\int_{\mathbb C^k} z^{\alpha +\gamma} \overline z^{\gamma} \ d\lambda = \int_{\mathbb C^k} z^\alpha \ d \lambda$ and so $$\begin{aligned}
\int_{\mathbb C^k} z^{\alpha +\gamma} \overline z^{\gamma} \ d\lambda
& = \int_{\mathbb C^k} z^\alpha \ d \lambda
= \frac 1 n \sum_{d \in \mathcal D} z ^\alpha(d) = \frac 1 n \sum_{d \in \mathcal D} |z^\gamma(d)|^2 z ^\alpha(d) \\
& = \frac 1 n \sum_{d \in \mathcal D} z^{\alpha + \gamma}(d) \, \overline z ^\gamma(d) \ ,\end{aligned}$$ where the third equality is due to the fact that $ |z^\gamma(d)|^2 =1$. We conclude that $(\mathcal D, w_s)$ is exact for $ z^{\alpha +\gamma} \overline z^{\gamma}$. Furthermore, from item 1 it follows that, since the cubature rule is exact for $ z^{\alpha +\gamma} \overline z^{\gamma}$, it is also exact for $ \overline{z^{\alpha +\gamma} \overline z^{\gamma}} $, that is for $ \overline z^{\alpha +\gamma}z^{\gamma}$.
The vice-versa is analogous.
Gaussian distribution {#Sc:Gauss}
---------------------
In this section we characterise the cubature rules $(\mathcal D, S)$ with equal weights and the Gaussian distribution. First of all, we present some results about integration with respect to the Gaussian measure.
### Gaussian measure
Let $Z^t=(Z_1, \dots, Z_p) $ be a $p$-variate Gaussian complex random variable. Let $Z_k=X_k + \mathbf{i} Y_k$, $k=1,\dots,p$, then the vector of real and imaginary parts $(X_1,Y_1,\dots, X_p,Y_p)$ is a $2p$-variate Gaussian real random vector. We assume the following relations among the expected values of the real and imaginary parts of the $Z$ variables. $$\begin{aligned}
\label{conditions}
E(X_k)=E(Y_k)=0 && \\
E(X_j X_j)=E(Y_jY_j)=\frac{\sigma^2_j}{2} && E(X_j Y_j)=0 \\
E(X_j X_k)=E(Y_jY_k)=\frac{\alpha_{jk}}{2} \textrm{ for } j\ne k && E(X_j Y_k)=-E(X_kY_j)=-\frac{\beta_{jk}}{2} \textrm{ for } j > k\end{aligned}$$ for $j,k=1,\dots,p$.
We denote by $\Sigma$ the matrix $E(Z \overline Z^t) = [ E(Z_j\overline Z_k)]_{j,k=1,\dots,p}$. Then, from the previous conditions, $$\begin{aligned}
\Sigma_{jk}=E(Z_j\overline Z_k) = \begin{cases}
\sigma_k^2 & \text{ if } j=k\\
\alpha_{jk}+\mathbf{i} \beta_{jk} & \text{ if } j< k\\
\alpha_{jk}-\mathbf{i} \beta_{jk} & \text{ if } j> k \ .
\end{cases} \end{aligned}$$ The probability density function of the zero mean $p$-variate complex Gaussian distribution is given by (see e.g. [@Goodman1963]) $$\begin{aligned}
\label{Gauss_distr}
p(z)=\frac{1}{\pi^p \det (\Sigma)} \exp(-\overline z^t \Sigma^{-1} z)\end{aligned}$$ where $z=[z_1,\dots,z_p]^t$.
We consider the complex measure $\nu$ such that $d\nu= p(z) \ d\mu$, where $\mu$ is the $\sigma$-finite measure of $\mathbb C^p$ identifiable with the Lebesgue measure of $\mathbb R^{2p}$, and so from the results presented in Appendix \[Sc:Complex\_Integ\], since $z_k=x_k+\mathbf{ i } y_k$, $k=1\dots p$, we have $\int_{\mathbb C^p} f(z) \ d\nu = \int_{\mathbb C^p} f(z) p(z) \ d \mu =\int_{\mathbb R^{2p}} f(x,y) p(x,y) \ dx \, dy $, where $x=[x_1,\dots,x_p]^t$ and $y=[y_1,\dots,y_p]^t$.
We denote by $\nu(n_1,m_1,\dots,n_p,m_p)$ the moment: $$\nu(n_1,m_1,\dots,n_p,m_p)=\frac{1}{\pi^p \det (\Sigma)} \int_{\mathbb C^p} z_1^{n_1}\ \overline z_1^{m_1}\ z_2^{n_2}\ \overline z_2^{m_2} \cdots z_p^{n_p} \ \overline z_p^{m_p} \exp(-\overline z^t \Sigma^{-1} z) \ dz_1 \cdots dz_p$$ In the vector $(n_1,m_1,\dots,n_p,m_p)$, consisting of the exponents of the moments, the $n_j$ indices are in odd entries and refer to $Z_j$ while the $m_j$ indices are in even entries and refer to $\overline Z_j$.
In [@FassinoPistoneRiccomagnoRogantin2017] the following theorem on the null moments is shown.
\[th:conditions\] The conditions for nullity of the moment $\nu(n_1,m_1,\dots,n_p,m_p)$ depend on the structure of independence of the variables.
1. If no variable $Z_j$, $j=1,\dots,p$, is independent from all the others, then the moment $\nu(n_1,m_1,\dots,n_p,m_p)$ is zero if $$\sum_{j=1}^p n_j\ne \sum_{j=1}^p m_j\ ;$$
2. If there exists a subset of variables $Z_r$, $r\in R \subset \{1,\dots,p\}$ such that $Z_r$ is independent from all the others, then $\nu(n_1,m_1,\dots,n_p,m_p)$ is zero if there exists $r \in R $ such that $$n_r\ne m_r \qquad \textrm{or} \qquad \sum_{j=1,j\notin R}^p n_j\ne \sum_{j=1,j\notin R}^p m_j \ .$$
3. If there exist $q$ subsets of variables $R_1\dots, R_q$ with variables dependent within each subset and independent between subsets, then $\nu(n_1,m_1,\dots,n_p,m_p)$ is zero if there exists $h \in \{1,\dots,q\} $ such that $$\sum_{j\in R_h}^p n_j\ne \sum_{j\in R_h} m_j \ .$$
From Theorem \[th:conditions\] we obtain the value of the integral of the monomials in $\mathbb T$ with respect to the Gaussian distribution.
\[cor:integral\] Let $z^\alpha=z_1^{n_1}\cdots z_p^{n_p}$ a monomial in $\mathbb T$. Then $$\begin{aligned}
\begin{cases}
\int_{\mathbb C^p} z^\alpha p(z) \ dz =0 & \text{if there exists } r \in \{1,\dots,p\} \ s.t. \ n_r \neq 0\\ \\
\int_{\mathbb C^p} z^\alpha p(z) \ dz =1 & \text{if } n_1=\dots = n_p= 0
\end{cases}\end{aligned}$$
In this case $m_1=\dots=m_p=0$, and so, if there exists an exponent $n_r\neq 0$, from Theorem \[th:conditions\] we have $ \nu(n_1,0,\dots,n_p,0)=\displaystyle \frac{1}{\pi^p \det (\Sigma)} \int_{\mathbb C^p} z_1^{n_1}\ \cdots z_p^{n_p} \ p(z) \ dz_1 \cdots dz_p =0$ and the first part of the thesis follows. The second part is an obvious result.
### Cubature rules with equal weights for Gaussian distribution
Let $\nu$ be the Gaussian distribution. Since, in this case $\int_{\mathbb C^k} z^\alpha \ d\nu =0$, if $\alpha \neq (0,\dots,0)$, Theorem \[teo\_pesi\_uguali\] can be reformulated as follows.
\[normale\_uguali\] Let $\nu$ be the Gaussian distribution. Let $\mathcal D \subset \Omega_m^k$ be a set of $n$ nodes and let $f$ its indicator function, as in Eq. . Let $S \subset S_m$ be a monomial set such that the pair $(\mathcal D, \operatorname{Span}(S))$ is correct and let $w_S$ as in Proposition \[pr:int-cub-r\]. Then $w_S= \frac 1 n 1_n$ if and only if $ S \cap \operatorname{Supp}(f) = \{ 1\}$.
From Theorem \[teo\_pesi\_uguali\] we have that $w_S= \frac 1 n 1_n$ if and only if $S\subset A$, where $$A=\left \{ z^\alpha \in\mathbb T, \ \alpha \in \mathbb Z^k_{\ge 0} \ \left| \ \int_{\mathbb C^k} z^\alpha \ d\nu = \frac {m^k} n \ b_{\overline{\alpha}}\right. \right\} \ ,$$ and so we first describe $A$ for the Gaussian distribution.
When $\alpha=[0,\dots,0]$ we have that $\int z^{\alpha} d\nu =1 $ and $b_{\overline \alpha}=\frac n {m^k}$ and so $1$ belongs to $A$. Furthermore, if $\alpha \neq [0,\dots, 0]$, then $\int_{\mathbb C^k} z^\alpha d\nu =0$, and so the set $A$ is given by $$\begin{aligned}
A=\{1\} \cup \left \{ z ^\alpha \in \mathbb T ,\ \alpha \in \mathbb Z^k_{> 0}\ \left| \ b_{\overline\alpha} =0\right. \right\}=\{1\} \cup \left \{ z ^\alpha \in \mathbb T ,\ \alpha \in \mathbb Z^k_{> 0}\ \left| \ b_{[\alpha]_m} =0\right. \right\}\end{aligned}$$ since $b_{[\alpha]_m}=\overline{ b_{\overline\alpha} }$. From Equation (\[ind\]), we have that $b_{[\alpha]_m}=0$ if and only if $z^{[\alpha]_m} \notin \operatorname{Supp}(f)$, and thus $$A =\{1\} \cup \left \{z^\alpha \in \mathbb T, \alpha \in \mathbb Z^k_{> 0}\;|\; z^{[\alpha]_m} \notin \operatorname{Supp}(f) \right \} \ .$$ We conclude that $S\subset A$ if and only if $ S \cap \operatorname{Supp}(f) = \{ 1\}$.
The following theorem characterises the precision basis of a cubature rule with equal weights, with respect to the gaussian distribution.
\[pesi\_uguali\] Let $\nu$ be the Gaussian distribution. Let $\mathcal D \subset \Omega_m^k$ be a set of $n$ nodes and let $f$ its indicator function, as in Eq. . Let $S \subset S_m$ be a monomial set such that the pair $(\mathcal D, \operatorname{Span}(S))$ is correct and let $w_S$ as in Proposition \[pr:int-cub-r\]. If $w_S= \frac 1 n 1_n$ then $\left \{1\right\} \cup \left \{z^\alpha \in \mathbb T\,|\, z^{[\alpha]_m} \notin \operatorname{Supp}(f) \right \} $ is the precision basis $\mathcal B$ for $(\mathcal D,w_S)$.
Given a set of nodes $\mathcal D$, it is possible to check if there exists a basis $S$ so that the corresponding cubature rule $(\mathcal D, S)$ has equal weights in an easier way than in the general case, since it is sufficient to consider only monomials $t \in \mathbb Z_m^k \setminus \operatorname{Supp}(f)$, as Corollary \[pesi\_uguali\] shows that $ S \cap \operatorname{Supp}(f) =\{1\}$ in order to have equal weights.
Let $\nu$ be the Gaussian distribution. If $\mathcal D$ is a regular fraction, for each basis $S$ such that the pair $(\mathcal D, \operatorname{Span}(S))$ is correct, the corresponding cubature rule has equal weights.
If $\mathcal D$ is a regular fraction, for each monomial $s\in \mathbb T$, the vector $[s(d)]_{d \in \mathcal D}$ is equal to $\gamma 1_n$, for a given $\gamma\in \mathbb C$, or orthogonal to $1_n$. Since $X_{\mathcal D, S}$ is a non singular matrix whose first column is the vector $1_n$, for each monomial basis $S=\{z^\alpha\}$, the columns of $X_{\mathcal D, S}$, except the first one, are orthogonal to $1_n$. It follows that $\Sigma_{d\in \mathcal D} z^\alpha(d) =0$ for each $z^\alpha \in S\setminus \{1\}$ and so $z^\alpha \notin \operatorname{Supp}(f)$. We conclude that, if $\mathcal D$ is a regular fraction, $S \cap \operatorname{Supp} (f)= \{1\}$ for each possible basis $S$ and so the corresponding cubature rule has equal weights.
The following example shows that a cubature rule can have equal weights even if $\mathcal D$ is not a regular fraction of $\Omega_m^k$.
Let $\mathbb C[z_1,\dots, z_4]$ be the polynomial ring with indeterminates $z_1,\dots,z_4$ and let $\mathcal D$ be the set of nodes contained in $\Omega_2^4$, $$\begin{aligned}
\mathcal D =& \, \{(1,1,1,1),(1,1,-1,1),(1,-1,1,-1),(1,-1,-1,-1), \\
& \quad (-1, 1, 1, 1), (-1, -1, -1, 1), (-1, -1, -1, -1), (-1, 1, 1, -1)\}\end{aligned}$$ whose indicator function is $f = (2 + z_2z_3 + z_2z_4-z_1z_2z_3 + z_1z_2z_4)/4$. The set $\mathcal D$ is not a regular fraction since $[z_1z_2z_4(d) ]_{d\in \mathcal D} = [1, 1, 1,1, -1, 1, -1,1]^t $, that is $[z_1z_2z_4(d) ]_{d\in \mathcal D} $ is not orthogonal nor parallel to $1_8$.
Given the monomial set $S = [1, z_1, z_2, z_3, z_4, z_1z_2, z_3z_4, z_1z_3z_4]$, the weights $w_S$ are the solution of the linear system $X_{ \mathcal D, S}^t w_S = [\int sd\nu]_{s\in S}$, where $\int s \,d\nu =0 $ if $s\neq 1$ and $\int s \,d\nu=1$ if $s=1$. Since the columns of the matrix $X_{ \mathcal D, S}$, except the first one, are orthogonal to $1_8$, the vector $w_S= \frac 1 8 1_8$ is the solution of the previous linear system, even if the set of nodes is not a regular fraction.
The following example illustrates a cubature rule with equal weights and its precision basis.
[Let $k=2$, $m=4$ and let $\omega_0=1$, $\omega_1= \mathbf{i}$, $\omega_2=-1$ and $\omega_3=-\mathbf{i}$ be the fourth root of the unit. In this case, the full factorial is $\mathcal F=[\omega_0, \omega_1, \omega_2, \omega_3 ]^2$. Let $\mathcal D= \{ (1,1), (\mathbf{i},-\mathbf{i}), (-1,\mathbf{i}), (-\mathbf{i},-1) \} $ be the set of nodes with indicator function $$\begin{aligned}
f= \frac1 8 \left(2+z_1z_2+(1+\mathbf{i})z_1z_2^2+ (1-\mathbf{i})z_1^2z_2 - \mathbf{i}z_1z_2^3 + \mathbf{i} z_1^3z_2 +(1+\mathbf{i})z_1^2z_2^3+(1-\mathbf{i})z_1^3z_2^2+z_1^3z_2^3\right) \, .
\end{aligned}$$ Given the monomial set $S = [1, z_2, z_1, z_2^3]$, we have $S \cap \operatorname{Supp}(f)=\{1\}
$ and so the nodes of the cubature rule $(\mathcal D, S)$ are $w_d = \frac 1 4$, for all $ d \in \mathcal D$. From Corollary \[pesi\_uguali\] it follows that the precision basis is $$\mathcal B_{\mathcal D,S} = \{1\} \cup \left \{ z^\alpha \in \mathbb T \;|\; [\alpha]_4 \in \{ (0,1), (1,0), (0,2), (2,0),(0,3), (3,0), (2,2) \} \right \} \, .$$ From Theorem \[monomio\_coniugato\] it follows that the cubature rule is also exact for $\left \{{\overline z}^\alpha \;|\; z^\alpha \in\mathcal B_{\mathcal D,S} \right \}$. Furthermore, since for each $\alpha \neq (0,\dots,0)$ and for each $\gamma \in \mathbb Z_{\ge 0}^4$ we have $\int_{\mathbb C^k} z^\alpha d \nu = 0 =\int_{\mathbb C^k} z^{\alpha +\gamma} \, \overline z^{\gamma} \, d\nu$ , the cubature rule is also exact for $\left \{\overline z^{\alpha+\gamma} z^\gamma, z^{\alpha+\gamma}\overline z^\gamma \;|\; z^\alpha \in \mathcal B_{\mathcal D,S} , \ \gamma \in \mathbb Z_{\ge 0}^4\right \}$. ]{}
The following example shows the case of a set of nodes $\mathcal D$ which does not generate any cubature rule with equal weights.
Let $k=2$, $m=3$ and let $\omega_0=1$, $\omega_1= \cos(2\pi/3) +\mathbf{i} \sin(2\pi/3)$ and $\omega_2=\overline{\omega}_1$ be the third root of the unit.
Let $\mathcal D =\{(1,\omega_2),\;(\omega_2,\omega_1) \}$ be the set of nodes with indicator function $$f= \frac1 9 \left(2 -z_2 - \omega_2z_1 - z_2^2 - \omega_2z_1z_2 - \omega_1z_1^2 + 2\omega_2z_1z_2^2 + 2\omega_1z_1^2z_2- \omega_1z_1^2z_2^2\right) \,.$$ Since $\operatorname{Supp}(f)=\{z^\alpha \,|\, \alpha \in \mathbb Z_3^2 \}$, there not exist a monomial basis $S\subset \mathbb Z_3^2$ such that $S\cap \operatorname{Supp}(f) = \{1\}$ and so there not exist a cubature rule $(\mathcal D,S)$ with equal weights.
Appendix A. Complex integration {#Sc:Complex_Integ .unnumbered}
===============================
Let $\mathcal M$ be a $\sigma$-algebra in a set $X$ and let $\{E_k\}$ be a countable partition of $E$, that is $E=\cup_k E_k$ and $E_k \cap E_j =\emptyset$, if $k\neq j$. A complex measure $\lambda$ on $\mathcal M$ is a complex-valued function on $\mathcal M$ such that $$\lambda (E) = \sum_{k=1}^\infty \lambda(E_k) < +\infty.$$ The total variation $|\lambda|$ of $\lambda$ is a real positive measure defined as $$|\lambda| (E)= \sup_{\{E_k\}_{k=1}^\infty}\sum_{k=1}^\infty |\lambda(E_k)| \qquad \text{ for all } E \in \mathcal M$$ where ${\{E_k\}_{k=1}^\infty}$ is a generic partition of $E$.
The following theorem is the Lebesbue-Radon-Nikodym Theorem presented in [@Rudin1987 Th. 6.10].
\[Lebesbue-Radon-Nikodym\] Let $\mu$ be a positive $\sigma$-finite measure on a $\sigma$ algebra $\mathcal M$ in a set $X$, and let $\lambda$ be a complex measure on $\mathcal M$.
[(a)]{} There is then a unique pair of complex measures $\lambda_a$ and $\lambda_s$ on $\mathcal M$ such that $$\lambda= \lambda_a +\lambda_s \qquad \lambda_a {\ll} \mu \qquad \lambda_s {\perp }\mu \, .$$
[(b)]{} There is a unique $ h \in L^1(\mu)$, called the Radon-Nikodym derivative w.r.t. $\mu$, such that $$\lambda_a(E) = \int_E h \ d\mu \, .$$
The choice $\mu=|\lambda|$ gives the following theorem [@Rudin1987 th. 6.12].
\[Th\_dlambda\] Let $\lambda$ be a complex measure on a $\sigma$-algebra $\mathcal M$ in $X$. Then there is a function $h\in L^1(|\lambda|)$, called the Radon-Nikodym derivative w.r.t. $|\lambda|$, such that $|h(x)| =1$ for all $x \in X$ and such that $ \lambda(E) = \int_E h \ d|\lambda|$ or, equivalently, that $d\lambda= h\ d|\lambda|$.
From Theorem \[Th\_dlambda\] it follows that $\lambda(X) = \int_X h d|\lambda|$; furthermore, it is possible to define $$\int_X f \ d\lambda \stackrel{def}{=} \int_X f h \ d|\lambda| \, .$$ Later on, we consider a special case. Let $\mu$ be a positive real measure on $\mathcal M$ and let $g: X \rightarrow \mathbb C$ be a function in $L^1(\mu)$. We can define a complex measure $\lambda$ on $\mathcal M$ in the set $X$ as follows: $$\lambda(E)=\int_E g \ d\mu \quad \text{ for all } \, E \in \mathcal M \, .$$ Since in this case $d\lambda= g \ d\mu$, from Theorem \[Th\_dlambda\] we have $g \ d\mu = d\lambda = h\ d|\lambda|$, with $|h|=1$, and so $\overline h g \ d\mu = \overline h h d|\lambda| = d|\lambda|$. We conclude that, in this case, $$\begin{aligned}
\label{integrale}
\int_X f \ d\lambda =\int_X f h \ d|\lambda| = \int_X f h \overline h g \ d\mu = \int_X f g |h|^2 \ d\mu = \int_X f g \ d\mu \, .
\end{aligned}$$
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